{ \centering{\begin{minipage}{4.5in}\footnotesize\baselineskip=10pt \parindent=0pt #1\par \parindent=15pt #2\par \parindent=15pt #3 \end{minipage}}\par}
gr-qc0202041.tex
\documentclass[twoside]{article}
%\documentstyle[twoside,psfig]{article}
%% *2**** Latex  Yu.V.Pavlov "Creation of the Nonconformal Particles *****
%% International Journal of Modern Physics A --- IJMPA
%--------------------NEW ADDITIONS TO EXISTING ARTICLE.STY------------------
\catcode`\@=11
\long\def\@makefntext#1{
\protect\noindent \hbox to 3.2pt {\hskip-.9pt
$^{{\eightrm\@thefnmark}}$\hfil}#1\hfill}       %CAN BE USED

\def\thefootnote{\fnsymbol{footnote}}
\def\@makefnmark{\hbox to 0pt{$^{\@thefnmark}$\hss}}    %ORIGINAL

\def\ps@myheadings{\let\@mkboth\@gobbletwo
\def\@oddhead{\hbox{}
\rightmark\hfil\eightrm\thepage}
\def\@oddfoot{}\def\@evenhead{\eightrm\thepage\hfil
\leftmark\hbox{}}\def\@evenfoot{}
\def\sectionmark##1{}\def\subsectionmark##1{}}

%--------------------START OF IJMPA1.STY------------------------------------
%THIS STYLE FILE (IJMPA1.STY) IS FOR REFERENCES FROM 1--9 ITEMS ONLY

%THIS STYLE FILE WILL PRODUCE
%$^1$ IN BODY TEXT AND 1. AT REFERENCE SECTION

\oddsidemargin=\evensidemargin
\addtolength{\oddsidemargin}{-30pt}
\addtolength{\evensidemargin}{-30pt}
\headsep=15pt
\baselineskip=13pt

%---------------------------------------------------------------------------
\newcommand{\symbolfootnote}{\renewcommand{\thefootnote}
    {\fnsymbol{footnote}}}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\newcommand{\alphfootnote}
    {\setcounter{footnote}{0}
     \renewcommand{\thefootnote}{\sevenrm\alph{footnote}}}

%---------------------------------------------------------------------------
%NEWLY-DEFINED SECTION COMMANDS
\newcounter{sectionc}\newcounter{subsectionc}\newcounter{subsubsectionc}
\renewcommand{\section}[1] {\vspace{12pt}\addtocounter{sectionc}{1}
\setcounter{subsectionc}{0}\setcounter{subsubsectionc}{0}\noindent
    {\tenbf\thesectionc. #1}\par\vspace{5pt}}
\renewcommand{\subsection}[1] {\vspace{12pt}\addtocounter{subsectionc}{1}
    \setcounter{subsubsectionc}{0}\noindent
    {\bf\thesectionc.\thesubsectionc. {\kern1pt \bfit #1}}\par\vspace{5pt}}
\renewcommand{\subsubsection}[1] {\vspace{12pt}\addtocounter{subsubsectionc}{1}
    \noindent{\tenrm\thesectionc.\thesubsectionc.\thesubsubsectionc.
    {\kern1pt \tenit #1}}\par\vspace{5pt}}
\newcommand{\nonumsection}[1] {\vspace{12pt}\noindent{\tenbf #1}
    \par\vspace{5pt}}

%NEW MACRO TO HANDLE APPENDICES
\newcounter{appendixc}
\newcounter{subappendixc}[appendixc]
\newcounter{subsubappendixc}[subappendixc]
\renewcommand{\thesubappendixc}{\Alph{appendixc}.\arabic{subappendixc}}
\renewcommand{\thesubsubappendixc}
    {\Alph{appendixc}.\arabic{subappendixc}.\arabic{subsubappendixc}}

\renewcommand{\appendix}[1] {\vspace{12pt}
        \refstepcounter{appendixc}
        \setcounter{figure}{0}
        \setcounter{table}{0}
        \setcounter{lemma}{0}
        \setcounter{theorem}{0}
        \setcounter{corollary}{0}
        \setcounter{definition}{0}
        \setcounter{equation}{0}
        \renewcommand{\thefigure}{\Alph{appendixc}.\arabic{figure}}
        \renewcommand{\thetable}{\Alph{appendixc}.\arabic{table}}
        \renewcommand{\theappendixc}{\Alph{appendixc}}
        \renewcommand{\thelemma}{\Alph{appendixc}.\arabic{lemma}}
        \renewcommand{\thetheorem}{\Alph{appendixc}.\arabic{theorem}}
        \renewcommand{\thedefinition}{\Alph{appendixc}.\arabic{definition}}
        \renewcommand{\thecorollary}{\Alph{appendixc}.\arabic{corollary}}
        \renewcommand{\theequation}{\Alph{appendixc}.\arabic{equation}}
%   \noindent{\tenbf Appendix \theappendixc. #1}\par\vspace{5pt}}
        \noindent{\tenbf Appendix \theappendixc #1}\par\vspace{5pt}}
\newcommand{\subappendix}[1] {\vspace{12pt}
        \refstepcounter{subappendixc}
        \noindent{\bf Appendix \thesubappendixc. {\kern1pt \bfit #1}}
    \par\vspace{5pt}}
\newcommand{\subsubappendix}[1] {\vspace{12pt}
        \refstepcounter{subsubappendixc}
        \noindent{\rm Appendix \thesubsubappendixc. {\kern1pt \tenit #1}}
    \par\vspace{5pt}}

%---------------------------------------------------------------------------
%FOLLOWING THREE COMMANDS ARE FOR `LIST' COMMAND.
\topsep=0in\parsep=0in\itemsep=0in
\parindent=15pt

%---------------------------------------------------------------------------
\newcommand{\textlineskip}{\baselineskip=13pt}
\newcommand{\smalllineskip}{\baselineskip=10pt}

%---------------------------------------------------------------------------
%MACRO FOR COPYRIGHT BLOCK
\def\eightcirc{
\begin{picture}(0,0)
\put(4.4,1.8){\circle{6.5}}
\end{picture}}
\def\eightcopyright{\eightcirc\kern2.7pt\hbox{\eightrm c}}

\newcommand{\copyrightheading}[1]
    {\vspace*{-2.5cm}\smalllineskip{\flushleft
    {\footnotesize International Journal of Modern Physics A #1}\\
    {\footnotesize $\eightcopyright$\, World Scientific Publishing
     Company}\\
     }}

%---------------------------------------------------------------------------
%MACRO FOR PUBLISHER INFORMATION SECTION
\newcommand{\pub}[1]{{\begin{center}\footnotesize\smalllineskip
    Received #1\\
    \end{center}
    }}

\newcommand{\publisher}[2]{{\begin{center}\footnotesize\smalllineskip
    Received #1\\
    Revised #2
    \end{center}
    }}

%---------------------------------------------------------------------------
%MARCO FOR ABSTRACT BLOCK
\def\abstracts#1#2#3{{
    \centering{\begin{minipage}{4.5in}\footnotesize\baselineskip=10pt
    \parindent=0pt #1\par
    \parindent=15pt #2\par
    \parindent=15pt #3
    \end{minipage}}\par}}

%---------------------------------------------------------------------------
%MARCO FOR KEYWORDS BLOCK
\def\keywords#1{{
    \centering{\begin{minipage}{4.5in}\footnotesize\baselineskip=10pt
    {\footnotesize\it Keywords}\/: #1
     \end{minipage}}\par}}

%---------------------------------------------------------------------------
%NEW MACRO FOR BIBLIOGRAPHY
\newcommand{\bibit}{\nineit}
\newcommand{\bibbf}{\ninebf}
\renewenvironment{thebibliography}[1]
    {\frenchspacing
     \ninerm\baselineskip=11pt
     \begin{list}{\arabic{enumi}.}
    {\usecounter{enumi}\setlength{\parsep}{0pt}
     \setlength{\leftmargin 12.7pt}{\rightmargin 0pt} %FOR 1--9 ITEMS
%    \setlength{\leftmargin 17pt}{\rightmargin 0pt}   %FOR 10--99 ITEMS
%    \setlength{\leftmargin 22pt}{\rightmargin 0pt}  %FOR 100+ABOVE ITEMS
     \setlength{\itemsep}{0pt} \settowidth
    {\labelwidth}{#1.}\sloppy}}{\end{list}}

%---------------------------------------------------------------------------
%LIST ENVIRONMENTS
\newcounter{itemlistc}
\newcounter{romanlistc}
\newcounter{alphlistc}
\newcounter{arabiclistc}
\newenvironment{itemlist}
        {\setcounter{itemlistc}{0}
     \begin{list}{$\bullet$}
    {\usecounter{itemlistc}
     \setlength{\parsep}{0pt}
     \setlength{\itemsep}{0pt}}}{\end{list}}

\newenvironment{romanlist}
    {\setcounter{romanlistc}{0}
     \begin{list}{$($\roman{romanlistc}$)$}
    {\usecounter{romanlistc}
     \setlength{\parsep}{0pt}
     \setlength{\itemsep}{0pt}}}{\end{list}}

\newenvironment{alphlist}
    {\setcounter{alphlistc}{0}
     \begin{list}{$($\alph{alphlistc}$)$}
    {\usecounter{alphlistc}
     \setlength{\parsep}{0pt}
     \setlength{\itemsep}{0pt}}}{\end{list}}

\newenvironment{arabiclist}
    {\setcounter{arabiclistc}{0}
     \begin{list}{\arabic{arabiclistc}}
    {\usecounter{arabiclistc}
     \setlength{\parsep}{0pt}
     \setlength{\itemsep}{0pt}}}{\end{list}}

%---------------------------------------------------------------------------
%FIGURE CAPTION
\newcommand{\fcaption}[1]{
        \refstepcounter{figure}
        \setbox\@tempboxa = \hbox{\footnotesize Fig.~\thefigure. #1}
        \ifdim \wd\@tempboxa > 5in
           {\begin{center}
        \parbox{5in}{\footnotesize\smalllineskip Fig.~\thefigure. #1}
            \end{center}}
        \else
             {\begin{center}
             {\footnotesize Fig.~\thefigure. #1}
              \end{center}}
        \fi}

%TABLE CAPTION
\newcommand{\tcaption}[1]{
        \refstepcounter{table}
        \setbox\@tempboxa = \hbox{\footnotesize Table~\thetable. #1}
        \ifdim \wd\@tempboxa > 5in
           {\begin{center}
        \parbox{5in}{\footnotesize\smalllineskip Table~\thetable. #1}
            \end{center}}
        \else
             {\begin{center}
             {\footnotesize Table~\thetable. #1}
              \end{center}}
        \fi}

%---------------------------------------------------------------------------
%MACROS FOR SETTING \cite{x} OR \citeup{x}
\def\@citex[#1]#2{\if@filesw\immediate\write\@auxout
    {\string\citation{#2}}\fi
\def\@citea{}\@cite{\@for\@citeb:=#2\do
    {\@citea\def\@citea{,}\@ifundefined
    {b@\@citeb}{{\bf ?}\@warning
    {Citation `\@citeb' on page \thepage \space undefined}}
    {\csname b@\@citeb\endcsname}}}{#1}}

\newif\if@cghi
\def\cite{\@cghitrue\@ifnextchar [{\@tempswatrue
    \@citex}{\@tempswafalse\@citex[]}}
\def\citelow{\@cghifalse\@ifnextchar [{\@tempswatrue
    \@citex}{\@tempswafalse\@citex[]}}
\def\@cite#1#2{{$\null^{#1}$\if@tempswa\typeout
    {IJCGA warning: optional citation argument
    ignored: `#2'} \fi}}
\newcommand{\citeup}{\cite}

%---------------------------------------------------------------------------
%FOR SUB/SUPERSCRIPT BOLDFACED + ITALICS
\def\pmb#1{\setbox0=\hbox{#1}
    \kern-.025em\copy0\kern-\wd0
    \kern.05em\copy0\kern-\wd0
    \kern-.025em\raise.0433em\box0}
\def\mbi#1{{\pmb{\mbox{\scriptsize ${#1}$}}}}
\def\mbr#1{{\pmb{\mbox{\scriptsize{#1}}}}}

%---------------------------------------------------------------------------
%FOR FNSYMBOL FOOTNOTE AND ALPH{FOOTNOTE}
\def\fnm#1{$^{\mbox{\scriptsize #1}}$}
\def\fnt#1#2{\footnotetext{\kern-.3em
    {$^{\mbox{\scriptsize #1}}$}{#2}}}

%---------------------------------------------------------------------------
%FOR CREATING THE OPENING PAGE NUMBER
\def\fpage#1{\begingroup
\voffset=.3in
\thispagestyle{empty}\begin{table}[b]\centerline{\footnotesize #1}
    \end{table}\endgroup}

%--------------------------------------------------------------------------
%MACRO FOR RUNNINGHEAD
%\def\runninghead#1#2{\pagestyle{myheadings}
%\markboth{{\protect\footnotesize\it{\quad #1}}\hfill}
%{\hfill{\protect\footnotesize\it{#2\quad}}}}
%\headsep=15pt

\def\thefootnote{\fnsymbol{footnote}}
\def\@makefnmark{\hbox to 0pt{$^{\@thefnmark}$\hss}}    %ORIGINAL

\def\ps@myheadings{%
    \let\@oddfoot\@empty\let\@evenfoot\@empty
    \def\@evenhead{\slshape\leftmark\hfil}%       %EVEN PAGE
    \def\@oddhead{\hfil{\slshape\rightmark}}%     %ODD PAGE
    \let\@mkboth\@gobbletwo
    \let\sectionmark\@gobble
    \let\subsectionmark\@gobble
    }
%
%---------------------------------------------------------------------------
\font\tenrm=cmr10
\font\tenit=cmti10
\font\tenbf=cmbx10
\font\bfit=cmbxti10 at 10pt
\font\ninerm=cmr9
\font\nineit=cmti9
\font\ninebf=cmbx9
\font\eightrm=cmr8
\font\eightit=cmti8
\font\eightbf=cmbx8
\font\sevenrm=cmr7
\font\fiverm=cmr5

%---------------------------------------------------------------------------
\newtheorem{theorem}{\indent Theorem}
%OR \newtheorem{theorem}{Theorem}[sectionc] WHICH HAS SECTION NUMBER

\newtheorem{lemma}{Lemma}
%OR USE \newtheorem{lemma}{Lemma}[sectionc]

\newtheorem{definition}{Definition}
\newtheorem{corollary}{Corollary}
%USERS CAN ADD ON THEIR OWN NEW THEOREM-LIKE ENVIRONMENTS.

\newcommand{\proof}[1]{{\tenbf Proof.} #1 $\Box$.}

%--------------------END OF IJMPA1.STY--------------------------------------


%--------------------START OF DATA FILE-------------------------------------
\textwidth=5truein
%\textheight=7.8truein
\textheight=7.26truein

%---------------------------------------------------------------------------
%DEFINITIONS
\def\qed{\hbox{${\vcenter{\vbox{            %HOLLOW SQUARE
   \hrule height 0.4pt\hbox{\vrule width 0.4pt height 6pt
   \kern5pt\vrule width 0.4pt}\hrule height 0.4pt}}}$}}

\renewcommand{\thefootnote}{\fnsymbol{footnote}}  %USE SYMBOLIC FOOTNOTE

%---------------------------------------------------------------------------
% **********************************************************************
% *********************** text of paper ********************************
\pagestyle{myheadings}

\begin{document}
\setlength{\textheight}{7.7truein}  %for 2nd page onwards

\thispagestyle{empty}

%\markboth{\protect{\footnotesize\it Instructions for Typesetting
%Manuscripts}}{\protect{\footnotesize\it Instructions for
%Typesetting Manuscripts}}

\normalsize\textlineskip

\setcounter{page}{1}

%\copyrightheading{}     %{Vol.~0, No.~0 (2000) 000--000}

%\vspace*{0.88truein}

%\fpage{1}
\centerline{\bf CREATION OF THE NONCONFORMAL SCALAR PARTICLES}
\vspace*{0.035truein}
\centerline{\bf IN NONSTATIONARY METRIC}
\vspace*{0.37truein}
\centerline{\footnotesize YU. V. PAVLOV\footnote{E-mail: pavlov@ipme.ru}}
\baselineskip=12pt
\centerline{\footnotesize\it Institute of Mechanical Engineering,
Russian Academy of Sciences,}
\baselineskip=10pt
\centerline{\footnotesize\it 61 Bolshoy, V.O., St.Petersburg, 199178,
Russia}
\vspace*{0.225truein}
%\pub{(received date)}

\vspace*{0.21truein}
\abstracts{
    The nonconformal scalar field is considered
in $N$-dimensional space-time with metric  which includes,
in particular, the cases of nonhomogeneous spaces and
anisotropic spaces of Bianchi type-I.
   The modified Hamiltonian is constructed.
   Under the diagonalization  of it the energy of quasiparticles is
equal to the oscillator frequency of the wave equation.
   The density of particles created by nonstationary metric is
investigated.
    It is shown  that the densities of conformal and nonconformal
particles created in Friedmann radiative-dominant Universe
coincide.}{}{}

\setcounter{footnote}{0}
\renewcommand{\thefootnote}{\alph{footnote}}

\vspace*{1pt}\textlineskip
% **************************** Introduction ******************************
\section{Introduction}
\vspace*{-0.5pt}
\noindent
   Quantum field theory in curved space-time was actively developed
in the 70-ties of the last century (see books\cite{1,2}).
   However, there are a number of unsolved problems in this theory.
   One of them is related to the definition of an elementary particle
and vacuum in a curved space-time.
   The reason is that in the case of a curved space  there is no group of
symmetries similar to the Poincar\'{e} group in the Minkowski space.
   If we believe that a particle is associated with a quantum of energy,
then according to quantum mechanics, the observation of particles at
an instant implies finding an eigenstate of the Hamiltonian.
    The Hamiltonian diagonalization method\cite{1} takes this into account
automatically.

    The use of the Hamiltonian constructed from the metrical
energy-momen\-tum tensor, first proposed by A.A.Grib and
S.G.Mamayev,\cite{3} was successful in the homogeneous isotropic
space-time for the conformal scalar fields.
   But such Hamiltonian leads to the difficulties related to an
infinite density of created quasiparticles in the nonconformal
case.\cite{4}
    The energy of such quasiparticles differs from the oscillator
frequency of the wave equation.\cite{5,6}
    The nonconformal case is important for investigation by many reasons.
    In particular,  the additional
nonconformal terms are dominant in the vacuum expectation values
of the energy-momentum tensor.\cite{7}
    Recently,\cite{8} the modified Hamiltonian was found  that the
density of the particles corresponding to its diagonal form and created
in the nonstationary homogeneous isotropic space-time is finite.

    The aim of this paper is the generalization of the method,
proposed in Ref.~8, for the cases of the nongomogeneous and anisotropic
spaces.
    In Sec.~2 we quantize the nonconformal scalar field
in curved space and construct the modified Hamiltonian.
    In Sec.~3 we diagonalize this Hamiltonian and investigate the
questions about the energies and the density of created particles.
%   We use the system of units where $\hbar=c=1$.


% *********************** section 2 **************************************
\section{Quantizing a Scalar Field in Curved Space}
\noindent
    We consider a complex scalar field  $\varphi(x)$ of the mass $m$
with Lagrangian
    \begin{equation}
L(x)=\sqrt{|g|}\ [\,g^{ik}\partial_i\varphi^*\partial_k\varphi -
(m^2+f(R))\, \varphi^* \varphi \,] \,,
\label{Lag}
\end{equation}
  and corresponding equation of motion
\begin{equation}
 ({\nabla}^i {\nabla}_{\! i} + f(R) + m^2) \varphi(x)=0 \, ,
\label{Eqm}
\end{equation}
   where ${\nabla}_{\! i}$ are the covariant derivatives in
$N$-dimensional space-time,
$ g\!=\!{\rm det}(g_{ik})$,  $\ f(R)$ is the function of the
invariant combinations $g_{ik}$ and curvature tensor $R^i_{\,klm}$.
    The equation~(\ref{Eqm}) is conformally invariant if $m=0$
and $f(R)=\xi_c R $, where $R$ is scalar curvature, and
$\xi_c = (N-2)/\,[\,4\,(N-1)] $ (conformal coupling).
   We investigate the case of metric which in some coordinate system
$(x)=(\eta, {\bf x})$ takes the form
    \begin{equation}
ds^2 = a^2(\eta) \left( d\eta^2 - \gamma_{\alpha \beta}(x)\,
d x^\alpha d x^\beta \right) \,,
\label{gikn}
\end{equation}
   where  $\alpha, \beta= 1, \ldots, N-1 $ and
$\ {\rm det}(\gamma_{\alpha \beta})=\gamma({\bf x})$.
    It realizes, in particular, for the homogeneous isotropic space-time;
\ for the nonhomogeneous spaces with
   \begin{equation}
ds^2=dt^2 - a^2(t) \, \gamma_{\alpha \beta}({\bf x}) \,
dx^\alpha d x^\beta  \ ;
\label{giknh}
\end{equation}
        for the anisotropic metric of Bianchi type-I
$
\left( ds^2 = dt^2 - a^2_\alpha(t)\,(d x^\alpha )^2 \right). \
$
     The equation~(\ref{Eqm}) for the functions
$ \tilde{\varphi}=a^{(N-2)/2}\varphi $
in coordinates $(\eta, {\bf x})$ takes the form
   \begin{equation}
\tilde{\varphi}''- \Delta_{N-1} \tilde{\varphi} +
\biggl[ \left(m^2 + f(R) \right) a^2 - \frac{N-2}{4} \left( 2c'+
(N-2) c^2 \right) \biggr] \, \tilde{\varphi}=0 \,,
\label{eft}
\end{equation}
    where the prime denotes the derivative with respect to time $\eta$,
$\ c \equiv a'/a \ $,
$\ \Delta_{N-1} \equiv \gamma^{-1/2} \partial_\alpha (\sqrt{\gamma}\,
\gamma^{\alpha \beta} \partial_{\beta} ) $.
    Let us assume that Laplace-Beltrami operator $\Delta_{N-1}$
has the complete orthonormal set of the eigenfunctions
$\Phi_{\!J}({\bf x})$ non-depending on time
    \begin{equation}
\Delta_{N-1} \Phi_{\!J}({\bf x}) = -\lambda^2(J,\eta)\,
\Phi_{\!J}({\bf x}) \,.
\label{DelPhi}
\end{equation}
    The full set of the motion equation solutions can be found in the form
$ \tilde{\varphi}(x) = g_J(\eta) \Phi_{\!J} ({\bf x}) $\,,
  where
      \begin{equation}
g_J''(\eta) + \Omega^2(\eta)\,g_J(\eta)=0 \,,
\label{gdd}
\end{equation}
$\Omega$ is the oscillator frequency:
\begin{equation}
\Omega^2(\eta)=\left( m^2 + f(R) \right) a^2 -
\frac{N-2}{4} \left( 2c'+(N-2) c^2 \right) +\lambda^2(J,\eta) \,.
\label{Ome}
\end{equation}
    For quantization, we expand the field $ \tilde{\varphi}(x) $
       \begin{equation}
\tilde{\varphi}(x)=\int \! d\mu(J)\,\biggl[ \tilde{\varphi}{}^{(+)}_J \,
a^{(+)}_J + \tilde{\varphi}{}^{(-)}_J \, a^{(-)}_J \,\biggr] \ ,
\label{fff}
\end{equation}
     where $d\mu(J)$ is the measure on the space of $\Delta_{N-1}$
eigenvalues,
\begin{equation}
\tilde{\varphi}{}^{(+)}_J (x) =\frac{1}{\sqrt{2}}\,
g_J(\eta)\,\Phi^*_{\!J}({\bf x}) \ ,
\ \  \ \
\tilde{\varphi}{}^{(-)}_J (x)=\biggl(\tilde{\varphi}{}_J^{(+)}({\bf x})
 \biggr)^* \,,
\label{fpm}
\end{equation}
     and impose the usual commutation relations on the operators
$ a_J^{(\pm)} , \ \stackrel{*}{a}\!{\!}_J^{(\pm)}$.

    We construct Hamiltonian as canonical one for variables
$\tilde{\varphi}(x)$ and $\tilde{\varphi}^*(x)$.
  If we add $N$-divergence $(\partial J^i/ \partial x^i)$ to Lagrangian
density~(\ref{Lag}), where in the coordinate system $(\eta, {\bf x})$
the $ N$-vector
$\ (J^i)=(\sqrt{\gamma}\,c\,\tilde{\varphi}^*\, \tilde{\varphi}\,(N-2)/2,
\, 0, \, \ldots \,, 0) $, \
   the motion equation~(\ref{Eqm}) is invariant under this addition.
    By using the Lagrangian density
$ L^{\Delta}(x)=L(x)+({\partial J^i}/{\partial x^i})$, \
we obtain by integration on hypersurface $\Sigma :\ \eta=const $
of Hamiltonian density
$ h(x)= \tilde{\varphi}'\,(\partial L^{\Delta})/
(\partial \tilde{\varphi}') +
\tilde{\varphi}^{* \prime}\,(\partial L^{\Delta})/
(\partial \tilde{\varphi}^{* \prime})-L^{\Delta}(x) $,
the modified Hamiltonian
     \begin{eqnarray}
H(\eta) &=& \int_\Sigma h(x) \, d^{N-1}x = \int_\Sigma d^{N-1}x \,
  \sqrt{\gamma} \, \Biggl\{
\tilde{\varphi}^{* \prime} \tilde{\varphi}{}'
+\gamma^{\alpha \beta} \partial_\alpha\tilde{\varphi}^*
\partial_\beta \tilde{\varphi}+
\nonumber                \\
&+& \biggl[ \left(m^2+f(R) \right) a^2 - \frac{N-2}{4} \left(2c'+
(N-2)c^2\right)
\biggr]\,  \tilde{\varphi}^* \tilde{\varphi} \Biggr\} \,.
\label{Hp}
\end{eqnarray}
      Hamiltonian~(\ref{Hp}) is expressed in terms of the operators
$ a_J^{(\pm)} , \ \stackrel{*}{a}\!{\!}_J^{(\pm)}$ by
    \begin{equation}
H(\eta)= \! \int \! \! d\mu(J)  \biggl\{ \! E_J(\eta)\!
\left(\stackrel{*}{a}\!{\!}^{(+)}_J a^{(-)}_J +
\stackrel{*}{a}\!{\!}^{(-)}_{\bar{J}} a^{(+)}_{\bar{J}} \right) +
F_J(\eta)  \stackrel{*}{a}\!{\!}^{(+)}_J a^{(+)}_{\bar{J}} +
F^*_J(\eta) \stackrel{*}{a}\!{\!}^{(-)}_{\bar{J}} a^{(-)}_J \! \biggr\},
\label{H}
\end{equation}
    where
\begin{equation}
E_J(\eta)=\frac{1}{2} \biggl[\, |g_J'|^2+ \Omega^2 |g_J|^2 \,
\biggr]  \ ,
\ \ \ F_J(\eta)=\frac{\vartheta_{\!J}}{2} \biggl[\, g_J'{}^{\! 2} +
\Omega^2  g_J^{\, 2} \, \biggr]  \,,
\label{EJFJ}
\end{equation}
   here we used the normalization condition
$ g_J\,g_J^*{}' - g_J^{\, \prime}\,g_J^* = -2 i \  $ and the
special choice of the eigenfunctions at which for an arbitrary $J$ such
$\bar{J}$ existed, that $\Phi_{\!J}^*({\bf x}) = \vartheta_{\!J}
\Phi_{\!\bar{J}}({\bf x})$.


% *********************** section 3 **************************************
\section{Hamiltonian Diagonalization and Particle Creation}
\noindent
The Hamiltonian diagonalization for an arbitrary instant $\eta$
is realized in terms of the operators
$ \stackrel{*}{b}\!{\!}^{(\pm)}_J, \  b^{(\pm)}_J $
  related to the operators
$ \stackrel{*}{a}\!{\!}^{(\pm)}_J, \  a^{(\pm)}_J $
   via the time-dependent Bogolubov transformations
\begin{equation}
a_J^{(-)}=\alpha^*_J(\eta) b^{(-)}_J(\eta) -
\beta_J(\eta) \vartheta_{\!J} b^{(+)}_{\bar{J}}(\eta), \ \ \
\stackrel{*}{a}\!{\!}_J^{(-)}=
\alpha^*_J(\eta) \! \stackrel{*}{b}\!{\!}^{(-)}_J\!(\eta) -
\beta_J(\eta) \vartheta_{\!J} \!
\stackrel{*}{b}\!{\!}^{(+)}_{\bar{J}}\!(\eta),
\label{db}
\end{equation}
     where the functions $\alpha_J, \ \beta_J $ satisfy
the initial conditions
$|\alpha_J(\eta_0)|=1, \ \beta_J(\eta_0)=0 $  and the identity
$ |\alpha_J(\eta)|^2-|\beta_J(\eta)|^2=1 $.
   Substituting expansion~(\ref{db}) in  (\ref{H})
and requiring that the coefficients at the nondiagonal terms
$ \stackrel{*}{b}\!{\!}^{(\pm)}_J  b^{(\pm)}_J $
vanish, we obtain
%the equalities
    \begin{equation}
-2\alpha_J\beta_J \vartheta_{\!J}E_J + \alpha_J^2 F_J
+\beta_J^2 \vartheta_{\!J}^2 F_J^* = 0   \,, \ \ \
|\beta_J|^2=
\frac{1}{4\Omega}\biggl( |g_J'|^2+\Omega^2 |g_J|^2 \biggr) -\frac{1}{2} \,,
\label{bFJ}
\end{equation}
     \begin{equation}
H(\eta) =\int d\mu(J) \,\Omega(\eta) \,
\left(\,\stackrel{*}{b}\!{\!}^{(+)}_J b^{(-)}_J +
\stackrel{*}{b}\!{\!}^{(-)}_{\bar{J}} b^{(+)}_{\bar{J}}\, \right) \,.
\label{Hbb}
\end{equation}
   Therefore, the energies of quasiparticles corresponding to the diagonal
form of the Hamiltonian~(\ref{H}) are equal to the oscillator frequency
$\Omega(\eta)$.

   The vacuum state for instant $\eta$ is defined by
$
b^{(-)}_J|\,0_\eta\!> \,=\, \stackrel{*}{b}\!{\!}^{(-)}_J|\,0_\eta\!>\,
= 0 \,.
$
    The state $|\,0\!>=|\,0_{\eta_0}\!>$ contains
$|\beta_J(\eta)|^2$ quasiparticle pairs corresponding to the operators
$b^{(\pm)}_J(\eta)$ in every mode.\cite{1,2}
    The density of created particles is proportional to
$\int d\mu(J)\, |\beta_J|^2 $.
    Using the first iteration in the integral equation for
$S(\eta)=|\beta_J(\eta)|^2$
    \begin{equation}
S(\eta)=\frac{1}{2}\,\int_{\eta_0}^\eta d\eta_1 \,
w(\eta_1)\, \int_{\eta_0}^{\eta_1} d\eta_2 \,w(\eta_2)\,
(1+2 S(\eta_2)) \cos[2\,\Theta(\eta_2,\eta_1)]  \,,
\label{iuS}
\end{equation}
    where $w(\eta)=\Omega'(\eta)/\,\Omega(\eta) \ $,
$\Theta(\eta_1, \eta_2)=\int_{\eta_1}^{\eta_2}
\Omega(\eta)\,d \eta$,
     we can see, that
if the eigenvalues $-\lambda^2(J)$ of the Laplace-Beltrami operator
    $\Delta_{N-1}$ don't depend on time, then
    $w \sim \lambda^{-2}$\ and \
    $S \sim \lambda^{-6} $ in  $\lambda(J) \to \infty $.

    Therefore, in the space-time with metric~(\ref{giknh}) the density
of created particles for $N=4$ is finite. (In Bianchi type-I the
result is infinite).
   From equation~(\ref{iuS}) for $S(\eta)$ we can see
that  the densities of conformal and nonconformal
(under $ f(R)=\xi R$)\ particles  created in Friedmann
radiative-dominant  Universe coincide.


\nonumsection{Acknowledgements}
\noindent
The author is grateful to Prof. A.A.Grib for helpful discussions.
This work was supported by Min. of Education of Russia, grant E00-3-163.


\nonumsection{References}
\begin{thebibliography}{000}
\bibitem{1}
A. A. Grib, S. G. Mamayev and V. M. Mostepanenko, {\bibit
Vacuum Quantum Effects in Strong Fields}
(Friedmann Laboratory Publishing, St.Petersburg, 1994).

\bibitem{2}
N. D. Birrell and P. C. W. Davies, {\bibit Quantum Fields in Curved Space}
(Cambridge University Press, Cambridge, 1982).

\bibitem{3}
A. A. Grib and S. G. Mamayev, {\bibit Yad. Fiz.}
{\bf 10}, (1969) 1276 (Engl. trans. in {\bibit Sov. J. Nucl. Phys.}
{\bf 10}, 722).

\bibitem{4}
S. A. Fulling, {\bibit Gen. Relat. Gravit.} {\bf 10}, (1979) 807.

\bibitem{5}
M. Castagnino and R. Ferraro, {\bibit Phys. Rev.} {\bf D34}, (1986) 497.

\bibitem{6}
V. B. Bezerra, V. M. Mostepanenko and C. Romero,
{\bibit Int. J. Mod. Phys.} {\bf D7}, (1998) 249.

\bibitem{7}
M. Bordag, J. Lindig, V. M. Mostepanenko and Yu. V. Pavlov,
{\bibit Int. J. Mod. Phys.} {\bf D6}, (1997) 449.

\bibitem{8}
Yu. V. Pavlov, {\bibit Theor. Math. Phys.} {\bf 126}, (2001) 92.
\end{thebibliography}
\end{document}

gr-qc0202041.extracted_bib
\begin{thebibliography}{000}
\bibitem{1}
A. A. Grib, S. G. Mamayev and V. M. Mostepanenko, {\bibit
Vacuum Quantum Effects in Strong Fields}
(Friedmann Laboratory Publishing, St.Petersburg, 1994).

\bibitem{2}
N. D. Birrell and P. C. W. Davies, {\bibit Quantum Fields in Curved Space}
(Cambridge University Press, Cambridge, 1982).

\bibitem{3}
A. A. Grib and S. G. Mamayev, {\bibit Yad. Fiz.}
{\bf 10}, (1969) 1276 (Engl. trans. in {\bibit Sov. J. Nucl. Phys.}
{\bf 10}, 722).

\bibitem{4}
S. A. Fulling, {\bibit Gen. Relat. Gravit.} {\bf 10}, (1979) 807.

\bibitem{5}
M. Castagnino and R. Ferraro, {\bibit Phys. Rev.} {\bf D34}, (1986) 497.

\bibitem{6}
V. B. Bezerra, V. M. Mostepanenko and C. Romero,
{\bibit Int. J. Mod. Phys.} {\bf D7}, (1998) 249.

\bibitem{7}
M. Bordag, J. Lindig, V. M. Mostepanenko and Yu. V. Pavlov,
{\bibit Int. J. Mod. Phys.} {\bf D6}, (1997) 449.

\bibitem{8}
Yu. V. Pavlov, {\bibit Theor. Math. Phys.} {\bf 126}, (2001) 92.
\end{thebibliography}