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\begin{document}
\large
\begin{center}{\large\bf PROBLEM OF CONSTRUCTING DISCRETE AND
FINITE QUANTUM THEORY}\end{center}
\vskip 1em \begin{center} {\large Felix M. Lev} \end{center}
\vskip 1em \begin{center} {\it Artwork Conversion Software Inc.,
1201 Morningside Drive, Manhattan Beach, CA 90266, USA 
(E-mail:  felixlev@hotmail.com)} \end{center}
\vskip 1em

{\it Abstract:}
\vskip 0.5em

We consider in detail an approach (proposed by the author earlier)
where quantum states are described by elements of a linear space
over a Galois field, and operators of physical quantities - by
linear operators in this space. The notion of Galois fields 
(which is extremely simple and elegant) is discussed in detail 
and we also discuss the conditions when our description gives 
the same predictions as the conventional one. In quantum theory 
based on a Galois field, all operators are well defined and 
divergencies cannot not exist in principle. A particle and its 
antiparticle are described by the same modular irreducible 
representation of the symmetry algebra. This automatically 
explains the existence of antiparticles and shows that a particle 
and its antiparticle are the different states of the same object. 
As a consequence, a new symmetry arises, and the structure of the 
theory is considerably simplified. In particular, one can work  
with only creation operators or only annihilation ones since they 
are no more independent. In our approach the problem arises
whether the existence of neutral elementary particles (e.g. the 
photon) is compatible with the usual relation between spin and 
statistics, or in other words whether neutral particles can be 
elementary or only composite. 

\vfill\eject

\tableofcontents

\vfill\eject

\chapter{Motivation}
\label{C1}

\section{Remarks on local quantum field theory}
\label{S1.1} 
 
The phenomenon of local quantum field theory (LQFT) has no 
analogs in the history of science. There is no branch of 
science where so 
impressive agreements between theory and experiment have been
achieved. In particular, the theoretical results for the 
electron and muon magnetic moments have been experimentally
confirmed with the accuracy of eight decimal points
(see e.g. Ref. \cite{g-2} and references therein). The theory
has successfully predicted the existence of many new particles
and even new interactions. For example, the prediction of W and
Z bosons have been made on the basis of the unified theory of
electromagnetic and weak interactions (electroweak theory).
It is hard to believe that these and many other achievements 
are only coincidences. At the same time, the level of 
mathematical rigor used in the LQFT is very poor and, as a 
result, the LQFT has several well known difficulties and 
inconsistencies. The absolute majority of physicists believe 
that agreement with experiment is much more important than the
lack of mathematical rigor, but not all of them think so.
For example, such a famous physicist as Dirac who made
a great contribution to the LQFT, writes in Ref. \cite{DirMath}: 
$'$The agreement
with observation is presumably by coincidence, just like the
original calculation of the hydrogen spectrum with Bohr orbits.
Such coincidences are no reason for turning a blind eye to the
faults of the theory. Quantum electrodynamics is rather like
Klein-Gordon equation. It was built up from physical ideas
that were not correctly incorporated into the theory and it
has no sound mathematical foundation.$'$  

In addition, the LQFT fails in quantizing gravity since
the gravitational constant has the dimension (length)$^2$
(in units where $c=1,\quad \hbar=1$), and as a result, 
quantum gravity is not renormalizable.

Usually there is no
need to require that the level of mathematical rigor in physics
should be the same as in mathematics. However physicists should
have a feeling that, at least in principle, mathematical
statements used in the theory can be substantiated. The absence
of a well-substantiated LQFT by no means can be treated as a
pure academic problem. This becomes immediately clear when one
wants to work beyond perturbation theory. The problem arises to 
what extent the difficulties of the LQFT can be removed 
in the framework of the LQFT itself or the LQFT can only be a 
special case of a more general theory based on essentially new 
ideas. The majority of physicists believes that the LQFT should
be treated \cite{Wein} $'$in the way it is$'$, but at the same
time it is \cite{Wein} a $'$low energy approximation to a
deeper theory that may not even be a field theory, but something
different like a string theory$'$. 

The main problem of course is the choice of strategy
for constructing a new quantum theory. Since 
nobody knows for sure which strategy is the best one, 
different approaches should 
be investigated. In the present paper we try to follow Dirac's 
advice given in Ref. \cite{DirMath}: $'$I learned to distrust all
physical concepts as a basis for a theory. Instead one should
put one's trust in a mathematical scheme, even if the scheme
does not appear at first sight to be connected with physics.
One should concentrate on getting an interesting mathematics.$'$ 

We believe that quantum theory will ultimately be based on 
mathematics involving Galois fields. The main goal of 
the paper is to convince the reader that this point of view is 
correct (According to Hilbert, every man has some outlook. When 
it is narrowing and becomes a point, the man says 'This is my 
point of view'). We first argue that the difficulties of the LQFT 
lead to serious problems in understanding the theory.       

 Typically the LQFT starts from a local Lagrangian for which, by
using the canonical Noether formalism, one can determine a set of
conserving physical observables. After quantizing the local fields
in question, these observables become quantum operators 
and the quantum Lagrangian obtained in such a way contains
products of local quantum fields at coinciding points. 
However interacting field operators can be treated only
as operator valued distributions \cite{Haag} and therefore their
product at coinciding points is not well defined.
One can consider the Lagrangian only as an auxiliary tool for
constructing the generators of the symmetry group and/or the 
S-matrix in the framework of the canonical Noether formalism 
(see e.g. Ref. \cite{AB,Schweber,Wein}). However for solving this 
problem we still have to treat somehow the problem of handling
the products of local operators at coinciding points.

It is known as the problem of constructing composite operators 
(see e.g. Ref. \cite{Zim}). Wilson has proposed a hypothesis 
called the operator product expansion (OPE) \cite{Wil}, which has 
turned out to be highly successful in describing experimental data. 
The essence of the OPE is that perturbative and nonperturbative
effects can be separated out but the proof beyond perturbation
theory has not been obtained. 
When perturbation theory does not apply, the usual prescriptions 
are to separate the arguments of the operators in question and 
define the composite operator as a limit of nonlocal 
operators when the separation goes to zero (see e.g.
Ref. \cite{J} and references therein).
However any attempt to separate the arguments of local
operators immediately makes the Lagrangian nonlocal and the 
whole edifice of the LQFT (including canonical formalism) 
becomes useless. 
Meanwhile the only known way of constructing the generators of 
the symmetry group in the LQFT is canonical 
formalism. As a consequence, the generators constructed in such a 
way are usually not well-defined (see e.g. Refs. \cite{JL,CCJ}). 



The fact that the most fundamental operators, including the 
momentum, angular momentum and Lorentz boost generators,
are not well-defined, means that there is no proof that 
the well-known local quantum field theories (including QED)
are Poincare invariant. It is often believed that a theory
is Poincare invariant if its equations can be written in a
covariant form. However this is always true only for classical
field theory while on the quantum level Poincare covariance
means (by definition) that the representation generators of the 
Poincare group should satisfy the commutation relations for
the generators of the Poincare group Lie algebra.   

It is also worth noting 
that there exists the well-known paradox: on the one hand, the 
renormalizable perturbation theory, which is formally based on 
interaction picture, has turned out to be highly
successful in electroweak theory and QCD, while on the other hand,
according to the Haag theorem \cite{Haag}, the interaction
picture does not exist.

The mostly known consequence of the poor definition of
local operators is the fact that the S-matrix 
contains divergencies already in the second order of perturbation
theory in coupling constant. The problem of handling 
divergencies is probably the main ingredient of the existing 
approach to the LQFT. It has been shown (see e.g.
the well-known textbooks \cite{AB,Schweber,Wein}) that for a class of 
theories called renormalizable, the series for the S-matrix can be
rearranged in such a way that all the divergencies dissappear
and the S-matrix is finite in each order of perturbation theory.
Such a procedure is called renormalization. It may seem not very
appealing from the aesthetics point of view but the physical meaning
of renormalization is very clear. 

\begin{sloppypar}
Indeed, consider, for example,
quantum electrodynamics (QED), i.e. the theory of 
interacting electrons,
positrons and photons. The initial Lagrangian should contain at 
least two
parameters: the "bare" electron mass $m_0$ and its "bare" charge $e_0$.
As a result of electromagnetic interactions, these quantities become
the real mass $m$ and charge $e$ respectively. By using the 
renormalization procedure one can express the S-matrix in terms of
$m$ and $e$ rather than $m_0$ and $e_0$. So the very idea of
renormalization is very reasonable. However if the renormalization
is not well defined (contains divergencies etc.) we cannot restore
the values of $m_0$ and $e_0$ from $m$ and $e$. It's reasonable
to believe that $m_0$ and $e_0$ should be defined from the first
principles; then the theory will make it possible to compute $m$
and $e$, and compare them with the corresponding experimental values.
However at present we have no such a possibility but can only take
the values of $m$ and $e$ from experiment. As noted by Dirac 
\cite{DirMath} $'$the renormalization idea would be sensible
only if it was applied with finite renormalization factors, not
infinite ones.$'$. In his textbook \cite{Wein} Weiberg describes
the problem of infinities as follows $'$Disappointingly this
problem appeared with even greater severity in the early days
of quantum theory, and although greatly ameliorated by
subsequent improvements in the theory, it remains with us to the
present day.$'$ A reader may also be interested in the discussion
of infinities given in $'$t Hooft's Nobel Lecture \cite{tHooft}.  
\end{sloppypar}

The fact that we can get rid off divergencies in the perturbation
expansion of the S-matrix, does not imply that the other operators 
(e.g. the Hamiltonian) automatically become well-defined. As a 
consequence, one can solve the bound state problem only in special 
cases when it's possible to analytically sum up the perturbation 
series for the S-matrix.
Also we still have the problem of anomalies when the operators
do not satisfy the commutation relations they should. A well known
example is the existence of Schwinger terms in the commutator of
current operators \cite{Schwinger}.

As mentioned above, the main problem in constructing well-defined
operators in the LQFT is that they involve the product of quantized 
fields at the same space-time point. If we accept the basic principle
of quantum theory that every physical quantity should be described
by a selfadjoined operator in the Hilbert space, then, even in  
nonrelativistic theory, the problem arises what operator corresponds
to time (see e.g. the discussion in Ref. \cite{time}). 
As it has been known already in 30th, in relativistic 
quantum theory there is no operator satisfying all the requirements
for the spatial position operator \cite{Schweber,GW}. As a result 
of evolution of these ideas, the majority of physicists working on 
quantum theory in 50th
and 60th believed that local quantized fields and Lagrangians are
the rudimentary notions which will disappear in the future theory
(see e.g. the discussion in Ref. \cite{BLP}). For example, in the
Heisenberg program it is claimed that only the S-matrix describing
the transition from a state at $t \rightarrow -\infty$ to a state at 
$t \rightarrow +\infty$ is meaningful (on quantum level) while the 
description of the evolution at each moment of time $t$ is meaningless.
A lot of efforts have been made to investigate what restrictions on
the S-matrix should be imposed (e.g. analyticity, causality, unitarity)
for constructing the theory of strong interactions. 
Impressive successes of local gauge theories in 70th and 80th has
shifted the general trend away from those ideas although no arguments
questioning them were given. So we have to make the decision on the 
following: 

\begin{itemize}

\item Should we accept the principle that every physical quantity
is described by a selfadjoint operator in the Hilbert space?

\end{itemize}

If the answer is yes then we have no choice but to acknowledge 
that the
quantity $x$ in the Lagrangian density $L(x)$ and in the arguments of
quantized fields is only a parameter which has the meaning
of the space-time coordinate only in the classical limit. In 
that case the
standard requirement of locality does not have any fundamental physical
meaning. So it is reasonable to believe that the future relativistic 
quantum theory either will not deal with space-time coordinates at 
all or the notion of space-time will be somehow modified.
In contrast to the situation existed in 60th and 70th, the
papers devoted to the former approach are practically absent
while the latter is investigated rather intensively.

The first paper in which it has been proposed to modify 
coordinates at small distances was probably that by Snyder
\cite{Snyder}. The state of the art of the investigations
involving discrete or noncommutative space-time coordinates 
can be found, for example, in Ref. \cite{noncomm} and 
references therein. 

At the beginning of 80th physicists were excited by the existence
of superstring theories without anomalies (see e.g. Ref. \cite{GSW}).
They have made important achievements during the last two decades
(see e.g. Ref. \cite{string} and references therein). Such
theories can be treated as versions of a nonlocal theory
because the product of quantized fields (now they describe strings 
rather then particles) enters the Lagrangian density at different 
space-time points. Several alternatives to the superstring theory
have been also proposed, for example, the loop quantum gravity,
the matrix theory and others
(see Ref. \cite{lqg} and references therein).

In the present paper we propose an approach where quantum theory 
is based on Galois fields. Such a possibility has been investigated
by several authors (see e.g. Ref. \cite{Galois} and references
therein) but they usually considered a replacement of the conventinal
space-time by Galois or P-adic fields. In contrast to those 
approaches, our one does not contain space-time coordinates at all 
(in the spirit of Heisenberg ideas) but (instead of the S-matrix),
the main ingredient of the theory is linear spaces and operators 
over Galois fields. The notion of Galois fields is discussed in 
the next section.  

\begin{sloppypar}
\section{What mathematics is most suitable for physics?}
\label{S1.2}
\end{sloppypar}

 The mathematics used in conventional physics has proven its 
relevence so
many times that the above question might seem to be purely academic.
This mathematics is based on the field of complex numbers which contains
an infinite number of elements.

Suppose, for example, that we wish to verify experimentally whether 
addition is commutative, i.e. whether a + b = b + a is always 
satisfied. If our Universe is finite and contains not more than N 
elementary particles then we shall not be able to do this if 
$a + b > N$. Even this simple example shows
that mathematics used in physics may be not valid when the numbers 
in question are very big.

Let us look at mathematics from the point of view of the famous 
Kronecker expression: 'The integers were invented by the God and 
all others were invented by people'. We would like to slightly 
change this expression as follows: 'The natural numbers were 
invented by the God and all others were
invented by people'. Indeed, the natural numbers 0, 1, 2... 
(we treat zero as
a natural number) have clear physical meaning. However only two operations
are always possible in the set of natural numbers: addition and 
multiplication. In order to make addition reversible, we introduce negative 
integers -1, -2 etc. Then, instead of the set of natural numbers we can work
with the ring of integers where three operations are always possible:
addition, subtraction and multiplication. However the negative numbers 
themselves have not physical meaning (we cannot say, for example, 'I have 
minus two apples'). Their only role is to make addition reversible. 

The next step is the transition to the field of rational numbers in which all
four operations excepting division by zero are always possible. One might think
that division is a natural operation since we know from everyday experience
that any macroscopic object can be divided by two, three and even a million
parts. But is it possible to divide by two or three the electron
or neutrino? It is obvious that if elementary particles do exist, then
division has only a limited sense. Indeed, let us consider, for example,
the gram-molecule of water having the mass 18 grams. It contains the
Avogadro number of molecules $6\cdot 10^{23}$. We can divide this 
gram-molecule by ten, million, billion, but when we begin to divide by 
numbers greater than the Avogadro one, the division operation loses its
sense. The obvious conclusion is that that the notion of division has a
sense only within some limits.

In mathematics the notion of linear space is widely used, and such important
notions as the basis and the dimension of this space are meaningful only
if the space is considered over a field or body. Therefore if we start from
natural numbers and wish to have a field, then we have to introduce negative
and rational numbers. However, if, instead of all natural numbers, we consider
only $p$ numbers 0, 1, 2, ... $p-1$ where $p$ is prime, then we can easily
construct a field without adding any new elements. This construction, called
Galois field, contains nothing that could prevent its understanding even by 
pupils of elementary schools. However the absolute majority of 
physicists is not familiar with Galois fields. 

Let us denote the set of numbers 0, 1, 2,...$p-1$ as $F_p$. Define addition
and multiplication as usual but take the final result modulo $p$. For simplicity,
let us consider the case $p=5$. Then $F_5$ is the set 0, 1, 2, 3, 4. Then
$1+2=3$ and $1+3=4$ as usual, but $2+3=0$, $3+4=2$ etc. Analogously, $1\cdot 2=2$,
$2\cdot 2=4$, but $2\cdot 3 =1$, $3\cdot 4=2$ etc. By definition, the element
$y\in F_p$ is called opposite to $x\in F_p$ and is denoted as $-x$ if $x+y=0$ in
$F_p$. For example, in $F_5$ we have -2=3, -4=1 etc. Analogously $y\in F_p$ is
called inverse to $x\in F_p$ and is denoted as $1/x$ if $xy=1$ in $F_p$. 
For example, in $F_5$ we have 1/2=3, 1/4=4 etc. It is easy to see that
addition is reversible for any natural $p>0$ but for making multiplication
reversible we should choose $p$ to be a prime. Otherwise the product of two
nonzero elements may be zero modulo $p$. If $p$ is chosen to be a prime then
indeed $F_p$ becomes a field without introducing any new objects (like negative
numbers as fractions). For example, in this field each element can obviously be 
treated as positive and negative {\bf simultaneously}! 

One might say: well, this is beautiful but impractical since in physics and 
everyday life 2+3 is always 5 but not 0. Let us suppose however that fundamental
physics is described not by 'usual mathematics' but by 'mathematics modulo $p$' 
where $p$ is a very large number.
Then, operating with numbers much smaller than $p$ we shall not notice this $p$,
at least if we only add and multiply. We will feel a difference between 
'usual mathematics' and 'mathematics modulo p' only while operating with numbers
comparable with $p$. 

We can easily extend the correspondence between $F_p$ and the ring of integers $Z$
in such a way that the subtraction will be also included. To make it clearer we note
the following. Since the field $F_p$ is cyclic (adding 1 successively, we will
obtain 0 eventually), it is convenient to visually depict its elements by the 
points of a circle of the radius $p/2\pi$ on the plane $(x,y)$. 
In Fig. 1.1 only a
part of the circle near the origin is depicted. 
\begin{figure}[!h]
\centerline{\scalebox{1.1}{\includegraphics{gr1.eps}}}%
\caption{%
  Relation between $F_p$ and the ring of integers%
}%
\label{Fig.1}
\end{figure}
Then the distance between
neighboring elements of the field is equal to unity, and the elements 
%\begin{figure}[tbh] 
%\special{psfile=fig1.ps angle=0. 
%voffset=-340. hoffset=20. hscale=70. vscale=70.}
%\begin{center} \vspace{4cm} \parbox{13cm} {\caption[Delta] 
%{\sl Correspondence between $F_p$ and the ring of integers} 
%\label{Fig}}
%\end{center} 
%\end{figure}
0, 1, 2,... are situated on the circle counterclockwise. At the same time we depict 
the elements of $Z$ as usual such that each element $z\in Z$ is depicted by 
a point with the
coordinates $(z,0)$. We can denote the elements of $F_p$ not only as 0, 1,... $p-1$
but also as 0, $\pm 1$, $\pm 2,$,...$\pm (p-1)/2$, and such a set is called the
set of minimal residues. Let $f$ be a map from $F_p$ to Z, such that the element
$f(a) \in Z$ corresponding to the minimal residue $a$ has the same notation as
$a$ but is considered as the element of $Z$. Denote $C(p) = p^{1/(lnp)^{1/2}}$ and
let $U_0$ be the set of elements $a\in F_p$ such that $|f(a)|<C(p)$. Then if
$a_1,a_2,...a_n\in U_0$ and $n_1,n_2$ are such natural numbers that
\begin{equation}
n_1<(p-1)/2C(p),\,\,n_2<ln((p-1)/2)/(lnp)^{1/2}
\end{equation}
\label{Eq. (1.1)}
then 
$$f(a_1\pm a_2\pm...a_n)=f(a_1)\pm f(a_2)\pm ...f(a_n)$$ 
if $n\leq n_1$ and
$$f(a_1 a_2...a_n)=f(a_1)f(a_2) ...f(a_n)$$ if $n\leq n_2$.
Thus though $f$ is not a homomorphism of rings $F_p$ and $Z$, 
but if $p$ is sufficiently large, then for a sufficiently 
large number of elements of $U_0$ the
addition, subtraction and multiplication are performed according 
to the same rules
as for elements $z\in Z$ such that $|z|<C(p)$. 
Therefore $f$ can be regarded as a 
local isomorphism of rings $F_p$ and $Z$.

The above discussion has a well known historical analogy. For a 
long time people
thought that our Earth is flat and infinite, and only after a big 
period of time they realized that it is finite and has a curvature. 
It is difficult to notice the curvature when we deal only with 
distances much less than the radius of the
curvature $R$. Analogously one might think that the set of 
numbers describing physics has a curvature defined by a very big 
number $p$ but we do not notice it until
we deal only with numbers much less than $p$.

Our arguments imply that the standard field of real numbers
will not be fundamental in future quantum physics (although
there is no doubt that it is relevant for macroscopic
physics). Let us discuss this question in a greater detail.
The notion of real numbers can be fundamental only if the
following property is valid: for any $\epsilon>0$, it is
possible (at least in principle) to build a computer
operating with a number of bits $N(\epsilon)$ such that
computations can be performed with the accuracy better
than  $\epsilon$.
It is clear that this is not the case if, for example, the
Universe is finite. Note also that the notion of equality
of two real numbers has a limited sense because they can be
calculated only with a finite accuracy. To illustrate this
point, consider a simple problem: let $x=0.2$, $y=1.2$, and
we want to ask a computer whether or not $6x=y$. Suppose,
for example that we are using a standard C or C++ compiler. 
Then the computer will answer 'no', and the reason is obvious.
The computer takes the number $0.2$ and represents it in the
only way it can: by using the available number of bits. It
is clear that at such a representation an accuracy is lost
and only a number close to $0.2$ will be written in the
computer memory. An analogous operation is performed with
$1.2$. After that it is clear that $6x\neq y$. This discussion
again shows that only finite integers are well defined. 
 
Let us note that even for elements from $U_0$ the result of 
division in the field
$F_p$ differs generally speaking, from the corresponding 
result in the field of rational number $Q$. For example the 
element 1/2 in $F_p$ is a very big number
$(p+1)/2$. For this reason one might think that physics based 
on Galois
fields has nothing to with the reality. We will see in the next chapter
that this is not so since the spaces describing quantum systems are
projective.

By analogy with the field of complex numbers, we can consider a 
set $F_{p^2}$
of $p^2$ elements $a+bi$ where $a,b\in F_p$ and $i$ is a formal element such
that $i^2=1$. The question arises whether $F_{p^2}$ is a field, i.e.
we can define all the four operations excepting division by zero.
The definition of addition, subtraction and multiplication in $F_{p^2}$
is obvious and, by analogy with the field of complex numbers, we can
try to define division as $1/(a+bi)\,=a/(a^2+b^2)\,-ib/(a^2+b^2)$.
This definition can be meaningful only if $a^2+b^2\neq 0$ in $F_p$
for any $a,b\in F_p$ i.e. $a^2+b^2$ is not divisible by $p$.
Therefore the definition is meaningful only if $p$ {\it cannot}
be represented as a sum of two squares and is meaningless otherwise.
Since the prime number $p$ is question is necessarily odd, we
have two possibilities: the value of $p\,(mod \,4)$ is either 1
or 3. The well known result of the number theory (see e.g. the 
textbooks \cite{VDW,IR} is that a prime number $p$ can be 
represented as a sum of two squares only in the former case
and cannot in the latter one. Therefore the above construction of
the field $F_{p^2}$ is correct only if $p\,(mod \,4)\,=\,3$.
The first impression is that if Galois fields can somehow 
replace the conventional field of complex numbers then this 
can be done only for $p$ satisfying $p\,(mod \,4)\,=\,3$ and
therefore the case $p\,(mod \,4)\,=\,1$ is of no interest for
this purpose. We will see in the next chapter that 
correspondence between complex numbers and Galois fields
containing $p^2$ elements can also be established if
$p\,(mod \,4)\,=\,1$. In general, it is well known (see e.g.
Ref. \cite{VDW,IR}) that any Galois field consists of $p^n$
elements where $p$ is prime and $n>0$ is natural. The numbers
$p$ and $n$ define the field $F_{p^n}$ uniquely up to 
isomorphism and $p$ is called the characteristic of the
Galois field.


\section{Poincare invariance vs. de Sitter invariance}
\label{S1.3}

The usual requirement to a new theory is that at certain
conditions it should give the same predictions as the old
one. Sometimes this is called the correspondence principle
between the new and old theories. We require that when 
$p\rightarrow \infty$ then there should exist a 
correspondence principle between the theory based on a 
Galois field of characteristic $p$ and conventional quantum
theory.

If a conventional quantum theory has a symmetry group, then
there exists a unitary representation of the group in the
Hilbert space describing the quantum system under 
consideration. For example, if the theory is Poincare
invariant then the representation generators satisfy the
well-known commutation relations
$$[P^{\mu},P^{\nu}]=0, \quad [M^{\mu\nu}, P^{\rho}]= 
-i(g^{\mu\rho}P^{\nu}-g^{\nu\rho}P^{\mu}),$$
\begin{equation}
[M^{\mu\nu},M^{\rho\sigma}]=-i (g^{\mu\rho}M^{\nu\sigma}+
g^{\nu \sigma}M^{\mu\rho}-g^{\mu\sigma}M^{\nu\rho}-g^{\nu\rho}
M^{\mu\sigma})
\label{1.2}
\end{equation}
where $\mu,\nu,\rho,\sigma=0,1,2,3$, $P^{\mu}$ are the 
four-momentum operators, $M^{\mu\nu}$ are the angular momentum
operators, the metric tensor in Minkowski space has the nonzero 
components $g^{00}=-g^{11}=-g^{22}=-g^{33}=1$, and we use the 
system of units with $\hbar=c=1$. The operators $M^{\mu\nu}$ are
antisymmetric: $M^{\mu\nu}=-M^{\nu\mu}$ and therefore there are
only six independent angular momentum operators. 

The question arises whether we can have {\it a direct transition}
from a Galois field based quantum theory (GFQT) to conventional 
Poincare invariant quantum theory when $p\rightarrow \infty$. 
The answer to this question is probably 'no' and the reason is
the following. The GFQT is discrete
and finite because the only numbers it can contain are the
elements of the Galois field in question. These numbers obviously 
have no dimension and all operators in the GFQT cannot have the 
continuous spectrum. In the Poincare 
invariant quantum theory the angular momentum operators are 
dimensionless (if $\hbar=c=1$) but the momentum operators have 
the dimension of the inverse length. In addition, the momentum
operators and the generators of the Lorentz boosts $M^{0i}$ 
$(i=1,2,3)$ contain the continuos spectrum. 

This observation might prompt a skeptical reader to immediately
conclude that any GFQT cannot describe the nature. However a 
simple way out of this situation is as follows.

\begin{sloppypar}
First we recall the well-known fact that the conventional
Poincare invariant theory is a special case of the
de Sitter invariant one. The symmetry group of the de Sitter
invariant quantum theory can be either SO(2,3) or SO(1,4)
(strictly speaking we should take the universal covering groups
but we will not go into details). The group SO(2,3) is the
symmetry group of the four-dimensional manifold in the 
five-dimensional space, defined by the equation
\begin{equation}
 x_5^2+x_0^2-x_1^2-x_2^2-x_3^2=R^2
\label{1.3}
\end{equation}
where a constant $R$ has the dimension of length.
We use $x_0$ to denote the conventional time coordinate and
$x_5$ to denote the fifth coordinate. The notation $x_5$
rather than $x_4$ is used because in the literature the
latter is sometimes used to denote $ix_0$. 
Analogously the group SO(1,4) is the symmetry group of the 
four-dimensional manifold in the five-dimensional space, 
defined by the equation 
\begin{equation}
 x_0^2-x_1^2-x_2^2-x_3^2-x_5^2=-R^2
\label{1.4}
\end{equation}
\end{sloppypar}

The quantity $R^2$ is often written as $R^2=3/\Lambda_c$ where
$\Lambda_c$ is the cosmological constant. The existing astronomical
data show that $\Lambda_c$ is very small and the usual estimates based
on popular cosmological models give $R>10^{26}cm$. On the other
hand, in models based on the de Sitter cosmology the quantity
$R$ is related to the Hubble constant $H$ as $R=1/H$, and in
this case $R$ is of order $10^{27}cm$ \cite{desitt}.

The nomenclature is such that $\Lambda_c < 0$ corresponds to the case
of Eq. (\ref{1.3}) while $\Lambda_c >0$ - to the case of Eq.
(\ref{1.4}). For this reason sometimes only the latter is called the
de Sitter (dS) space while the former is called the anti de Sitter 
(AdS) one. Analogously, some authors prefer to call only SO(1,4)
as the dS group while SO(2,3) is called the AdS one in this case.  
In view of recent cosmological investigations it is now
believed that $\Lambda_c > 0$ (see e.g. Ref. \cite{Perlmutter}).

The both de Sitter groups are ten-parametric, as well as the
Poincare group. However, 
in contrast to the Poincare group, all the representation 
generators of the de Sitter groups are angular momenta, and in 
the units $\hbar=c=1$ they are
dimensionless. The commutation relations now can be written in
the form of one tensor equation
\begin{equation}
[M^{ab},M^{cd}]=-i (g^{ac}M^{bd}+g^{bd}M^{cd}-
g^{ad}M^{bc}-g^{bc}M^{ad})
\label{1.5}
\end{equation}
where $a,b,c,d$ take the values 0,1,2,3,5 and the operators 
$M^{ab}$ are antisymmetric. The diagonal metric tensor has the 
components $g^{00}=-g^{11}=-g^{22}=-g^{33}=1$ as usual,
while $g^{55} =1$ for the group SO(2,3) and $g^{55}=-1$ for the 
group SO(1,4).  

When $R$ is very large then the transition from the de Sitter
symmetry to the Poincare one (this procedure is called 
contraction \cite{IW}) is performed as follows. We introduce 
the operators
$P^{\mu} = M^{\mu 5}/R$. Then, when $R\rightarrow \infty$,
Eq. (\ref{1.5}) splits into the set of expressions given by
Eq. (\ref{1.2}). 

The de Sitter symmetry implies that even an empty space has a
nonzero curvature and this was a subject of the well-known
discussion between Einstein and de Sitter. In the spirit of
general relativity, the existence of the nonzero curvature should
reflect the presence of matter and so one might think that the
Poincare symmetry is more relevant than the de Sitter one.
However, as we discussed above, in quantum theory the space-time
does not have a fundamental meaning and the symmetry is expressed
in terms of the commutation relations between the representation
generators. In quantum theories of gravity, one usually starts from
a Minkowski background, and the cosmological constant is given
by a contribution of certain vacuum diagrams. Then the problem
arises that the result of estimations of $\Lambda_c$ is usually
much greater than its  experimental value. 

If the de Sitter symmetry is taken from the beginning, then all the 
ten generators are
on equal footing, they are dimensionless and Eq. (\ref{1.5})
looks much more attractive than Eq. (\ref{1.2}). The 
{\it conventional} de Sitter invariant quantum theory can be 
constructed in such a way that it operates only with 
dimensionless quantities and does not contain
the parameter $R$ at all (see the next chapter). Roughly speaking
this is a consequence of the fact that all the representation
generators are angular momenta and therefore their action can
be described in terms of angular variables. The parameter $R$
appears in this case only when we contract the dS or AdS group
to the Poincare one. In addition, the {\it conventional} de Sitter
invariant quantum theory can be constructed in such a way (see Chapters
\ref{C2} and \ref{C3}) that all the generators have only the discrete
spectrum. For all these reasons conventional de Sitter
invariant quantum theories are good candidates for generalizing
them to the GFQT (see the preceding section).

In the preceding section we noted that the characteristic of the
Galois fiels $p$ can be treated as a radius of the curvature of 
the number set, and if it is big, we do not notice this curvature.
Analogously we could say that when $R$ is big, we do not notice 
that our space has a curvature. However, as we stressed above, in
quantum theory space-time is not fundamental.

Let us now discuss the following question. If we accept the
de Sitter symmetry from the beginning, then we should accept 
that energy and
momentum are no longer fundamental physical quantities. We can use 
them at certain conditions but only as approximations. The same
is true (even in the greater extent) with respect to the 
energy-momentum tensor. Meanwhile, in General Relativity and 
quantum theories in curved spaces (see e.g. Ref. \cite{BirDav}), 
this tensor is used in situations where the space-time is much 
more complicated than the dS or AdS space. In the literature
there have been a lot of discussions about the role of the 
energy-momentum tensor. It is well-known that Einstein was not
satisfied by the presence of this tensor in his theory. The way
it is used in General Relativity was criticized in papers by
Logunov with coworkers (see e.g. Ref. \cite{Logunov} and references
therein) but it turn Logunov's approach was criticized by other
authors. We will not go into the details of this discussion.  A 
thorough investigation of the validity of energy-momentum tensor
in General Relativity has been carried out by Faddeev \cite{Faddeev}. 
His conclusion is that it has a clear physical meaning at least when 
space-time is asymptotically flat at infinity but in other cases
the question remains open.

In conventional Poincare invariant theories, the energy and momentum
can be written as integrals from energy-momentum tensor over a
space-like hypersurface, while the angular momentum operators can
be written as integrals from the angular momentum tensor over the
same hypersurface. In conventional de Sitter invariant theories 
all the representation generators are angular momenta. Therefore it is
reasonable to think that the energy-momentum tensor in that case is
not needed at all and all the representation generators can be written
as integrals from a five-dimensional angular momentum tensor over some
hypersurface. In Chapter \ref{C4} we discuss this question in detail.

Finally we briefly discuss the differences between SO(2,3) and SO(1,4)
invariant theories. There exists a vast literature devoted to the both
of them. 

The former has many features analogous to those in Poincare
invariant theories. There exist representations of the AdS group
(or its Lie algebra) where the operator $M^{05}$ (which is the de Sitter
analog of the energy operator) is bounded below by some positive
value which can be treated as the AdS analog of the mass (see also
Chap. \ref{C2}). With such an interpretation, a system of free particles
with the AdS masses $m_1,m_2...m_n$ has the minimum energy equal
to $m_1+m_2+...m_n$. There also exist representations with negative
energies where the energy operator is bounded above. In the 
standard approach to the LQFT they can be used for the description of 
antiparticles. SO(2,3) invariant theories can be easily generalized
if supersymmetry is required while supersymmetric generalizations
of SO(1,4) invariant theories is problematic. 
In SO(2,3) invariant theories there is no S-matrix in its usual 
sense but there exists a variable (see Chap. \ref{C3}) which 
has many common features with conventional time.     

On the contrary, the latter has many unusual features. For 
example, even in
the representation describing a free particle, the operator 
$M_{05}$ (which is
the SO(1,4) analog of the energy operator) has the  
spectrum in the
interval $(-\infty,+\infty)$. The dS mass operator of the 
system of free
particles with the dS masses  $m_1,m_2...m_n$  is not 
bounded below by
the value of  $m_1+m_2+...m_n$ and also has the spectrum in the 
interval $(-\infty,+\infty)$ \cite{lev1,lev2}. 

For these and other reasons there existed the opinion that de 
Sitter invariant
quantum theories can be based only on the group SO(2,3) and not 
SO(1,4).
However, as noted above, in view of recent cosmological observations 
it is now believed that $\Lambda_c > 0$, and the opinion has been 
dramatically changed. As noted in our papers \cite{lev1,lev2,lev3} 
(done before these developments), SO(1,4) invariant theories 
have many 
interesting properties, and the fact that they are unusual does 
not mean 
that they contradict the experiment. For example, a particle 
described by a free-particle representation has only an 
exponentially small probability to 
have the energy less than its mass (see e.g. Ref. \cite{lev2}). 
The fact that the mass operator is not
bounded below by  the value of  $m_1+m_2+...m_n$ poses an 
interesting
question whether some interactions (even gravity) can be a direct 
consequence of
the de Sitter symmetry \cite{lev2,lev3}. A very interesting property of 
the SO(1,4) invariant theories is that they do not contain bound states 
at all and, as a consequence, the free and interacting operators are 
unitarily equivalent \cite{lev3}. This poses the problem
whether the notion of interactions is needed at all. 

As argued by Witten \cite{Witten}, the Hilbert space for quantum 
gravity in the
dS space is finite-dimensional and therefore there are problems in
quantizing General Relativity in that space (because any unitary
representation of the SO(1,4) group is necessarily 
infinite-dimensional).
However this difficulty disappears in our approach since 
linear spaces in Galois field quantum theories are 
finite-dimensional. 

In the present paper the conventional quantum theory is 
considered by using the standard approach because our main goal 
is to discuss how the GFQT becomes the conventional theory in 
the limit $p\rightarrow \infty$. For this
reason we consider only SO(2,3) invariant theories. The problem
of the Galois field generalization of SO(1,4) invariant theories 
has been discussed in Ref. \cite{lev2}.   
    
The paper is organized as follows. In Chap. \ref{C2} we 
discuss in detail how to
describe free elementary particles in the theory based on 
a Galois field and how this
description becomes the standard one in the limit when the 
characteristic of the Galois field in question goes to infinity.
In the standard approach one also needs the description of
elementary particles in terms of wave functions on the AdS
space. Such a description is discussed in Chapt. \ref{C3}.
Finally, in Sect. \ref{C4} it is shown how to construct the
GFQT if the representation
generators of the standard theory are known. For illustration
we consider the case of QED because this is the most known 
quantum theory, but the way of generalizing any quantum theory
to the corresponding GFQT is rather general. 


\vfill\eject

\chapter{Elementary particles in Galois field based theories}
\label{C2}

\section{Modular and conventional representations of Lie algebras}
\label{S2.1}

In conventional quantum theory the state of a system is 
described by a vector $\tilde x$ from a separable Hilbert 
space $H$. We will use a "tilde"  to denote elements
of Hilbert spaces and complex numbers while elements
of linear spaces over Galois fields and elements of the
fields will be denoted without a "tilde". 

Let $(\tilde e_1,\tilde e_2,...)$ be a basis in $H$. This
means that $\tilde x$ can be represented as
\begin{equation}
\tilde x =\tilde c_1 \tilde e_1+\tilde c_2 \tilde e_2+...
\label{eq2.1}
\end{equation}
where $(\tilde c_1,\tilde c_2,...)$ are complex numbers. It is assumed
that there exists a complete set of commuting selfadjoint 
operators $(\tilde A_1,\tilde A_2,...)$ in $H$ such that
each $\tilde e_i$ is the eigenvector of all these operators:
$\tilde A_j\tilde e_i =\tilde \lambda_{ji}\tilde e_i$. Then the
elements $(\tilde e_1,\tilde e_2,...)$ are mutually orthogonal:
$(\tilde e_i,\tilde e_j)=0$ if $i\neq j$ where (...,...) is
the scalar product in $H$. In that case the coefficients can
be computed as
\begin{equation}
\tilde c_i = \frac{(\tilde e_i,\tilde x)}{(\tilde e_i,\tilde e_i)}
\label{eq2.2}
\end{equation}
Their meaning is that 
$|\tilde c_i|^2(\tilde e_i,\tilde e_i)/(\tilde x,\tilde x)$
represents the probability to find $\tilde x$ in the state
$\tilde e_i$. In particular, when $\tilde x$ and the basis
elements are normalized to one, the probability is exactly
equal to $|\tilde c_i|^2$.

Let us note first that the Hilbert space contains a big
redundancy of elements, and we do not need to know all of
them. Indeed, with any desired accuracy we can approximate
each $\tilde x\in H$ by a finite lnear combination
\begin{equation}
\tilde x =\tilde c_1 \tilde e_1+\tilde c_2 \tilde e_2+...
\tilde c_n\tilde e_n 
\label{eq2.3}
\end{equation}
where $(\tilde c_1,\tilde c_2,...\tilde c_n)$ are
rational complex numbers. In turn, the set Eq. (\ref{eq2.3})
is reduntant too. Indeed, we can use the fact that Hilbert
spaces in quantum theory are projective: $\psi$ and 
$c\psi$ represent the same physical state. Then we can
multiply both parts of Eq. (\ref{eq2.3}) by a common
denominator of the numbers 
$(\tilde c_1,\tilde c_2,...\tilde c_n)$. As a result, we
can always assume that in Eq. (\ref{eq2.3}) 
$\tilde c_j=\tilde a_j +i\tilde b_j$ where $\tilde a_j$
and $\tilde b_j$ are integers. If it is convenient, we can
approximate all physical states by another sets of elements.
For example, we can use Eq. (\ref{eq2.3}), where
$\tilde c_j=\tilde a_j +\sqrt{2}i\tilde b_j$ and
$\tilde a_j$ and $\tilde b_j$ are integers. 

The meaning of the fact that Hilbert
spaces in quantum theory are projective is very clear.
The matter is that not the probability itself but the
relative probabilities of different measument outcomes
have a physical meaning. 
We believe, the notion of probability is a
good illustration of the Kronecker expression about
integers (see Sect. \ref{S1.2}). Indeed, this notion 
arises as follows. Suppose that conducting 
experiment $n$ times we have seen the first event $n_1$ times, 
the second event $n_2$ times etc. such that $n_1 + n_2 + ... = n$. 
We introduce the quantities $w_i(n) = n_i/n$ (these quantities 
depend on $n$) and $w_i = lim\, w_i(n)$ when $n \rightarrow 
\infty$. Then $w_i$ is called the probability of the 
$ith$ event. We see that 
all information about the experiment under consideration is 
given by a set of integers (and even natural numbers). 
However, in order to define 
probability, people introduce additionally the notion of 
rational numbers and the notion of limit. Of course, we can 
use conventional probability even if quantum theory is based 
entirely on integers, but by doing so we should realize that 
it is only a convenient (or common?) way to describe the 
measurement outcomes.

The Hilbert space is an example of a linear space over the
field of complex numbers. Roughly speaking this means that
one can multiply the elements of the space by the elements 
of the field and use the properties
$\tilde a(\tilde b\tilde x)=(\tilde a\tilde b)\tilde x$
and $\tilde a(\tilde b\tilde x+\tilde c\tilde y)= 
\tilde a\tilde b\tilde x +\tilde a\tilde c\tilde y$ where
$\tilde a,\tilde b,\tilde c$ are complex numbers and
$\tilde x,\tilde y$ are elements of the space. The fact that 
complex numbers form a field is important for such notions
as linear dependence and the dimension of spaces over complex
numbers. 

In general, it is possible to consider linear
spaces over any fields (see any textbook on modern
algebra, e.g. \cite{VDW}). It is natural to assume that
in the GFQT the states of physical
systems should be described by elements of linear spaces over
a Galois field. Since we wish to have a correspondence
with the conventional theory, we assume that the Galois field
in question contains $p^2$ elements where $p$ is the
characteristic of the field. In Sect. \ref{S1.2} we discussed
the correspondence between the ring of integers and
the field $F_p$. It has been also noted that if 
$p=3\,(mod\,4)$ then the elements of $F_{p^2}$ can be
written as $a+bi$ where $a,b\in F_p$. We will now discuss
the construction of the field $F_{p^2}$ in more general
cases.

The field $F_{p^2}$ can be constructed by means of the 
standard extension of the field $F_p$. Let the equation
$\kappa^2=-a_0$ $(a_0\in F_p)$ have no solutions in $F_p$.
Then $F_{p^2}$ can be formally described as a set of
elements $a+b\kappa$, where $a,b\in F_p$ and $\kappa$ 
satisfies the condition $\kappa^2=-a_0$. The actions in
$F_{p^2}$ are defined in the natural way. The condition
that the equation $\kappa^2=-a_0$ has no solutions in 
$F_p$ is important in order to ensure that any nonzero 
element from $F_{p^2}$ has an inverse.
Indeed, the definition 
$(a+b\kappa)^{-1}=(a-b\kappa)/(a^2+b^2a_0)$ is correct 
since the denominator can be equal to zero only if both,
$a$ and $b$ are equal to zero.

Consider three cases.

(1) $a_0=1$. In that case it is natural to write $i$
instead of $\kappa$. We must be sure that the element -1
in $F_p$ cannot be represented as a square of another
element from $F_p$. In the number theory this is
expressed by saying that -1 is not a quadratic residue
in $F_p$. It is well known (see e.g. Ref. \cite{VDW,IR})
that this is the case when $p=3\,(mod\,4)$ and is not
the case when $p=1\,(mod\,4)$. This fact is closely 
related to one mentioned in Sect. \ref{S1.2} that a
prime $p$ can be represented as a sum of two squares 
only if $p=1\,(mod\,4)$. 

(2) $a_0=2$. Assume that $p=1\,(mod\,4)$. Since -1 is
a quadratic residue in $F_p$ in this case, then the problem
is that 2 should not be a quadratic residue in this field.
This is the case (see e.g. Ref. \cite{IR}) 
if $p=5\,(mod\,8)$.

(3) $a_0=3$. Again, assume that $p=1\,(mod\,4)$. 
Then 3 should not be a quadratic residue in $F_p$. 
This is, for example, the case (see e.g. Ref. 
\cite{IR}),
when $p$ is the prime of Fermat's type, i.e. $p=2^n+1$.

In the general case, the field $F_p$ can be extended to a field
containing $p^2$ elements as follows. First we note a well known
fact \cite{VDW,IR} that $F_p$ is a cyclic group with respect
to multiplication. There exists at least one element $r\in F_p$
such that $(r,r^2,...r^{p-1}=1)$ represent the set $(1,2,...p-1)$
in some order. The element $r$ is called the primitive root of
the field $F_p$. It is clear from this observation that $F_p$ 
contains the equal number $(p-1)/2$ of quadratic residues and
nonresidues: the elements represented as odd powers of $r$ are
nonresidues while those represented by even powers of $r$ ---
residues. In particular, $r$ is not a quadratic residue. Now
we can formally introduce $\kappa$ as an element satisfying
the condition $\kappa^2=r$, and $F_{p^2}$ as a set of
elements $a+b\kappa$ $(a,b\in F_p)$. In this case the
definition
$(a+b\kappa)^{-1}=(a-b\kappa)/(a^2-b^2r)$ is correct since
$a^2-b^2r\neq 0$ if $a\neq 0$ or $b\neq 0$.

Consider, for example, the case $p=7$. Then $r=3$ is the
primitive root in $F_p$. We can write the elements of $F_{p^2}$
as $a+bi$ or $a+b\kappa$ where $a,b\in F_p$ and $\kappa^2=3$.
The relation between $\kappa$ and $i$ is $\kappa = 2i$. Indeed,
$\kappa^2=3$ and $(2i)^2=-4=3$ in $F_p$.  

The above observation shows that if $F_p$ is extended to
$F_{p^2}$ then any element of $F_p$ has a square root
belonging to $F_{p^2}$. 

It is well known (see e.g. Ref. \cite{VDW,IR}) that the 
field of $p^2$ elements has only one nontrivial
automorphism. In the above cases it can be defined as
$a+b\kappa\rightarrow a-b\kappa$ or as 
$a+b\kappa\rightarrow (a+b\kappa)^p$. We will use a bar
to denote this automorphism. This means that if
$c=a+b\kappa$ $(a,b\in F_p)$ then $\bar{c}=a-b\kappa$.

By analogy with conventional quantum theory, we
require that linear spaces V over $F_{p^2}$, used for
describing physical states, are supplied by a
scalar product (...,...) such that for any $x,y\in V$
and $a\in F_{p^2}$, $(x,y)$ is an element of $F_{p^2}$
and the following properties are satisfied: 
\begin{equation}
(x,y) =\overline{(y,x)},\quad (ax,y)=\bar{a}(x,y),\quad 
(x,ay)=a(x,y)
\label{eq2.4}
\end{equation}  

In Sect. \ref{S1.2} we discussed a map $f$ from $F_p$
to the ring of integers $Z$. We can extend the map in such
a way that it maps $F_{p^2}$ to a subset of complex numbers.
For example, if $p=3\,(mod\,4)$, $c\in F_{p^2}$, $c=a+bi$
$(a,b\in F_p)$ then we can define $f(c)$ as $f(a)+f(b)i$.
If $a_0=-2$ and $c=a+b\kappa$, we define 
$f(c)=f(a)+\sqrt{2}f(b)i$. Analogously, if $a_0=-3$ and 
$c=a+b\kappa$, we define $f(c)=f(a)+\sqrt{3}f(b)i$.
Let $U$ be a subset of elements $c=a+b\kappa$ where 
$a,b\in U_0$ (see Sect. \ref{S1.2}). Then, for the elements
from $U$, addition, subtraction and multiplication look 
the same as for the
elements $f(c)$ of the corresponding subset in the field of 
complex numbers. This subset is of the form $\tilde a +
\tilde b i$ in the first case, $\tilde a + \sqrt{2}
\tilde b i$ in the second case and $\tilde a + \sqrt{3}
\tilde b i$ in the third one. Here $\tilde a, \tilde b \in Z$.    

We will always consider only finite dimensional spaces $V$ over
$F_{p^2}$. Let $(e_1,e_2,...e_N)$ be a basis in such a space. 
Consider subsets in $V$ of the form $x=c_1e_1+c_2e_2+...c_ne_n$ 
where for any $i,j$
\begin{equation}
 c_i\in U,\quad (e_i,e_j)\in U
\label{eq2.5}
\end{equation}
On the other hand, as noted above, in conventional quantum
theory we can describe quantum states by subsets of the form   
Eq. (\ref{eq2.3}). If $n$ is much less than $p$,
\begin{equation}
f(c_i)=\tilde c_i,\quad f((e_i,e_j))=(\tilde e_i,\tilde e_j)
\label{eq2.6}
\end{equation}
then we have the correspondence between the description of
physical states in projective spaces over $F_{p^2}$ on the
one hand and projective Hilbert spaces on the other. This means
that if $p$ is big, then for a big number of elements from $V$,
linear combinations with the coefficients belonging to $U$ and
scalar products look in the same way as for the elements from
a corresponding subset in the Hilbert space. 

In the general case a scalar product in $V$ does not define
any positive definite metric, and thus there is no
probabilistic interpretation for all elements from $V$. 
In particular, $(e,e)=0$ does not necessarily imply that $e=0$. 
However, the probabilistic interpretation exists for such
subsets that the conditions (\ref{eq2.6}) are satisfied.
In Sects. \ref{S2.2} and \ref{S2.3} we will explicitly construct 
a basis $(e_1,...e_N)$ such that $(e_j,e_k)=0$ for $j\neq k$ and
$(e_j,e_j)\neq 0$ for all $j$. If $x=c_1e_1+...c_Ne_N$ $(c_j\in
F_{p^2})$ then the coefficients are uniquely defined by 
$c_j=(e_j,x)/(e_j,e_j)$.

 As usual, if $A_1$ and $A_2$ are linear operators in $V$ such that
\begin{equation}
(A_1x,y)=(x,A_2y)\quad \forall x,y\in V
\label{eq2.7}
\end{equation}
they are said to be conjugated: $A_2=A_1^*$. It is easy to see
that $A_1^{**}=A_1$ and thus $A_2^*=A_1$. If $A=A^*$ then the
operator $A$ is said to be Hermitian.

 If $(e,e)\neq 0$, $Ae=ae$, $a\in F_{p^2}$, and $A^*=A$, then it
is obvious that $a\in F_p$. At the same time, there exist 
possibilities (see e.g. Ref. \cite{lev4}) when $e\neq 0$, 
$(e,e)=0$ and the element $a$ is imaginary, i.e. $a=b\kappa$, 
$b\in F_p$. Further, if
\begin{eqnarray}
&&Ae_1=a_1e_1,\quad Ae_2=a_2e_2,\quad (e_1,e_1)\neq 0,\nonumber\\
&& (e_2,e_2)\neq 0, \quad a_1\neq a_2
\label{eq2.8}
\end{eqnarray}
then as in the usual case, one has $(e_1,e_2)=0$. At the same time,
the situation when
\begin{eqnarray}
&&(e_1,e_1)=(e_2,e_2)=0,\quad (e_1,e_2)\neq 0,\nonumber\\
&& a_1=b_1\kappa, \quad a_2=b_2\kappa\quad (b_1,b_2\in F_p)
\label{eq2.9}
\end{eqnarray}
is also possible \cite{lev4}.

Let now $(A_1,...A_k)$ be a set of Hermitian commuting operators
in $V$, and $(e_1,...e_N)$ be a basis in $V$ with the  properties
described above, such that $A_je_i=\lambda_{jl}e_i$. Further, let
$({\tilde A}_1,...{\tilde A}_k)$ be a set of Hermitian  commuting
operators  in  some  Hilbert  space  $H$,   and
$(\tilde e_1,\tilde e_2,...)$ be some basis in $H$ such that
$\tilde A_je_i={\tilde \lambda}_{ji}\tilde e_i$.
Consider a subset $c_1e_1+c_2e_2+...c_ne_n$ in $V$ such that,
in addition to the conditions (\ref{eq2.6}), the elements $e_i$
are the eigenvectors of the operators $A_j$ with $\lambda_{ji}$
belonging to $U$ and such that 
$f(\lambda_{ji})={\tilde \lambda}_{ji}$. Then the action of the 
operators on such
elements have the same form as the action of corresponding 
operators on the subsets of elements in Hilbert spaces discussed
above. 

Summarizing this discussion, we conclude that if $p$ is big then
there exists a correspondence between the description of physical
states on the language of Hilbert spaces and selfadjoint operators
in them on the one hand, and on the language of linear spaces over
$F_{p^2}$ and Hermitian operators in them on the other (in 
Hilbert spaces a Hermitian operator is not necessarily selfadjoint, 
but we will not discuss this question). 

The field of complex numbers is algebraically closed (see 
standard textbooks on modern algebra, e.g. Ref. \cite{VDW}).
This implies that any equation of the $nth$ order in this field
always has $n$ solutions. This is not, generally speaking, the
case for the field $F_{p^2}$. As a consequence, not every linear
operator in the finite-dimensional space over $F_{p^2}$ has an 
eigenvector (because the characteristic equation may have no 
solution in this field). One can define a field of
characteristic $p$ which is algebraically closed and contains
$F_{p^2}$. However such a field will necessarily be infinite and 
we will not use it. We will see in Sects. \ref{S2.2} and \ref{S2.3},
that uncloseness of the field $F_{p^2}$ does not prevent one
from constructing physically meaningful representations
describing elementary particles in the GFQT.

In physics one usually considers Lie  algebras  over  $R$ and
their representations by  Hermitian  operators  in
Hilbert  spaces.  It  is  clear   that   the   analogs   of   such
representations  in  our  case  are  the  representations  of  Lie
algebras over $F_p$  by Hermitian operators in spaces over $F_{p^2}$.
Representations  in  spaces  over  a  field   of   nonzero
characteristics are called modular representations.
There exists wide literature devoted to such representations;
detailed references can be found for example in Ref.
\cite{FrPa} (see also Ref. \cite{lev4}). In particular, it has
been shown by Zassenhaus \cite{Zass} that all modular irreducible
representations (IRs) are finite-dimensional and many papers have
dealt with the maximum dimension of such representations.
For classical Lie algebras this has been done in Refs.
\cite{BN,Zass,Rud,Rud1,Kac}. The complete classification of IRs
has been obtained only for $A_1$ algebras \cite{Rud}; important results
for $A_2$ and $B_2$ algebras are obtained in Refs. \cite{Braden,Jan};
results for $A_n$ algebras can be found in Refs. \cite{Jan,Pan,Dip},
and results obtained for some other algebras can be found in Ref.
\cite{Jan}. At the same time, it is worth noting that usually 
mathematicians consider only representations
over an algebraically closed field. 

From  the previous, it is natural to expect that the
correspondence  between
ordinary and modular representations of two Lie algebras over $R$ 
and $F_p$, respectively, can be obtained if the structure 
constants of the Lie algebra
over $F_p$ - $c_{kl}^j$, and the structure constants of the Lie
algebra over $R$ - ${\tilde c}_{kl}^j$, are connected by the relations
$f(c_{kl}^j)={\tilde c}_{kl}^j$ (the Chevalley basis \cite{Chev}), and
all the $c_{kl}^j$ belong to $U_0$. In Refs. \cite{lev4,lev2} 
there have been
considered modular analogs of irreducible representations
of su(2), sp(2), so(2,3), so(1,4) algebras and the osp(1,4)
superalgebra. Also modular representations describing strings
have been briefly mentioned. In all these cases the quantities 
${\tilde c}_{kl}^j$ take only the values $0,\pm 1,\pm 2$ and there 
have been shown that the correspondence under consideration does 
have a place.

In this paper we need the modular analogs of IRs of su(2), sp(2)
and so(2,3) algebras. Since we want to make this paper 
self-contained as possible, we describe these representations 
in detail. 

\section{Modular representations of algebras su(2) and sp(2)}
\label{S2.2}

Let ${\bf L}=(L_1, L_2, L_3)$ be the operator of ordinary
three-dimensional rotations. The eigenvalues of the operators
$L_i$ $(i=1,2,3)$ are always multiple to $\hbar /2$. Therefore,
if $\hbar /2$ rather than $\hbar$ is taken as a unit of measurement
of angular momentum then the commutation relations for the
components of ${\bf L}$ have the form
\begin{equation}
[L_1,L_2]=2iL_3,\quad [L_3,L_1]=2iL_2,\quad [L_2,L_3]=2iL_1
\label{2.10}
\end{equation}

We can introduce the operators $L_{\pm}$ such that
\begin{equation}
L_1 = L_++L_- \quad L_2 = -i(L_+-L_-)
\label{2.11}
\end{equation}
Then Eq. \ref{2.10} can be rewritten as
\begin{equation}
[L_3,L_+]=2L_+\quad [L_3,L_-]=-2L_-\quad [L_+,L_-]=L_3
\label{2.12}
\end{equation}
The operator ${\bf L}$ should be selfadjoint, and this implies
that $L_3$ is selfadoint while $L_+^*=L_-$.

If $p=3\,(mod\, 4)$ then we can consider the implementation of
these relations in the modular case. However, if $p=1\, (mod\, 4)$
then the problem arises what are the modular analogs of Eqs. 
(\ref{2.10}) and (\ref{2.11}) because the imaginary unity $i$ 
does not exist in this case (see the preceding section). However 
there is no problem with the modular analog of Eqs. (\ref{2.12}),
and for constructing modular IRs we need just this relation.
Our goal now is to construct IRs of the relations (\ref{2.12})
in spaces $F_{p^2}$. We will see that the construction does not
depend on the fact that the representation space is $F_{p^2}$,
it can be $F_{p^n}$ with any $n\geq 1$. We need $F_{p^2}$ only
for correspondence with conventional quantum theory.

As follows from Eq. (\ref{2.12}), the operator 
\begin{equation}
K=L_3^2-2L_3+4L_+L_-=L_3^2+2L_3+4L_-L_+
\label{2.13}
\end{equation}
is the Casimir operator for algebra ${\bf L}$, i.e. $K$ commutes
with all the components of ${\bf L}$. It is known that in the 
ordinary case all the Casimir operators are functions of $K$
but in the modular case this is not so (see e.g. Ref. 
\cite{Kozerenko}).

Consider only the representations containing a vector 
$e_0$ such that
\begin{equation}
L_+e_0=0,\quad L_3e_0=se_0,\quad (e_0,e_0)\neq 0
\label{2.14}
\end{equation}
Then (see Sect. \ref{S2.1}) $s\in F_p$ and, as follows from
Eq. (\ref{2.13}), $e_0$ is the eigenvector of the operator
$K$ with the eigenvalue $s(s+2)$: $Ke_0=s(s+2)e_0$. This result
differs from the well known one $s(s+1)$ because we have chosen
$\hbar /2=1$. 

Denote $$e_n=(L_-)^ne_0,\quad n=0,1,2...$$
Since $K$ is the Casimir operator, all the $e_n$ are its
eigenvectors with the same eigenvalue $s(s+2)$, and, as follows
from Eq. (\ref{2.12}), $L_3e_n=(s-2n)e_n$ for any $n$. Hence it 
follows from Eq. (\ref{2.13}) that
\begin{equation}
L_+L_-e_n=(n+1)(s-n)e_n,
\label{2.15}
\end{equation}
and since $L_+^*=L_-$ it follows from this expression that
\begin{equation}
(e_{n+1},e_{n+1})=(n+1)(s-n)(e_n,e_n)
\label{2.16}
\end{equation} 
It is clear from this expression that for $n=0,1,...s$ the
vectors $e_n$ are mutually orthogonal and $(e_n,e_n)\neq 0$.
If we also require $L_-e_s=0$ then the representation is
irreducible (as well as in the ordinary case) and its dimension
is equal to $s+1$ (the ordinary result is $2s+1$ because in the
ordinary case $\hbar = 1$). There exists the general statement
\cite{Rud2} that the maximum dimension of modular IR of 
classical Lie algebra of dimension $n$ and rank $r$ is equal to
$p^{(n-r)/2}$. In the given case $(n=3,\, r=1)$ it is equal to 
$p$. This "nonphysical" IR corresponds to the choice $s=p-1$,
i.e. $s=-1$ in $F_p$. 

We conclude that the construction of IRs of the su(2) algebra
in the modular case is almost fully identical to that in the
ordinary one. The only difference is that in the ordinary case
$s$ can be any natural number while in the modular case $s$
can have only values of $0,1,...p-1$. If $s\in U_0$ (and thus
$s$ is much less than $p$) then we have the full correspondence
between ordinary and modular IRs of the su(2) algebra in the
sense of Sect. \ref{S2.2}.

Our choice $\hbar /2=1$ implies that bosons are described by
representations with even $s$ while fermions --- with odd ones.
Then we have the correspondence between ordinary and modular
representations for both, bosons and fermions. It would not
be the case if $\hbar =1$ because 1/2 in the field $F_p$ is
a very big number $(p+1)/2$. 

We now consider the modular analogs of representations of the
sp(2) algebra. Let $a',a",h$ be such operators that
\begin{equation}
[h,a']=-2a'\quad [h,a"]=2a"\quad [a',a"]=h
\label{2.17}
\end{equation}
$h^*=h$ and $(a')^*=a"$. The Casimir operator of the second order
now has the form
\begin{equation}
K=h^2-2h-4a"a'=h^2+2h-4a'a"
\label{2.18}
\end{equation}

We will consider representations with the vector $e_0$, such that
\begin{equation}
a'e_0=0,\quad he_0=q_0e_0,\quad (e_0,e_0)\neq 0 
\label{2.19}
\end{equation} 
Then (see Sect. \ref{S2.1}) $q_0\in F_p$. Denote $e_n =(a")^ne_0$.
Then it follows from Eqs. (\ref{2.17}) and (\ref{2.18}),
that for any $n=0,1,2,...$
\begin{equation}
he_n=(q_0+2n)e_n,\quad Ke_n=q_0(q_0-2)e_n, 
\label{2.20}
\end{equation} 
and instead of Eqs. (\ref{2.15}) and (\ref{2.16}) we have
\begin{eqnarray}
&&a'a"e_n=(n+1)(q_0+n)e_n\nonumber\\
&&(e_{n+1},e_{n+1})=(n+1)(q_0+n)(e_n,e_n)
\label{2.21}
\end{eqnarray}

The case $q_0=0$ is trivial and corresponds to zero representation,
so we assume that $q_0\neq 0$. Then we have the case when ordinary 
and modular representations considerably differ each other. 
Consider first the ordinary case when $q_0$ is any real positive 
number. Then IR is infinite-dimensional, $e_0$ is a vector
with a minimum eigenvalue of the operator $h$ (minimum weight)
and there are no vectors with the maximum weight. This is in 
agreement with the well known fact that unitary IRs of
noncompact groups are infinite dimensional. However in the
modular case $q_0$ is one of the numbers $1,...p-1$. 
The set $(e_0,e_1,...e_N)$ will be a basis
of IR if $a"e_i\neq 0$ for $i<N$ and $a"e_N=0$. These conditions
must be compatible with $a'a"e_N=0$. Therefore, as follows from
Eq. (\ref{2.21}), $N$ is defined by the condition $q_0+N=0$ in
$F_p$. As a result, $N=p-q_0$ and the dimension of IR is equal
to $p-q_0+1$. The maximum dimension of modular IR is again 
equal to $p$ as it should be because the $sp(2)$ algebra has the 
dimension 3 and rank 1. 

In contrast to the standard case, where the eigenvalue of the 
Casimir operator $K$ fully characterizes IR in question, in the
modular case this is not so. Indeed, IRs with minimum weights
$q_0$ and $p+2-q_0$ has the same value of $K$ but different
dimensions.

It is interesting to derive the explicit relation between 
$(e_0,e_0)$ and $(e_N,e_N)$. As follows from Eq. (\ref{2.21}),
\begin{equation}
(e_N,e_N)=N!(q_0+N-1)!(e_0,e_0)/(q_0-1)!
\label{Wilson1}
\end{equation}
Since $q_0+N=p$ and $(p-1)!=-1$ in $F_p$ (this is the well
known Wilson theorem \cite{VDW,IR}), it follows from Eq. 
(\ref{Wilson1}) that
\begin{equation}
(e_N,e_N)=(-1)^{q_0-1}(e_0,e_0)/[(q_0-1)!]^2
\label{Wilson2}
\end{equation}

One might say that $e_0$ is the vector with the
minimum weight while $e_N$ is the vector with the maximum weight.
However the notions of "less than" or "greater than" have only
a limited sense in $F_p$. It is easy to see that $e_N$ is the
eigenvector of the operator $h$ with the eigenvalue $-q_0$ in
$F_p$. If $q_0\in U$ and $f(q_0)>0$, then we can treat $q_0$ as
a positive number such that $q_0$ is much less than $p$. In this
case, for linear combinations $c_1e_1+c_2e_2+...c_ne_n$
we have a correspondence with the ordinary representation
if $n<< p$ and $c_i\in U$ for all $i=1,2,...n$. In particular,
as follows from Eq. (\ref{2.21}), $(n!)^2$ should be much
less than $p$. For example, we can choose $(e_0,e_0)=1$, 
$n<(lnp)^{1/4}$ and then we
will have the correspondence in the sense of Sect. \ref{S2.1}. 
It is clear that the correspondence takes place only for a 
small subset of the full representation space. However, if
$p$ is very big and the existing AdS energies are
much less than $p$, then this "small subset" (which in fact
contains a very big number of elements) is sufficient for
describing experimental data, and the results in ordinary and
modular cases are indistinguishable. The fact that $q_0$ in
the ordinary case is not necessarily integer is of no 
importance because the quantum of energy in usual units is
$\hbar /R$. This is so tiny number that no experiment can
determine whether the value of $q_0$ is exactly integer or not. 

The existing approach in the LQFT is such
that antiparticles are described by representations with 
negative energies. In the conventional approach negative 
energies have no direct physical
meaning but the wave functions with negative energies are
used in the technique of second quantization (see also Chapter
\ref{C4}). We can construct ordinary IRs of sp(2) algebra
with negative energies by analogy with the above construction,
but now the cyclic vector $e_0'$ should be chosen in such a 
way that 
\begin{equation}
he'_0=-q_0e'_0,\quad a"e'_0=0,\quad (e'_0,e'_0)\neq 0
\label{2.22}
\end{equation} 
$(q_0>0)$ and, $e'_n$ should be defined as $e'_n=(a')^ne'_0$. 

Analogously we can construct modular IRs which correspond
to ordinary IRs with negative energies. Now we should treat
$q_0$ as an element of $F_p$. If $q_0\in U_0$
then for linear combinations $c_1e'_1+c_2e'_2+...c_ne'_n$
we have the correspondence with the ordinary case if
$n<<p$, for example if $n<(lnp)^{1/4}$. However, this
modular IR is the same as constructed above.
The proof is as follows.

The modular IR constructed above is fully defined by a
vector $e_0$ and a number $q_0\in F_p$ satisfying Eq. (\ref{2.19}).
If we show that IR defined by Eq. (\ref{2.22}) contains a
vector satisfying Eq. (\ref{2.19}) then IRs defined by Eqs.
(\ref{2.19}) and (\ref{2.22}) are the same. By analogy with
the above construction, one can show that the vector $e'_N$
indeed satisfies Eq. (\ref{2.19}).  
We see that, while in the ordinary case, IRs with positive and
negative energies are different, in the modular case they
the same modular IR. Let us note that for the group SO(1,4) 
ordinary IRs for particles and antiparticles are the same but the 
space-time description of
particles and antiparticles is different 
(see e.g. Ref. \cite{Mensky}). In the next section we use
modular IRs of the sp(2) algebra for constructing 
modular IRs of the so(2,3) algebra and discuss in
detail the fact that modular "positive energy" and "negative
energy" IRs are the same.   

\section{Modular representations of the AdS algebra so(2,3)}
\label{S2.3}

Since we now assume that $\hbar /2=1$ (see the preceding section)
then, instead of Eq. \ref{1.5}, the commutation relations 
for the representation generators of the so(2,3) algebra have
the form 
\begin{equation}
[M^{ab},M^{cd}]=-2i (g^{ac}M^{bd}+g^{bd}M^{cd}-
g^{ad}M^{bc}-g^{bc}M^{ad})
\label{2.23}
\end{equation}

As noted in Sect. \ref{S2.1}, in the modular case one can 
directly use this relation in $F_{p^2}$ only if 
$p=3\, (mod\, 4)$. However, 
instead of the ten operators $M^{ab}$ we will work with another
set of ten operators. Let $(a_j',a_j",h_j)$ $(j=1,2)$ be two
independent sets of operators satisfying the commutation 
relations (\ref{2.17}). They are independent in the sense that
for different $j$ they mutually commute with each other. 
We denote additional four operators as $b', b",L_+,L_-$.
The meaning of $L_+,L_-$ is as follows. The operators 
$L_3=h_1-h_2,L_+,L_-$ satisfy the commutation relations
(\ref{2.12}). The other commutation relations are as follows:
\begin{eqnarray}
&[a_1',b']=[a_2',b']=[a_1",b"]=[a_2",b"]=\nonumber\\
&[a_1',L_-]=[a_1",L_+]=[a_2',L_+]=[a_2",L_-]=0\nonumber\\
&[h_j,b']=-b'\quad [h_j,b"]=b"\quad 
[h_1,L_{\pm}]=\pm L_{\pm},\nonumber\\
&[h_2,L_{\pm}]=\mp L_{\pm}\quad [b',b"]=h_1+h_2\nonumber\\
&[b',L_-]=2a_1'\quad [b',L_+]=2a_2'\quad [b",L_-]=-2a_2"\nonumber\\
&[b",L_+]=-2a_1",\quad [a_1',b"]=[b',a_2"]=L_-\nonumber\\
&[a_2',b"]=[b',a_1"]=L_+,\quad [a_1',L_+]=[a_2',L_-]=b'\nonumber\\
&[a_2",L_+]=[a_1",L_-]=-b"
\label{2.24}
\end{eqnarray}  

At first glance these relations might seem to be rather 
chaotic but in fact they are very natural in the Weyl basis
of the so(2,3) algebra. For constructing modular IRs we need just 
these relations, and the way of constructing IRs applies for 
spaces over $F_{p^n}$
with any $n\geq 1$. We need the space $F_{p^2}$ only for
correspondence with conventional representations describing
elementary particles in the AdS space. 
If $p=3\, (mod\, 4)$ then the relation between the both sets of 
operators is as follows
\begin{eqnarray}
&M_{10}=i(a_1"-a_1'-a_2"+a_2')\quad M_{15}=a_2"+a_2'-a_1"-a_1'\nonumber\\
&M_{20}=a_1"+a_2"+a_1'+a_2'\quad M_{25}=i(a_1"+a_2"-a_1'-a_2')\nonumber\\
&M_{12}=L_3\quad M_{23}=L_++L_-\quad M_{31}=-i(L_+-L_-)\nonumber\\
&M_{05}=h_1+h_2\quad M_{35}=b'+b"\quad M_{30}=-i(b"-b')
\label{2.25}
\end{eqnarray}

There exists a vast literature on ordinary IRs of the so(2,3) algebra in 
Hilbert spaces. The representations relevant for elementary particles
in the AdS space have been constructed for the first time in Refs. 
\cite{Fronsdal,Evans}, while modular representations of algebra 
(\ref{2.24}) have been investigated for the first time by Braden 
\cite{Braden}. We will reformulate his investigation in such a way that the 
correspondence between modular and ordinary IRs will be 
straightforward. By analogy with Braden's work \cite{Braden}
and with Ref. \cite{Evans}, we use the basis in which the operators 
$(h_j,K_j)$ $(j=1,2)$ are diagonal. Here $K_j$ is the Casimir operator 
(\ref{2.18}) for algebra $(a_j',a_j",h_j)$. 

By analogy with Braden's work \cite{Braden} we introduce the
operators
\begin{eqnarray}
&A^{++}=b"(h_1-1)(h_2-1)-a_1"L_-(h_2-1)-\nonumber\\
&a_2"L_+(h_1-1)+a_1"a_2"b' \quad A^{--}=b'\nonumber\\
&A^{+-}=L_+(h_1-1)-a_1"b'\quad A^{-+}=L_-(h_2-1)-a_2"b'
\label{2.26}
\end{eqnarray}
and consider their action only on the space of "minimal" 
sp(2)$\times$sp(2) vectors, i.e. such vectors $x$ that $a_j'x=0$
for $j=1,2$ (Braden \cite{Braden} works in the space of "maximal"
vectors such that $a_j"x=0$; on the other hand, the very instructive
approach of Refs. \cite{Evans} cannot be directly generalized to the
modular case since it uses irrational numbers explicitly).

It is easy to see that if $x$ is a minimal vector such that
$h_jx=\alpha_jx$ then $A^{++}x$ is the minimal eigenvector of the
operators $h_j$ with the eigenvalues $\alpha_j+1$, $A^{+-}x$ - 
with the eigenvalues $(\alpha_1+1,\alpha_2-1)$, 
$A^{-+}x$ - with the eigenvalues $(\alpha_1-1,\alpha_2+1)$, 
and $A^{--}x$ - with the eigenvalues $\alpha_j-1$.

The algebra so(2,3) possesses the Casimir operators of the second
and the fourth orders and in the modular case there exist additional 
independent invariants. The expression for the Casimir operator
of the second order can be chosen in the form
\begin{equation}
I_2=\frac{1}{4}\sum_{ab}M_{ab}M^{ab}
\label{I2}
\end{equation} 
We need only the
action of $I_2$ on the space of minimal vectors:
\begin{equation}
I_2=h_1^2+h_2^2-2(h_1+2h_2)+2(L_-L_+-b"b')
\label{2.27}
\end{equation}

It follows from Eqs. (\ref{2.24}) and (\ref{2.27}) that
\begin{eqnarray}
&[A^{--},A^{-+}]=[A^{--},A^{+-}]=[A^{++},A^{-+}]=
  [A^{++},A^{+-}]=0\nonumber\\
&[A^{--},A^{++}]=(h_1+h_2-2)[\frac{1}{2}(h_1+h_2)^2-2(h_1+h_2)-
\frac{1}{2}I_2]\nonumber\\
&[A^{+-},A^{-+}]=(h_1-h_2)[2+\frac{1}{2}I_2-\frac{1}{2}(h_1-h_2)^2]
\label{2.28}
\end{eqnarray}

By analogy with the construction of ordinary representations with
positive energy \cite{Fronsdal,Evans} we require the existence
of the vector $e_0$ satisfying the conditions
\begin{eqnarray}
&a_j'e_0=b'e_0=L_+e_0=0\quad h_je_0=q_je_0\nonumber\\
&(e_0,e_0)\neq 0\quad (j=1,2)
\label{2.29}
\end{eqnarray}
Then (see Sect. \ref{S2.1}) $q_1,q_2\in F_p$ and, as follows from Eq.
(\ref{2.27}), the invariant $I_2$ assumes in the representation
space the numerical value
\begin{equation}
I_2=q_1^2+q_2^2-2(q_1+2q_2)
\label{2.30}
\end{equation}

As follows from Eq. (\ref{2.24}), the operators 
$(a_1',a_2',b')$ reduce the eigenvalue of the "energy" operator
$M_{05}=h_1+h_2$ by two units. Therefore in the conventional theory
$e_0$ is the state with the minimum energy. In this theory spin
is equal to the maximum value of the operator $L_3=h_1-h_2$ in the
"rest state". For this reason we use $s$ to denote $q_1-q_2$.
As noted in Sect. \ref{S1.3}, the operator $M_{05}$ is the AdS
analog of the energy operator. Since in the usual theory 
the mass is the minimum value of the energy, we use $m$ to denote 
$q_1+q_2$.

As follows from Eq. (\ref{2.28}), in the space of minimal vectors
the basis can be chosen in the form $e_{nk}=(A^{++})^n(A^{-+})^ke_0$
$(n,k=0,1,2...)$. We will see below that these numbers are 
bounded above and the limits will be explicitly indicated.
As follows from Eqs. (\ref{2.28}) and (\ref{2.30})
\begin{equation}
A^{--}A^{++}e_{nk}=(n+1)(m+n-2)(q_1+n)(q_2+n-1)e_{nk}
\label{2.31}
\end{equation}
\begin{eqnarray}
&(e_{n+1,k},e_{n+1,k})=(q_1+n-k-1)(q_2+n-k-1)(n+1)(m+n-2)\nonumber\\
&\times (q_1+n)(q_2+n-1)(e_{nk},e_{nk})
\label{2.32}
\end{eqnarray}
\begin{equation}
A^{+-}A^{-+}e_{nk}=(k+1)(s-k)(q_1-k-2)(q_2+k-1)e_{nk}
\label{2.33}
\end{equation}
\begin{eqnarray}
&(e_{n,k+1},e_{n,k+1})=(q_2+n+k-1)(q_1-k-2)(q_2+k-1)\nonumber\\
&\times (k+1)(s-k)(e_{nk},e_{nk})/(q_1+n-k-2)
\label{2.34}
\end{eqnarray}
The presence of division in the last expression does not mean that
there is no correspondence with the conventional case because by
writing down the explicit expression for $(e_{nk},e_{nk})$ one
can see that the numerator contains terms which cancel out
the denominator.

The full basis of the representation space can be chosen in the
form 
\begin{equation}
e(n_1n_2nk)=(a_1")^{n_1}(a_2")^{n_2}e_{nk} 
\label{en}
\end{equation}
where, as follows from the
results of the preceding section, 
\begin{eqnarray}
&n_1=0,1,...N_1(n,k)\quad n_2=0,1,...N_2(n,k)\nonumber\\
&N_1(n,k)=p-q_1-n+k\quad N_2(n,k)=p-q_2-n-k
\label{2.35}
\end{eqnarray}
As follows from Eq. (\ref{Wilson1}) and the results of this
section, 
\begin{eqnarray}
&Norm(n_1n_2nk)==(e(n_1n_2nk),e(n_1n_2nk))=\nonumber\\
&n_1!n_2![(q_1+n-k+n_1-1)!(q_2+n+k+n_2-1)!/\nonumber\\
&((q_1+n-k-1)!(q_2+n+k-1)!)](e_{nk},e_{nk})
\label{Norm}
\end{eqnarray}

For conventional IRs the following classification exists: 
massive case $(m-s>2)$ \cite{Fronsdal,Evans}, massless case 
$(m-s=2)$ \cite{Evans,FronsdalB} and two Dirac singletons 
\cite{Ehrman,DiracS,Castell} with $m=1$, $s=0$ and $m=2$, $s=1$. 
Let us note that even if the conventional mass of an elementary 
particle is very small, its
AdS mass is very big. The lightest known elementary
particle with nonzero mass is the electron whose Compton
wave length $\hbar /(m_e c)$ is equal to $4\cdot 10^{-11}cm$. If 
$R$ is say, of the order of $10^{27}cm$ then the AdS 
mass of the electron is of the order of $10^{38}$ (maybe 
this fact in an indication that electron is not an 
elementary particle?). Therefore even for the electron the AdS
mass is much greater than the spin. However the problem 
exists what is the relation between the AdS mass and spin 
for a particle which is treated as massless in Poincare invariant 
theories (say the photon).  

The exceptional values of singleton masses prompted several
authors to investigate the possibility that sigletons are true
elementary particles \cite{DiracS,Castell}. The Dirac paper
\cite{DiracS} in which the singleton representations have been
investigated in detail, is called "A remarkable
representation of the 3 + 2 de Sitter group". Moreover, in the
supersymmetric theory the both singletons represent the bosonic
and fermionic parts of one superparticle \cite{FronsdalS}.
The modular analog of singletons and supersingleton can be 
investigated very simply \cite{lev4} and we will not discuss 
this case. We first consider the massive case and the 
massless one will be discussed in the next section.

In the massive case, as follows from Eqs. (\ref{2.33}) and
(\ref{2.34}), $k$ can assume only the values $0,1,...s$, as well
as in the ordinary case. At the same time, it follows from Eqs.
(\ref{2.31}) and (\ref{2.32}), that, in contrast to the ordinary 
case where $n=0,1,...\infty$, in the modular one $n=0,1,...N$ 
where $N=p+2-m$. Hence the space of minimal vectors has the 
dimension $(s+1)(N+1)$,
and IR turns out to be finite-dimensional and even finite since
the field $F_{p^2}$ is finite. If $D(m,s)$ is the dimension of IR
with the given $m$ and $s$, then, by using Eq. (\ref{2.35}), it
can be shown by a direct calculation that
\begin{eqnarray}
&D(m,s)=\sum_{k=0}^s \sum_{n=0}^{p+2-m} (p+1-q_1-n+k)
(p+1-q_2-n-k)=\nonumber\\ 
&\frac{1}{12}(s+1)(p+3-m)[4p^2-3p(m-1)+\nonumber\\
&(m-2)(5m-4)-s(s+2)]
\label{2.36}
\end{eqnarray}

The "maximum" value of the energy is equal to $m+2N$, but
this value is equal to $-m+4$ in $F_p$. Nevertheless, if we 
consider only linear combinations
$$\sum_{n_1n_2nk}c(n_1,n_2,n,k)e(n_1n_2nk)$$
where nonnegative integers $(n_1,n_2,n,k)$ are much less 
than $p$, the action of the operators $h_j,a_j',a_j"$ 
$(j=1,2)$ on such combinations and the action of the operators 
$A^{++},A^{-+},A^{+-},A^{--}$ on minimal vectors with 
$n,k<<p$, then, as follows from the above expressions, we 
have the correspondence with the conventional case in the sense
described in Sect. \ref{S2.1}.      

\begin{sloppypar}
As follows from Eq. (\ref{2.26}), the action of the operators
$(b",b',L_+,L_-)$ on minimal vectors is given by
\begin{eqnarray}
&b"=(A^{++}-a_1"A^{-+}-a_2"A^{+-}-a_1"a_2"b')
[(h_1-1)(h_2-1)]^{-1}\nonumber\\
&b'=A^{--} \quad L_+=(A^{+-}+a_1"b')(h_1-1)^{-1}\nonumber\\
& L_-=(A^{-+}+a_2"b')(h_2-1)^{-1} 
\label{2.37}
\end{eqnarray} 
By using the commutation relations (\ref{2.24}) we can find
how these operators act on any vector. For the massive case
the representations constructed above do not contain elements
with the eigenvalues of the operators $h_1$ or $h_2$ equal to
1. Therefore the operators inverse to $(h_1-1)$ and $(h_2-1)$
are well defined. On the other hand, the problem arises whether
we should require the compatibility with the conventional
case for all the representation generators. As seen from Eq.
(\ref{2.37}), the expressions for $(L_{\pm},b")$ contain 
division and so the compatibility is questionable. These 
operators are not diagonal and so the compatibility in the
sense of Sect. \ref{S2.1} is not violated. 
\end{sloppypar}

One can construct another nonorthogonal basis 
where the compatibility takes place for all the ten generators.
Indeed, the elements 
\begin{equation}
E(n_1n_2nk)=(a_1")^{n_1}(a_2")^{n_2}(b")^n(L_-)^ke_0
\label{nonortho}
\end{equation}
where $(n_1n_2nk)$ take the same values as above, form a
basis where the correspondence with the standard case exists
for all the representation generators (see Ref. \cite{lev4}
for details). 

Let us now consider the following question. Is it possible
to choose the basis in such a way that it is not only 
orthogonal but also normalized to one? As noted in Sect.
\ref{S2.1}, such a normalization is not necessary for the
probabilistic interpretation, however it might be convenient. 
Let us define $e(n_1n_2nk)=\eta(n_1n_2nk)E(n_1n_2nk)$ where
we assume now that 
\begin{equation}
(E(n_1'n_2'n'k'),E(n_1n_2nk))=\delta_{n_1'n_1}\delta_{n_2'n_2}
\delta_{n'n}\delta_{k'k}
\label{orthonorm}
\end{equation}
Then, as follows from Eq. (\ref{Norm}), one should have
\begin{equation}
\eta(n_1n_2nk)\bar{\eta}(n_1n_2nk)=Norm(n_1n_2nk)
\label{root}
\end{equation}
If $Norm(n_1n_2nk)$ is a quadratic residue in $F_p$ (see Sect.
\ref{S2.1}) then we can define $\eta(n_1n_2nk)$ as a square root
from $Norm(n_1n_2nk)$ in $F_p$. In that case 
${\bar\eta}(n_1n_2nk)=\eta(n_1n_2nk)$ and the problem is solved.
However if $Norm(n_1n_2nk)$ is a quadratic nonresidue in $F_p$
and  $\eta(n_1n_2nk)$ is defined as explained in Sect. \ref{S2.1},
then ${\bar\eta}(n_1n_2nk)=-\eta(n_1n_2nk)$ and the problem
remains. In any case, a solution of Eq. (\ref{root}) 
might significantly differ from the one given by the standard 
square root. For this reason we prefer to work with the basis 
$e(n_1n_2nk)$ rather than $E(n_1n_2nk)$. 

We can construct modular analogs of representations with the
negative energies as follows. Let $q_1,q_2$ be positive integers
which are much less than $p$, and $q_1\geq q_2$. Consider $q_1,q_2$
as elements of $F_p$. Instead of Eq. (\ref{2.29}) we now require 
the existence of the vector $e_0'$ such that
\begin{eqnarray}
&a_j"e_0'=b"e_0'=L_-e_0'=0\quad h_je_0'=-q_je_0'\nonumber\\
&(e_0',e_0')\neq 0\quad (j=1,2)
\label{2.38}
\end{eqnarray}
We again denote $s=q_1-q_2$ and $m=q_1+q_2$. Then we can construct
IR by analogy with the above construction. 

The meaning of the first expression in Eq. (\ref{2.38}) is as
follows. Since the operators $(a_j",b")$ increase the energy
by two units, they should annihilate the vector which (in the
usual approach) has the maximum energy. We require $L_-e_0'=0$
instead of $L_+e_0=0$ in Eq. (\ref{2.29}) because $e_0'$ is the
eigenvector of the operator $L_3$ with the negative value (in the
usual sense), so we require it to be the minimal vector of the 
su(2) representation with the spin s.  

By analogy with the above construction, we can define 
maximal vectors $e_{nk}'$, construct the basis by using
the elements 
$$e(n_1n_2nk)'=(a_1')^{n_1}(a_2')^{n_2}e_{nk}'$$
and consider only linear combinations
$$\sum_{n_1n_2nk}c(n_1n_2nk)e{n_1n_2nk}'$$
where nonnegative integers $(n_1,n_2,n,k)$ are much less 
than $p$. Then we will have the correspondence with the 
ordinary negative energy IR corresponding to the maximum
value of the energy equal to $-m$.

A very interesting property of the modular analogs of IRs
corresponding to positive and negative energies, is that 
{\bf the modular analog of the negative energy IR with the 
maximum energy $-m$ and the modular analog of the positive 
energy IR with the minimum energy $m$ represent the same
modular IR!} The proof of this statement is as follows.

Denote $N_1=p-q_1$ and $N_2=p-q_2$. According to the results
of the preceding section, the vector 
$x=(a_1')^{N_1}(a_2')^{N_2}e_0'$ is the minimal vector of
the sp(2)$\times$sp(2) representation with the eigenvalues
of the operators $h_1,h_2$ equal to $q_1$ and $q_2$, 
respectively. This means that $a_1'x=a_2'x=0$. 

Let us now prove that $b'x=0$. Since $b'$ commute with 
$a_1'$ and $a_2'$, we can write $b'x$ in the form
\begin{equation}
b'x = (a_1')^{N_1}(a_2')^{N_2}b'e_0'
\label{2.39}
\end{equation}
As follows from Eqs. (\ref{2.24}) and (\ref{2.38}), 
$a_2"b'e_0'=-L_-e_0'=0$ and $b'e_0'$ is the eigenvector
of the operator $h_2$ with the eigenvalue $-q_2-1$.
Therefore, $b'e_0$ is the maximal vector of the sp(2)
representation which has the dimension $p-q_2=N_2$.
Therefore $(a_2')^{N_2}b'e_0'=0$ and $b'x=0$.

The next stage of the proof is to show that $L_+x=0$.
As follows from Eqs. (\ref{2.24}) and the definition of
$x$,
\begin{equation}
L_+x = (a_2')^{N_2}(a_1')^{N_1}L_+e_0'-
N_1(a_2')^{N_2}(a_1')^{N_1-1}b'e_0'
\label{2.40}
\end{equation}
We have already shown that $(a_2')^{N_2}b'e_0'=0$,
and therefore it suffice to prove that the first term
in the r.h.s. of Eq. (\ref{2.40}) is equal to zero. As follows
from Eqs. (\ref{2.24}) and (\ref{2.38}), $a_2"L_+e_0'=-b"e_0'=0$
and $L_+e_0'$ is the eigenvector of the operator $h_2$ with the
eigenvalue $-q_2-1$. Therefore $(a_2')^{N_2}L_+e_0'=0$ and we
have proved that $L_+x=0$.

The fact that $(x,x)\neq 0$ immediately follows from the
definition of the vector $x$ and the results of the preceding
section. Moreover, as follows from Eq. (\ref{Wilson2}), the
explicit relation between $(x,x)$ and $(e_0',e_0')$ is
\begin{equation}
(x,x)=(-1)^m[(q_1-1)!(q_2-1)!]^2(e_0',e_0') 
\label{2.41}
\end{equation} 
 
Therefore the vector $x$ satisfies the same conditions as
the vector $e_0$ in Eq. (\ref{2.29}). We conclude that the 
representation containing a vector satisfying Eq. (\ref{2.38}) 
also contains a vector satisfying Eq. (\ref{2.29}) and our 
statement is proved.

The fact that the same modular IR can be treated
as the modular analog of ordinary positive energy and negative 
energy IRs simultaneously, does not mean of course that
modular IRs of the so(2,3) algebra contradict experiment.
At one end of the spectrum there exists the correspondence
between the modular IR and ordinary positive energy IR with
the minimum de Sitter energy $m$ while at the other end - the 
correspondence between the same modular IR and ordinary 
negative energy IR with the maximum energy $-m$. A naive
explanation of this fact is that if 2 is added to $m$ 
$(2p-m)$ times then we obtain $-m$ in $F_p$. It is also 
interesting
to note that while the ordinary positive and negative IRs
for a particle with the mass $m$ do not have eigenvectors
of the energy operator in the interval $(-m,m)$ (this 
phenomenon is called the energy gap), the modular analog
of this IRs contains the eigenvectors with such eigenvalues.
However this takes place only for elements 
$e(n_1n_2nk)$ where at least some of the
numbers $(n_1,n_2,n)$ are comparable with $p$.

In the standard theory, negative energies have no direct
physical meaning but they are associated with antiparticles
in the formalism of second quantization. In Sect. \ref{S4.3}
we will consider this question for both, the standard theory
and the GFQT.

The matrix elements of the operator $M^{ab}$ in the basis
$e(n_1n_2nk)$ are defined as
\begin{equation}
M^{ab}e(n_1n_2nk)=\sum_{n_1'n_2'n'k'}
M^{ab}(n_1'n_2'n'k',n_1n_2nk)e(n_1'n_2'n'k')
\label{2.42}
\end{equation}
Taking into account the fact that our basis is orthogonal but
not normalized, it is easy to see, that, as follows from the
definition of the adjoint operator, the matrix elements of
$M^{ab}$ and its adjoint $M^{ab*}$ are related to each other as
\begin{equation}
M^{ab*}(n_1n_2nk,n_1'n_2'n'k')=\frac{Norm(n_1'n_2'n'k')}
{Norm(n_1n_2nk)}\overline{M^{ab}(n_1'n_2'n'k',n_1n_2nk)}
\label{2.43}
\end{equation}
where the bar is used to denote the usual complex conjugation. 

In the modular case the trace of each operator $M^{ab}$ is
equal to zero. For the operators $(a_j',a_j",L_{\pm},b',b")$
this is clear immediately: since they do not contain
nonzero diagonal elements at all, they necessarily change
one of the quantum numbers $(n_1n_2nk)$. The proof for the
diagonal operators $h_1$ and $h_2$ is as follows. For each 
IR of the sp(2) algebra with the minimal weight $q_0$ and the 
dimension $N+1$, the eigenvalues of the operator $h$ are 
$(q_0,q_0+2,...q_0+2N)$. The sum of these eigenvalues is
equal to zero in $F_p$ since $q_0+N=0$ in $F_p$ (see the
preceding section). Therefore we conclude that
\begin{equation}
\sum_{n_1n_2nk} M^{ab}(n_1n_2nk,n_1n_2nk)=0
\label{2.45}
\end{equation} 
It is obvious that such a property does not exist for 
conventional IRs.   

\section{Modular analogs of massless irreducible representations}
\label{S2.4}

In the case $m-s=2$, which is treated in the literature as
massless, the following technical problem arises. Since $e_0$ 
is the eigenvector of the operator $h_2$ with the eigenvalue 1,
then Eq. (\ref{2.37}) cannot be directly used. One might think that 
this problem is not very important since the AdS mass $m$ 
corresponds to the ordinary mass $m/2R$.
Therefore the cases $m-s=2$ and, say, $m-s=3$ are practically
indistinguishable. However, in the conventional approach, the
fact whether a particle is exactly massless or has a small
nonzero mass, may be very important in gauge theories because
the mass term in the Lagrangian breaks gauge invariance. It is
well known, for example, that QED survives if the photon has a
small nonzero mass, while in the Yang-Mills theory with colored
gluons, their mass must be exactly equal to zero. The exact value
of the mass is also important in conformal theories which are now
intensively investigated.

A simple way of generalizing the above results to the massless
case is as follows. To resolve ambiguities 0/0 in matrix elements 
arising in the case $q_2=1$, we can write $q_2$ in the form 
$q_2=1+\epsilon$ and take the limit $\epsilon\rightarrow 0$ at the
final stage of computations. This confirms a well known fact that
analytical methods can be very useful in problems involving only
integers. The results obtained below can also be justified by using
only algebraic methods, but we will not dwell on this. 

Instead of the operators (\ref{2.26}), we now work with the operators
\begin{eqnarray}
&B^{++}=A^{++}[(h_1-1)(h_2-1)]^{-1}=b"-a_1"L_-(h_1-1)^{-1}\nonumber\\
&-a_2"L_+(h_2-1)^{-1}+a_1"a_2"b'[(h_1-1)(h_2-1)]^{-1}\nonumber\\
&B^{+-}=A^{+-}(h_1-1)^{-1}=L_+-a_1"b'(h_1-1)^{-1}\nonumber\\
&B^{-+}=A^{-+}(h_2-1)^{-1}=L_--a_2"b'(h_2-1)^{-1}\nonumber\\
&B^{--}=A^{--}=b'
\label{2.46}
\end{eqnarray}
and again require that IR in question is defined by a cyclic vector $e_0$
satisfying Eq. (\ref{2.29}). Then the problem arises how to define the
action of the operators $B^{++}$ and $B^{-+}$ on $e_0$ which is the 
eigenvector of the operator $h_2$ with the eigenvalue $q_2=1$. By using the
above prescription, we require that
\begin{eqnarray}
B^{++}e_0=[b"-a_1"L_-(h_1-1)^{-1}]e_0\quad B^{-+}e_0=L_-e_0
\label{2.47}
\end{eqnarray}
if $s\neq 0$ (and thus $h_1\neq 1$), and
\begin{equation}
B^{++}e_0=b"e_0\quad B^{+-}e_0=B^{-+}e_0=e_0
\label{2.48}
\end{equation}
if $s=0$. As follows from the previous results, so defined operators
transform minimal vectors to minimal ones, and therefore the element
\begin{equation}
e_{nk}=(B^{++})^n(B^{-+})^ke_0
\label{2.49}
\end{equation}
is the minimal sp(2)$\times$sp(2) vector with the eigenvalues of the
operators $h_1$ and $h_2$ equal to $q_1+n-k$ and $q_2+n+k$, respectively.
Note that the operators $B^{++}$ and $B^{-+}$ do not commute 
with each other
and therefore their order in Eq. (\ref{2.49}) is important.

Once the vectors $e_{nk}$ are found, we can construct the full 
basis of IR as
in Eq. (\ref{en}). Therefore the specifics of the massless case 
is manifested
only in the fact that the range of the numbers $n$ and $k$ 
differs from that in 
the preceding section, and our goal is to determine this range.

By analogy with the massive case, we define $a_{nk}$ and 
$b_{nk}$ such that
\begin{equation}
B^{--}B^{++}e_{nk}=a_{nk}e_{nk}\quad B^{+-}B^{-+}e_{nk}=b_{nk}e_{nk} 
\label{2.50}
\end{equation} 
Then, by analogy with the massive case, one can show that
\begin{equation}
(e_{n+1,k},e_{n+1,k})=a_{nk}(e_{nk},e_{nk})\quad 
(e_{n,k+1},e_{n,k+1})=b_{nk}(e_{nk},e_{nk})
\label{2.51}
\end{equation}
where, in contrast to Eqs. (\ref{2.31}) and (\ref{2.34})
\begin{eqnarray}
&a_{nk}=\frac{(n+1)(q_1+q_2+n-2)(q_1+n)(q_2+n-1)}
                  {(q_1+n-k-1)(q_2+n+k-1)}\nonumber\\
&\nonumber\\
&b_{nk}=\frac{(k+1)(q_1-q_2-k)(q_1-k-2)(q_2+k-1)}
{(q_1+n-k-2)(q_2+n+k-1)}
\label{2.52}
\end{eqnarray}

When $s=0$, we should require for consistency that 
$b_{n0}=0$ for all values
of $n$, and the expression for $a_{n0}$ becomes
\begin{equation}
a_{n0}=(n+1)^2
\label{spinless}
\end{equation}
since $q_1=1$ if $s=0$. Therefore in the spinless case there exist only 
minimal sp(2)$\times$sp(2) vectors $e_{n0}$, as well as in the massive 
case. The difference with the massive case manifests only in the 
fact that now
the maximum value of $n$ is equal to $N=p-1$, not $N=p$ as one might
expect from the results of the previous section.

Let us now assume that $s\neq 0$. 
Since $q_2=1$ in the massless case, the expressions in Eq. (\ref{2.52})
contain ambiguities 0/0 at some values of $n$ and $k$. We will treat them 
as explained above. Consider first the case $n=0$. Then, as follows from 
Eq. (\ref{2.52}),
\begin{equation}
b_{0k}=(k+1)(s-k)
\label{2.53}
\end{equation}
Therefore, if $n=0$, the quantity $k$ can take the values of $0,1,...s$ in
full analogy with the massive case. On the other hand, if $n\neq 0$ then it
it is obvious that $b_{nk}=0$ if $k\geq 1$. As a result, we should require
for consistency that 
\begin{equation}
B^{-+}e_{n0}=0\quad (if\quad n\neq 0)
\label{2.54}
\end{equation}

Consider now the expression for $a_{nk}$ if $n=0$. It is clear that 
if $q_2=1$ then $a_{nk}\neq 0$ only if the value of $k$ is either 0 or $s$.
Therefore we should require for consistency that
\begin{equation}
B^{++}e_{0k}=0\quad (if\quad k=1,2,..s-1)
\label{2.55}
\end{equation}
For spinless particles and particles with the spin 1/2 in usual units
(i.e. $s=1$ in our system of units), Eq. (\ref{2.55}) does not impose any
restriction at all. For this reason some authors treat as massless only
the cases when (in addition to $q_2=1$) the value of spin is greater or
equal to unity in usual units. However, as it will be clear soon, in the
modular case it is reasonable to treat a case as massless if $q_2=1$
regardless of the value of $s$.

So far our discussion of the case $s\neq 0$ was not using any specifics 
of the fact that we consider a modular IR and not a conventional one. At 
this end we have arrived at the same conclusions as in the conventional 
case \cite{Evans}. Namely, in the massless case the quantities $(n,k)$ can 
take only the following values: If $n=0$ then $k=0,1,...s$ and $n\neq 0$ 
only if $k$ is either 0 or $s$. The last property makes it clear why 
it is natural to treat IRs with $q_2=1$ as massless. Indeed, in the
standard Poincare invariant theory, a massless particle with spin $s$ 
can have only two values of the spin projection onto the axis directed
along the momentum: $s$ or $-s$, and helicity is a conserving physical
quantity. An analogous situation takes place for a massless particle in
the AdS space but with one exception: in the state with the minimum AdS
energy, the value of $k$ can be $0,1,...s$ as usual, but for any energy
greater than $m$, the value of $k$ can be either 0 or $s$.

In the standard case the quantity $n$ can take any value in the range
$n=0,1,2...$. However in the modular case $n=0,1,2,...N$ where 
$N=p-q_1=p-1-s$. This follows from the fact that if $k=0$ or $k=s$
then (see Eq. (\ref{2.52}))      
\begin{equation}
a_{nk}=(n+1)(q_1+n)
\label{2.56}
\end{equation}
The result differs from that in the massive case where $N=p+2-m$.
For the dimension of the massless modular IR one can derive a
formula analogous to that in Eq. (\ref{2.36}) but we will not
go into details.


Since the expressions in Eq. (\ref{2.52}) contain division, one might
wonder whether there exists the correspondence with the standard case.
Let us note first that for the values of $n$ and $k$ such that 
$e_{nk}\neq 0$, the quantities $a_{nk}$ and $b_{nk}$ do not contain
division (see Eqs. (\ref{2.53}) and (\ref{2.56})). Note also that 
the 0/0 ambiguities in Eq. (\ref{2.52}) take place only if the AdS energy
is exactly equal to $m=q_1+1$ (i.e. $n=0$) while for all other values
of $n$ we can use the basis elements $e_{nk}$ normalized as in the
massive case. 

As shown in the preceding section, modular IRs have at least two
important distinct properties. First, the modular analog of a standard
IR with the positive energy is simultaneously the modular analog of the
same IR with the negative energy. Second, the trace of any representation 
generator is equal to zero (see Eq. (\ref{2.45})). The proof of these
statements is based on the properties of modular IRs of sp(2) algebra
and does not involve any information about the $(nk)$ content of the
modular IR under consideration. For these reasons the statements remain
valid in the case of massless IRs.  


\vfill\eject

\begin{sloppypar}
\chapter{Space-time description of elementary particles in SO(2,3)
invariant theories}
\label{C3}
\end{sloppypar}

\section{Kinematics of AdS space}
\label{S3.1}

In Poincare invariant quantum theories there are two ways of
describing elementary particles. On the one hand, a particle can 
be described by a unitary IR of the Poincare group which is 
implemented by functions on a three-dimensional momentum
hyperboloid. On the other, it can be described by a nonunitary
reducible representation which is implemented by function on the
four-dimensional Minkowski space. The relation between the two
is implemented by constructing an intertwinning operator and
requiring that the functions on Minkowski space should satisfy
some covariant equation (Klein-Gordon, Dirac etc.).

An analogous program has been implemented for theories with the
de Sitter symmetries SO(2,3) and SO(1,4) and there exists a vast
literature devoted to this problem. The description of modern 
approaches can be found, for example in Ref. \cite{Metsaev} and
references therein, while in Ref. \cite{BirDav} the
description in arbitrary curved spaces is discussed.  

There are two essentially different approaches for writing
down covariant equations in the de Sitter spaces.
The first one is to use the formalism of differential geometry 
on the four-dimensional spaces defined by Eqs. (\ref{1.3}) and 
(\ref{1.4}). In particular, one defines four-dimensional 
connections, four-dimensional covariant derivatives etc. 
The philosophy of such an approach is that when the curvature 
of space-time is small, there should exist a smooth transition 
to the space-time description in Minkowski space.

In Sect. \ref{1.3} we argued that Poincare invariant and de Sitter
invariant theories have essentially different features. For
example, the four-dimensional energy-momentum operator is
fundamental only if space-time is exactly flat. If the
cosmological constant is very small but not zero, then the
operator is no more the representation generator of the
symmetry group. In this sense there is no smooth transition from 
de Sitter invariant to Poincare invariant theories.
For this reason we prefer another approach which uses
explicitly the fact that the dS and AdS spaces are embedded into
a five-dimensional space. In such an approah
the derivative operator $D_a$ has five 
components, a vector particle is described by a five-component
function $A_a(x)$ etc. For the first time this approach
has been probably considered in the paper by Dirac 
\cite{DirCE} where the electron equation in the dS space
has been discussed, and was further developed by several authors 
(see e.g. Ref. \cite{FronsdalB} and references therein). We will see 
in the next chapter that such an approach provides a clear covariant 
way for computing the angular momentum operators $M_{ab}$ for systems 
of interacting fields.
Let us stress once more (see the discussion in Sect. \ref{S1.3})
that the energy-momentum tensor does not play any fundamental
role in standard de Sitter invariant theories. In those theories
it is reasonable to have only the angular momentum tensor such
that its integrals over some hypersurface produce all the ten
operators $M_{ab}$.

In Chapt. \ref{C1} we argued that in future quantum theories
space-time will not be so fundamental as in the existing ones,
and in the GFQT the notion of space-time is not present at all. 
However, as we
stressed several times, at some conditions the description
given by the GFQT should be compatible with the
conventional one. In the present chapter we show that for
modular IRs considered in the preceding chapter, this is indeed
the case. 

\begin{sloppypar}
As pointed out in Sect. \ref{S1.3}, the AdS space
is a four-dimensional manifold defined by Eq. (\ref{1.3}) in the
five-dimensional space. The metric on the AdS space is
inherited from the metric of the five-dimensional space whose
metric tensor $g_{ab}$ is diagonal and has the components
$g_{55}=g_{00}=-g_{11}=-g_{22}=-g_{33}=1$. Although we will not
use the metric on the AdS space, we display it for completeness.
\end{sloppypar}

Consider a vicinity of the point $(x_5,x_0,{\bf r})=(-R,0,{\bf 0})$.
Here ${\bf r}$ is the usual spatial three-dimensional vector
${\bf r}=(x^1,x^2,x^3)$. We assume that raising and lowering the 
indices is accomplished by using the tensor $g_{ab}$. We choose
the vicinity in such a way that $(x_0,{\bf r})$ can be taken as
independent variables and $x_5$ is a function obtained from Eq.
(\ref{1.3}): $x_5=-\sqrt{R^2+{\bf r}^2-x_0^2}$. Then the
expression for the interval $ds$ in this vicinity of the AdS
space is
\begin{equation}
ds^2=dx_0^2-d{\bf r}^2+\frac{(x_0dx_0-{\bf r}d{\bf r})^2}
{R^2+{\bf r}^2-x_0^2}
\label{3.1}
\end{equation}
When $R\rightarrow \infty$, the vicinity becomes the ordinary
Minkowski space with the metric $ds^2=dx_0^2-d{\bf r}^2$. We could
also consider the transition to Minkowski space by choosing a
vicinity of the point $(R,0,{\bf 0})$ but the above choice is
more convenient for us.

In the general case we parametrize the points of the AdS 
space $(x_5,x_0,{\bf r})$ by ${\bf r}$ and $\tau\in [-\pi ,\pi ]$
such that
\begin{equation}
x_5=-\sqrt{R^2+{\bf r}^2}cos\tau\quad 
x_0=\sqrt{R^2+{\bf r}^2}sin\tau
\label{3.2}
\end{equation}
Then the transition to Minkowski space takes place when 
$R>>|{\bf r}|$ and $\tau$ is small, such that $x_0\approx R\tau$. 

The SO(2,3) invariant volume element on the AdS space has the
form
\begin{equation}
d\Omega(x) =dx_5dx_0d^3{\bf r}\delta (x_5^2+x_0^2-{\bf r}^2-R^2)=
\pi d^3{\bf r} d\tau
\label{3.3}
\end{equation}
The SO(2,3) transformations are described by $5\times 5$ real
matrices $g$ satisfying the condition $g^T\eta g=\eta$ where 
$\eta$ is the matrix representing the metric tensor and $g^T$ 
is the matrix transposed to $g$.

A matrix $g\in SO(2,3)$ is fully defined by an antisymmetric 
tensor $\omega_{ab}$ such that
\begin{equation}
g=exp(-\frac{i}{2}\omega_{ab}V^{ab}),\quad
(V^{cd})^a_b=-2i(\delta^c_b g^{da}-\delta^d_b g^{ca})
\label{3.4}
\end{equation}
Here and henceforth we assume that the summation over repeated
indices should be performed. One might wonder why we use the
expression containing $i$ though $g$ is real. The convenience
of such a choice is that the matrices $V^{ab}$ are Hermitian
and satisfy the same
commutation relations as the representation generators of the
AdS group in the general case (see Eq. (\ref{2.23})):  
\begin{equation}
[V^{ab},V^{cd}]=-2i (g^{ac}V^{bd}+g^{bd}V^{cd}-
g^{ad}V^{bc}-g^{bc}V^{ad})
\label{3.5}
\end{equation}
Usually the expression (\ref{3.4}) is written without the
multiplier $2i$ and in that case the matrices $V^{ab}$ are
anti-Hermitian.

For a representation of the AdS group in the space of 
functions $f(x)$ on the five-dimensional space, the generators
can be chosen in the form
\begin{equation}
L_{ab}=2i(x_a\frac{\partial}{\partial x_b}-
x_b\frac{\partial}{\partial x_a})
\label{3.6}
\end{equation}
They also satisfy the commutation relations
\begin{equation}
[L_{ab},L_{cd}]=-2i (g_{ac}L_{bd}+g_{bd}L_{cd}-
g_{ad}L_{bc}-g_{bc}L_{ad})
\label{3.7}
\end{equation}
However usually one considers representations in the space of
functions defined only on the AdS space (\ref{1.3}).
Therefore we have to modify the derivatives for that case.
If instead of $(x_5,x_0,{\bf r})$ we use the variables 
$(Z,\tau, {\bf r})$ such that
\begin{equation}
x_5=-\sqrt{Z^2+{\bf r}^2}cos\tau\quad 
x_0=\sqrt{Z^2+{\bf r}^2}sin\tau
\label{3.8}
\end{equation}
then the derivatives are modified as
\begin{eqnarray}
&\partial/\partial x_5\rightarrow 
(x_5/Z)(\partial/\partial Z)+(sin\tau/\sqrt{Z^2+{\bf r}^2})
(\partial/\partial \tau)\nonumber\\
&\partial/\partial x_0\rightarrow 
(x_0/Z)(\partial/\partial Z)+(cos\tau/\sqrt{Z^2+{\bf r}^2})
(\partial/\partial \tau)\nonumber\\
&\partial/\partial {\bf r}\rightarrow
\partial/\partial {\bf r} - ({\bf r}/Z)(\partial/\partial Z)
\label{3.9}
\end{eqnarray}
The AdS space is defined by the condition $Z^2=R^2$ and
so, when the derivatives are reduced on functions defined
only on the AdS space, they should not act over the
variable $Z$, i.e. they should act only in tangent directions
with respect to the AdS space. 

We choose the following 
expressions for the tangent derivatives:
\begin{eqnarray}
D_a=(\frac{sin\tau}{\sqrt{Z^2+{\bf r}^2}}
\frac{\partial}{\partial \tau},
\frac{cos\tau}{\sqrt{Z^2+{\bf r}^2}}
\frac{\partial}{\partial \tau},
\frac{\partial}{\partial {\bf r}})-
\frac{x_a}{R^2}({\bf r}\frac{\partial}{\partial {\bf r}})
\label{3.10}
\end{eqnarray} 
where the first term lists the derivatives in the order
$5,0,i$ $(i=1,2,3)$. At such a choice the tangent derivatives
satisfy the condition 
\begin{equation}
x^aD_a=0
\label{3.11}
\end{equation}
At the same time, they are not anti-Hermitian in the sense that
\begin{equation}
\int f_1({\bf r},\tau)(D_af_2({\bf r},\tau))d\Omega\neq
-\int (D_af_1({\bf r},\tau))f_2({\bf r},\tau)d\Omega
\label{3.12}
\end{equation}
We will discuss this property a bit later.

A comparison of Eqs. (\ref{3.9}) and (\ref{3.10}) shows that
the difference $(\partial /\partial x^a)-D_a$ is a vector
proportional to $x_a$. Therefore when the operators (\ref{3.6})
are reduced on functions defined only on the AdS space,
they can be written in the form
\begin{equation}
L_{ab}=2i(x_aD_b-x_bD_a)
\label{3.13}
\end{equation}
A direct computation shows that 
\begin{equation}
[D_a,x_b]=g_{ab}-x_ax_b/R^2\quad [D_a,D_b]=(x_aD_b-x_bD_a)/R^2
\label{3.14}
\end{equation}
As follows from these relations, the operators defined by Eq.
(\ref{3.13}) satisfy the same commutation relations as in 
(\ref{3.7}),
and $x_a$ and $D_a$ are true vector operators since their
commutation relations with the operators $L_{ab}$ are given by
\begin{equation}
[L_{ab},D_c]=-2i(g_{ac}D_b-g_{bc}D_a)\quad 
[L_{ab},x_c]=-2i(g_{ac}x_b-g_{bc}x_a)
\label{3.15}
\end{equation}

Our next goal is to derive the Euler-Lagrange equations in the
Lagrangian formalism. Suppose that our system is defined by
a set of fields $u_j(x)$ $(j=1,2,..n)$ where $x$ is a point of
the AdS hyperboloid (\ref{1.3}). We use $u(x)$ to denote this set.
In the general case the Lagrangian density $L(u(x),Du(x),x)$ is 
a function of the
fields, their derivatives and $x$. It should satisfy the
minimum action principle: the action
\begin{equation}
S=\int_{\Lambda} L(u(x),Du(x),x)d\Omega(x)
\label{3.16}
\end{equation} 
should have a minimum. Here the integration is performed over 
a region in the AdS space confined by a surface $\Lambda$. 

As follows from the definition of the differential operators
(\ref{3.10}) and the volume element (\ref{3.3}), the integral
\begin{equation}
\int_{\Lambda}(D_a-4x_a/R^2)f({\bf r},\tau)d\Omega
\label{3.17}
\end{equation}
can be transformed to a surface integral from the function 
$f({\bf r},\tau)$ over the hypersurface $\Lambda$. 

Let $\delta u(x)$ be a small variation of the fields $u(x)$.
Then the variation of the action is equal to
\begin{equation}
\delta S=\int_{\Lambda}( \frac{\partial L}{\partial u_j}
\delta u_j(x)+ \frac{\partial L}{\partial u_{j,a}}
\delta D_au_j(x))d\Omega
\label{3.18}
\end{equation} 
Here $u_{j,a}$ means $D_au_j$. We can interchange $\delta$ and
$D$ and write this expression in the form
\begin{eqnarray}
&\delta S=\int_{\Lambda}[ \partial L/\partial u_j\,
-(D_a-4x_a/R^2)(\partial L /\partial u_{j,a})]
\delta u_j(x)d\Omega+\nonumber\\
&\int_{\Lambda}(D_a-4x_a/R^2)[(\partial L/
\partial u_{j,a})\delta u_j(x)]d\Omega
\label{3.19}
\end{eqnarray}
In the minimum action principle, we should compare the action
only for those fields which are the same on the hypersurface
$\Lambda$. Therefore, the second term disappears and, taking
into account the fact that the variations $\delta u_j(x)$ are
arbitrary, we arrive at the Euler-Lagrange equations
\begin{equation}
\partial L/\partial u_j\,=\,
(D_a-4x_a/R^2)(\partial L /\partial u_{j,a})\quad (j=1,2,...n)
\label{3.20}
\end{equation}

We are now in position to consider such a description of
elementary particles, when the corresponding representations 
of the group SO(2,3) are realized in the space of functions on
the hyperboloid (\ref{1.3}). A general scheme of constructing
such representations is as follows. Let $g\rightarrow D(g)$ be
a finite-dimensional (nonunitary) representation of the group
SO(2,3). Let $\Gamma_{ab}$ be the generators of this
representations such that 
\begin{equation}
[\Gamma_{ab},\Gamma_{cd}]=-2i (g_{ac}\Gamma_{bd}+
g_{bd}\Gamma_{cd}-g_{ad}\Gamma_{bc}-g_{bc}\Gamma_{ad})
\label{3.21}
\end{equation}
Consider a representation $g\rightarrow T(g)$ of the group
SO(2,3), which acts on functions $f(x)$, having the range in
the space of the representation $g\rightarrow D(g)$, as
follows:
\begin{equation}
T(g)\psi(x)=D(g)\psi(g^{-1}x)
\label{3.22}
\end{equation}
The generators of such a representation 
$M_{ab}=L_{ab}+\Gamma_{ab}$ satisfy the commutation relations
\begin{equation}
[M_{ab},M_{cd}]=-2i (g_{ac}M_{bd}+g_{bd}M_{cd}-
g_{ad}M_{bc}-g_{bc}M_{ad})
\label{3.23}
\end{equation}
since the operators $L_{ab}$ and $\Gamma_{ab}$ commute with
each other. It is clear that the condition (\ref{3.22}) is not
sufficient for the description of an elementary particle
because it contains no information about the particle mass.
Therefore, in addition to (\ref{3.22}), we should add a
covariant equation containing the mass. 

The meaning of the statement that an equation for $\psi(x)$ is 
covariant is as follows. Let $\psi(x)$ be a wave function
describing a particle in the reference frame K. Let K' be
another reference frame such that the transformation from
K to K' is performed by an element $g\in SO(2,3)$. This
means that the same physical point of the de Sitter
hyperboloid which has the coordinates $x$ in K, will have
the coordinates $x'=gx$ in K'. In the reference frame K', the
same physical state is now described by a function $\psi'(x')$. 
Eq. (\ref{3.22}) means that the relation between $\psi(x)$
and $\psi'(x')$ is $\psi'(x')=D(g)\psi(g^{-1}x')=D(g)\psi(x)$.
Then the equation is covariant if $\psi'(x')$ satisfies an
equation which has {\bf the same form} as the equation for
$\psi(x)$, i.e. the equation for $\psi'(x')$ can be obtained
from the equation for $\psi(x)$ by simply replacing $x$ by
$x'$. 

In the next sections
we consider the implementation of this program for particles
with the spin 0, 1/2 and 1 (in the usual units). 

In Poincare invariant theories the meaning of space-time
representations has been widely considered in the literature.
There exists a well-known Pauli statement \cite{Pauli} 
(see also Ref. \cite{AB,Schweber,Wein}) that for bosons it is
impossible to define a positive definite charge density,
and for fermions - a positive definite energy density.
However in the technique of second quantizations the 
space-time wave functions play only the auxiliary role
(see also Chap. \ref{C4}). 

\section{Spinless particle}
\label{S3.2}

For spinless particles, the representation $g\rightarrow D(g)$
is trivial, i.e. $D(g)=1$ for any $g\in$SO(2,3). Consider the
representation (\ref{3.22}) in the space of real functions 
$\psi (x)$ satisfying the condition
\begin{equation}
\int \psi (x)^2d\Omega \, < \,\infty
\label{3.24}
\end{equation}
where the integration is performed over the whole de Sitter
space. The representation cannot be
irreducible because the functions $\psi (x)$ are defined
on the four-dimensional space, and the representation does
not contain any information about the particle mass.

By analogy with the conventional case, we choose the
Lagrangian density as
\begin{equation}
L(x)=-\frac{1}{2}(D_a\psi(x))(D^a\psi(x))+
\frac{m^2}{8R^2}\psi (x)^2
\label{3.25}
\end{equation}
Then, as follows from Eqs. (\ref{3.20}) and (\ref{3.11}),
the AdS analog of the Klein-Gordon equation is
\begin{equation}
(D_aD^a+\frac{m^2}{4R^2})\psi(x)=0
\label{3.26}
\end{equation}
When $R$ is very large and $\tau <<1$, the operator $D_5$
is negligible and $D_aD^a$ becomes the standard D'Alambertian
if $t=R\tau$ (see Eq. (\ref{3.10})). In our system of units
$m/2R$ becomes the conventional mass and therefore Eq. 
(\ref{3.26}) becomes the standard Klein-Gordon equation. 
 
The following remark is in order. If, instead of the Hermitian
generators $L_{ab}$, we choose anti-Hermitian ones, which
do not contain $i$, then the action of such generators on
real functions $\varphi(x)$ results in real functions too.
However it will be convenient for us to work with Hermitian
generators and also have an opportunity to work with the
set of ten operators considered in Sect. \ref{S2.3}. 
Therefore it is convenient to consider a complexification of
our space of functions such that it becomes a usual Hilbert
space with the usual scalar product on it.

The question arises how the representation in question
is related to the conventional IR discussed in Sect. 
\ref{S2.3}. As shown in that section, the latter is defined
by a vector $e_0$ satisfying (in the spinless case) the
conditions
\begin{equation}
a_1'e_0=a_2'e_0=b'e_0=L_+e_0=L_{12}e_0=0 
\quad L_{05}e_0=me_0
\label{3.27}
\end{equation}
As shown in Sect. \ref{S2.3}, the basis in the space of IR
can be obtained by multiple actions of the representation
generators on the vector $e_0$. Therefore we can intertwin
the both representations as follows. First we should find
a function $\psi_0(x)$ which plays the role of $e_0$ for 
the representation
in the space of functions. This function should be the 
eigenvector of the operator $(D_aD^a\, +\, m^2/4R^2)$ with
the eigenvalue 0 (see Eq. (3.26)). As follows from Eq.
(\ref{3.15}), this operator commutes with all the representation
generators. Therefore, all the functions obtained by multiple
actions of the representation generators on $\psi_0(x)$
will be the eigenvectors of the operator 
$(D_aD^a\, +\, m^2/4R^2)$ with the eigenvalue 0. This will
complete the insertion of IR into the representation (\ref{3.22}).        

As follows from Eqs. (\ref{3.10}) and (\ref{3.13}), the AdS 
energy operator in the space of functions is equal to
\begin{equation}
L_{05}^{(0)}=2i\partial/\partial \tau
\label{3.28}
\end{equation}
where the superscript $(0)$ indicates that we consider the
operator in the spinless case. The result shows that $\tau$ 
is a natural evolution parameter in the de Sitter space. If 
$\tau$ is small and we denote $R\tau =t$, then the 
conventional energy operator $L_{05}^{(0)}/2R$ is equal to 
$i\partial/\partial t$ as it should be.

Instead of $x^1$ and $x^2$, we will work with the polar angle
$\varphi$ and $r_{\bot}$ - the magnitude of the projection of 
the vector ${\bf r}$ on the plane $x^1,x^2$. The relation
between these variables is
\begin{equation}
x^1 =r_{\bot}cos\varphi\quad x^2=r_{\bot}sin\varphi 
\label{3.29}
\end{equation}
The operators $a_1',a_2',b',L_3,L_+$ can be expressed in terms of the
operators $L_{ab}$ by using Eq. (\ref{2.25}) where $M_{ab}$
should be replaced by $L_{ab}$. The result is
\begin{eqnarray}
&a_1^{(0)'}=\frac{1}{2}exp(i(\tau-\varphi))
[\frac{r_{\bot}}{\sqrt{R^2+r^2}}\frac{\partial}
{\partial \tau}-
i\sqrt{R^2+r^2}(\frac{\partial}{\partial r_{\bot}}-\frac{i}{r_{\bot}}
\frac{\partial}{\partial \varphi})]\nonumber\\
&a_2^{(0)'}=\frac{1}{2}exp(i(\tau+\varphi))
[\frac{-r_{\bot}}{\sqrt{R^2+r^2}}\frac{\partial}
{\partial \tau}+
i\sqrt{R^2+r^2}(\frac{\partial}{\partial r_{\bot}}+\frac{i}{r_{\bot}}
\frac{\partial}{\partial \varphi})]\nonumber\\
&b^{(0)'}=exp(i\tau)(\frac{-z}{\sqrt{R^2+r^2}}\frac{\partial}
{\partial \tau}+i\sqrt{R^2+r^2}\frac{\partial}{\partial z})
\quad L_3^{(0)}=-2i\frac{\partial}{\partial \varphi}\nonumber\\
&L_+^{(0)}=exp(i\varphi)(z\frac{\partial}{\partial r_{\bot}}-
r_{\bot}\frac{\partial}{\partial z}+\frac{iz}{r_{\bot}}
\frac{\partial}{\partial \varphi})
\label{3.30}
\end{eqnarray}
where $r=|{\bf r}|$ and $z=x^3$. Note that the expressions for the
operators $L_3$ and $L_+$ are the same as in the standard theory.

As follows from Eqs. (\ref{3.27}) and (\ref{3.30}) for $L_{12}$,
the function $\psi_0(x)$ which plays the role of $e_0$ for our
representation, does not depend on the polar angle $\varphi$.
Then, as follows from Eq. (\ref{2.27}) and the expression for
$L_+$, it also does not depend on the angle between ${\bf r}$
and the $z$ axis. So the dependence on ${\bf r}$ is reduced to
the dependence on $r$ only. Further, the condition that 
$\psi_0(x)$ is the eigenvector of the operator (\ref{3.28})
with the eigenvalue $m$, means that $\psi_0(x)$ depends on
$\tau$ as $exp(-im\tau /2)$. By using all these conditions
and Eq. (\ref{3.30}), we now conclude that
\begin{equation}
\psi_0(\tau,{\bf r})=exp(-im\tau/2)\psi_0(r)
\label{3.31}
\end{equation} 
and the function $\psi_0(\tau,{\bf r})$ is annihilated by
the operators $a_1',a_2',b'$ only if the following condition
is satisfied:
\begin{equation}
mr\psi_0(r)+2(R^2+r^2)\frac{d\psi_0(r)}{dr}=0
\label{3.32}
\end{equation}

We conclude that the only solution for the function 
$\psi_0(\tau,{\bf r})$ satisfying the conditions (\ref{3.27})
is 
\begin{equation}
\psi_0(\tau,{\bf r})=const\,exp(\frac{-im\tau}{2})
(\frac{R^2}{R^2+r^2})^{m/4}
\label{3.33}
\end{equation} 
where the value of $const$ can be chosen if we require some
normalization condition. For example, we can require that
the normalization integral for the function 
$\psi_0(\tau,{\bf r})$ is equal to one:
\begin{equation}
\int |\psi_0(\tau,{\bf r})|^2d\Omega=1
\label{3.34}
\end{equation}
In contrast to the Poincare invariant theory, where the state
with the minimum energy is not normalizable, the normalization
integral (\ref{3.34}) exists if $m>3$. In particular,
it does not exist in the case $m=2$ which is treated as 
massless if $s=0$.

By using Eq. (\ref{3.10}), we now can directly verify that
the function $\psi_0(\tau,{\bf r})$ given by Eq. (\ref{3.33})
indeed satisfies the AdS analog of the Klein-Gordon equation
(\ref{3.26}). Therefore, as noted above, all the functions
obtained from $\psi_0(\tau,{\bf r})$ by multiple actions
of the operators $L_{ab}$ on $\psi_0(\tau,{\bf r})$ also
satisfy this equation. We can therefore interpret 
Eq. (\ref{3.26}) as a condition which selects among the functions
belonging to the representation (\ref{3.22}) only those
corresponding to particle with the mass $m$.  

Analogously we can consider the case of negative energy
representations. Now the function $\psi_0'(x)$, which is
the analog of $e_0'$ in Sect. \ref{S2.3}, should be the
state with the maximum energy and satisfy the conditions  
(see Eq. (\ref{2.28})
\begin{equation}
a_1"\psi_0'=a_2"\psi_0'=b"\psi_0'=L_-\psi_0'=L_{12}\psi_0'=0 
\quad L_{05}\psi_0'=-m\psi_0'
\label{3.35}
\end{equation}
A direct calculation shows that in the spinless case
\begin{eqnarray}
&a_1^{(0)"}=\frac{1}{2}exp(-i(\tau-\varphi))
[-\frac{r_{\bot}}{\sqrt{R^2+r^2}}\frac{\partial}
{\partial \tau}-
i\sqrt{R^2+r^2}(\frac{\partial}{\partial r_{\bot}}+\frac{i}{r_{\bot}}
\frac{\partial}{\partial \varphi})]\nonumber\\
&a_2^{(0)'}=\frac{1}{2}exp(-i(\tau+\varphi))
[\frac{r_{\bot}}{\sqrt{R^2+r^2}}\frac{\partial}
{\partial \tau}+
i\sqrt{R^2+r^2}(\frac{\partial}{\partial r_{\bot}}-\frac{i}{r_{\bot}}
\frac{\partial}{\partial \varphi})]\nonumber\\
&b^{(0)"}=exp(-i\tau)(\frac{z}{\sqrt{R^2+r^2}}\frac{\partial}
{\partial \tau}+i\sqrt{R^2+r^2}\frac{\partial}
{\partial z})\nonumber\\
&L_-^{(0)}=exp(-i\varphi)(-z\frac{\partial}{\partial r_{\bot}}+
r_{\bot}\frac{\partial}{\partial z}+\frac{iz}{r_{\bot}}
\frac{\partial}{\partial \varphi})
\label{3.36}
\end{eqnarray}
By analogy with the case of positive energies one can show
that the function with the maximum negative energy is
\begin{equation}
\psi_0'(\tau,{\bf r})=const\,exp(\frac{im\tau}{2})
(\frac{R^2}{R^2+r^2})^{m/4}
\label{3.37}
\end{equation}

In Sect. \ref{S2.3} we have considered a purely algebraic
construction of modular and conventional IRs. There exists
the correspondence between these representations in the sense
of Sect. \ref{S2.1}. In this section we have shown that (in the
spinless case) the conventional IR can also be implemented by 
operators in the space of functions on the AdS 
hyperboloid (\ref{1.3}) such that the functions satisfy Eq. 
(\ref{3.26}). Therefore there also exists the correspondence
between the spinless modular IR and the representation considered
in this section. 

\section{Spin 1/2 particle}
\label{S3.3}

In our system of units, IR describing a particle with the
conventional spin 1/2 is characterized by the value of the
parameter $s$ equal to one. As noted in Sect. \ref{S3.1},
for constructing a representation describing a particle in
the space of functions on the AdS hyperboloid, one
should specify a finite-dimensional representation 
$g\rightarrow D(g)$. What is a relevant choice of such a
representation for a spin 1/2 particle? To answer this
question, we note the following. In Poincare invariant
theories, a spin 1/2 particle is described by the famous
Dirac equation. This is an equation for a bispinor, i.e.
a spinor $\psi$ which can be represented as a column with
four components $(\psi^1,\psi^2,\psi^3,\psi^4)$. The Dirac
equation involves the matrices $\gamma^{\mu}$ 
$(\mu=0,1,2,3)$ which can be chosen in the form
\begin{equation}
\gamma^0=
\left\|\begin{array}{cc}
0&1\\
1&0
\end{array}\right\|\quad 
\gamma^i=\left\|\begin{array}{cc}
0&-\sigma_i\\
\sigma_i&0
\end{array}\right\|\quad (i=1,2,3)
\label{3.38}
\end{equation}
where in turn, the elements of the $\gamma$-matrices 
are 2$\times$2 matrices and the $\sigma_i$ are the Pauli 
matrices which we display for completeness:
\begin{equation}
\sigma_1=
\left\|\begin{array}{cc}
0&1\\
1&0
\end{array}\right\|\quad 
\sigma_2=\left\|\begin{array}{cc}
0&-i\\
i&0\end{array}\right\|\quad 
\sigma_3=\left\|\begin{array}{cc}
1&0\\
0&-1\end{array}\right\| 
\label{3.39}
\end{equation}

To prove Poincare invariance of the Dirac equation, it is 
convenient, in addition to the $\gamma$-matrices, to use also  
the matrices
\begin{equation}
\sigma^{\mu\nu}=\frac{i}{2}[\gamma^{\mu},\gamma^{\nu}]\quad 
(\mu,\nu=0,1,2,3)
\label{3.40}
\end{equation}
As it has been noted in the literature, if we define 
$\Gamma^{5\mu}=\gamma^{\mu}$, 
$\Gamma^{\mu 5}=-\Gamma^{5\mu}$,
$\Gamma^{\mu\nu}=\sigma^{\mu\nu}$, then one can directly verify 
that the set $\Gamma^{ab}$ satisfies the commutation relations 
(\ref{3.21}).
We believe, this property makes the Dirac equation in the SO(2,3) 
theory even more natural than in the standard case (see below).

We now choose the representation $g\rightarrow D(g)$ in such a 
way, that the $\Gamma^{ab}$ are its representation generators.
Since not all the operators $\Gamma^{ab}$ are Hermitian, the
representation $g\rightarrow D(g)$ is not unitary. Let $^*$ be
the operation of conventional Hermitian conjugation. Instead
of the standard scalar product $(\psi_1,\psi_2)=\psi_1^*\psi_2$,
we now define the new one (which is standard for Dirac spinors)
$$(\psi_1,\psi_2)=\bar{\psi_1}\psi_2=\psi_1^*\gamma^0\psi_2$$ 
where $\bar{\psi}=\psi^*\gamma^0$.
Then the new Hermitian
conjugation of the operator $A$ is $A^+=\gamma^0A^*\gamma^0$.
It is easy to see that with such a definition, the operators 
$\Gamma^{ab}$ are Hermitian but the metric is not positive
definite. However we will see that for elements corresponding
to IR with a definite mass, the scalar product is always 
positive definite.

As follows from the above discussion, it is natural to implement
the representation (\ref{3.22}) in the space of functions
$\psi(x)$ such that for each $x$ belonging to the AdS
hyperboloid, $\psi(x)$ is a Dirac spinor, and the scalar
product is defined as
\begin{equation}
(\psi_1,\psi_2)=\int \bar{\psi_1}(x)^* \psi_2(x) d\Omega(x)
\label{3.41}
\end{equation}
The generators of such a representation are 
$M_{ab}=L_{ab}+\Gamma_{ab}$. We also have to choose a 
covariant equation which singles out only particles with a
given mass. As an AdS analog of the Dirac equation we
choose 
\begin{equation}
(\frac{1}{2}L_{ab}\Gamma^{ab}+m+1)\psi(x)=0
\label{3.42}
\end{equation}
whose AdS covariance is obvious. This equation can be
obtained from the Lagrangian density 
\begin{equation}
L(x)=\bar{\psi(x)}(\frac{1}{2}L_{ab}\Gamma^{ab}+m+1)\psi(x)
\label{3.43}
\end{equation}
by using Eq. (\ref{3.20}) and considering $\psi(x)$ and 
$\bar{\psi(x)}$ as independent fields. The Lagrangian
density (\ref{3.43}) is not Hermitian but, by analogy with
the standard case \cite{AB,Wein}, it can be replaced by a
Hermitian one for which the Euler-Lagrange equation are
the same. 

If $R$ is very large and we consider Eq. (\ref{3.42}) 
in a vicinity of the point $(x_5=-R,0,0,0,0)$, it becomes the
standard Dirac equation. This is seen by dividing this
expression by $2R$, using Eqs. (\ref{3.10}), (\ref{3.13}),
the condition $\Gamma^{5\mu}=\gamma^{\mu}$ and the fact that
$m/2R$ becomes the standard mass when $R\rightarrow \infty$.

For the case of positive energies, the Dirac spinor 
$\psi_0(x)$ which is an analog of $e_0$ in Eq. (\ref{2.29}),
should satisfy the conditions
\begin{equation}
a_j'\psi_0=b'\psi_0=L_+\psi_0=0\quad 
M_{05}\psi_0=m\psi_0\quad M_{12}\psi_0=\psi_0 
\label{3.44}
\end{equation}
where the operators $(a_1',a_2',b')$ can be expressed in
terms of $M_{ab}$ by using Eq. (\ref{2.25}). We can
represent these operators, as well as the operators 
$(M_{05},M_{12})$, as sums of two terms - the orbital parts,
corresponding to the spinless case, and spin parts 
originated from $\Gamma_{ab}$. The orbital parts of
the operators in question have been already calculated
in the preceding section. The spin parts have the form
\begin{eqnarray}
&\Gamma_{05}=\left\|\begin{array}{cc}
0&-1\\
-1&0
\end{array}\right\|\quad
\Gamma_{12}=\left\|\begin{array}{cc}
\sigma_3&0\\
0&\sigma_3
\end{array}\right\|\quad
b^{(s)'}=\frac{1}{2}\left\|\begin{array}{cc}
\sigma_3&-\sigma_3\\
\sigma_3&-\sigma_3
\end{array}\right\|\nonumber\\
&a_1^{(s)'}=-\frac{1}{4}\left\|\begin{array}{cc}
\sigma_-&-\sigma_-\\
\sigma_-&-\sigma_-
\end{array}\right\|\quad
a_2^{(s)'}=\frac{1}{4}\left\|\begin{array}{cc}
\sigma_+&-\sigma_+\\
\sigma_+&-\sigma_+
\end{array}\right\|\nonumber\\
&L_+^{(s)}=-\frac{1}{2}\left\|\begin{array}{cc}
\sigma_+&0\\
0&\sigma_+
\end{array}\right\|   
\label{3.45}
\end{eqnarray}
where $\sigma_{\pm}=\sigma_1\pm i \sigma_2$ and the
superscript $(s)$ means that the spin part of the
operator in question is considered.

We will seek the solution of the conditions (\ref{3.44})
in the form 
$$\psi_0(x)=\left\|\begin{array}{c}
\chi(x)\\
\chi(x)
\end{array}\right\|$$
where $\chi(x)$ is an ordinary two-dimensional spinor.
Then, as follows from Eq. (\ref{3.45}),
the spin parts of the operators $(a_1',a_2',b')$ do not
contribute to (\ref{3.44}). As we have seen in the 
preceding section, the ground state
for the spinless particle is spherically symmetric. 
It is reasonable to assume that this is the case for $s=1$
too. Then the orbital parts of the three-dimensional
rotation operators do not contribute to Eq. (\ref{3.44}). 
The conditions $L_+\psi_0=0$ and $\Gamma_{12}\psi_0=\psi_0$ 
will now be satisfied if
the spinor $\chi(x)$ has only the upper component, i.e.
\begin{equation}
\psi_0(x)=\left\|\begin{array}{c}
f(\tau,r)\\
0\\
f(\tau,r)\\
0
\end{array}\right\|
\label{3.46}
\end{equation}
The rest of calculations can be carried out by analogy with the
spinless case with one exception. Since the state (\ref{3.46})
is the eigenvector of the operator $\Gamma_{05}$ with the
eigenvalue -1, we now have
\begin{equation}
L_{05}\psi_0(x)=(m+1)\psi_0(x),
\label{3.47}
\end{equation}
i.e. $\psi_0(x)$ is the eigenvector of $L_{05}$ with the
eigenvalue $(m+1)$, not $m$ as in the spinless case.
The final result for $f(\tau,r)$ is
\begin{equation}
f(\tau,r)=const\,exp(\frac{-i(m+1)\tau}{2})
(\frac{R^2}{R^2+r^2})^{(m+1)/4}
\label{3.48}
\end{equation}

It is easy to see that the state (\ref{3.46}) with the
condition (\ref{3.48}) satisfies the AdS analog of
the Dirac equation (\ref{3.42}). Indeed, both, orbital
and spin parts of the operators $(a_1',a_2',b',L_+)$, acting
on $\psi_0(x)$ give zero and the same is true for the
orbital parts of the remaining rotation operators. Therefore
$$L_{ab}\Gamma^{ab}\psi_0(x)=2L_{05}\Gamma^{05}\psi_0(x)=
-2L_{05}\psi_0(x)$$
The result now follows from Eq. (\ref{3.47}).  
Since $\psi_0$ is the eigenvector of the Dirac operator 
$L_{ab}\Gamma^{ab}/2\,\, +m+1$ with the eigenvalue zero, and
this operator commutes with all the operators $M_{ab}$, any 
function obtained by their multiple actions on $\psi_0(x)$ is
the eigenstate of the Dirac operator with the eigenvalue 0.
This means that every such function satisfies Eq. (\ref{3.42}).

\begin{sloppypar}
We now consider negative energy representations. The bispinor
function $\psi_0'(x)$ which is the analog of the vector $e_0'$
in Sect. \ref{S2.3}, should satisfy the conditions
\begin{equation}
a_j"\psi_0'=b"\psi_0'=L_-\psi_0'=0\quad 
M_{05}\psi_0'=-m\psi_0'\quad M_{12}\psi_0'=-\psi_0 
\label{3.49}
\end{equation}
A direct calculation gives the following expressions for the
spin parts of the operators $(a_j",b",L_-)$ 
\begin{eqnarray}
&a_1^{(s)"}=\frac{1}{4}\left\|\begin{array}{cc}
\sigma_+&\sigma_+\\
-\sigma_+&-\sigma_+
\end{array}\right\|\quad
a_2^{(s)"}=\frac{1}{4}\left\|\begin{array}{cc}
-\sigma_-&-\sigma_-\\
\sigma_-&\sigma_+
\end{array}\right\|\nonumber\\
&b^{(s)"}=\frac{1}{2}\left\|\begin{array}{cc}
-\sigma_3&-\sigma_3\\
\sigma_3&\sigma_3
\end{array}\right\|\quad
L_-^{(s)}=-\frac{1}{2}\left\|\begin{array}{cc}
\sigma_-&0\\
0&\sigma_-
\end{array}\right\|   
\label{3.50}
\end{eqnarray}
By using these expressions, one can show by analogy with the
above calculations that the solution can be found in the form
\begin{equation}
\psi_0(x)'=\left\|\begin{array}{c}
0\\
f(\tau,r)\\
0\\
-f(\tau,r)
\end{array}\right\|
\label{3.51}
\end{equation}
where $L_{05}\psi_0'(x)=-(m+1)\psi_0'(x)$ and
\begin{equation}
f(\tau,r)=const\,exp(\frac{i(m+1)\tau}{2})
(\frac{R^2}{R^2+r^2})^{(m+1)/4}
\label{3.52}
\end{equation}
It is easy to show that the bispinor function $\psi_0'(x)$
also satisfies Eq. (\ref{3.42}). If one requires that 
scalar products of elements belonging to the negative
energy representation should be the same as for IRs
considered in Sect. \ref{S2.3}, then the scalar product
should be defined as
\begin{equation}
(\psi_1,\psi_2)=-\int \bar{\psi_1}(x)^* \psi_2(x) d\Omega(x)
\label{3.53}
\end{equation}
because in that case it is positive definite.
\end{sloppypar}

We have shown that ordinary IRs with $s=1$ (corresponding to 
spin 1/2 particles) can be implemented in the space of Dirac
spinors satisfying the AdS analog of the Dirac
equation (\ref{3.42}). The AdS covariance of this
equation immediately follows from the commutation relations
for the representation generators. In turn, as shown in
Sect. \ref{S2.3}, there exists the correspondence between
ordinary and modular IRs. It has been also shown that 
ordinary positive energy IRs with the minimum energy $m$
and negative energy IRs with the maximum energy $-m$
correspond to the same modular IR. Therefore the same
correspondence takes place for modular IRs on the one hand,
and ordinary representations implemented in the space of  
Dirac spinors, on the other.

\section{Spin 1 particle}
\label{S3.4}

In the standard approach, spin 1 particles in 
four-dimensional space-time are described by the vector
potential $A^{\mu}(x)$ $(\mu=0,1,2,3)$, satisfying the
condition $\partial_{\mu} A^{\mu}(x)=0$. If space-time
is four-dimensional but not flat, $A^{\mu}(x)$ is often treated
as a vector field with respect to allowable transformations
of that space-time and $\partial_{\mu}$ is replaced by the
covariant derivative. However, by analogy with Ref. \cite{DirCE}
and the works developing these results (see e.g. Ref. \cite{FronsdalB}
and references therein), 
it is possible to adopt another approach in which spin 1 
particles in the AdS space are described by a five-dimensional 
vector function $A^a(x)$ on that space
such that the following constraints are satisfied:
\begin{equation}
x_aA^a(x)=0\quad D_aA^a(x)=0
\label{3.54}
\end{equation}

The representation $g\rightarrow D(g)$ is now the standard 
vector representation acting on five-dimensional columns in 
such a way that for $g\in SO(2,3)$, $D(g)=g$. 
We can complexify the representation space for convenience
and then the representation $g\rightarrow T(g)$ can be 
considered in the space of complex vector functions 
$A^a(x)$ with the scalar product 
\begin{equation}
(A_1,A_2)=\int A_1^{a*}(x)A_2^b(x)g_{ab} d\Omega(x)
\label{3.55}
\end{equation}

At the same time (if the particle is chargeless), the 
Lagrangian density is written for a real vector field. 
In the standard theory, there exists a well known problem,
about the best choice of the Lagrangian density, whether
it should contain $(\partial_{\nu} A^{\mu}(x))^2$ or
$(\partial_{\nu} A_{\mu}(x)-\partial_{\mu} A_{\nu}(x))^2$.
We accept the simplest choice: 
\begin{equation}
L(x)=-\frac{1}{2}(D_aA^b(x))(D^aA_b(x))+
\frac{m^2}{8R^2}A^c(x)A_c(x)
\label{3.56}
\end{equation}
Then, by analogy with the scalar case,
\begin{equation}
(D_bD^b+\frac{m^2}{4R^2})A^a(x)=0
\label{3.57}
\end{equation}
When $R$ is very large, $\tau <<1$ and $t=R\tau$,
we again have the standard D'Alambertian equation for
each component of $A^a(x)$. 

The case of vector particle corresponds to $s=2$ in Sect.
\ref{S2.3}. By analogy with the cases $s=0$ and $s=1$,
considered in the previous sections, we will first seek
a vector field $E^a(x)$ which plays the role of the vector
$e_0$ in Sect. \ref{S2.3}. This means that $E^a(x)$ should
satify the conditions:  
\begin{equation}
a_j'E=b'E=L_+E=0\quad 
M_{05}E=mE\quad M_{12}E=2E 
\label{3.58}
\end{equation}

The generators of the representation $g\rightarrow T(g)$
can be written as sums of orbital and spin parts. The former
are the same as in the spinless case, and the corresponding
expressions have been written in Sect. \ref{S3.2} (see Eqs.
(\ref{3.28}), (\ref{3.30}) and (\ref{3.36})). The latter are
given by the matrices $V_{ab}$ in Eq. (\ref{3.4}). Each 
matrix contains 25 elements but only a few of them are not
zero. Below we display the nonzero elements of the  
spin parts of the operators in Eq. (\ref{3.58}):
\begin{eqnarray}
&(V_{05})^5_0=-(V^{05})^0_5=-2i\quad 
(V_{12})^1_2=-(V^{12})^2_1=-2i\nonumber\\
&(a_1^{(s)'})^5_1=-(a_1^{(s)'})^1_5=-(a_1^{(s)'})^0_2=
(a_1^{(s)'})^2_0=-\frac{i}{2}\nonumber\\
&(a_1^{(s)'})^5_2=-(a_1^{(s)'})^2_5=(a_1^{(s)'})^0_1=
-(a_1^{(s)'})^1_0=-\frac{1}{2}\nonumber\\
&(a_2^{(s)'})^5_1=-(a_2^{(s)'})^1_5=(a_2^{(s)'})^0_2=
-(a_2^{(s)'})^2_0=\frac{i}{2}\nonumber\\
&(a_2^{(s)'})^5_2=-(a_2^{(s)'})^5_2=-(a_2^{(s)'})^0_1=
(a_2^{(s)'})^1_0=-\frac{1}{2}\nonumber\\
&(b^{(s)'})^0_3=-(b^{(s)'})^3_0=1\quad
(b^{(s)'})^5_3=-(b^{(s)'})^3_5=i
\label{3.59}
\end{eqnarray}
By using these expression, one can verify that the
solution of Eq. (\ref{3.58}) is unique up to a constant factor:
\begin{equation}
E(x)=const\,exp(-\frac{im\tau}{2})f(r)\left\|\begin{array}{c}
-exp(i(\varphi-\tau))r_{\bot}/\sqrt{R^2+r^2}\\
iexp(i(\varphi-\tau))r_{\bot}/\sqrt{R^2+r^2}\\
1\\
i\\
0
\end{array}\right\|
\label{3.60}
\end{equation}
where
\begin{equation}
f(r)=(\frac{R^2}{R^2+r^2})^{(m+1)/4}
\label{3.61}
\end{equation}

Let us stress that that the result (\ref{3.60}) is the
solution of Eq. (\ref{3.59}) only. Now we can verify that
the solution indeed satisfies the restrictions (\ref{3.54})
and (\ref{3.57}). When $R\rightarrow \infty$, the spin
part of $E(x)$ becomes the standard spin 1 function with
the $z$ projection of the spin equal to 1.

If the spin is equal to 1 in usual units, then $s=2$ and 
$m\geq 4$, where the equality takes place in the massless case.
We see that the function defined by Eq. (\ref{3.61}) is
quadratically integrable in both, massive and massless case.
Note that in the standard Poincare invariant quantum theory, 
the ground state wave function in the massive case is spherically symmetric
while a massless particle does not have rest states and cannot be 
described by spherically symmetric functions (see e.g. Ref. \cite{AB,Wein}). 
On the other hand, in SO(2,3) invariant quantum theories, a massless particle
has an exceptional value of the AdS energy corresponding to the rest 
states (see Sect. \ref{S2.4}). A detailed discussion 
of massless wave equations in the AdS space can be found in
Ref. \cite{FronsdalB}.    

Since the photon is the neutral particle, its quantized field
can be expressed only in terms of wave functions with
positive energies. For this reason we do not discuss the
photon wave functions with negative energies.

\chapter{Interacting fields in SO(2,3) invariant theories}
\label{C4}

\section{Angular momentum tensor in SO(2,3) invariant theories}
\label{S4.1}

In the preceding chapter, the requirement that the
action is SO(2,3) invariant was used only implicitly.
Namely, as a consequence of this invariance, Euler-Lagrange
equations derived from the minimum action principle are
covariant. As a consequence of Poincare invariance of the
action in the standard theory, not only the Euler-Lagrange
equations are Poincare covariant but also the 
energy-momentum operator and generators of the Lorentz group
are conserving physical quantities. The former can be written
in terms of integrals from the energy-momentum tensor over 
some space-like hypersurface while the latter - as integrals 
from the angular momentum tensor over the same hypersurface. 
As argued in Sect. \ref{S1.3}, in SO(2,3) invariant theories,
where all the representation generators are angular momenta,
there is no need to have the energy-momentum tensor and it
is reasonable to expect that all the generators can be expressed
in terms of a five-dimensional angular momentum tensor. In this
section we derive the expression for this tensor in the
Lagrangian formalism. As usual, we first consider classical
(non quantized fields) and the quantization is discussed 
after we are done with the classical construction.  

Let $K$ and $K'$ be two reference frames connected by an
element $g\in SO(2,3)$, such that if a point of the AdS
hyperboloid has the coordinates $x$ in K, it has the 
coordinates $x'=gx$ in K'. If the action in $K$ has the form
of Eq. (\ref{3.16}), then in $K'$ it reads
\begin{equation}
S'=\int_{\Lambda'} L(u'(x'),D'u'(x'),x')d\Omega(x')
\label{4.1}
\end{equation}

The difference between Eqs. (\ref{3.16}) and (\ref{4.1}) is as
follows. $\Lambda'$ is the same surface as $\Lambda$ but in $K'$
its equation is written in terms of $x'$. The relation between
the set of $n$ fields $u(x)$ in $K$ and $u'(x')$ in $K'$ 
should be specified by choosing a representation of the SO(2,3)
group by $n\times n$ matrices $g\rightarrow G(g)$ such that 
\begin{equation}
u'(x')=G(g)u(g^{-1}x')=G(g)u(x) 
\label{4.2}
\end{equation}
The symbol $D'$ means a set of derivatives $D_a'$ in $K'$, and
the volume element $d\Omega(x')$ in $K'$ has the same form
as $d\Omega(x)$ but in terms of $x'$. We assume that the
form of the Lagrangian in the both reference frames is the same,
i.e. $L=L'$.

In Eq. (\ref{4.2}) we rename the integration variable $x'$ by
$x$ and then, as follows fro Eq. (\ref{4.2}),
\begin{equation}
u'(x)=G(g)u(g^{-1}x) 
\label{4.3}
\end{equation}
We also assume that the parameters $\omega_{ab}$ in Eq. (\ref{3.4})
are so small that it is sufficient to consider the difference
$S'-S$ only in the first order in these parameters. 
As follows from Eq. (\ref{3.4}), in this order
\begin{equation}
(g^{-1}x)^c=x^c+\omega_{ab}(x^ag^{bc}-x^bg^{ac})
\label{4.4}
\end{equation}
The requirement
of SO(2,3) invariance implies that $S'=S$ and therefore
in the first approximation in $\omega_{ab}$ the quantity
$\delta S= S'-S$ should be equal to zero. As seen from
Eq. (\ref{4.3}), in this approximation one can represent
$\delta S$ as a sum of three terms. The first term which
is usually called the orbital one, is a consequence
of the fact that $g^{-1}x\neq x$ and $D'\neq D$. The 
second term, called the spin one, is a consequence of the
fact that $G(g)\neq 1$. Finally, the third term is a
consequence of the fact that the equations of the surface 
$\Lambda$ in $K$ and $K'$ are not the same.  

For computing the orbital term we neglect the differences
between $G(g)$ and 1 and between $\Lambda$ and $\Lambda'$,
take into account Eqs. (\ref{4.3}), (\ref{4.4}) and
the fact that $D_a'$ transforms as a covariant vector.
Therefore in the first order in $\omega_{ab}$
\begin{equation}
D_c'=D_c-\omega^{ab}(D_ag_{bc}-D_bg_{ac})
\label{4.5}
\end{equation}
and for our purpose we can write $S'$ as
\begin{eqnarray}
&S'=\int_{\Lambda}\,L(u(x)+\omega_{ab}(x^aD^b-x^bD^a)u,\nonumber\\
&u_{,c}(x)-\omega^{ab}(D_ag_{bc}-D_bg_{ac})u(x)+\nonumber\\
&\omega_{ab}(x^aD^b-x^bD^a)D_cu(x),x)d\Omega(x)
\label{4.6}
\end{eqnarray}

Using the fact that $\omega_{ab}$ is the antisymmetric tensor,
we now can write $\delta S$ in the form
\begin{eqnarray}
&\delta S^{orb}=2\omega_{ab}\int_{\Lambda} (\frac{\partial L}
{\partial u}x^aD^bu(x)-\frac{\partial L}{\partial u_{,c}}
D^a\delta_c^bu(x)+\nonumber\\ 
&\frac{\partial L}{\partial u_{,c}}x^aD^bD_cu(x))d\Omega(x)  
\label{4.7}
\end{eqnarray}
As follows from Eq. (\ref{3.14}),
\begin{equation}
\omega_{ab}x^aD^bD_c=\omega_{ab}D_cx^aD^b-\omega_{ab}D^b\delta_a^c
\label{4.8}
\end{equation} 
and therefore Eq. (\ref{4.7}) can be simplified:
\begin{equation}
\delta S^{orb}=2\omega_{ab}\int_{\Lambda} (\frac{\partial L}
{\partial u}x^aD^bu(x)+ 
\frac{\partial L}{\partial u_{,c}}D_cx^aD^bu(x))d\Omega(x)  
\label{4.9}
\end{equation}
We can represent this expression in the form
\begin{eqnarray}
&\delta S^{orb}=2\omega_{ab}\int_{\Lambda} (\frac{\partial L}
{\partial u}-(D_c-\frac{4x_c}{R^2})
\frac{\partial L}{\partial u_{,c}})x^aD^bu(x))d\Omega(x)+\nonumber\\ 
&2\omega_{ab}\int_{\Lambda} (D_c-\frac{4x_c}{R^2})
\frac{\partial L}{\partial u_{,c}}(x^aD^bu(x))d\Omega(x)
\label{4.10}
\end{eqnarray}
We now take into account the fact that $\delta S^{orb}$ is
computed for fields satisfying the Euler-Lagrange equations.
Therefore, as follows from Eq. (\ref{3.20}),
the first term in Eq. (\ref{4.10}) does not
contribute to $\delta S^{orb}$ and we can write the final
expression for $\delta S^{orb}$ in the form 
\begin{eqnarray}
\delta S^{orb}=\omega_{ab}\sum_{i=1}^{n}
\int_{\Lambda} (D_c-\frac{4x_c}{R^2})
[\frac{\partial L}{\partial u_{i,c}}(x^aD^b-x^bD^a)u_i(x)]d\Omega(x)
\label{4.11}
\end{eqnarray}

The computation of the spin part of $\delta S$ is much simplier.
We neglect the difference between $x$ and $x'$, $D$ and $D'$,
and $\Lambda$ and $\Lambda'$. Then by using the Euler-Lagrange
equations we obtain analogously that
\begin{equation}
\delta S^{spin}=-\frac{i}{2}\omega_{ab}\sum_{i,j=1}^n
\int_{\Lambda} (D_c-\frac{4x_c}{R^2})
[\frac{\partial L}{\partial u_{i,c}}(\Gamma^{ab})_j^iu^j(x)]d\Omega(x) 
\label{4.12}
\end{equation} 
where the $\Gamma^{ab}$ are the generators of the representation
$g\rightarrow G(g)$.

To derive the contribution related to the fact that 
$\Lambda\neq \Lambda'$, we should have an analog of the 
Gauss-Ostrogradsky theorem for the relation between 
surface and volume integrals. The derivation does not depend 
on the AdS structure of the theory but only on the
explicit expressions for the derivatives $D_a$. For readers
who do not want to go into details, we note that below we will
use only the choice of $\Lambda$ which seems to be the most
natural: since $\tau$ is a natural AdS analog of 
time, we will consider only the case when the integration for
$S$ is performed over the whole conventional three-dimensional 
space while $\tau$ is in the range $[\tau_1,\tau_2]$. 
In this case the derivation is straightforward. In the general
case the result is similar to that in the standard theory:
\begin{equation}
\delta S^{surf}=-\omega_{ab} \int_{\Lambda}
(D_c-\frac{4x_c}{R^2})[L(u,Du,x)(x^ag^{bc}-x^bg^{ac})]d\Omega(x)
\label{4.13}
\end{equation}

We can combine Eqs. (\ref{4.11})-(\ref{4.13}) as follows.
Denote
\begin{eqnarray}
&M^{abc}(x)=\sum_{i,j=1}^n 
[\frac{\partial L}{\partial u_{i,c}}(x^aD^b-x^bD^a)u_i(x)-
\frac{i}{2}\frac{\partial L}{\partial u_{i,c}}
(\Gamma^{ab})_j^iu^j(x)]-\nonumber\\
&L(x)(x^ag^{bc}-x^bg^{ac})
\label{4.14}
\end{eqnarray}    
Then 
\begin{equation}
\delta S= \omega_{ab}\int_{\Lambda}(D_c-\frac{4x_c}{R^2})
M^{abc}(x)d\Omega(x)
\label{4.15}
\end{equation}

As noted in Chap. \ref{C3}, one can transform such an integral
into the integral over the surface $\Lambda$. In the case when
the integration in (\ref{4.15}) is carried out over the whole
three-dimensional space and $\tau$ is in the range 
$[\tau_1,\tau_2]$, it follows from Eq. (\ref{3.10}) that
\begin{equation}
\delta S=\omega_{ab}(M^{ab}(\tau_2)-M^{ab}(\tau_1))
\label{4.16}
\end{equation}
where
\begin{equation}
M^{ab}(\tau)=\int (sin\tau M^{ab5}({\bf r},\tau)+
cos\tau M^{ab0}({\bf r},\tau))\frac{d^3{\bf r}}{\sqrt{R^2+r^2}}
\label{4.17}
\end{equation}

We conclude that, as a consequence of AdS invariance,
$M^{ab}(\tau)$ does not depend on $\tau$ and there exist 
ten conserving quantities which can be chosen as 
$M^{ab}(0)$. However these quantities do depend on initial
conditions. For example, if they are such that at $\tau=\tau_0$
the interacting fields are the same as the free ones than the
operators $M^{ab}$ depend on the choice of $\tau_0$. The tensor 
$M^{abc}(x)$ in Eq. (\ref{4.14}) has the meaning of the angular 
momentum tensor because $M^{abc}(x)$
and $M^{ab}(\tau)$ are related by Eq. (\ref{4.17}). It is
also interesting to note that in the coordinates 
$({\bf r},\tau)$, the description looks closer to the
nonrelativistic case than to the relativistic one. Indeed,
in the nonrelativistic case the time is distinguished while
in the relativistic one the temporal and spatial coordinates 
are on equal footing.

\section{QED in AdS space}
\label{S4.2}

Our expressions for the Lagrangians of the electron-positron
and photon fields are given by Eqs. (\ref{3.25}) and
(\ref{3.56}). If the fields do not interact with each other,
the total Lagrangian is a sum of these Lagrangians and we
have
\begin{eqnarray}
&L_{free}(x)=-\frac{1}{2}(D_a\psi(x))(D^a\psi(x))+
\frac{m^2}{8R^2}\psi (x)^2-\nonumber\\
&\frac{1}{2}(D_aA^b(x))(D^aA_b(x))+
\frac{m_0^2}{8R^2}A^c(x)A_c(x)
\label{4.18}
\end{eqnarray}
where $m$ is the AdS mass of the electron and $m_0$ is
the AdS mass of the photon.

Now we have to choose the form of the interaction Lagrangian.
Our choice is
\begin{equation}
L_{int}(x)= -\frac{e}{2}\bar{\psi}(x)(x_aA_b(x)-x_bA_a(x))
\Gamma^{ab}\psi(x)
\label{4.19}
\end{equation}
where $\Gamma^{ab}$ are the matrices described in Sect. 
\ref{S3.3}. This interaction Lagrangian is obviously SO(2,3)
invariant, and when $R\rightarrow \infty$, it becomes 
proportional to the standard interaction Lagrangian
$$-e\bar{\psi}(x)A_{\mu}(x))\gamma^{\mu}\psi(x)$$
The quantity $e$ has the dimenion $R^{-1/2}$. We can redefine
all the fields and instead of ${\bf r}$ we can work with 
dimensionless variables ${\bf r}/R$. Then the theory will not
contain any dependence on $R$ and all the fields will be
dimensionless. However the transition to the flat space
will not be so obvious. 

As follows from Eqs. (\ref{3.20}), (\ref{4.18}) and 
(\ref{4.19}), the Euler-Lagrange equations now have the
form
\begin{eqnarray}
&[\frac{1}{2}L_{ab}\Gamma^{ab}+m+1-\frac{e}{2}
(x_aA_b-x_bA_a)\Gamma^{ab}]\psi(x)=0\nonumber\\
&(D_cD^c+\frac{m_0^2}{R^2})A^a(x)=ex_b\bar{\psi}(x)
\Gamma^{ba}\psi(x)
\label{4.20}
\end{eqnarray}
and we can add an equation for $\bar{\psi}(x)$ obtained
by the transposition of the first equation. The initial
condition should specify the fields 
$(\psi(x),\bar{\psi}(x),A^a(x))$ on some surface and the
normal derivative of $A^a(x)$ with respect to this surface.
For example, we can specify 
\begin{equation}
(\psi({\bf r},\tau_0),\bar{\psi}({\bf r},\tau_0),
A^a({\bf r},\tau_0),\partial A^a({\bf r},\tau_0)/
\partial \tau)
\label{4.21}
\end{equation}
by requiring that these quantities are the same as for
free fields.

Our next goal is to represent the angular momenta  as
\begin{equation}
M^{ab}=M^{ab}_{free} + M^{ab}_{int}
\label{4.22}
\end{equation}
where the first term represents the angular momenta
for the free electron-positron and photon fields while the second
term is the contribution of interaction. Since the interaction
Lagrangian does not contain derivatives of the fields (see 
Eq. (\ref{4.19})), one might expect that, as a consequence of
Eq. (\ref{4.14})
\begin{equation}
M^{abc}(x)=M^{abc}_{free}(x)+L_{int}(x)(x^ag^{bc}-x^bg^{ac})
\label{4.23}
\end{equation}
Suppose the initial conditions are specified on the surface
$\tau=\tau_0$ such that the fields (\ref{4.21}) are the
same as free ones. Since the equations for the fields contains
the interaction, their values at $\tau\neq \tau_0$ will not
be the same as for free fields. Therefore the decomposition
(\ref{4.23}) has a chance to be correct only if  
$\tau= \tau_0$. The problem is that even when
$\tau= \tau_0$, the derivative $\partial 
\psi({\bf r},\tau_0)/\partial \tau$ is interaction dependent
(see Eq. (\ref{4.20})). However, one can verify that, as
follows from Eqs. (\ref{4.14}) and (\ref{4.18}), in 
$M^{abc}_{free}$ the derivatives $\partial 
\psi({\bf r},\tau)/\partial \tau$ cancel out. Therefore,
Eq. (\ref{4.23}) is correct when $\tau= \tau_0$. As a result,
by using Eqs. (\ref{4.17}) we obtain
\begin{equation}
M^{ab}=M^{ab}_{free}+M^{ab}_{int}
\label{4.24}
\end{equation} 
where
\begin{eqnarray}
&M^{ij}_{int}=0\quad M^{i5}_{int}=-\int (x^isin\tau_0/\sqrt{R^2+r^2})
L_{int}({\bf r},\tau_0)d^3{\bf r}\nonumber\\
&M^{0i}_{int}=\int (x^icos\tau_0/\sqrt{R^2+r^2})
L_{int}({\bf r},\tau_0)d^3{\bf r}\quad  \quad (i,j=1,2,3)\nonumber\\
&M^{05}_{int}=-\int L_{int}({\bf r},\tau_0)d^3{\bf r}
\label{4.25}
\end{eqnarray}   

Let us note that the quantities $M^{ab}$ depend on $\tau_0$ 
because $\tau=\tau_0$ is the surface where the initial conditions
are specified. At the same time, being the conserving quantities
they are the same at different values of $\tau \geq \tau_0$.  

So far only classical fields have been considered and now the 
problem arises, how to generalize the above 
construction for quantum fields. 

By analogy with the conventional theory, we first introduce
creation and annihilation operators as follows. Suppose the 
electron is described by a positive-energy IR. Then its
state is a superposition of ones having quantum numbers 
$(n_1n_2nk)$ (see Sect. \ref{S2.3}). On the language of
IRs, the state with the quantum numbers $(n_1n_2nk)$
is described by the vector $e(n_1n_2nk)$ (see Sect.
\ref{S2.3}) while, as discussed in Sect. \ref{S3.3}, we
can also describe such a state by a solution of the Dirac
equation with the positive energy, which we denote as
$\psi_{n_1n_2nk}^{(+)}(x)$. Now we introduce a new
description such that the state with the quantum numbers 
$(n_1n_2nk)$ is represented as 
$a(n_1n_2nk)^*\Phi_0$ where $\Phi_0$ is the vacuum state
and $a(n_1n_2nk)^*$ is the operator of the electron
creation in the state with the quantum numbers 
$(n_1n_2nk)$. We also define $a(n_1n_2nk)$ as the
operator annihilating the state $a(n_1n_2nk)^*\Phi_0$.
Then the operators $a(n_1n_2nk)$ and $a(n_1n_2nk)^*$
will correspond to their meaning if $a(n_1n_2nk)\Phi_0=0$,
their commutator or anticommutator is a $c$-number, and they
are adjoint to each other. As usual, in the case of fermions
we assume anticommutator relations while in the case of
bosons - commutator ones. We assume that
\begin{eqnarray}
&\{a(n_1n_2nk),a(n_1'n_2'n'k')^*\}=\nonumber\\
&Norm(n_1n_2nk)
\delta_{n_1n_1'}\delta_{n_2n_2'}\delta_{nn'}\delta_{kk'}
\label{4.26}
\end{eqnarray}
where $Norm(n_1n_2nk)$ is defined in Eq. (\ref{Norm}).

If we assume that the positron is the antiparticle for the
electron, then in the standard approach it is described by
the solutions of the Dirac equation with negative energies.
However they do not have a direct physical meaning, and for
this reason the following formalism is adopted. The positron
states are described by the operators 
$b(n_1n_2nk)$ and $b(n_1n_2nk)^*$ which have the
same meaning as $a(n_1n_2nk)$ and $a(n_1n_2nk)^*$,
respectively, but for the creation and annihilation of
positrons rather than electrons. In particular, 
\begin{eqnarray}
&\{b(n_1n_2nk),b(n_1'n_2'n'k')^*\}=\nonumber\\
&Norm(n_1n_2nk)
\delta_{n_1n_1'}\delta_{n_2n_2'}\delta_{nn'}\delta_{kk'}
\label{4.27}
\end{eqnarray}
and $b(n_1n_2nk)\Phi_0=0$. At the same time, the positron
is described by the solutions of the Dirac equation with
negative energies. We use $\psi_{n_1n_2nk}^{(-)}(x)$ to 
denote the solution corresponding to the basis element
$e(n_1n_2nk)'$ in the IR with the negative energy (see
Sect. \ref{S2.3}) and require that electron and positron
operators anticommute with each other:
\begin{eqnarray}
&\{a(n_1n_2nk),b(n_1'n_2'n'k')^*\}=\{a(n_1n_2nk)^*,b(n_1'n_2'n'k')^*\}=\nonumber\\
&\{a(n_1n_2nk)^*,b(n_1'n_2'n'k')\}=\{a(n_1n_2nk),b(n_1'n_2'n'k')\}=0
\label{4.28}
\end{eqnarray}

The final step in quantizing the free fields $\psi(x)$ and
$\bar{\psi}(x)$ is to write them as
\begin{eqnarray}
&\psi(x)=\sum_{n_1n_2nk}[\psi_{n_1n_2nk}^{(+)}(x)a(n_1n_2nk)+
\psi_{n_1n_2nk}^{(-)}(x)b(n_1n_2nk)^*]\nonumber\\
&\bar{\psi}(x)=\sum_{n_1n_2nk}[\bar{\psi}_{n_1n_2nk}^{(+)}(x)
a(n_1n_2nk)^*+\nonumber\\
&\bar{\psi}_{n_1n_2nk}^{(-)}(x)b(n_1n_2nk)]
\label{4.29}
\end{eqnarray}
Then, as follows from Eqs. (\ref{4.26}-\ref{4.28})
\begin{eqnarray}
&G_{\alpha\beta}(x,x')==\{\psi_{\alpha}(x),
\psi_{\beta}^*(x')\}=\sum_{n_1n_2nk}Norm(n_1n_2nk)\nonumber\\
&[\psi_{n_1n_2nk\alpha}^{(+)}(x)\psi_{n_1n_2nk\beta}^{(+)*}(x')+
\psi_{n_1n_2nk\alpha}^{(-)}(x)\psi_{n_1n_2nk\beta}^{(-)*}(x')]
\label{4.30}
\end{eqnarray}

In the standard case the form of the anticommutation relations 
is not postulated but derived from the canonical formalism.
Namely, $\psi(x)_{\alpha}^*$ is the canonical momentum
corresponding to $\psi(x)_{\alpha}$ since 
$$\psi(x)_{\alpha}^*=
\partial L(x)/\partial (\partial\psi_{\alpha}(x)/\partial t)$$ 
Therefore in the standard approach it is required that the
equal-time anticommutator 
$$\{\psi_{\alpha}({\bf r},t),\psi_{\beta}^*({\bf r}',t)\}$$ 
should be proportional to $\delta^{(3)}({\bf r}-{\bf r}')$
and the anticommutation relations for the creation and 
annihilation operators follow from this requirement.

In the AdS case, $\psi(x)_{\alpha}^*$ is proportional to
$$\partial L(x)/\partial (\partial\psi_{\alpha}(x)/\partial \tau)$$
only in the limit $R\rightarrow \infty$. Therefore
$\psi(x)_{\alpha}$ and $\psi(x)_{\alpha}^*$ are not canonically
conjugated to each other if the role of time is played by
$\tau$. Nevertheless it is still possible that  
\begin{equation}
G_{\alpha\beta}({\bf r},\tau,{\bf r}',\tau)=const 
\delta_{\alpha\beta}\delta^{(3)}({\bf r}-{\bf r}')
\label{4.31}
\end{equation}
Indeed, as follows from Eq. (\ref{4.30}), this condition
will be satisfied if the functions
$\psi_{n_1n_2nk}^{(+)}({\bf r},\tau),
\psi_{n_1n_2nk}^{(-)}({\bf r}',\tau)$ forms a complete set
in the Hilbert space of Dirac spinors considered as functions
of ${\bf r}$ and such that the scalar product is defined as
\begin{equation}
(\psi_1({\bf r},\tau),\psi_2({\bf r},\tau))=
\sum_{\alpha}\int \psi_{1\alpha}^*({\bf r},\tau)
\psi_{2\alpha}({\bf r},\tau) d^3{\bf r} 
\label{4.32}
\end{equation}

We did not succeed in proving the completeness but note that
this is a problem of the standard quantum theory which has
nothing to do with Galois fields. So we assume that the
completeness is satisfied.

Analogously we can introduce creation and annihilation operators
for the photon field. We denote them as $c(n_1n_2nk)$ and
$c(n_1n_2nk)^*$, respectively, require that they satisfy the
commutation relations
\begin{eqnarray}
&[c(n_1n_2nk),c(n_1'n_2'n'k')^*]=\nonumber\\
&Norm(n_1n_2nk)
\delta_{n_1n_1'}\delta_{n_2n_2'}\delta_{nn'}\delta_{kk'}
\label{4.33}
\end{eqnarray}
and the condition $c(n_1n_2nk)\Phi_0=0$. Let $A_{n_1n_2nk}^a(x)$
be a photon wave function corresponding to $e(n_1n_2nk)$ 
(see Sect. \ref{S3.4}). Since the photon is a neutral particle,
its field should be Hermitian and can be expressed only in terms
of functions with the positive energy. We can represent the
free quantized photon field as
\begin{eqnarray}
A^a(x)=\sum_{n_1n_2nk}[A_{n_1n_2nk}^a(x)c(n_1n_2nk)+
A_{n_1n_2nk}^{a*}(x)c(n_1n_2nk)^*]
\label{4.34}
\end{eqnarray}

In the formalism of second quantization the wave function of
the system of electron-positron and photon fields is a vector
in the Fock space. This space can be obtained as follows. We
take the creation operators $a(n_1n_2nk)*$, $b(n_1n_2nk)*$
and $c(n_1n_2nk)*$ at different possible values of $(n_1n_2nk)$.
Then we act by these operators on the vacuum vector $\Phi_0$
and take all possible linear combinations of such obtained
elements.

By using Eqs. (\ref{4.14}), (\ref{4.17}-\ref{4.19}), we can now
try to write down a quantum operator $M_{free}^{ab}$. However
in that case we will have a problem how to justify 
manipulations with quantum fields (see the discussion in
Sect. \ref{S1.1}). At the same time, as far as the {\bf free} 
operator $M^{ab}$ is concerned, its form is clear from very 
general considerations. First of all, it is clear that this
operator can be represented as a sum of operators corresponding
to free electrons, positrons and photons separately. Therefore
it is sufficient to construct the operator $M^{ab}$ for the
field corresponding to an arbitrary IR.

For the electron field we can write
\begin{eqnarray}  
&M^{ab}_{electron}=\sum_{n_1'n_2'n'k'n_1n_2nk}
M^{ab}_{electron}(n_1'n_2'n'k',n_1n_2nk)\nonumber\\
&a(n_1'n_2'n'k')^*a(n_1n_2nk)/Norm(n_1n_2nk)
\label{4.35}
\end{eqnarray}
where the $M^{ab}_{electron}(n_1'n_2'n'k',n_1n_2nk)$ are the
matrix elements of the generators of IR for the electron. 
Taking into account Eq. (\ref{4.26}) and the fact that the
matrix elements satisfy the proper commutation relations, one
can verify that the operators defined by Eq. (\ref{4.35})
satisfy the commutation relations (\ref{2.23}). 

As follows from Eq. (\ref{4.35}), the operator adjoint to
$M^{ab}_{electron}$ is given by
\begin{eqnarray}  
&M^{ab*}_{electron}=\sum_{n_1'n_2'n'k'n_1n_2nk}
\overline{M^{ab}_{electron}(n_1'n_2'n'k',n_1n_2nk)}\nonumber\\
&a(n_1n_2nk)^*a(n_1'n_2'n'k')/Norm(n_1n_2nk)
\label{4.36}
\end{eqnarray}
As follows from Eq. (\ref{2.43}), we can express this operators
in terms of the matrix elements of the operator which is adjoint
to $M^{ab}$ in IR:
\begin{eqnarray}  
&M^{ab*}_{electron}=\sum_{n_1n_2nkn_1'n_2'n'k'}
M^{ab*}_{electron}(n_1n_2nk,n_1'n_2'n'k')\nonumber\\
&a(n_1n_2nk)^*a(n_1'n_2'n'k')/Norm(n_1'n_2'n'k')
\label{4.37}
\end{eqnarray}
In particular, if $M^{ab}$ is selfadjoint in IR, the 
corresponding secondly quantized operator is selfadjoint too.

Analogously, the secondly quantized operators for the positron
and photon fields can be written as
\begin{eqnarray}  
&M^{ab}_{positron}=\sum_{n_1'n_2'n'k'n_1n_2nk}
M^{ab}_{electron}(n_1'n_2'n'k',n_1n_2nk)\nonumber\\
&b(n_1'n_2'n'k')^*b(n_1n_2nk)/Norm(n_1n_2nk)
\label{4.38}
\end{eqnarray}
\begin{eqnarray}  
&M^{ab}_{photon}=\sum_{n_1'n_2'n'k'n_1n_2nk}
M^{ab}_{photon}(n_1'n_2'n'k',n_1n_2nk)\nonumber\\
&c(n_1'n_2'n'k')^*c(n_1n_2nk)/Norm(n_1n_2nk)
\label{4.39}
\end{eqnarray}
Note that as far as free operators are concerned, the positron
field should be described in the same way as electron one,
because on the level of secondly quantized operators the 
negative energy IRs have no physical meaning
in the standard theory, and the requirement of C-invariance 
means that the theory should be invariant under the transformation
\begin{eqnarray}
&a(n_1n_2nk)^*\rightarrow {\bar \eta}_Cb(n_1n_2nk)^*\quad 
b(n_1n_2nk)^*\rightarrow a(n_1n_2nk)^*/{\bar \eta}_C\nonumber\\
&a(n_1n_2nk)\rightarrow \eta_Cb(n_1n_2nk) \quad 
b(n_1n_2nk)\rightarrow a(n_1n_2nk)/\eta_C
\label{CC}
\end{eqnarray}
where $\eta_C$ is the charge parity such that $\eta_C{\bar \eta}_C=1$.
Therefore Eq. (\ref{4.38}) follows from Eq. (\ref{4.36}) and
the requirement of C-invariance.

The operator $M^{ab}_{free}$ can now be written as
\begin{equation}
M^{ab}_{free}=M^{ab}_{electron}+M^{ab}_{positron}
+M^{ab}_{photon}
\label{4.40}
\end{equation} 
and it obviously satisfies the commutation relations (\ref{1.5}).

One can try to obtain the interacting part of the $M^{ab}$ as
follows. First, the quantum operator $L_{int}(x)$ can be
written as in Eq. (\ref{4.19}), but for $\psi(x)$ and $A^a(x)$
we should take their quantum expressions (\ref{4.29}) and
(\ref{4.34}) at $\tau = \tau_0$ (at this value of $\tau$ the
fields are supposed to be the same as free ones). Then the
expression for $M^{ab}_{int}$ in the quantum case can be
formally written in the form of Eq. (\ref{4.25}). 

However in
doing so we are immediately facing several problems. The first one
is whether the expressions (\ref{4.25}) in the quantum case
represent well defined operators. Then, if this is the case,
the problem arises whether they satisfy the commutation
relations (\ref{2.23}). Indeed, some of the terms in the commutators 
contain the product of six quantum fields at the same point!
All the operators $M^{ab}_{int}$ should commute with each other
since any commutator should be linear in $M^{ab}$ and therefore
should not contain terms quadratic in $e$. If Eq. (\ref{4.31})
is correct then formally  
\begin{equation}
[L_{int}({\bf r},\tau_0),L_{int}({\bf r'},\tau_0)]=0
\label{4.41}
\end{equation}
but the proper commutation of free and interacting parts still
is to be proved. 

Let us note that in Poincare invariant theories even a formal
computation of the commutation relations between the representation
generators of the Poincare group (momenta and angular momenta)
is very difficult and the standard textbooks (see e.g. Refs.
\cite{AB,Schweber,Wein}) do not contain that computation for QED
(although, as noted in Sect. \ref{S1.1}, only the correct
commutation relations represent the fact that a theory is
Poincare invariant). Typically it is proved (on a formal level)
that only the S-matrix is Poincare invariant. In the spirit
of the Heisenberg program there is no need to introduce
interacting momenta and angular momenta at all, and Poincare
invariance of the S-matrix is sufficient. However since in
standard theories (e.g. QED) the S-matrix is expressed in terms
of full representation generators (containing interaction),
they must satisfy proper commutation relations. 

To the best of our knowledge, in de Sitter invariant QED
(in the both cases - SO(2,3) and SO(1,4)) even a formal proof
of the correct commutation relations does not exists. If one
succeeds in performing integration in Eq. (\ref{4.25}) then the
expression for $M^{ab}_{int}$ will be represented in the form
\begin{eqnarray}
&M^{ab}_{int}=e_{eff}\sum \{
[C^{ab}_1(n_1"n_2"n"k"n_1'n_2'n'k'n_1n_2nk)c(n_1n_2nk)+\nonumber\\
&\overline{C^{ab}_1(n_1'n_2'n'k'n_1"n_2"n"k"n_1n_2nk)}
c(n_1n_2nk)*]\times\nonumber\\
&[a(n_1"n_2"n"k")^*a(n_1'n_2'n'k')-b(n_1"n_2"n"k")^*
b(n_1'n_2'n'k')]+\nonumber\\
&C^{ab}_2(n_1"n_2"n"k"n_1'n_2'n'k'n_1n_2nk)
[a(n_1"n_2"n"k")^*b(n_1'n_2'n'k')^*\nonumber\\
&c(n_1n_2nk)-a(n_1"n_2"n"k")b(n_1'n_2'n'k')c(n_1n_2nk)^*] \}
\label{4.42}
\end{eqnarray}
where $e_{eff}$ is an effective dimensionless electric charge,
in each of the sets, $(n_1"n_2"n"k")$ and $(n_1'n_2'n'k')$,
the indices take the same values as in the electron IR while
$(n_1n_2nk)$ take the same values as in the photon IR. As follows
from the Hermiticity and C-invariance,
\begin{equation}
C^{ab}_2(n_1"n_2"n"k"n_1'n_2'n'k'n_1n_2nk)=
C^{ab}_2(n_1'n_2'n'k'n_1"n_2"n"k"n_1n_2nk)
\label{4.43}
\end{equation}
and this function is real.

The full representation generators can now be written in the form
\begin{equation}
M^{ab}=M^{ab}_{free}+M^{ab}_{int}
\label{4.44}
\end{equation}
It is necessary to prove that they are well-defined and satisfy
the commutation relations (\ref{2.23}). The actual computation
of the functions $C^{ab}_1$ and $C^{ab}_2$ in Eq. (\ref{4.42})
is expected to be much more complicated than the corresponding
computation in the standard QED. Since the sum in Eq. (\ref{4.42})
is taken over a discrete set of indices (rather than the 
continuous one in the standard QED), we believe that the 
operators $M^{ab}$ have much more chances to be well-defined
than the momenta and angular momenta in the standard QED.
At the the same time, since the indices take the values in an
infinite set, this cannot be guaranteed for granted.

\section{Finite analog of QED in AdS space}
\label{S4.3}

In Chapt. \ref{C2} we discussed how to describe elementary
particles in theories based on Galois fields. It has been shown
that modular IRs describing elementary particles in such theories
correspond to standard IRs describing elementary particles in the 
AdS space, the meaning of the correspondence being described in Sect. 
\ref{S2.1}. In Chapt. \ref{C3} it has been shown how the standard 
IRs can be embedded into representations where elementary particles
are described by wave functions on the AdS space. By using the results
of Chapts. \ref{C2} and \ref{C3}, it has been shown in this chapter
how QED in the AdS space can be constructed in the framework of the
standard approach. 

We can generalize this description to the
case when the theory is based on a Galois field $F_{p^2}$ 
with some value of $p$. In this section we describe a naive way of
generalizing and in the next one we discuss which new features arise 
in the GFQT.

In the naive approach we define creation and annihilation 
operators which satisfy the same commutation or anticommutation 
relations as in Eqs. (\ref{4.26}),
(\ref{4.27}) and (\ref{4.33}). The Fock space in Galois field based
theory can be obtained by analogy with the standard case, i.e. by
acting by the creation operators on the vacuum vector $\Phi_0$.
While in the standard case the quantum numbers $(n_1n_2n)$ take an
infinite number of values, in the modular one they can take only a 
finite number of values as shown in Sect. \ref{S2.3}. The operators 
$M^{ab}_{free}$ can be written as in Eq. (\ref{4.40}) where the
contributions of the electron, positron and photon fields can be
written in the form (\ref{4.36}), (\ref{4.37}) and (\ref{4.39}),
respectively. The expression for the interacting part of $M^{ab}$ can
be written in the same form as in Eq. (\ref{4.42}) but $e_{eff}$ is now
an element of $F_p$, and instead of complex conjugation one should use
conjugation in the sense of $F_{p^2}$ described in Sect. \ref{S2.1}. As
it has been noted in that section, if $p=3\,(mod\, 4)$, we still can use
the element which has the meaning of the imaginary unity $i$ while
if $p=1\,(mod\, 4)$, the conjugation should be understood as the only
nontrivial automorphism of the field $F_{p^2}$. The difference between
the Galois field based theories and standard ones is that in the former
all the indices $(n_1n_2n)$ belong to a finite set, in contrast to the
latter where they belong to an infinite set. Therefore in theories based
on Galois fields, the operators $M^{ab}$, being defined only by finite 
sums,
are always well-defined, and infinities cannot exist in principle. 

If the expressions for the functions $C^{ab}_1$ and $C^{ab}_2$ in 
the standard theory were known, then it would be possible to
replace them by functions with the range in $F_{p^2}$ such that 
the proper commutation relations are satisfied. If
$p=3\, (mod\, 4)$ these relations can be still written in the form
of Eq. (\ref{1.5}) while in the general case one should use
Eqs. (\ref{2.12}) and (\ref{2.24}) (see the discussion in Sect.
\ref{S2.3}). 

To prove this statement, consider for simplicity the case $p=3
\,(mod\, 4)$ and assume that $\tau_0=0$ in Eq. (\ref{4.25}).
Then the interaction is present only in the operators $M^{0b}$
$(b=1,2,3,5)$ (by analogy with Dirac's classification of the
dynamics forms in Poincare invariant theories \cite{Dir}, we
can say in that case that the dynamics is specified in 
the instant form). The interactions in the operators of 
Lorentz boosts $M^{0j}$ $(j=1,2,3)$ are fully defined by ones
in $M^{05}$ since, as follows from Eq. (\ref{1.5}),
$M^{0j}=[M^{05},M^{j5}]/2i$ and the operators $M^{j5}$ are free.
Our construction of the free operators is such that all their
matrix elements are rational. We can assume that with 
such an implementation of the free operators, 
the functions $C^{05}_1$ and $C^{05}_2$ in the standard theory
are expressed only in terms of rational numbers too. If this is 
not the case we can always approximate the functions by
rational ones with any desired accuracy. We now require that
all the quantum numbers $(n_1n_2nk)$ should have the range 
in the sets defined by the modular IRs for the electron and
photon, replace any integer by its value modulo $p$ and 
require that all the arithmetic operations should be understood 
in the sense of $F_{p^2}$. Then all the commutation relations 
(\ref{2.23}) will be satisfied in the sense of operators in linear 
spaces over $F_{p^2}$. Strictly speaking, such a procedure can
be used only in the basis where the correspondence between the
standard and modular matrix elements exists for all the
operators $M^{ab}$, e.g. in the basis defined by Eq. 
(\ref{nonortho}).  

One might think that the standard QED cannot be generalized to
the theory based on a Galois field because the main ingredient
of the standard QED is the dimensionless fine structure constant
which is approximately equal to 1/137. The nearest integer to
this value is zero in which case the interaction does not exist.
However at least two objections can be made. First, the quantity 
$e_{eff}$ is the bare electric charge, not the physical one. The
second (and more important) objection is as follows. The electric 
charge or the fine structure constant can be measured not
directly but only by their contribution to the energy, momentum
or angular momentum. For example, in the standard nonrelativistic
classical electrostatics, the interaction energy of two particles  
with the electric charges $e_1$ and $e_2$ is equal to 
$E=e_1e_2/r$ where $r$ is the distance between them. Therefore
the AdS interaction energy is equal to $E_{AdS}=Re_1e_2/r$.
If we want to write this expression as $E_{AdS}=e_{1eff}e_{2eff}/r$
then the effective electric charges will not be dimensionless
and their values in the usual units will be very large. On the
contrary, as noted in the preceding section, if the interaction
Lagrangian is chosen in the form of Eq. (\ref{4.19}) then the
effective electric charge has the dimension $R^{-1/2}$. So the
notion of the effective electric charge in de Sitter invariant
theories has no fundamental meaning. 

The naive way of generalizing the standard theory to the GFQT is
expected to be realistic only for states with quantum numbers 
$(n_1n_2n)$ which are much less than $p$. For greater values
the two approaches might considerably differ each other. We 
now discuss the features of the GFQT which have no analogs in 
the standard theory.   

\section{Symmetry properties of Galois field based theories}
\label{S4.4}


As shown in Sect. \ref{S2.3}, the modular analog of a standard
IR with the positive energy is simultaneously the modular 
analog of the corresponding standard IR with the negative
energy. In other words, the representation describing a
particle simultaneously describes its antiparticle. Moreover
we can now say that even the notion of particle and 
antiparticle is not fundamental. We have one object which
is manifested itself as a particle at the one end of the
spectrum and as an antiparticle at the other. 

Let us first recall how the notion of antiparticle arises in 
the standard Poincare invariant theory. 

After realizing that the Dirac equation has solutions
with both, positive and negative energies, we should somehow
interpret this fact, since in the standard theory negative
energies have no direct physical meaning. There exists a vast 
literature where different interpretations of negative energy 
solutions of the Dirac equation have been proposed. The
approach adopted in the modern theory is as follows. 

We first decompose the Dirac field as 
\begin{eqnarray}
&\psi(x)=\sum_{\sigma}\int [\psi({\bf p},\sigma,x)^{(+)}
a({\bf p},\sigma)^{(+)}+\nonumber\\
&\psi(-{\bf p},-\sigma,x)^{(-)}a(-{\bf p},-\sigma)^{(-)}]
d^3{\bf p}/\omega({\bf p})
\label{P1}
\end{eqnarray}
where $\sigma=\pm 1/2$ is the spin projection onto the $z$ axis,
$$\omega({\bf p})=\sqrt{(m^2+{\bf p}^2)},$$ 
$m$ is the mass in the
Poincare invariant theory, $\psi({\bf p},\sigma,x)^{(+)}$ is 
the solution of the Dirac
equation with the three-momentum ${\bf p}$, spin projection
$\sigma$ and the positive energy $\omega({\bf p})$,
$\psi(-{\bf p},-\sigma,x)^{(-)}$ is 
the solution of the Dirac equation with the three-momentum 
$-{\bf p}$, spin projection $-\sigma$ and the negative energy 
$-\omega({\bf p})$, and $a({\bf p},\sigma)^{(+)}$ and
$a(-{\bf p},-\sigma)^{(-)}$ are the decomposition coefficients. 
We can simplify the notations by omitting the subscript $(+)$
in $a({\bf p},\sigma)^{(+)}$ and introducing a function
$b({\bf p},\sigma)$ such that
\begin{equation}
a(-{\bf p},-\sigma)^{(-)} = b({\bf p},\sigma)^*
\label{P2}
\end{equation}
where $^*$ means the usual complex conjugation since it is
applied to a $c$-valued function.

The next stage is the computation of the dynamical quantities.
In particular, the result for the Hamiltonian is
\begin{equation}
H=\sum_{\sigma}\int [a({\bf p},\sigma)^*a({\bf p},\sigma) -
b({\bf p},\sigma)b({\bf p},\sigma)^*]d^3{\bf p}
\label{P3}
\end{equation}
and the contribution of the negative energy solutions to the
Hamiltonian is negative as expected.

A possible way out of this difficulty is as follows. We now
say that the Dirac equation simutaneously describes two
particles - electron and positron. The electron is 
associated with positive energy solutions while the
positron - with negative energy ones. We also require that
$$(a({\bf p},\sigma),a({\bf p},\sigma)^*,
b({\bf p},\sigma),b({\bf p},\sigma)^*)$$
now represent operators rather than $c$-valued functions.
If $a({\bf p},\sigma)$ is the operator of the electron
annihilation and $a({\bf p},\sigma)^*$ is the operator
of the electron creation then, as follows from Eq. (\ref{P2}),
$b({\bf p},\sigma)^*$ has the meaning of the positron
creation. Indeed, since the states with negative energies
do not exist in the standard approach, the annihilation
operator for a particle with the negative energy should
now be interpreted as the creation operator for an
antiparticle with the positive energy. Analogously
$b({\bf p},\sigma)$ should be interpreted as the operator
of the positron annihilation. The well-known minus sign
in Eq. (\ref{P3}) is the reason why we should require
anticommutation (rather than commutation) relations
\begin{equation}
\{a({\bf p},\sigma),a({\bf p}',\sigma')^*\}=
\omega({\bf p})\delta^{(3)}({\bf p}-{\bf p}')\delta_{\sigma\sigma'}
\label{P4}
\end{equation}
\begin{equation}
\{b({\bf p},\sigma),b({\bf p}',\sigma')^*\}=
\omega({\bf p})\delta^{(3)}({\bf p}-{\bf p}')\delta_{\sigma\sigma'}
\label{P5}
\end{equation}
By using these relations we now can rewrite Eq. (\ref{P3}) in
the form 
\begin{equation}
H=\sum_{\sigma}\int [a({\bf p},\sigma)^*a({\bf p},\sigma)^* +
b({\bf p},\sigma)^*b({\bf p},\sigma)]d^3{\bf p}+Const
\label{P6}
\end{equation}
where $Const$ is an indefinite negative constant. It can be
removed by requiring that all secondly quantized operators
should be written in the so called normal order (see e.g. the
textbooks \cite{AB,Schweber,Wein}).

In the theory based on a Galois field we can go farther.
Let us first discuss some differences between the standard
theory and our approach.
In the standard theory, IRs for a particle and its antiparticle
are decoupled and therefore the sets $(a,a^*)$ on the one hand,
and $(b,b^*)$ on the other are independent. In our approach,
where a particle and its antiparticle are described by the same
modular IR, only one of these sets is independent. Therefore
one might work only with the set $(a,a^*)$ and not 
introduce the operators $(b,b^*)$ at all. Introducing these
operators is only a matter of convenience if one would like
to interpret modular IRs in terms of particles and antiparticles.
In the standard theory we can consider different transformations
relating the sets $(a,a^*)$ and $(b,b^*)$. We have already
mentioned the C-transformation (see Eq. (\ref{CC})). Another
example is the CPT-transformation in Schwinger's formulation,
which transforms $a^*$ to $b$ \cite{AB,Schweber,Wein}. In our 
approach we can discuss analogous transformations but first
of all we should {\bf define} the set $(b,b^*)$ in terms of 
$(a,a^*)$ or vice versa. We now assume that the operators 
$(a(n_1n_2nk),a(n_1n_2nk)^*,b(n_1n_2nk),b(n_1n_2nk)^*)$
describe the creation and annihilation operators for a
field with an arbitrary spin $s$ and use $M^{ab}$ to
denote the secondly quantized generators for this field. 
Therefore our first goal is to express 
$(a(n_1n_2nk),a(n_1n_2nk)^*)$ in terms of
$(b(n_1n_2nk),b(n_1n_2nk)^*)$ or vice versa.

We now can treat Eq. (\ref{4.35})
as representing the generators of the particle-antiparticle field,
i.e. we can write 
\begin{eqnarray}  
&M^{ab}=\sum M^{ab}(n_1'n_2'n'k',n_1n_2nk)\nonumber\\
&a(n_1'n_2'n'k')^*a(n_1n_2nk)/Norm(n_1n_2nk)
\label{P13}
\end{eqnarray}
It is clear that the conventional analog of this expression 
correctly describes the contribution of particles if the 
$a$-operators satisfy either the anticommutation relations
\begin{eqnarray}
&\{a(n_1n_2nk),a(n_1'n_2'n'k')^*\}=\nonumber\\
&Norm(n_1n_2nk)
\delta_{n_1n_1'}\delta_{n_2n_2'}\delta_{nn'}\delta_{kk'}
\label{anticomm}
\end{eqnarray}
or the commutation relation
\begin{eqnarray}
&[a(n_1n_2nk),a(n_1'n_2'n'k')^*]=\nonumber\\
&Norm(n_1n_2nk)
\delta_{n_1n_1'}\delta_{n_2n_2'}\delta_{nn'}\delta_{kk'}
\label{comm}
\end{eqnarray}
The problem arises whether Eq. (\ref{P13}) correctly describes 
the contribution of antiparticles in the GFQT. Indeed, the
operators given by Eq. (\ref{P13}) satisfy the proper
anticommutation or commutation relations if Eqs. 
(\ref{anticomm}) or (\ref{comm})
are valid for all the values of $(n_1n_2nk)$. At the same time,
for such values of $(n_1n_2n)$ when the AdS energy is negative,
the operator $a(n_1n_2nk)$ cannot be treated as the annihilation
operator and $a(n_1n_2nk)^*$ cannot be treated as the creation
operator. 

Let us recall (see Sect. \ref{S2.3}) that the vector $e_{nk}$ is the
minimal vector of the sp(2)$\times$sp(2) representation with the 
eigenvalue of the operator $h_1$ equal to $Q_1=q_1+n-k$ and the
eigenvalue of the operator $h_2$ equal to $Q_2=q_2+n+k$. 
The first IR
of the sp(2) algebra has the dimension $N_1+1=p-Q_1+1$ and the 
second IR has the dimension $N_2+1=p-Q_2+1$. The vector 
$$e(n_1n_2nk)=(a_1")^{n_1}(a_2")^{n_2}e_{nk}$$ 
has the eigenvalue of the operator $h_1$ equal to $Q_1+2n_1$ 
and the eigenvalue 
of the operator $h_2$ equal to $Q_2+2n_2$. If $n_1=N_1$ then the first 
eigenvalue is equal to $-Q_1$ in $F_p$, and if $n_2=N_2$ then 
the second eigenvalue is equal to $-Q_2$ in $F_p$.
We use ${\tilde n}_1$ to denote $N_1-n_1$ and ${\tilde n}_2$ to denote 
$N_2-n_2$. Then $e({\tilde n}_1{\tilde n}_2nk)$ is the eigenvector 
of the operator $h_1$ with the eigenvalue $-(Q_1+2n_1)$ and the 
eigenvector of the operator $h_2$ with the eigenvalue 
$-(Q_2+2n_2)$. It is now reasonable to think that the 
operator $b(n_1n_2nk)$ should be defined in such a way that
it is proportional to $a({\tilde n}_1,{\tilde n}_2,n,k)^*$
and $b(n_1n_2nk)^*$ should be defined in such a way that
it is proportional to $a({\tilde n}_1,{\tilde n}_2,n,k)$. 

As follows from Eq. (\ref{Norm}),   
\begin{equation}
Norm(n_1n_2nk)=F(n_1n_2nk)G(nk)
\label{4.45}
\end{equation}
where 
\begin{eqnarray}
&F(n_1n_2nk)= n_1!(Q_1+n_1-1)!n_2!(Q_2+n_2-1)!\nonumber\\
&G(nk)=Norm(00nk)/[(Q_1-1)!(Q_2-1)!]
\label{4.46}
\end{eqnarray}
As follows from the Wilson
theorem $(p-1)!=-1$ in $F_p$ (see e.g. \cite{VDW,IR})
and Eq. (\ref{4.46})
\begin{equation}
F(n_1n_2nk)F({\tilde n}_1{\tilde n}_2nk) = (-1)^{q_1+q_2}=(-1)^s
\label{4.47}
\end{equation}   
since $q_1=q_2+s$. 

We now {\bf define} the $b$-operators as follows. 
\begin{equation}
a(n_1n_2nk)^*=\eta(n_1n_2nk) b({\tilde n}_1{\tilde n}_2nk)/
F({\tilde n}_1{\tilde n}_2nk) 
\label{4.48}
\end{equation}
where $\eta(n_1n_2nk)$ is some function. 
This expression looks similar to Eq. (\ref{P2}), but in contrast to 
the standard theory, where only the r.h.s. of Eq. (\ref{P2}) is 
meaningful, both sides of Eq. (\ref{4.48}) are meaningful. It 
represents the fact that the annihilation of the antiparticle
with the negative energy can be treated as the creation of the
particle. Note that in the standard theory the 
CPT-transformation in Schwinger's formulation also transforms 
$a^*$ to $b$ \cite{AB,Schweber,Wein}, but in that case the 
both operators refer only to positive energies, in contrast to 
Eq. (\ref{4.48}). 

As a consequence of this definition, 
\begin{eqnarray}
&a(n_1n_2nk)=\bar{\eta}(n_1n_2nk) b({\tilde n}_1{\tilde n}_2nk)^*/
F({\tilde n}_1{\tilde n}_2nk)\nonumber\\ 
&b(n_1n_2nk)^*=a({\tilde n}_1{\tilde n}_2nk)
F(n_1n_2nk)/{\bar \eta}({\tilde n}_1{\tilde n}_2nk)\nonumber\\ 
&b(n_1n_2nk)=a({\tilde n}_1{\tilde n}_2nk)^*
F(n_1n_2nk)/\eta({\tilde n}_1{\tilde n}_2nk)
\label{adjoint1}
\end{eqnarray}

Eq. (\ref{4.48}) defines a transformation when the set $(a,a^*)$
is replaced by the set $(b,b^*)$. To understand whether it is
a new symmetry, we should investigate when so defined $(b,b^*)$
operators satisfy the same commutation or anticommutation
relations as the $(a,a^*)$ operators. Therefore, as follows from Eqs. 
(\ref{anticomm}) and (\ref{comm}), the $b$-operators should 
satisfy either 
\begin{eqnarray}
&\{b(n_1n_2nk),b(n_1'n_2'n'k')^*\}=\nonumber\\
&Norm(n_1n_2nk)
\delta_{n_1n_1'}\delta_{n_2n_2'}\delta_{nn'}\delta_{kk'}
\label{Banticomm}
\end{eqnarray}
in the case of anticommutators or
\begin{eqnarray}
&[b(n_1n_2nk),b(n_1'n_2'n'k')^*]=\nonumber\\
&Norm(n_1n_2nk)
\delta_{n_1n_1'}\delta_{n_2n_2'}\delta_{nn'}\delta_{kk'}
\label{Bcomm}
\end{eqnarray}
in the case of commutators.

Now, as follows from Eqs. (\ref{anticomm}),
(\ref{4.47}), (\ref{4.48}) and (\ref{adjoint1}), 
Eq. (\ref{Banticomm}) is satisfied if
\begin{equation}
\eta(n_1n_2nk) {\bar \eta}(n_1,n_2,nk)=(-1)^s
\label{etaanti}
\end{equation}
At the same time, in the case of commutators it 
follows from Eqs. (\ref{comm}), (\ref{4.47}), (\ref{4.48}),
and (\ref{adjoint1}) that Eq. (\ref{Bcomm}) is satisfied if
\begin{equation}
\eta(n_1n_2nk) {\bar \eta}(n_1,n_2,nk)=(-1)^{s+1}
\label{etacomm}
\end{equation}

We now define a new function $f(n_1n_2nk)$ such that
\begin{equation}
\eta(n_1n_2nk)=\alpha f(n_1n_2nk)
\label{f}
\end{equation}
where
\begin{equation}
f(n_1n_2nk) {\bar f}(n_1,n_2,nk)=1
\label{f1}
\end{equation}
Then 
\begin{equation}
\alpha {\bar \alpha}=\pm (-1)^s
\label{al}
\end{equation}
where the plus sign refers to anticommutators and the minus
sign to commutators, respectively.
If the normal relations between spin and
statistics is satisfied, i.e. we have anticommutators for
odd values of $s$ and commutators for even ones (this is the 
well known Pauli theorem in LQFT \cite{Pauli}) then the r.h.s. 
of Eq. (\ref{al}) is equal to -1. 

This relation is a consequence of the fact that our basis is
not normalized to one (see Sect. \ref{S2.3} for a discussion of
Eq. (\ref{orthonorm})). In the standard theory such a relation is 
impossible but if $\alpha\in F_{p^2}$, a solution of Eq. (\ref{al}) 
exists. Indeed, we can use the fact that any Galois field is 
cyclic with respect to 
multiplication (in Sect. \ref{S2.1} this fact was mentioned 
only for $F_p$). Let $r$ be a primitive root of $F_{p^2}$ 
and $\alpha =r^k$.
As mentioned in Sect. \ref{S2.1}, $F_{p^2}$ has only one
nontrivial automorphism which is defined as 
$\alpha\rightarrow {\bar \alpha}=\alpha^p$. 
Therefore $\alpha{\bar \alpha}=
r^{(p+1)k}$. On the other hand, since $r^{(p^2-1)}=1$, we
conclude that $r^{(p^2-1)/2}=-1$. Therefore there exists at
least a solution with $k=(p-1)/2$. 

For a reason which will be clear soon, we choose the function
$f(n_1n_2nk)$ in the form
\begin{equation}
f(n_1n_2nk)=(-1)^{n_1+n_2}
\label{f2}
\end{equation}
and in that case Eq. (\ref{f1}) is obviously satisfied.

Let us consider the operators (\ref{P13}) and
use the fact that in the modular case the trace of the
generators of IRs is equal to zero (see Eq. (\ref{2.45})).   
Therefore, as follows from Eqs. (\ref{anticomm}) and (\ref{comm}), 
we can rewrite Eq. (\ref{P13}) as
\begin{eqnarray}  
&M^{ab}=\mp \sum
M^{ab}(n_1'n_2'n'k',n_1n_2nk)\nonumber\\
&a(n_1n_2nk)a(n_1'n_2'n'k')^*/Norm(n_1n_2nk)
\label{4.50}
\end{eqnarray}
where the minus sign refers to anticommutators and the plus sign -
to commutators. Then by using Eqs. (\ref{4.45}-\ref{adjoint1}), 
(\ref{f}) and (\ref{al}), we obtain in the both cases
\begin{eqnarray}  
&M^{ab}=-\sum
M^{ab}({\tilde n}_1{\tilde n}_2nk,
{\tilde n}_1'{\tilde n}_2'n'k')
f({\tilde n}_1{\tilde n}_2nk)
{\bar f}({\tilde n}_1'{\tilde n}_2'n'k')\nonumber\\
&b(n_1'n_2'n'k')^*b(n_1n_2nk)/[F(n_1n_2nk)G(n'k')]
\label{4.51}
\end{eqnarray}

In particular, as follows from the above remarks about the
eigenvalues of the operators $h_1$ and $h_2$, and from Eqs.
(\ref{4.46}) and (\ref{f2})  
\begin{eqnarray}  
&M^{05}=\sum_{n_1n_2nk}
[m + 2(n+n_1+n_2)]b(n_1n_2nk)^*\times\nonumber\\
&b(n_1n_2nk)/Norm(n_1n_2nk)
\label{4.52}
\end{eqnarray}
As follows from Eq. (\ref{P13}), {\bf the same}
operator can also be written as 
\begin{eqnarray}  
&M^{05}=\sum_{n_1n_2nk}
[m + 2(n+n_1+n_2)]a(n_1n_2nk)^*\times\nonumber\\
&a(n_1n_2nk)/Norm(n_1n_2nk)
\label{4.53}
\end{eqnarray}
In Eqs. (\ref{4.52}) and (\ref{4.53}), the sum is taken over
all the values of $(n_1n_2nk)$ relevant to the particle modular IR.
At the same time, for the correspondence with the standard
case, we should consider only the values of the $(n_1n_2nk)$ which
are much less than $p$ (see Chapt. \ref{C2}). The derivation
of Eq. (\ref{4.52}) demonstrates that the contribution of
those $(n_1n_2nk)$ to that expression originates from such a
contribution of $(n_1,n_2)$ to Eq. (\ref{4.53}) that
$({\tilde n}_1,{\tilde n}_2)$ are much less than $p$.
In this case the $(n_1,n_2)$ are comparable to $p$. Therefore,
if we consider only such states that the $(n_1n_2nk)$ are
much less than $p$ then the AdS Hamiltonian can be
written in the form
\begin{eqnarray}  
&M^{05}=\sum_{n_1n_2nk}' 
[m + 2(n+n_1+n_2)][a(n_1n_2nk)^*\times\nonumber\\
&a(n_1n_2nk)+b(n_1n_2nk)^*b(n_1n_2nk)]/Norm(n_1n_2nk)
\label{4.54}
\end{eqnarray}
where $\sum_{n_1n_2nk}'$ means that the sum is taken only
over the values of the $(n_1n_2nk)$ which are much less than $p$.
In this expression the contributions of particles and
antiparticles are written explicitly and the corresponding
standard AdS Hamiltonian is positive definite.

The above results show that as far as the operator $M^{05}$ is
concerned, Eq. (\ref{4.48}) indeed defines a new symmetry since
$M^{05}$ has the same form in terms of $(a,a^*)$ and $(b,b^*)$
(compare Eqs. (\ref{4.52}) and (\ref{4.53})). Note that we did
not assume that the theory is C-invariant (see Eq. (\ref{CC})).
It is well known that C-invariance is not a fundamental symmetry.
For example, Eq. (\ref{CC}) does not apply to neutrino. On the
other hand, in our approach one modular IR decribes neutrino and
antineutrino simultaneously, and Eq. (\ref{4.48}) is valid in
this case too. In the standard theory only CPT-invariance is
fundamental since, according to the famous CPT-theorem \cite{GLR}, 
any local Poincare invariant theory is automatically CPT-invariant. 
Our assumption is that Eq. (\ref{4.48}) defines a fundamental
symmetry in the GFQT.

To understand the properties of the new symmetry, one has to 
investigate not only $M^{05}$ but other representation generators as 
well.  

Consider first the operator $a_1"$ (see Sect. \ref{S2.3}).
As follows from Eqs. (\ref{en}), (\ref{2.42}) and (\ref{P13}),
the secondly quantized form of this operator is
\begin{eqnarray}
a_1"=\sum_{n_1=0}^{N_1-1}\sum_{n_2nk}a(n_1+1,n_2nk)^*
a(n_1n_2nk)/Norm(n_1n_2nk)
\label{a1plus}
\end{eqnarray}
We have to prove that in terms of $(b,b^*)$ this operator
has the same form, i.e.
\begin{eqnarray}
a_1"=\sum_{n_1=0}^{N_1-1}\sum_{n_2nk}b(n_1+1,n_2nk)^*
b(n_1n_2nk)/Norm(n_1n_2nk)
\label{a1plusB}
\end{eqnarray}
This can be shown by using either Eq. (\ref{a1plus}) directly
(taking into account Eqs. (\ref{4.48}) and (\ref{adjoint1}))
or Eq. (\ref{4.51}). In the both cases it is clear that Eq.
(\ref{a1plusB}) follows from Eq. (\ref{a1plus}) if $f(n_1n_2nk)$
is chosen as in Eq. (\ref{f2}).

Since the action of the operator $a_1'$ can be written as
$$a_1'e(n_1n_2nk)=a_1'a_1"e(n_1-1,n_2nk)$$
then, as follows
from Eqs. (\ref{2.21}), (\ref{en}), (\ref{2.42}) and (\ref{P13}),
the secondly quantized form of this operator is
\begin{eqnarray}
&a_1'=\sum_{n_1=1}^{N_1}\sum_{n_2nk}n_1(Q_1+n_1-1)
a(n_1+1,n_2nk)^*\nonumber\\
&a(n_1n_2nk)/Norm(n_1n_2nk)
\label{a1minus}
\end{eqnarray}
By analogy with the proof of Eq. (\ref{a1plusB}), one
can prove that in terms of $(b,b^*)$ this operator
has the same form, i.e.
\begin{eqnarray}
&a_1'=\sum_{n_1=1}^{N_1}\sum_{n_2nk}n_1(Q_1+n_1-1)
b(n_1+1,n_2nk)^*\nonumber\\
&b(n_1n_2nk)/Norm(n_1n_2nk)
\label{a1minusB}
\end{eqnarray}
Note that in the process of derivation $n_1$ transforms to
$N_1+1-n_1$ and therefore
$$n_1(Q_1+n_1-1)\rightarrow (N_1+1-n_1)(Q_1+N_1-n_1)=n_1(Q_1+n_1-1)$$
in $F_p$ since $N_1+Q_1=0$ in $F_p$. This derivation clearly
has no analog in the standard theory.  

Analogously we can prove that the secondly quantized operators 
$a_2"$ and $a_2'$ also have the same form in terms of $(a,a^*)$
and $(b,b^*)$, and therefore Eq. (\ref{4.48}) indeed defines a
symmetry compatible with the algebra sp(2)$\times$sp(2).

The problem arises whether it is possible to extend the symmetry 
in such a way that it will become compatible with the full algebra 
so(2,3). This problem will be considered elsewhere and here we 
only briefly outline its essence. Eq. (\ref{4.48}) relates the 
operators characterized by the same values of $(n,k)$. As shown 
in Sect. \ref{S2.3}, in the massive case the quantity 
$n$ can take the values $0,1,...p+2-m$, and the quantity $k$ -
the values $0,1,...s$, while in the massless case 
(see Sect. \ref{S2.4}) - $n=0,1,...p+1-m$ and $k$ is either $0$ or
$s$ if $n\neq 0$. One might expect that
the theory has an additional symmetry $n\rightarrow (p+2-m-n)$ 
and $k\rightarrow (s-k)$ in the massive
case and analogously $n\rightarrow (p+1-m-n)$ and 
$k\rightarrow (s-k)$ in the massless one. To exploit this symmetry,
it is convenient to enumerate the $(b,b^*)$
operators not by $(nk)$ but by another pair of integers such that 
the remaining operators $(b',b",L_+,L_-)$ should have the same 
form in terms of $(a,a^*)$ and $(b,b^*)$.

The following remarks are now in order. As already noted, the
discussion in this section does not involve C-invariance. 
The question arises whether we can apply the transformations in 
Eq. (\ref{CC})
to the both parts of Eqs. (\ref{4.48}) and (\ref{adjoint1}). 
If it were possible then we would be able to obtain new
restrictions on the function $\eta(n_1n_2nk)$. However the
requirement of C-invariance means that the transformations
(\ref{CC}) can be applied only to the operators of observable
quantities which are biliniar in $(a,a^*)$ or $(b,b^*)$.
Suppose now that the particle in question is neutral, i.e.
the particle coincides with its antiparticle. On the language
of the operators $(a,a^*)$ and $(b,b^*)$ this means that these
sets are the same, i.e. $a(n_1n_2nk)=b(n_1n_2nk)$ and
$a(n_1n_2nk)^*=b(n_1n_2nk)^*$. 

If we denote now
the annihilation and creation operators as $c(n_1n_2nk)$ and
$c(n_1n_2nk)^*$, respectively, then Eq. (\ref{4.48}) has
the form  
\begin{equation}
c(n_1n_2nk)^*=\eta(n_1n_2nk) c({\tilde n}_1{\tilde n}_2nk)/
F({\tilde n}_1{\tilde n}_2nk) 
\label{P30}
\end{equation}
and therefore 
\begin{equation}
c(n_1n_2nk)={\bar \eta}(n_1n_2nk) c({\tilde n}_1{\tilde n}_2nk)^*/
F({\tilde n}_1{\tilde n}_2nk) 
\label{P31}
\end{equation}
As follows from Eqs. (\ref{4.47}), (\ref{f}) and (\ref{al}),
these expressions are
compatible with each other only if
\begin{equation}
f(n_1n_2nk){\bar f}({\tilde n}_1,{\tilde n}_2,nk)=\pm 1
\label{ftilde}
\end{equation}
where the plus sign refers to anticommutators and the minus
sign to commutators, respectively. Therefore the problem arises
whether Eqs. (\ref{f1}) and (\ref{ftilde}) are compatible with
each other. The compatibility will take place only if
\begin{equation}
f({\tilde n}_1,{\tilde n}_2,nk)=\pm f(n_1n_2nk)
\label{stat}
\end{equation}

It is obvious that if the function $f(n_1n_2nk)$ is chosen in the
form (\ref{f2}) then Eq. (\ref{stat}) is valid only in the case
of the broken relation between spin and statistics. Indeed,
\begin{eqnarray}
&{\tilde n}_1+{\tilde n}_2=N_1+N_2-n_1-n_2=
2p-Q_1-Q_2-n_1-n_2=\nonumber\\
&2p-2n-2q_2-s-n_1-n_2
\label{stat1}
\end{eqnarray}
and therefore in that case
\begin{equation}
f({\tilde n}_1,{\tilde n}_2,nk)=(-1)^s f(n_1n_2nk)
\label{stat2}
\end{equation}
what contradicts Eq. (\ref{stat}) for the normal relation
between spin and statistics.

The above observation poses the problem whether the existence
of neutral elementary particles in the GFQT is compatible with
the normal relation between spin and statistics. We can
reformulate the problem in this way: is it natural that the
requirement about the normal relation between spin and
statistics excludes the existence of neutral elementary
particles in the GFQT? If this is the case then it sheds a new
light on the nature of the Pauli theorem \cite{Pauli}. To give 
a definite answer to this question, one has to investigate a 
symmetry which is compatible not only with the sp(2)$\times$sp(2) 
algebra, but with the full AdS so(2,3) algebra. We believe that 
the existence of neutral elementary particles in the GFQT is not 
natural and the main reason is as follows. Since one modular IR
simultaneously describes a particle and its antiparticle, the
AdS energy operator necessarily contains the contribution of
the both parts of the spectrum, corresponding to the particle
and its antiparticle (see Eq. (\ref{4.54})). If a particle
were the same as its antiparticle then Eq. (\ref{4.54}) would 
contain two equal contributions and thus the value of the AdS
energy would be twice as big as necessary.

If neutral elementary particles in the GFQT cannot exist
then the photon cannot be an elementary particle in our approach 
and in that case the $c(n_1n_2nk)$ and $c(n_1n_2nk)^*$ can be 
treated as effective operators satisfying the commutation 
relations (\ref{4.33}) only in some approximation.

On the other hand, if there is a way out of this difficulty,
then the structure of QED in the modular case is much simpler
than in the standard one. Indeed, in the standard theory
there are six independent sets of operators ---
$$(a(n_1n_2nk),a(n_1n_2nk)^*,b(n_1n_2nk),b(n_1n_2nk)^*,
c(n_1n_2nk),c(n_1n_2nk)^*)$$
while in our approach there are only three independent
sets, for example
$$(a(n_1n_2nk),a(n_1n_2nk)^*,c(n_1n_2nk))$$
or
$$(a(n_1n_2nk),b(n_1n_2nk),c(n_1n_2nk))$$
The free part of the $M^{ab}$ operators can be written as
\begin{equation}
M^{ab}_{free}=M^{ab}_{electron-positron}+M^{ab}_{photon} 
\label{4.65}
\end{equation}
where the electron-positron contribution is given by Eq.
(\ref{P13}) and in Eq. (\ref{4.39}) the operator 
$c(n_1n_2nk)^*$ can be expressed in terms of $c(n_1n_2nk)$
by using Eq. (\ref{P30}).  
The interacting part can be written as 
\begin{eqnarray}
&M^{ab}_{int}=e_{eff}\sum 
C^{ab}(n_1"n_2"n"k"n_1'n_2'n'k'n_1n_2nk)\times\nonumber\\
&a(n_1"n_2"n"k")^*a(n_1'n_2'n'k')c(n_1n_2nk)
\label{4.66}
\end{eqnarray}
what is much simpler than Eq. (\ref{4.42}).

\section{Discussion}
\label{S4.5}

One might wonder what physical results can be obtained in the theory
containing only the operators $M^{ab}$ and not containing space-time 
and/or S-matrix. First, since all the operators are well-defined, the
bound state problem is well-defined too. Therefore one can investigate
bound states, the relations between the bare and physical charges and
masses etc. All these problems can be considered beyond perturbation 
theory and the relations between the bare and physical charges and 
masses do not involve any infinite renormalization constants. As
argued in Chap. \ref{C1}, the notion of time is well defined only
on the classical level. At some conditions the classical time may
be a good approximate parameter and in that case we can use the AdS 
Hamiltonian $M^{05}$ for describing the evolution. Another
possible approach is to construct the S-matrix by analogy with
the stationary approach in the standard scattering theory (see
e.g. Ref. \cite{Kato} and references therein). In this approach
the S-matrix is defined as an operatorial function of interacting
and free Hamiltonians without mentioning time at all. It 
is not clear whether the S-matrix can be correctly defined in  
SO(2,3) invariant theories. However for comparison with the data, 
a partial S-matrix describing the transition say from 
$\tau=-\pi/2$ to $\tau=\pi/2$ is quite sufficient. 

One of the main obstacles in constructing GFQT is that 
standard quantum theories in dS or AdS spaces
are not well developed yet. In particular, the values of the
coefficients in Eq. (\ref{4.43}) are not known. It is very
important to know these values, since at some conditions the
GFQT should give the 
same predictions as the standard QED. On the other hand, it is 
also important to seek approaches where the operators $M^{ab}$ 
in the GFQT can be defined from principles other than those
based on local Lagrangian (see below).

Suppose however that the above difficulties are overcome
and the GFQT is constructed. Will it have a strong
predictive power? The main feature of the present Poincare
invariant quantum theory is that particles are described by
their three-momenta which commute with each other, and each
component of the three-momentum has the range in the 
interval $(-\infty,\infty)$. As a consequence, we can use a 
well developed mathematical formalism for computing the
S-matrix. In the GFQT the particles should be described only
in the basis characterized by the $(n_1n_2nk)$ or
equivalent quantum numbers.
The GFQT formalism can involve neither derivatives or
integrals, but only finite sums. Taking into account the fact
that each of the quantum numbers $(n_1n_2n)$ can have the number
of values which is of order $p$, and $p$ is probably very big,
it is difficult to imagine that any computer will be able to
compute physical quantities. However in many cases a finite
sum may be approximated by an integral with a good accuracy.
In the GFQT this integral will necessarily contain a cutoff
and this does not indicate to any inconsistency.
On the contrary, in the standard approach, the dependence of 
physical quantities on the cutoff means that the theory is 
inconsistent. Indeed, the standard theory works with usual 
spatial derivatives, i.e. it is assumed that there is no 
fundamental length, while the presence of the cutoff $P_{max}$ in 
momentum representation implies that we introduce a 
fundamental length of order $\hbar /P_{max}$. 

In view of these observations, a pragmatic physicist might
now conclude that introducing a Galois field instead of the
field of complex numbers means only an artificial definition
of some cutoff related to the characteristic of the Galois field $p$. 
However, as shown in this paper, the fact that the arithmetic of 
Galois fields considerably differs from that in the field of complex 
numbers, leads to several features which are not present in the 
standard theory. 

First, it gives a very simple explanation of the existence 
of antiparticles, since each modular IRs describes a particle 
and its antiparticle simultaneously. In the present quantum
theory the existence of antiparticles is explained as a
consequence of the fact that the Dirac equation has solutions
with positive and negative energies. However in that case
the problem arises how to interpret the negative energy 
solutions. On the contrary, in the GFQT the existence of
antiparticles follows only from the algebraic properties of
modular IRs and in particular from the existence of the
number $p$.

Since in our approach, a particle and its antiparticle 
represent different states of the same object, these operators 
can be directly expressed in terms of each other. This imposes 
additional restrictions on the structure of the theory. 
In particular, as shown in Sect. \ref{S4.4}, one can work  
with only creation operators or only annihilation ones since they 
are no more independent. 

A very interesting problem in our approach is whether neutral
elementary particles can exist, or in other words, whether
neutral particles (say the photon, the graviton etc.) can be
be elementary or only composite. As argued in Sect.
\ref{S4.4}, there are indications that the existence of neutral
elementary particles is incompatible with the normal relation
between spin and statistic. If it were the case then the 
nature of the relation should be reconsidered. As noted in
Sect. \ref{S4.4}, a definite answer can be
obtained in the framework of a symmetry compatible with the
full AdS algebra, while the symmetry constructed in that
section is explicitly compatible only with the 
sp(2)$\times$sp(2) algebra. At the same time, as argued in
Sect. \ref{S4.4}, the existence of neutral elementary
particles in our approach seems to be rather unnatural. 

This problem arises for both, bosons and fermions,
but the case of bosons is probably of greater
importance. A possibility that bosons can be only composite, 
has been already discussed in the literature. For example, in Refs. 
\cite{Castell,FronsdalS} a model where the photon is 
composed of two singletons has been investigated. We believe that 
such a possibility is rather attractive and an additional 
argument in its favor is as follows.

Suppose we require
that in quantum theory infinities should not be present in
any form. Then can we say that we have achieved this goal?
As shown in Sect. \ref{S2.3}, IR describing a particle with the
mass $m$ and spin $s$ has the dimension $D(m,s)$ given by Eq.
(\ref{2.36}), and an analogous result can be obtained in the
massless case. Since the field $F_{p^2}$ is finite, the set of 
all possible states for an elementary particle
is finite too. The operators $M^{ab}$ also are defined by a finite
number of elements. Consider now fermions of a given type.
Since they obey the Fermi-Dirac statistics, there can be not
more that $D(m,s)\approx (s+1)p^3/3$ such fermions in the
Universe. At the same time, there are no limitations on the
number of bosons. Although no quantity describing any system
of bosons can have a value greater than $p$, the presence of
infinite number of bosons seems to be not attractive. 

In the standard quantum theory, the final goal is
to construct the S-matrix, but for this purpose one should use
Noether's formalism in the Lagrangian approach. Meanwhile
at the final stage, all local quantities disappear. As we
already mentioned in Chap. \ref{C1}, in 50th and 60th a lot
of efforts have been made to construct the S-matrix from the
first principles. There also exist approaches where the 
representation generators satisfying the commutation relations 
of the Poincare group Lie algebra are constructed
without using local Lagrangians. These investigations are
based on the Dirac paper \cite{Dir} where the notion of
dynamics forms has been proposed. 

For the system of two
interacting particles, the first construction has been
implemented by Bakamdjian and Thomas \cite{BT} and for
the system of three interacting particles - by Coester
\cite{Coester1} (see also Ref. \cite{BKT}). If the number
of particles $N\geq 4$, the construction is much more 
complicated since one has to satisfy not only correct
commutation relations but also cluster separability.
The solution has been found by Sokolov in his method 
of packing operators \cite{sok}. It has been also 
shown by Sokolov and Shatny \cite{SoSh} that all Dirac's
forms of dynamics are unitarily equivalent. By using
Sokolov's method, the explicit construction of the
$N$-body generators for arbitrary (but fixed) $N$ has been
proposed independently by Coester and Polyzou \cite{CP}
and Mutze \cite{M}; (see also Ref. \cite{lev6}). A review
of the results in this field can be found e.g. in Ref. 
\cite{KeiPol}. The methods developed in these works
have been also used for constructing electromagnetic and
weak current operators for systems of interacting particles
(see e.g. Ref. \cite{lev5}). We believe that these methods
can be successfully applied for investigating GFQT. 

Another possible way of investigations is as follows. As
we have noted in Sect. \ref{S1.3}, the conventional theory 
based on the SO(1,4) group has several unusual 
features. As it has been already mentioned in Sect. \ref{S1.3},
here one IR describes a particle and its antiparticle 
simultaneously (see e.g. Ref. \cite{Mensky}), and this was
usually treated as a drawback. However, in view of the above results,
this feature seems to be more natural than 
the standard one when a particle and its antiparticle are 
described by different IRs of the symmetry group (see also
the discussion in Ref. \cite{lev2}). Another interesting
feature of the SO(1,4) invariant theory is that the free and 
interacting operators are unitarily equivalent 
\cite{lev3}. A qualitative explanation of this fact is as
follows. It is well known that in the dS space there exists
the universal antigravity: the force of repulsion between
any two particles is proportional to the distance between
them. At large distances this interaction is so strong that
no bound states can exist. Therefore the free and interacting
operators have the same spectrum and, as a result, they are
unitarily equivalent. This means that any interaction can be
introduced by an appropriate unitary transformation, and the
problem arises whether the notion of interactions is needed 
at all. It is interesting to investigate how this property
can be formulated in the framework of the GFQT. Moreover,
when the field of complex numbers is replaced by a Galois
field, this can be treated as an effective interaction. Indeed,
the spectra of observables change and the rules of computing
probabilities change too. Such a possibility has been
discussed in Ref. \cite{lev2}, and it requires a further study.

{\it Acknowledgements: } I have greatly benefited from 
discussions with many physicists and mathematicians, and it's 
difficult to mention all of them. A collaboration with L.A. 
Kondratyuk considerably improved my understanding of basics of 
quantum theory. He pointed out that the theory should not 
necessarily be based on a local Lagrangian, and symmetry on 
quantum level means that proper commutation
relations are satisfied. Discussions with S.N. Sokolov about
his (very elegant) method of packing operators and mathematical 
structure of quantum theory were very encouraging. 
E.G. Mirmovich has proposed an idea that only angular momenta 
are fundamental physical quantities \cite{Mirmovich}. This 
encouraged me to study the properties of de Sitter invariant 
theories. A collaboration with E. Pace and G. Salme  
helped me to understand how relativistic dynamics 
based only on correct commutation relations and cluster 
separability can be applied for describing different 
experimental data. I am also grateful to F. Coester, 
S. Dolgobrodov, B. Fridman, M. Fuda, B. Hikin, B. Kopeliovich, 
S. Kremlev, Z. Lipkin, A. Makarov, W. Polyzou 
and H. Weber for numerous discussions.

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\end{document}
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