Some Mathematical Bases for Non-Commutative Field Theories\thanks{To be presented at the $7^a$ Reuni\'on Nacional de F\'{\i}sica y Matematica, ESFM IPN, D.F., M\'exico, May 13-17, 2002.}
Valeri V. Dvoeglazov
Misconceptions have recently been found in the definition of a partial derivative (in the case of the presence of both explicit and implicit dependencies of the function subjected to differentiation) in the classical analysis. We investigate the possible influence of this discovery on quantum mechanics and the classical/quantum field theory. Surprisingly, some commutators of operators of space-time 4-coordinates do {not} equal to zero. Thus, we provide the bases for new-fashioned noncommutative field theory.
math-ph0204043.tex

\documentstyle[aps,12pt]{revtex}

\begin{document}
\title{Some Mathematical Bases for Non-Commutative Field Theories\thanks{To be presented at the $7^a$ Reuni\'on Nacional de F\'{\i}sica y Matematica,
ESFM IPN, D.F., M\'exico, May 13-17, 2002.}}
\author{Valeri V. Dvoeglazov}
\address{Universidad de Zacatecas\\
Apartado Postal 636, Suc. UAZ\\
Zacatecas 98062, Zac., M\'exico\\
E-mail: valeri@ahobon.reduaz.mx}
\date{April 21, 2002}
\maketitle

\begin{abstract}
Misconceptions have recently been found in the definition of a partial
derivative (in the case of the presence of both explicit and implicit
dependencies of the function subjected to differentiation) in the classical
analysis. We investigate the possible influence of this discovery on quantum
mechanics and the classical/quantum field theory. Surprisingly, some
commutators of operators of space-time 4-coordinates do {\it not} equal to zero.
Thus, we provide the bases for new-fashioned noncommutative field theory.
\end{abstract}

To the best of my knowledge, the assumption that the operators of
coordinates do {\it not} commute $[\hat{x}_{\mu },\hat{x}_{\nu }]_{-}\neq 0$
has been made by H. Snyder~\cite{snyder}. The Lorentz symmetry thus may be
broken. Much interest has recently been attracted to this idea~\cite
{noncom,kruglov} in the context of ��brane theories''.

Moreover, the famous Feynman-Dyson proof of Maxwell equations~\cite{FD}
contains intrinsically the non-commutativity of velocities $[\dot x_i (t), 
\dot x_j (t)]_- \neq 0$ that also may be considered as a contradiction with
the well-accepted theories.

On the other hand, it was recently discovered that the concept of partial
derivative is {\it not} well defined in the case of both explicit and
implicit dependence of the corresponding function, which the derivatives act
upon~\cite{chub,brown} (see also the discussion in~\cite{chja}). The
well-known example of such a situation is the field of an accelerated
charge~ \cite{landau}.\footnote{%
First, Landau and Lifshitz wrote that the functions depended on $t^{\prime }$
and only through $t^{\prime }+R(t^{\prime })/c=t$ they depended implicitly
on $x,y,z,t$. However, later (in calculating the formula (63.7)) they used
the explicit dependence of $R$ on the space coordinates of the
observation point too. Chubykalo and Vlayev claimed that the time derivative
and curl do {\it not} commute in their case. Jackson, in fact, disagreed
with their claim on the basis of the definitions (``the equations
representing Faraday's law and the absence of magnetic charges ... are satisfied automatically''; see his Introduction in~[6b]). But,
he agrees with~\cite{landau} that one should find ``a contribution to the
spatial partial derivative for fixed time $t$ from explicit spatial
coordinate dependence (of the observation point).'' \v{S}kovrlj and Ivezi\'{c}~[6c] calls this partial derivative as `{\it complete} partial
derivative'; Chubykalo and Vlayev~[6a], as `{\it total} derivative with respect to
a given variable'; the terminology suggested by Brownstein~\cite{brown} is
`the {\it whole}-partial derivative'.}

Let us study the case when we deal with explicite and implicite dependencies 
$f ({\bf p}, E ({\bf p}))$. It is well known that the energy in the
relativism is connected with the 3-momentum as $E=\pm \sqrt{{\bf p}^2 +m^2}$%
; the unit system $c=\hbar=1$ is used. In other words, we must choose the
3-dimensional hyperboloid from the entire Minkowski space and the energy is 
{\it not} an independent quantity anymore. Let us calculate the commutator
of the whole derivative $\hat\partial /\hat\partial E$ and $\hat\partial / 
\hat\partial p_i$.\footnote{%
In order to make distinction between differentiating the explicit function
and that which contains both explicit and implicit dependencies, the `whole
partial derivative' may be denoted as $\hat\partial$.} In the general case
one has 
\begin{equation}
{\frac{\hat\partial f ({\bf p}, E({\bf p})) }{\hat\partial p_i}} \equiv {%
\frac{\partial f ({\bf p}, E({\bf p})) }{\partial p_i}} + {\frac{\partial f (%
{\bf p}, E({\bf p})) }{\partial E}} {\frac{\partial E}{\partial p_i}}\, .
\end{equation}
Applying this rule, we surprisingly find 
\begin{eqnarray}
&&{\frac{[\hat\partial }{\hat\partial p_i}},{\frac{\hat\partial }{\hat%
\partial E}}]_- f ({\bf p},E ({\bf p})) = {\frac{\hat\partial }{\hat\partial
p_i}} {\frac{\partial f }{\partial E}} -{\frac{\partial }{\partial E}} ({%
\frac{\partial f}{\partial p_i}} +{\frac{\partial f}{\partial E}}{\frac{%
\partial E}{\partial p_i}}) =  \nonumber \\
&=& {\frac{\partial^2 f }{\partial E\partial p_i}} + {\frac{\partial^2 f}{%
\partial E^2}}{\frac{\partial E}{\partial p_i}} - {\frac{\partial^2 f }{%
\partial p_i \partial E}} - {\frac{\partial^2 f}{\partial E^2}}{\frac{%
\partial E}{\partial p_i}}- {\frac{\partial f }{\partial E}} {\frac{\partial%
}{\partial E}}({\frac{\partial E}{\partial p_i}})\,.  \label{com}
\end{eqnarray}
So, if $E=\pm \sqrt{m^2+{\bf p}^2}$ 
and one uses the generally-accepted 
representation form of $\partial E/\partial p_i
=  p_i/E$,
one has that the expression (\ref{com})
appears to be equal to $(p_i/E^2) {\frac{\partial f({\bf p}, E ({\bf p}))}{%
\partial E}}$. Within the choice of the normalization the coefficient is the
longitudinal electric field in the helicity basis (the electric/magnetic
fields can be derived from the 4-potentials which have been presented in~ 
\cite{hb}). On the other hand, the commutator 
\begin{equation}
[{\frac{\hat\partial}{\hat\partial p_i}}, {\frac{\hat\partial}{\hat\partial
p_j}}]_- f ({\bf p},E ({\bf p})) = {\frac{1}{E^3}} {\frac{%
\partial f({\bf p}, E ({\bf p}))}{\partial E}} [p_i, p_j]_-\,.
\end{equation}
This may be considered to be zero unless we would trust to the genious
Feynman. He postulated that the velocity (or, of course, the 3-momentum)
commutator is equal to $[p_i,p_j]\sim i\hbar\epsilon_{ijk} B^k$, i.e., to
the magnetic field.\footnote{%
In fact, if we put in the corespondence to the momenta their
quantum-mechanical operators (of course, with the appropriate clarification $%
\partial \rightarrow \hat\partial$), we obtain again that, in general, the
derivatives do {\it not} commute $[{\frac{\hat\partial}{\hat\partial x_\mu}}%
, {\frac{\hat\partial}{\hat\partial x_\nu}}]_- \neq 0$.}

Furthermore, since the energy derivative corresponds to the operator of time
and the $i$-component momentum derivative, to $\hat x_i$, we put forward the
following anzatz in the momentum representation: 
\begin{equation}
[\hat x^\mu, \hat x^\nu]_- = \omega ({\bf p}, E({\bf p})) \,
F^{\mu\nu}_{\vert\vert}{\frac{\partial }{\partial E}}\,,
\end{equation}
with some weight function $\omega$ being different for different choices of
the antisymmetric tensor spin basis.

In the modern literature, the idea of the broken Lorentz invariance by this
method concurrs with the idea of the {\it fundamental length}, first
introduced by V. G. Kadyshevsky~\cite{kadysh} on the basis of old papers by
M. Markov. Both ideas and corresponding theories are extensively discussed,
e.g.~\cite{amelino}. In my opinion, the main question is: what is the space
scale, when the relativity theory becomes incorrect.

\section*{Conclusions}

We found that the commutator of two derivatives may be {\it not} equal to
zero. As a consequence, for instance, the question arises, if the derivative 
$\hat\partial^2 f/\hat\partial p^\nu\hat\partial p^\mu$ is equal to the
derivative $\hat\partial^2 f/\hat\partial p^\mu\hat\partial p^\nu$ in all
cases?\footnote{%
The same question can be put forward when we have differentiation with
respect to the coordinates too, that may have impact on the correct
calculations of the problem of accelerated charge in classical
electrodynamics.} The presented consideration permits us to provide some
bases for non-commutative field theories and induces us to look for further
development of the classical analysis in order to provide a rigorous
mathematical basis for operations with functions which have both explicit
and implicit dependencies.

\acknowledgements
I am grateful to Profs. A. E. Chubykalo, L. M. Gaggero Sager, V. Onoochin,
F. Rohrlich, S. Vlayev, Lic. R. Flores Alvarado and participants of the
EF-UAZ seminar, where this idea has arisen, for discussions. Old papers
(cited here) and old discussions with their authors helped me to realize new
trends in modern physics.

\begin{references}
\bibitem{snyder}  H. Snyder, Phys. Rev. {\bf 71}, 38 (1947); ibid. {\bf 72},
68 (1947).

\bibitem{noncom}  S. Doplicher {\it et al.}, Phys. Lett. B{\bf 331}, 39
(1994); S. Majid and H. Ruegg, Phys. Lett. B{\bf 334}, 348 (1994); J.
Lukierski, H. Ruegg and W. J. Zakrzewski, Ann. Phys. {\bf 243}, 90 (1995);
N. Seiberg and E. Witten, JHEP {\bf 9909}, 032 (1999), hep-th/9908142; I. F.
Riad and M. M. Sheikh-Jabbari, JHEP {\bf 08}, 045 (2000), hep-th/0008132; M.
M. Sheikh-Jabbari, Phys. Rev. Lett. {\bf 84}, 5265 (2000), hep-th/0001167;
J. Madore {\it et al.}, Eur. Phys. J. C{\bf 16}, 161 (2000).

\bibitem{kruglov}  S. I. Kruglov, hep-th/0110059 [To be published in a
special issue of ''Ann. Fond. Broglie'' on Contemporary Electrodynamics]; B.
G. Sidharth, physics/0110040 [To be published in a special issue of ''Ann.
Fond. Broglie'' on Contemporary Electrodynamics].

\bibitem{FD}  F. Dyson, Am. J. Phys. {\bf 58}, 209 (1990). S. Tanimura, Ann.
Phys. {\bf 220}, 229 (1992); M. Land, N. Shnerb and L. Horwitz,
hep-th/9308003.

\bibitem{chub}  A. E. Chubykalo and R. Smirnov-Rueda, Mod. Phys. Lett. A{\bf %
12}, 1 (1997), hep-th/9608038; A. E. Chubykalo and R. Alvarado, Hadronic J.
[To be published].

\bibitem{chja}  A. E. Chubykalo and S. Vlayev, Int. J. Mod. Phys. A{\bf 14},
3789 (1999); J. D. Jackson, hep-ph/0203076, L. \v{S}kovrlj and T. Ivezi\'{c}%
, hep-ph/0203116.

\bibitem{brown}  K. R. Brownstein, Am. J. Phys. {\bf 67}, 639 (1999).

\bibitem{landau}  L. D. Landau and E. M. Lifshitz, {\it The Classical Theory
of Fields.} 4th revised English ed. [Translated from the 6th Russian
edition] (Pergamon Press, 1979). 402p., \S 63

\bibitem{hb}  M. Jacob and G. C. Wick, Ann. Phys. {\bf 7}, 404 (1959); H. M.
Ruck and W. Greiner, J. Phys. G: Nucl. Phys. {\bf 3}, 657 (1977).

\bibitem{kadysh}  V. G. Kadyshevsky, Nucl. Phys. B{\bf 141}, 477 (1978); V.
G. Kadyshevsky, M. D. Mateev, R. M. Mir-Kasimov and I. P. Volobuev, Theor.
Math. Phys. {\bf 40}, 800 (1979) [Teor. Mat. Fiz. {\bf 40}, 363 (1979)]; V.
G. Kadyshevsky and M. D. Mateev, Phys. Lett. B{\bf 106}, 139 (1981); V. G.
Kadyshevsky and M. D. Mateev, Nuovo Cim. A{\bf 87}, 324 (1985); A. D.
Donkov, R. M. Ibadov, V. G. Kadyshevsky, M. D. Mateev and M. V. Chizhov,
ibid. {\bf 87}, 350 (1985); ibid. {\bf 87}, 373 (1985). See also the old
work: V. G. Kadyshevsky, R. M. Mir-Kasimov and N. B. Skachkov, Yad. Fiz. 
{\bf 9}, 212 (1969). See also references to the old works of M. Markov in
the above-cited papers.

\bibitem{amelino}  A. Kempf, G. Mangano and R. B. Mann, Phys. Rev. D{\bf 52}%
, 1108 (1995); G. Amelino-Camelia, Nature {\bf 408}, 661 (2000);
gr-qc/0012051; hep-th/0012238; gr-qc/0106004; J. Kowalski-Glikman,
hep-th/0102098; G. Amelino-Camelia and M. Arzano, hep-th/0105120; N. R.
Bruno, G. Amelino-Camelia and J. Kowalski-Glikman, hep-th/0107039.
\end{references}

\end{document}

math-ph0204043.extracted_bib
\bibitem{snyder}  H. Snyder, Phys. Rev. {\bf 71}, 38 (1947); ibid. {\bf 72},
68 (1947).


\bibitem{noncom}  S. Doplicher {\it et al.}, Phys. Lett. B{\bf 331}, 39
(1994); S. Majid and H. Ruegg, Phys. Lett. B{\bf 334}, 348 (1994); J.
Lukierski, H. Ruegg and W. J. Zakrzewski, Ann. Phys. {\bf 243}, 90 (1995);
N. Seiberg and E. Witten, JHEP {\bf 9909}, 032 (1999), hep-th/9908142; I. F.
Riad and M. M. Sheikh-Jabbari, JHEP {\bf 08}, 045 (2000), hep-th/0008132; M.
M. Sheikh-Jabbari, Phys. Rev. Lett. {\bf 84}, 5265 (2000), hep-th/0001167;
J. Madore {\it et al.}, Eur. Phys. J. C{\bf 16}, 161 (2000).


\bibitem{kruglov}  S. I. Kruglov, hep-th/0110059 [To be published in a
special issue of ''Ann. Fond. Broglie'' on Contemporary Electrodynamics]; B.
G. Sidharth, physics/0110040 [To be published in a special issue of ''Ann.
Fond. Broglie'' on Contemporary Electrodynamics].


\bibitem{FD}  F. Dyson, Am. J. Phys. {\bf 58}, 209 (1990). S. Tanimura, Ann.
Phys. {\bf 220}, 229 (1992); M. Land, N. Shnerb and L. Horwitz,
hep-th/9308003.


\bibitem{chub}  A. E. Chubykalo and R. Smirnov-Rueda, Mod. Phys. Lett. A{\bf %
12}, 1 (1997), hep-th/9608038; A. E. Chubykalo and R. Alvarado, Hadronic J.
[To be published].


\bibitem{chja}  A. E. Chubykalo and S. Vlayev, Int. J. Mod. Phys. A{\bf 14},
3789 (1999); J. D. Jackson, hep-ph/0203076, L. \v{S}kovrlj and T. Ivezi\'{c}%
, hep-ph/0203116.


\bibitem{brown}  K. R. Brownstein, Am. J. Phys. {\bf 67}, 639 (1999).


\bibitem{landau}  L. D. Landau and E. M. Lifshitz, {\it The Classical Theory
of Fields.} 4th revised English ed. [Translated from the 6th Russian
edition] (Pergamon Press, 1979). 402p., \S 63


\bibitem{hb}  M. Jacob and G. C. Wick, Ann. Phys. {\bf 7}, 404 (1959); H. M.
Ruck and W. Greiner, J. Phys. G: Nucl. Phys. {\bf 3}, 657 (1977).


\bibitem{kadysh}  V. G. Kadyshevsky, Nucl. Phys. B{\bf 141}, 477 (1978); V.
G. Kadyshevsky, M. D. Mateev, R. M. Mir-Kasimov and I. P. Volobuev, Theor.
Math. Phys. {\bf 40}, 800 (1979) [Teor. Mat. Fiz. {\bf 40}, 363 (1979)]; V.
G. Kadyshevsky and M. D. Mateev, Phys. Lett. B{\bf 106}, 139 (1981); V. G.
Kadyshevsky and M. D. Mateev, Nuovo Cim. A{\bf 87}, 324 (1985); A. D.
Donkov, R. M. Ibadov, V. G. Kadyshevsky, M. D. Mateev and M. V. Chizhov,
ibid. {\bf 87}, 350 (1985); ibid. {\bf 87}, 373 (1985). See also the old
work: V. G. Kadyshevsky, R. M. Mir-Kasimov and N. B. Skachkov, Yad. Fiz. 
{\bf 9}, 212 (1969). See also references to the old works of M. Markov in
the above-cited papers.


\bibitem{amelino}  A. Kempf, G. Mangano and R. B. Mann, Phys. Rev. D{\bf 52}%
, 1108 (1995); G. Amelino-Camelia, Nature {\bf 408}, 661 (2000);
gr-qc/0012051; hep-th/0012238; gr-qc/0106004; J. Kowalski-Glikman,
hep-th/0102098; G. Amelino-Camelia and M. Arzano, hep-th/0105120; N. R.
Bruno, G. Amelino-Camelia and J. Kowalski-Glikman, hep-th/0107039.
