\renewcommand{\theequation}{XXII.\arabic{equation}}
 \begin{center} 
\vspace*{8mm}{\LARGE \bf APPENDIX} 
\end{center}
\vspace{2mm}
\begin{center} 
{\LARGE\bf{SUMS  OF BESSEL FUNCTIONS}} 
\end{center}

\vspace{3mm}

\begin{center} 
{\Large  \bf I.  Type I sums of the modified Bessel functions} 
\end{center}
\vspace{5mm}

\setcounter{equation}{0}

The integral representations for series of modified Bessel
functions  may be calculated  from the following
summation formula:
\bd
\sum\limits_{n=-\infty }^\infty  
{\left[ {y^2 + 
				\left( {n + 1\over 2} \right)^2}
    \right]^{-1}}
={{\pi}\over{y}} {\tanh (\pi y)}.\label{aa1}
\ed 
In proper time representation the left side of  (\ref{aa1})  
can be written in the form
\bd
\sum\limits_{n=-\infty }^\infty 
				{\left[ {y^2
				+ \left( {n+1\over 2}\right)^2}
				\right]^{-1}} \nonumber
\ed
\veb
\bd
= \int\limits_0^\infty  {d\alpha \exp
\left( {-\alpha y^2} \right)}
\sum\limits_{n=-\infty }^\infty  
{\exp \left( {-\alpha\left( {n+{1 \over 2}} \right)^2} \right)}. \label{aa2}
\ed 
Taking into account the  equation
\bd
\sum\limits_{n=-\infty }^\infty  {\exp \left\{ {-\alpha \left( {n-z}
\right)^2}
\right\}}=\sum\limits_{n=-\infty }^\infty  {\left( {{\pi  \over \alpha }}
\right)}^{1\over 2}
\exp \left( {-{{\pi ^2} \over \alpha }n^2-2\pi izn} \right),\label{aa3}
\ed 
we can write (\ref{aa2}) as
\bd
 \sum\limits_{n=-\infty }^\infty 
    {\left[ {y^2 + \left( {n+1\over 2}\right)^2}
\right]^{-1}}
=\sum\limits_{n=-\infty }^\infty 
{(-1)^n}\int\limits_0^\infty 
{d\alpha \left( {{\pi  \over \alpha }} \right)^{1\over 2}\exp \left( {-\alpha
y^2-{{\pi ^2} \over \alpha }n^2} \right)} \nonumber
\ed
\veb
\bd
={\pi  \over y}+2\sum\limits_{n=1}^\infty  {(-1)^n\int\limits_0^\infty  {d\alpha
\left( {{\pi  \over \alpha }} \right)^{1\over 2}\exp \left( {-\alpha y^2-{{\pi ^2}
\over \alpha }n^2} \right)}}.\label{aa4}
\ed 
The right side of (\ref{aa1}) is
\bd
{\pi  \over y}\tanh (\pi y)={\pi  \over y}-{{2\pi } \over {y\left(
{e^{2\pi y}+1}
\right)}}. \label{aa5}
\ed 
Therefore, we get from (\ref{aa4}) and (\ref{aa5})  the following useful equation 
\bd
{1 \over {z\left( {e^z+1} \right)}}=-{1 \over {2\pi
^2}}\sum\limits_{n=1}^\infty  {(-1)^n\int\limits_0^\infty  {d\alpha \left( {{\pi 
\over \alpha }} 
\right)^{1\over 2}\exp \left( {-\alpha {{z^2} \over {4\pi ^2}}-{{\pi
^2} \over \alpha }n^2}
\right)}}. \label{aa6}
\ed 
Moreover, we may consider that $z^2=g_a(x^2)$ is the function of
variable $x\in R^3$ with a parameter $a$, namely $z^2=x^2+a^2$.

Integrating  with respect to $x$ one can get
\bd
\int {{{d^3x} \over {\left( {2\pi } \right)^3}}}\left[ {\sqrt
{x^2+a^2}\left( {\exp \left( {\sqrt {x^2+a^2}} \right)+1} \right)}
\right]^{-1} \nonumber
\ed
\veb
\bd
=-{1 \over 2}\sum\limits_{n=1}^\infty 
{(-1)^n\int\limits_0^\infty  {d\alpha \alpha ^{-2}\exp \left( {-\alpha
{{a^2} \over {4\pi ^2}}-{{\pi ^2} \over
\alpha }n^2} \right)}}.\label{aa7}
\ed 
The modified Bessel function may be written as
\bd
\int\limits_0^\infty  {d\alpha \cdot \alpha ^{\nu -1}\cdot \exp \left(
{-\gamma
\alpha -{\delta \over \alpha} } \right)}=2\left( {\delta \over \gamma }
\right)^{\nu \over 2}K_\nu \left(2\sqrt {\delta \,\gamma }\right).\label{aa8}
\ed 
Then  (\ref{aa7}) will be
\bd
\int {{{d^3x} \over {\left( {2\pi } \right)^3}}}\left[ {\sqrt
{x^2+a^2}\left( {\exp \left( {\sqrt {x^2+a^2}} \right)+1} \right)} \nonumber
\right]^{-1}
\ed
\veb
\bd
= -{1 \over 2}\left(
{{a \over {\pi ^2}}} \right)\sum\limits_{n=1}^\infty  {{{(-1)^n} \over
n}K_1}(an). \label{aa9}
\ed 
Scaling $x$ and $a$ with a parameter $\beta $ as $(x,a)=\beta
(k,m)$  write (\ref{aa9}) in the form
\bd
\int {{{d^3k} \over {\left( {2\pi } \right)^3}}\,{2 \over {\varepsilon
\left( {\exp
\left( {\beta \varepsilon } \right)+1} \right)}}}=-{m \over {\beta \pi
^2}}\sum\limits_{n=1}^\infty  {{{(-1)^n} \over n}K_1}(\beta mn), \label{aa10}
\ed 
where $\varepsilon =\sqrt {\vec{k}^2+m^2}$. 

Differentiating the equation  (\ref{aa7}) with   respect to parameter $a$ we get 
\bd
\int {{{d^3x} \over {\left( {2\pi } \right)^3}}\left(
{-{\partial 
\over {\partial a^2}}} \right)}\left[ {\sqrt {x^2+a^2}\left( {\exp \left( {\sqrt
{x^2+a^2}}
\right)+1} \right)} \right]^{-1}\hfil \nonumber
\ed
\veb
\bd
\hfil = -{1 \over {4\pi ^2}}\sum\limits_{n=1}^\infty 
{(-1)^nK_0}(an), \label{aa11} 
\ed 
or, in new variables,
\bd
\int {{{d^3k} \over {\left( {2\pi } \right)^3}}\left( {{\partial 
\over {\partial m^2}}} \right){1 \over {\varepsilon \left( {\exp \left( {\beta
\varepsilon }
\right)+1} \right)}}}={1 \over {4\pi ^2}}\sum\limits_{n=1}^\infty 
{(-1)^nK_0}(\beta mn). \label{aa12}
\ed 
Integrating in (\ref{aa7}) with respect to parameter $(a)$ and
using the equation 
\bd
\int\limits_{a^2}^\infty  {da^2\left[ {\sqrt {x^2+a^2}\left( {\exp
\left( {\sqrt {x^2+a^2}} \right)+1} \right)} \right]^{-1}} \nonumber
\ed
\veb
\bd
=2\ln \left(
{1+\exp\left(-
\sqrt {x^2+a^2}\right)}
\right) \label{aa13}
\ed  
we find with  new variables the following equation 
\bd
-{1 \over \beta }\int {{{d^3k} \over {\left( {2\pi } \right)^3}}}\ln
\left( {1+\exp (-\beta \varepsilon )} \right)={{m^2} \over {2(\beta \pi
)^2}}\sum\limits_{n=1}^\infty  {{{(-1)^n} \over {n^2}}K_2}(\beta mn). \label{aa14}
\ed 
High temperature asymptotes $(\beta{m}\ll{1})$ of the equations 
(\ref{aa9}), (\ref{aa11}) and (\ref{aa14}) are 
\bd
{{2m^2} \over {(\beta \pi )^2}}\sum\limits_{n=1}^\infty  {{{(-1)^n}\over
n^2}K_2}(\beta mn)=-{{7\pi ^2} \over {180\beta ^4}}
+{{m^2} \over {12\beta ^2}}+{1
\over 8}m^4\left( {\ln {{\beta m} \over {4\pi }}+\gamma -{3 \over 4}}
\right) \label{aa15}
\ed
\veb
\bd
-{m \over {\beta \pi ^2}}\sum\limits_{n=1}^\infty  {{{(-1)^n} \over
n}K_1}(\beta mn)={1 \over {12\beta ^2}}+{1 \over 4}m^2\left( {\ln \left(
{{{\beta m} \over {4\pi }}} \right)+\gamma -{1 \over 2}}
\right) \label{aa16}
\ed
\veb
\bd
{1 \over {2\pi ^2}}\sum\limits_{n=1}^\infty 
{(-1)^nK_0}(\beta mn)={1
\over 4}\left( {\ln \left( {{{\beta m} \over {4\pi }}} \right)+\gamma }
\right). \label{aa17}
\ed

\newpage


\begin{center} 
{\Large  \bf II.  Type II sums of the modified Bessel functions} 
\end{center}

Let us start with  the sum:
\bd
\sum\limits_{n=-\infty }^\infty  
 \left(y^2 + 	n^2 \right)^{-1}
={{\pi}\over{y}} {\coth (\pi y)}.\label{bb1}
\ed 
The   proper time representation the left side of (\ref{bb1}) will be
\bd
\sum\limits_{n=-\infty }^\infty 
				\left( y^2
				+ n^2\right)^{-1}
= \int\limits_0^\infty  {d\alpha \exp
\left( {-\alpha y^2} \right)}
\sum\limits_{n=-\infty }^\infty  
{\exp \left( {-\alpha n^2} \right)}.\label{bb2}
\ed 
Taking into account the  equation (\ref{aa3}) and putting $(z=0)$ we find
\bd
\sum\limits_{n=-\infty }^\infty  {\exp \left\{ {-\alpha n^2}
\right\}}=\sum\limits_{n=-\infty }^\infty  {\left( {{\pi  \over \alpha }}
\right)}^{1\over 2}\exp \left( -{{\pi ^2} \over \alpha }n^2 \right),\label{bb3}
\ed 
Then (\ref{bb2}) may be written in the form
\bd
 \sum\limits_{n=-\infty }^\infty 
    \left( y^2 + n^2\right)^{-1}
=\sum\limits_{n=-\infty }^\infty 
\int\limits_0^\infty 
d\alpha \left( \frac{\pi}{\alpha } \right)^{1\over 2}\exp \left( -\alpha
y^2-\frac{\pi ^2}{\alpha} n^2 \right) \nonumber
\ed
\veb
\bd
={\pi  \over y}+2\sum\limits_{n=1}^\infty  
\int\limits_0^\infty  d\alpha
\left( \frac{\pi}{\alpha } \right)^{1\over 2}\exp \left( -\alpha y^2-\frac{\pi ^2}
{\alpha }n^2 \right).\label{bb4}
\ed 
The right side of  (\ref{bb1}) is
\bd
{\pi  \over y}\coth (\pi y)={\pi  \over y}+{{2\pi } \over {y\left(
{e^{2\pi y}-1}
\right)}}.\label{bb5}
\ed 
Therefore, we get from (\ref{bb4}) and (\ref{bb5}) the following useful equation 
\bd
{1 \over {z\left( {e^z-1} \right)}}={1 \over {2\pi
^2}}\sum\limits_{n=1}^\infty  {\int\limits_0^\infty  {d\alpha \left( {{\pi 
\over \alpha }} 
\right)^{1\over 2}\exp \left( {-\alpha {{z^2} \over {4\pi ^2}}-{{\pi
^2} \over \alpha }n^2}
\right)}}. \label{bb6}
\ed 
Let $z^2=g_a(x^2)$ be a function of
variable $x\in R^3$ with a parameter $a$ of the form $z^2=x^2+a^2$.

After  integration with  respect to  $x$ we get
\bd
\int {{{d^3x} \over {\left( {2\pi } \right)^3}}}\left[ {\sqrt
{x^2+a^2}\left( {\exp \left( {\sqrt {x^2+a^2}} \right)-1} \right)}
\right]^{-1}\\[.15in] \nonumber
\ed
\veb
\bd
={1 \over 2}\sum\limits_{n=1}^\infty 
{\int\limits_0^\infty  {d\alpha \alpha ^{-2}\exp \left( {-\alpha
{{a^2} \over {4\pi ^2}}-{{\pi ^2} \over
\alpha }n^2} \right)}}.\label{bb7}
\ed 
The modified Bessel function is written as
\bd
\int\limits_0^\infty  {d\alpha \cdot \alpha ^{\nu -1}\cdot \exp \left(
{-\gamma
\alpha -{\delta \over \alpha} } \right)}=2\left( {\delta \over \gamma }
\right)^{\nu \over 2}K_\nu \left(2\sqrt {\delta \,\gamma }\right).\label{bb8}
\ed 
and (\ref{bb7}) will be
\bd
\int {{{d^3x} \over {\left( {2\pi } \right)^3}}}\left[ {\sqrt
{x^2+a^2}\left( {\exp \left( {\sqrt {x^2+a^2}} \right)-1} \right)}
\right]^{-1} \nonumber
\ed
\veb
\bd
= {1 \over 2}\left(
\frac{a}{\pi ^2} \right)\sum \limits_{n=1}^\infty  \frac{1}{n}K_1(an).\label{bb9}
\ed 
Scaling $x$ and $(a)$ with a parameter $\beta $ as $(x,a)=\beta
(k,m)$    write (\ref{bb9}) in the form
\bd
\int {{{d^3k} \over {\left( {2\pi } \right)^3}}\,{2 \over {\varepsilon
\left( {\exp
\left( {\beta \varepsilon } \right)-1} \right)}}}={m \over {\beta \pi
^2}}\sum\limits_{n=1}^\infty  \frac{1}{n}K_1(\beta mn),\label{bb10}
\ed 
where $\varepsilon =\sqrt {\vec{k}^2+m^2}$. 

Differentiating the equation  (\ref{bb7}) with respect to parameter $a$ we find 
\bd
\int {{{d^3x} \over {\left( {2\pi } \right)^3}}\left(
{-{\partial 
\over {\partial a^2}}} \right)}\left[ {\sqrt {x^2+a^2}\left( {\exp \left( {\sqrt
{x^2+a^2}}
\right)-1} \right)} \right]^{-1}\hfil \nonumber
\ed
\veb
\bd
\hfil = {1 \over {4\pi ^2}}\sum\limits_{n=1}^\infty 
{K_0}(an),\label{bb11} 
\ed 
or, in new variables,
\bd
\int {{{d^3k} \over {\left( {2\pi } \right)^3}}\left( {{\partial 
\over {\partial m^2}}} \right){1 \over {\varepsilon \left( {\exp \left( {\beta
\varepsilon }
\right)-1} \right)}}}=-{1 \over {4\pi ^2}}\sum\limits_{n=1}^\infty 
{K_0}(\beta mn).\label{bb12}
\ed 
Integrating  (\ref{bb7}) with respect to parameter $(a)$ and
using the equation 
\bd
\int\limits_{a^2}^\infty  {da^2\left[ {\sqrt {x^2+a^2}\left( {\exp
\left( {\sqrt {x^2+a^2}} \right)-1} \right)} \right]^{-1}} \nonumber
\ed
\veb
\bd
=2\ln \left(
{1-\exp\left(-
\sqrt {x^2+a^2}\right)}
\right) \label{bb13}
\ed  
we find with  new variables the following equation 
\bd
-{1 \over \beta }\int {{{d^3k} \over {\left( {2\pi } \right)^3}}}\ln
\left( {1-\exp (-\beta \varepsilon )} \right)={{m^2} \over {2(\beta \pi
)^2}}\sum\limits_{n=1}^\infty  \frac{1}{n^2}K_2(\beta mn).\label{bb14}
\ed 
High temperature asymptotes $(\beta{m}\ll{1})$ of the equations (\ref{bb9}),
 (\ref{bb11}) and (\ref{bb14}) are
\bd
{{2m^2} \over {(\beta \pi )^2}}\sum\limits_{n=1}^\infty  
\frac{1}{n^2}K_2(\beta mn) \nonumber
\ed
\veb
\bd
=\frac{\pi ^2}{90 \beta ^4}-\frac{m^3}{24 \beta^2}
+\frac{m^3}{12 \pi \beta}+\frac{m^4}{64}\left[ \ln \frac{m^2 \beta^2}{16 \pi^2}
-\frac{3}{2}+2\gamma\right] \nonumber
\ed
\veb
\bd
{m \over {4\beta \pi ^2}}\sum\limits_{n=1}^\infty  
\frac{1}{n}K_1 (\beta mn), \label{bb15}
\ed
\veb
\bd
=\frac{1}{24\beta^2}-\frac{m}{8 \pi \beta}-\frac{m^2}{32 \pi^2}
\left[ \ln \frac{m^2 \beta^2}{16 \pi ^2}-1+2\gamma \right] \label{bb16}
\ed
and
\bd
{1 \over {4\pi ^2}}\sum\limits_{n=1}^\infty 
K_0(\beta mn) \nonumber
\ed
\veb
\bd
=\frac{1}{16 \pi m \beta}+\frac{1}{32 \pi^2}
 \left[ \ln \frac{m^2 \beta^2}{16 \pi ^2}-\frac{1}{2}+2\gamma \right] \label{bb17}
\ed

\begin{thebibliography}{Longname}

\bibitem[Abrikosov   et al. 1963]{key13}
 A.A. Abrikosov, L.P. Gorkov, I.E. Dzyaloshinskii\\
{\it "Methods of Quantum Field Theory in Statistical Mechanics"}\\
Prentice-Hall, 1963

\bibitem[Altaie   \& Dowker  1978]{alt1}Altaie M.B. \& Dowker J.S.\\
{\it "Spinor Fields in an Einstein Universe:\\
Finite Temperature Effects"}\\
Physical Review {\bf D18} (1978) 3557-3564

\bibitem[Alvarez-Gaume  et al. 1985]{gcs1} Alvarez-Gaume L., 
Della Piera S., Moore G.\\
{\it "Anomalies and Odd Dimensions "}\\
 Annals of  Physics (N.Y.) {\bf 163} (1985)   288-317


\bibitem[Andreev  1981]{key5} I.V. Andreev\\ 
{\it "Chromodynamics and High Energy Processes"}\\
Moskva, "Nauka" 1981 

\bibitem[Arovas   et al. 1985]{arov1} Arovas D., Schrieffer J.R. \& Wilczek F.\\
{\it "Fractional Statistics and the Quantum Hall Effect"}\\
Physical Review Letters {\bf 53} (1984)   722-723



\bibitem[Babu  et al. 1987]{vcs3} Babu K.S., Ashok Das , Prasanta
Panigrahi\\
 {\it "Derivative axpansion and 
the induced Chern-Simons term \\
at finite temperature in 2+1 dimensions"}\\
Physical Review {\bf D36} (1987)  3725-3730

\bibitem[Bekenstein  1973]{i22}Bekenstein J.D.\\
{\it "Black Holes and Entropy"}\\
Physical Review {\bf D7} (1973)  2333-2346;\\
{\it " Generalized Second Law of Thermodynamics in Black-Hole Physics"}\\
Physical Review {\bf D9} (1973)  3292-3300

\bibitem[Bender et al.1977]{top4} C.Bender, F. Cooper, G. Guralnic\\
{\it"Path Integral Formalulation of Mean-Field Perturbation Theory."}\\
Annals of Physics {\bf 109} (1977) 165-209

\bibitem[ Bernard   1974]{key21} Claude W. Bernard\\ 
{\it "Feynman rules for gauge theories at finite temperature"}\\ 
Physical Review {\bf D9} (1974) 3312-3320

\bibitem[Binder 1979]{top7} K. Binder\\ 
{\it "Monte-CarloMethods in Statistical Physics"}\\
 Springer-Verlag. New-York, 1979

\bibitem[Binegar 1982]{int2}Binegar B.\\
{\it "Relativistic Theories in Three Dimensions"}\\
 Journal of Mathematical  Physics {\bf 23} (1982) 1511-1517


\bibitem[Birrell  \& Davies  1982]{key16} N. D. Birrell, 
P.S.W. Davies\\ {\it "Quantum Fields in Curved Space"}\\
Cambridge University Press,1982

\bibitem[Bjorken  \& Drell  1965]{key6} J.D. Bjorken and S.D.
Drell\\  {\it "Relativistic Quantum Fields"}\\
New York: McGraw-Hill, 1965


\bibitem[Bogoliubov   \& Shirkov   1980]{key7}
 N.N. Bogoliubov and D.V. Shirkov\\ 
{\it "Introduction to the Theory of 
Quantized Fields"}\\  
New York: Wiley \& Sons, 1980


\bibitem[Bonch-Bruevich   \& Tyablikov   1962]{i8}
Bonch-Bruevich V.L. \& Tyablikov S.V.\\
{\it "The Green Function Method in Statistical Mechanics"}\\
North-Holland Publishing Company 1962 

\bibitem[Brown   1992]{key2} Lowell S. Brown\\
 {\it "Quantum Field Theory"}\\ 
Cambridge University Press,1992

\bibitem [Brown  1979]{brown1}Brown L.S.\\ 
{\it "Stress Tensor Anomaly in a Gravitational Metric: Scalar Fields"}\\
Physical Review {\bf D15} (1977)  1469-1483

\bibitem [Bunch   \& Parker   1979]{bunch1}Bunch T.S. \& Parker L.\\ 
{\it "Feynman Propagator in Curved Space-Time:\\
 A Momentum Representation"}\\
Physical Review {\bf D20} (1979)  2499-2510

\bibitem[Chernikov  1962]{i1} Chernikov N.A.\\
{\it "Equilibrium Distribution of Relativistic Gas"}\\
 Preprint , Dubna P-1159, 28 p.

\bibitem [Christensen  1978]{christ1}Christensen S.M.\\ 
{\it "Regularization, Renormalization, and Geodesic Point Separation"}\\
Physical Review {\bf D17} (1978)  946-963


\bibitem[Coleman   \& Weinberg   1974]{top11} Coleman S,  Weinberg E.\\ 
{\it "Radiative Corrections as  the Origion of Spontaneous Symmetry Breaking."}\\ 
Physical Review {\bf D10} (1974) 1888-1910

\bibitem[Collins  1974]{col1} Collins J.C.\\
{\it "Scaling Behavior of $\lambda \phi^4$ \\
Theory and Dimensional Regularization"}\\
Physical Review {\bf D10} (1974)  1213-1218


\bibitem[Denardo   \& Spalucci   1983]{i16} Denardo G. \& Spalucci E.\\
{\it "Finite Temperature Spinor Pregeometry "}\\
Physics Letters {\bf B130} (1983)  43-46

\bibitem[Denardo   \& Spalucci   1983]{i17} Denardo G. \& Spalucci E.\\
{\it "Finite Temperature Scalar Pregeometry "}\\
Nuovo Cimento {\bf A74}  (1983)  450-460

\bibitem [ Deser  et al. 1982]{int6}
Deser S., Jackiw J., Templeton S.\\ 
1) {\it "Topologically massive Gauge Theories"}\\
Annals of  Physics (NY) {\bf 140} (1982) 372.

2) {\it "Three Dimensional Massive Gauge Theories"}\\
Physical Review. Letters {\bf 48} (1982) 975-978


\bibitem [DeWitt   1965]{dewitt1}DeWitt B.S.\\ 
{\it "Dynamical Theory of Group and Fields"}\\
Gordon and Breach; Science Publishers, Inc. (1965) 

\bibitem [DeWitt   1975]{dt2}DeWitt B.S.\\ 
{\it "Quantum Field Theory in Curved Space-Time"}\\
Physics Reports {\bf 19}  (1975)  295-357



\bibitem [Ditrich  1978]{dit1}Ditrich W. \\ 
{\it "Effective Lagrangians at Finite Temperature"}\\
Physical Review {\bf D19} (1978)  1385-1390 

\bibitem[Dolan   \& Jackiw   1974]{i15} Dolan L. \& Jackiw R.\\
{\it "Symmetry Behavior at Finite Temperature "}\\
Physical Review {\bf D9} (1974)  3320-3340

\bibitem[Donoghue  et al. 1984]{don1}Donoghue J.F., Holstein B.R., 
Robinett R.W.\\
{\it "Renormalization of the Energy-Momentum Tensor\\
and the Validity of the Equivalence Principle\\
at Finite Temperature"}\\
Physical Review {\bf 30} (1984)  2561-2572
\bibitem [Dowker   \& Kennedy   1978]{dow1}Dowker J. \& Kennedy G.
\\  {\it "Finite Temperature and Boundary Effects in Static Space-Time"}\\
Journal of Physics {\bf A11} (1978) 895-920

\bibitem [Dowker   \& Critchley   1977]{dow2} Dowker J. \& Critchley
R.\\  {\it "Vacuum Stress Tensor in an Einstein Universe:\\
Finite Temperatuire Effects"}\\
Physical Review {\bf D15} (1977)  1484-1493

 \bibitem[ Faddeev   \& Slavnov   1991]{key1}  FaddeevL.D.\&
Slavnov  A. A.\\  {\it"Gauge Fields: An Introduction 
to Quantum Theory"}\\
Addison-Wesley Publishing Company, 1991

 \bibitem[ Fetter   \& Walecka    1971]{fw1} Fetter A.L. \& Waleska J.D.\\ 
{\it"Quantum Theory of Many-Particle Systems"}\\
McGraw-Hill, 1971


\bibitem[Feynman  1972 ]{key15} R. P. Feynman\\ 
{\it "Statistical Mechanics"}\\
W.A. Benjamin, Inc. Advanced Book Program Reading, Massachusetts 1972

\bibitem[Feynman R.1953]{i5}Feynman R.\\
{\it "Atomic Theory of the Transition in Helium"}\\
Physical Review {\bf 91} (1953) 1291-1301

\bibitem[Feynman   \& Hibbs    1965]{fh1}Feynman R.\& Hibbs \\
{\it " Quantum Mechanics and Path Integrals"}\\
New York: McGraw-Hill,1965

\bibitem[Floratos et al. 1984]{top9} Floratos E., 
 Papantonoulos E., Zoupanos\\
{\it "The Casimir Effect as a source of chiral symmetry breaking in QCD"}\\
Preprint CERN-TH.4020/84

\bibitem[Ford  1980]{ford1}Ford L.H.\\
{\it "Twisted Scalar and Spinor Fields in Minkowski Spacetime"}\\
Physical Review {\bf D21} (1980) 949-957

\bibitem[Fradkin   1965]{i10}
Fradkin E.\\
{\it "Green's function Method in Quantum Field Theory \\
and Quantum Statistics"}\\
Fiz.Inst. Akad. Nauk, Moscow {\bf 29} (1965) 1-138 

\bibitem[Fujimoto  1985]{gcs5} Fujimoto Y.\\
{\it "Thermo Propagators in General Relativity"}\\
Classical and  Quantum Gravity {\bf 3} (1983)  921-926

\bibitem[Fulling   1989]{key17} S.A. Fulling\\
 {\it "Aspects of Quantum Field Theory 
in Curved Space-Time"}\\
Cambridge University Press, 1989 

\bibitem[  Garsia-Sucre 1985]{i3}  Garsia-Sucre M. \\
{\it "Quantum Statistics in a Simple Model of Spacetime"}\\
Journal of Theoretical Physics {\bf 24} (1985) 441-455

\bibitem[Goldstone 1961]{top1} Goldstone \\ 
{\it "Field Theories with "Superconductor" Solutions"}\\
Nuovo Chimento {\bf 19} (1961) 154-164

\bibitem[Gross   \& Neveu   1974]{top10} D.J. Gross, A. Neveu\\
{\it "Dynamical Symmetry Breaking in asymptotically Free  Field Models"}\\
Physical Review {\bf D10} (1974)  3235-3253

\bibitem[Guidry   1991]{guid1} Guidry M. \\ 
{\it "Gauge Field Theories"}\\
John Willey \& Sons, Inc.  1991

\bibitem[Guth  1966]{i12}
Guth A. H.\\
{\it "Inflationary Universe: A possible solution 
to the Horizon \\
and flatness problems"}\\
Physical Review {\bf  D23} (1981) 347-356 

\bibitem[Hale   \& Kocak    1991]{hk1} Hale J. \& Kocak H.\\
{\it "Dynamics and Bifurcations"}\\
Springer-Verlag 1991


\bibitem[Hawking    1975]{i21} Hawking S.W.\\
{\it "Black Holes and Thermodynamics "}\\
Physical Review {\bf D13} (1976)  191-197

\bibitem[Hawking    1977]{haw1} Hawking S.W.\\
{\it "Zeta Function Regularization of Path Integrals in Curved
Space-Time"}\\
Communication of Mathematical Physics {\bf 55} (1977)  133-148

\bibitem[Hu   1981-3]{i19} Hu B.L.\\
{\it "Quantum Fields in Expanding Universe"}\\
Physics Letters {\bf B103} (1981)  331-337;\\
{\it "Finite Temperature Effective Potential for $\phi^4$ in F-R-W Universe"}\\
Physics Letters {\bf B123} (1983)  189-193

\bibitem[Hu   1982]{i20} Hu B.L.\\
{\it "Finite Temperature Quantum Fields in Expanding Universe"}\\
Physics Letters {\bf B108} (1982)  19-29

\bibitem[Hu  1983]{gcs6} Hu B.L.\\
{\it "Finite Temperature Effective Potential for $\lambda \phi^4$\\ 
 Theory in Robertson-Walker Univers"}\\
Physics Letters {\bf B123} (1983) 189-196

\bibitem [Huang 1963]{huang1}Huang K.\\ 
{\it "Statistical Mechanics"}\\
John Wiley \& Sons, Inc. 1963

\bibitem[Huang   1992]{key8} Kerson Huang\\  
{\it "Quarks, Leptons and Gauge Fields"}\\ 
World Scientific, 1992

\bibitem[Hurt   1983]{hurt1} Hurt N.\\  
{\it "Geometric Quantization in Action"}\\
 D. Reidel Publishing Company, 1983

\bibitem[Ignat'ev  1985]{i4}Ignat'ev A.Ju.\\
{\it "Statistical dynamics of ensembles of the elementary particles"}\\
Collection "Gravity and Theory of Relativity", Kazan 1983\\

\bibitem[Ito   \& Yasue   1984]{ito1} Ito I.\& Yasue M.\\
{\it "Masses of Composite Fermions in \\
a Fermion-Boson-Symmetric Preon Model"}\\
Physical Review {\bf D29} (1984)  547-554

\bibitem[Itzykson   \& Zuber  1980]{its1}Itzykson C. \& Zuber J.B.
\\  {\it ""Quantum field theory""}\\  McGraw-Hill Inc. 1980

\bibitem[Ishikawa   1983]{ish1} Ishikawa K.\\
{\it "Gravitational Effect on Effective Potential"}\\
Physical Review {\bf D28} (1983)  2445-2454

\bibitem[Isham   1978]{isham1} Isham C.J.\\
{\it "Twisted Quantum Fields in Qurved Space-Time"}\\
Proc. Roy. Soc. London {\bf A362} (1978)  384-404

\bibitem[Israel   \& Kandrup  1984]{i2}Israel W. \& Kandrup H.\\
{\it "Nonequilibrium Statistical Mechanics in \\
the General Theory of Relativity"}\\
Annals of Physics {\bf 152} (1984)  30-84;\\
Annals of Physics {\bf 153} (1984)  44-102. 

\bibitem[Ivanenko  1959]{top12} Ivanenko D.D.  Preciding:\\
{\it "Nonlinear Quantum Theory of Field"} I.L.1959

\bibitem[Jackiw  1974]{jak1} Jackiw R.\\
{\it "Functional Evaluation of the Effective Potential"}\\
Physical Review {\bf D9} (1974) 1686-1701


\bibitem[Jackiw   Templeton  1981]{int1} Jackiw R. Templeton S.\\
{\it "How Super-renotmalizable Interactions Cure Their
Infrared Divergencies "}\\
Physical Review {\bf D23} (1981) 2291-2304

\bibitem[Kaku   1988]{kaku3}Kaku M.\\ 
{\it "Introduction to Superstrings"}\\ 
Springer-Verlag New York Inc. 1988 


\bibitem[Kapusta   1989]{key9} Joseph.I. Kapusta\\ 
{\it "Finite Temperature Field Theory"}\\ 
Cambridge University Press, 1989 

\bibitem[Kapusta   1979]{kap1} Kapusta J. \\
{\it "Quantum Chromodynamics at High Temperatures"}\\
 Nuclear Physics {\bf B148} (1979)  461-498

\bibitem [Kennedy   et al. 1980]{ken1} Kennedy G.,Critchley R.,
Dowker J. \\  {\it "Finite Temperature Field Theory with Boundaries:\\
Stress Tensor and Surface Action Renormalization"}\\
Annals of Physics (N.Y.) {\bf 125} (1980) 346-400

\bibitem[Kislinger   \& Morley   1976]{kis1}Kislinger M.B. \& Morley
P.D.\\ {\it "Collective Phenomena in Gauge Theories.\\
Renormalization in Finite-Temperature Field Theory"}\\
Physical Review {\bf D13} (1976)  2771-2777

\bibitem[Kawati   \& Miyata  1981]{km1}Kawati S. \& Miyata N.\\ 
{\it "Phase Transition in a Four-Dimensional Fermion model"}\\
Physical Review {\bf D23} (1981)  3010-3024

\bibitem[Kulikov   \& Pronin   1987]{lcs1}Kulikov I.K. \& Pronin
P.I.\\  {\it "Local Quantum Statistics in Arbitrary Curved Space-Time"}\\
Pramana Journal of Physics {\bf 28} (1987)  355-359

\bibitem[Kulikov   \& Pronin   1988]{gcs4} Kulikov I.K. \& Pronin P.I.\\
{\it "Finite Temperature Contributions to \\
the Renormalized Energy-Momentum Tensor for \\
an Arbitrary Curved Space-Time"}\\
Czechoslovak  Journal of Physics {\bf B38} (1988) 121-128

\bibitem[Kulikov   \& Pronin   1989]{top13}Kulikov I.K. \& Pronin
P.I.\\  {\it "Topology and Chiral Symmetry Breaking in Four-Fermion
Interaction"}\\ Acta Physica Polonica {\bf B20} (1989)  713-722 

\bibitem[Kulikov   \& Pronin  1993]{i23} Kulikov I.K. \& Pronin
P.I.\\ {\it "Bose Gas in Gravitational Field"}\\
International Journal of Theoretical Physics {\bf 32} (1993)  1261-1273

\bibitem[Kulikov   \& Pronin  1995]{kul1} Kulikov I.K. \& Pronin
P.I.\\ {\it "Low Temperature Properties of a Quantum Fermi Gas\\
in Curved Space-Time"}\\
International Journal of Theoretical Physics {\bf 34} (1995)  1843-1854

\bibitem[Landau   \&  Lifshitz   1959]{key12}
  Landay L.D. \&   Lifshitz E.M.\\ 
{\it "Statistical Physics"}\\ 
Addison-Wesley Publishing Company, 1959

\bibitem[ Landau  \&  Lifshitz  ]{land2}  Landay L.D.\&  Lifshitz E.M. \\ 
{\it "Theory of Field"} {\bf v.2} 19

\bibitem[Laughlin   1988]{lau1} Laughlin R.B. \\
{\it "The relationship Between High-Temperature Superconductivity \\
and the Fractional Hall Effect"}\\ 
Science {\bf 242} (1988)  11413-11423

\bibitem[Lee   \& Fisher   1989]{lfis1}
Lee   \& Fisher\\
{\it "The Inflationary Universe "}\\
Physical Review Letters {\bf 63} (1989) 903-906

\bibitem[Linde   1984]{i14}
Linde A.D.\\
{\it "The Inflationary Universe "}\\
Reports on Progress in Physics {\bf 47} (1984) 925-986

\bibitem[Martin   \& Schwinger  1959]{i6} Martin P. \& Schwinger J.\\
{\it "Theory of Many-Particles Systems"}\\
Physical Review {\bf 115} (1959) 1342-1373

\bibitem[Matsumoto H.et. al.1983]{mats1}Matsumoto H., 
 Ojima I. Umezawa H.
\\ {\it "Perturbation and Renormalization in Thermo Field Dynamics"}\\
Nuclear Physics {\bf 115} (1959) 348-375


\bibitem[Matsubara  1955]{i7} Matsubara T.\\
{\it "A New Approach to Quantum-Statistical Mechanics"}\\
Progress of Theoretical Physics  {\bf 14} (1955) 351-378

\bibitem[Mattuck  1992]{key11}  Mattuck Richard D.\\ 
{\it "A Guide to Feynman Diagrams in the Many-Body Problem"}\\
Dover Publicatios, Inc. New York, 1992

\bibitem[Misner et al. 1973]{miz1} Misner C.W., Torn K.S., Willer J.A. \\
{\it "Gravitation"}\\
W.H.Freeman and Company 1973

\bibitem[Morley  1978]{mor1} Morley P.D.\\
{\it "Thermodynamical Potential in Quantum Electrodynamics"}\\
Physical Review {\bf D17} (1978)  598-610

\bibitem[Nambu  \& Jona-Lasinio 1961]{top3} Nambu J.,Jona-Lasinio\\ 
{\it "Dynamical Model of Elementary Particles Based on \\
an Analogy with Superconductivity.I."}\\
Physical Review {\bf 122} (1961) 345-358

\bibitem[Nakahara  1992]{nak1}Nakahara M.\\
{\it "Geometry, Topology and Physics"}\\
Institute for Physics Publishing\\Bristol and Philadelphia, 1992


\bibitem[Nakazava  \& Fukuyama   1985]{i18} Nakazava N. \& Fukuyama T.\\
{\it "On energy Momentum Tensor at Finite Temperature\\
in Curved Space-Time"}\\
Nuclear Physics {\bf B252} (1985)  621-634

\bibitem[Ojima  1989]{ojima1} Ojima S.\\
{\it "Derivation of Gauge and Gravitational Induced\\
Chern-Simons Terms in Three Dimensions"}\\
Progress of  Theoretical  Physics  {\bf 81} (1989) 512-522

\bibitem [Panangaden   1981]{panan1}Panangaden P.\\ 
{\it "One-Loop Renormalizaton of Quantum \\
Electrodynamics in Curved 
Space-Time"}\\ Physical Review {\bf D23} (1981)  1735-1746

\bibitem [Parker 1979]{pr1}Parker L.\\ 
{\it "Aspects of Quantum Field Theory in Curved Space-Time:\\
Effective Action and Energy Momentum Tensor"}\\ 
Resent developments in Gravitation: Cartese 1978, ed. by M.Levy and
S.Deser Plenum, New York, pp.219-273 

\bibitem [Peressutti   \& Skagerstam   1982]{per1}Peressutti G. 
\& SkagerstamB.S. \\  {\it "Finite Temperature Effects 
in Quantum Field Theory"}\\
Physics Letters  {\bf 110B} (1982)  406-410


\bibitem [Petrov  1969]{petrov1}Petrov A.Z.\\ 
{\it "Einstein Spaces"}\\
Pergamon, Oxford (1969) 

\bibitem[Popov   1987]{key10} V.N. Popov\\ 
{\it "Functional Integrals and Collective Exitations"}\\
Cambridge University Press,1987

\bibitem [Ramond   1983]{ramond1}Ramond Pierre.\\ 
{\it "Field theory. A modern Primer"}\\
The Benjiamin Cummings Publishing Company, Inc. (1983) 

\bibitem[ Redlich   1984]{vcs1} Redlich A.\\
{\it "Gauge Noninvariance and Parity Nonconservation\\ 
of Three-Dimentional Fermions"}\\
1) Physical Review Letters {\bf 52} (1984)  18-21;

2){\it "Parity Violation and Gauge Noninvariance of \\
the Effective Gauge Field Action in Three Dimensions"}\\
Physical Review {\bf D29} (1984) 2366-2374

\bibitem[Rivers   1990]{rs1}Rivers R.J.\\ 
{\it "Path Integral Methods in Quantum Field Theory"}\\
Cambridge University Press,1990

\bibitem[ Rizzo   1983]{ro1}  Rizzo T.\\
{\it "Grand Unification of Preon Models"}\\
 Physical Review {\bf D28} (1983)  2258-2265

\bibitem[Ryder   1985]{key4} Lewis H. Ryder\\ 
{\it "Quantum Field Theory"}\\
Cambridge University Press,1985

\bibitem [Schwinger  1951]{schw1} Schwinger J.\\ 
{\it "On Gauge Invariance and Vacuum Polarization"}\\
Physical Review {\bf 82} (1951)  664-679

\bibitem[Shyriak  1983]{top8} E. Shyriak\\
 {\it "The role of Instantons 
in Quantum Chromodynamics.\\
I. Physical Vacuum. II. Hadronic Structure.\\
III. Quark-Gluon Plasma."}\\
Nuclear  Physics  {\bf B203} (1983)  93-154

\bibitem[Faddeev   \& Slavnov   1980]{sf1} Faddeev L. D. \& Slavnov A.A.\\ 
{\it "Gauge Fields: Introduction to Quantum Theory"}\\
Benjamin/Commutings

\bibitem[Starobinsky  1982]{i13}
Starobinsky A.A. \\
{\it "Dynamics of Phase Transition in the New Inflationalry\\
 Universe Scenario  and Generation of Perturbations "}\\
Physics Letters {\bf B117} (1982) 175-178 

\bibitem[Sterman   1993]{key3} George Sterman\\ 
{\it "An Introduction to Quantum Field Theory"}\\ 
Cambridge University Press,1993

\bibitem[ Farhi  \& Susskind   1980]{sd1} Farhi E. \&  Susskind L. \\ 
{\it "Technicolour"}\\ 
Physics Report {\bf 74 } (1981), 279-321

\bibitem[Symanzik   1966]{i11}
Symanzik K.\\
{\it "Euclidean Quantum Field Theory. I. Equations for Scalar Model"}\\
Journal of Mathematical Physics {\bf 7} (1966) 510-525 

\bibitem[Tamvakis   \& Guralnik  ]{top5} Tamvakis T. \& Guralnik G.\\ 
{\it"Renormalization of Four-Fermion Thories in a Mean-Field Expension"} 
Preprint BROWN-HET-362,54p.

\bibitem[Taylor 1976 ]{top6}  Teilor J.C.\\ 
{\it "Gauge Theories of Weak Interactions"}\\ 
Cambridge University Press 1976

\bibitem[Terazawa   1980]{ta1} Terazawa H.\\ 
{\it "Subquark Model of Leptons and Quarks"}\\ 
Physical Review {\bf D22} (1980), 184-199

\bibitem[Toms  1980]{toms1} Toms D.\\ 
{\it "Symmetry Breaking and Mass Generation by Space-Time Topology"}\\ 
Physical Review {\bf D21} (1980), 2805-2817

\bibitem['t Hooft  1971]{hooft1} 't Hooft G. \\
{\it"Renormalization of Massless Yang-Mills Fields "}\\
Nuclear Physics {\bf B33} (1971)  173-199
{\it"Renormalisable Lagrangians for Massive Yang-Mills Fields"}\\
Nuclear Physics {\bf B35} (1971)  167-188

\bibitem['t Hooft   \& Veltman   1972]{hv1} 't Hooft G. \& Veltman M. \\
{\it"Regularization and Renormalization of Gauge Fields"}\\
Nuclear Physics {\bf B44} (1972)  189-213

\bibitem[Tolmen 1934]{tol1}Tolmen \\
{\it "Relativity, Thermodynamics and Cosmolology"}\\
Clarendon,Oxford 1934

\bibitem[Utiyama   1956]{key19} R. Utiyama\\  
{\it "Invariant Theories Interpretation of Interaction "}\\
Physical Review {\bf 101} (1956), 1597-1607

\bibitem [Vladimirov   1976]{vl1}
Vladmirov V.S.\\ 
{\it "Equations of Mathematical Physics"}

\bibitem[Vuorio  1986]{gcs3} Vuorio I.\\
{\it "Parity Violation and the Effective Gravitational\\ 
Action in Tree Dimensions"}\\
Physics Letters {\bf B175} (1986) 176-178

\bibitem[Weinberg  1974]{wein1} Weinberg S.\\
{\it "Gauge and Global Symmetries at High Temperature"}\\
Physical Review {\bf D9}  (1974)   3357-3378

\bibitem [Wilczek   1977]{int4} Wilczek F.\\

1) {\it "Quantum Mechanics of Fractional-Spin Particles"}\\
 Physical Review Letters  {\bf 49} (1982) 957-959 

2) {\it "Fractional Statistics and Anyon Superconductivity"}\\
World Scientific Publishing Inc., 1990

\bibitem[Yang  \& Mills   1954]{key18} C. N. Yang and R. L. 
Mills\\  {\it "Conservation of Isotopic Spin and Gauge Invariance "}\\ 
Physical Review {\bf 96} (1954), 191-195

\bibitem[Zinn-Justin   1989]{key14}  Zinn-Justin J.\\ 
{\it "Quantum Field Theoryand Critical Phenomena"}\\
Oxford: Oxford University Press, 1989

\end{thebibliography} \begin{center} 
\vspace*{8mm}{\LARGE \bf PART I} 
\end{center}
\vspace{2mm}
\begin{center} 
{\LARGE\bf{LOCAL QUANTUM STATISTICS}} 
\end{center}
\begin{center} 
{\LARGE\bf{ AND THERMODYNAMICS }} 
\end{center}
\begin{center} 
{\LARGE\bf{IN CURVED SPACE-TIME}} 
\end{center}
\vspace{1mm}
\begin{center} 
{\Large\bf  Introduction } 
\end{center}

\vspace{4mm}
The statistics and thermodynamics of 
quantum systems in gravitational fields have  the  attracted attention of physicists
for a long time. 
The  equilibrium distribution
 of a relativistic quantum 
gas in a gravitational field has been  studied using the kinetic Boltzmann equation,
 and it
was found that the solution of a functional Boltzmann equation is a relativistic Maxwell 
distribution for a certain type of gravitational field \cite{i1}.
Later,   different  variants of 
non-equilibrium statistical mechanics in classical general relativity
as a generalization of the standard statistical equations for 
classical systems  in curved space-time were proposed \cite{i2,i4}.


At the same time field-theoretical methods were developed and 
applied to statistical mechanics  and thermodynamics. 
Since the middle of the 50's,  significant progress was made in   quantum 
 theory  of many body systems, and 
  was connected with the  development of finite 
temperature quantum field methods in statistical physics.
 Feynman \cite{i5} studied the  $\lambda$-transition
in Helium using the partition function in the form of a path  integral
 in  quantum mechanics.
Martin and Schwinger \cite{i6} considered a many-particle system in the context of 
quantum field theory in order
to treat multiparticle system from the quantum field-theoretical point of view.
They described the microscopic behavior 
of a multiparticle system using 
 Green's functions, and  found that the finite temperature Green's  functions are 
 related to intensive macroscopic variables when the energy and number of 
particles are large.
 Matsubara  \cite{i7}  proposed  finite temperature perturbation 
techniques similar to perturbative quantum field theory
and calculated the grand partition function by introducing Green's 
functions in imaginary time formalism.
The works of Feynman, Matsubara, and  Martin \& Schwinger 
created the basis for  understanding the close connection between 
Euclidean field theory and statistical mechanics. 


 In the  60's    convenient methods  using 
finite temperature calculations for interacting thermal systems 
 with a propagator formalism in the perturbative regime were proposed in   
\cite{i8}, \cite{key13}, \cite{i10},\\ 
and \cite{i11}. 
These works stressed the analogy between Euclidean Green functions and
distribution functions in statistical mechanics.
Analogies between statistical physics and  quantum field theory 
at finite temperatures were further strengthened in the study of infinite 
equilibrium systems of scalar and spinor fields \\
 \cite{i15}.  These works  provided an opportunity to construct the  
statistical mechanics and thermodynamics of quantum systems in the language of  
finite temperature field theory, and pointed the way towards a construction of  a 
statistical mechanics  and thermodynamics in curved space-time.   

In the  70's, interest in quantum statistical processes in general relativity 
was aroused 
by  the works of Hawking \cite{i21}, and Bekenstein \cite{i22} on the thermodynamics 
of black holes, where  an intimate connection between 
thermodynamics and  the structure of space-time was  pointed out. 

 Parallel to the research on the thermodynamics of black holes,   quantum effects 
present in the   early Universe  were thoroughly discussed by 
Guth et al.  \cite{i12,i13,i14}. 
From these works  one  may conclude that, for
  early time and high temperatures, 
the quantum statistical properties of matter and strong curvature of space-time 
become significant. 

With the development of a quantum field theory using functional integral methods
 at finite temperature  \cite{wein1,key1}, \cite{key9},\\
 \cite{key10} and of  
a formalism of  a quantum theory of field in curved space-time\\
\cite{dewitt1,key17},
 new possibilities for studing the behaviour 
of thermal systems with non-trivial 
geometry and topology were discussed. This approach was developed by
Denardo et al.  \cite{i16}, \cite{i18}.
 
 Hu  \cite{i19} introduced time dependent temperature  $T(t)=a(t)/T$
for an equilibrium gas of  scalar particles which is described by a conformal 
scalar field in a closed Robertson-Walker (RW) Universe in a different way, by   
 using the equilibrium temperature of a flat space-time $T$. 


The problems  of  a  thermodynamical description of a  non-equilibrium 
thermal quantum gas  and conditions for thermal equilibrium 
in an expanding Universe were studied by Hu \cite{i20}.
However,  in general this problem has not been solved satisfactorily.  
Difficulties in the construction of a thermodynamics of thermal systems in dynamical
 space-time are connected with   such problems as  
the definitions of temperature, energy spectrum, and the vacuum state of a thermal
system.   

In Part I of this  work we extend the results of finite temperature 
field theory in order to construct a   statistical mechanics 
and thermodynamics of
bosonic and fermionic quantum systems  in an external curved space-time. 


The methods developed here are based on using  the language of 
a finite temperature Green's 
function in  an arbitrary curved space time for the
definition of the thermodynamical potentials of thermal quantum systems. 

Green's functions of matter fields in curved space-time are nonlocal objects that are 
 well defined in a small range 
of the space-time manifold \cite{dewitt1}\\
\cite{i18}. 
Thermal properties
of quantum systems may be considered  the properties of the set of 
quasi-equilibrium sub-systems, which are  elements  of the whole quantum system 
\cite{i23}.
This suggestion will lead us  to the definition of  distribution functions of bose and
fermi systems, and  some interesting thermodynamical consequences.

The division  of the entire system into quasi-equilibrium sub-systems may be done 
in a simple way 
using  Riemann normal coordinates \cite{petrov1}. These coordinates permit us
 to rewrite Green's functions in a  momentum space representation, and
to introduce a local temperature in order to write thermodynamical potentials through
these Green's functions. As a result, the  thermodynamical potentials 
will be written as a series expansion in 
 powers of the curvature tensor at a selected point of the curved space-time 
manifold. Therefore,  at any point in space-time the coefficients of the series will 
 change because the curvature will change for all other points on the manifold.   

Therefore, in accordance with our suggestions, densities of thermodynamic 
potentials will be functionals of curvature, and they will be directly connected to the 
temperature Green's functions by a simple relation. The aim of this part will be 
to develop the  mathematical formalism of temperature Green's functions in 
external curved space-time and to use the formalism to construct a  quantum statistics 
and thermodynamics of thermal bosonic and fermionic  ideal quantum systems and a
thermodynamics of a photon gas in an arbitrary curved space-time. 

Part I is organized in the following way:
A short review of statistical mechanics is presented in chapter I.   
In chapter II functional integral methods are  applied to non-relativistic
and relativistic many-body systems in Euclidean space-time. 
The purpose will be 
to generalize  these methods for a  description of  statistical 
systems in curved space-time.
Finite temperature  Green's functions are introduced in chapter III
for computation of  the thermodynamical potentials of quantum gases. 
Finite temperature gauge fields are  studied in chapter IV.
An introduction to  quantum fields in  curved space-time is presented in chapter V.
The bosons and fermions in external gravitational fields are studied 
in chapters VI and VIII.
The thermodynamics of bose  and fermi gases in curved space-time 
is studied in chapters VII and IX.
The thermodynamics of vector bosons is considered in chapter X.
Renormalization problems are considered in chapter XI.
The phenomenon of Bose-Einstein condensation and the  low temperature properties 
of fermi gas are studied in chapters XII and XIII. \begin{center} 
\vspace*{8mm}{\LARGE \bf PART II} 
\end{center}
\vspace{2mm}
\begin{center} 
{\LARGE\bf{INTERACTING FIELDS}} 
\end{center}
\begin{center} 
{\LARGE\bf{AT FINITE TEMPERATURE}} 
\end{center}
\vspace{2mm}
\begin{center} 
{\Large\bf  Introduction } 
\end{center}

\vspace{5mm}
In Part I of this  work we studied ideal quantum systems 
(systems in which particles 
don't interact with each other, but only with external fields) 
at definite temperatures.  We described statistical properties
 and thermodynamical behavior 
of quantum ensembles of these particles in an arbitrary curved space-time.
However  can develop a formalism of  finite temperature field theory for
applications to the systems of interacting particles (bosons or fermions)
and study properties of such systems in external gravitational fields.

Part II is devoted to studying  thermal interacting  quantum systems 
in gravitational fields. 
 In this part we study renormalizability of the finite temperature
self-interacting scalar  $\lambda\varphi^4$ model, and  
consider the phenomenon of non-equality between inertial and
gravitational masses of the  boson in perturbative regime at finite temperature.

In Part II  the following topics are developed:
In chapter XIV  $\lambda\varphi^4$  model at
two-loop perturbative regime is considered. The concepts of renormalizations
and  all necessary counterterms  are described.  
In chapter XV a complete renormalization procedure 
for finite temperature model $\lambda\varphi^4$ in two loop approximation  
of the perturbative scheme is   developed. 
The  Green's function of a boson in a heat bath is computed.

In chapter XVI the  finite temperature 
Hamiltonian of a  boson is constructed.
The  phenomenon  of non-equality between
inertial and gravitational masses of a  boson in non-relativistic aproximation at
high temperature in the heat bath is  described. 

\chapter{TWO-LOOP RENORMALIZATIONS IN}
\centerline{\Large \bf  $\lambda \phi^4$ MODEL}
\vs

We will start with the problem of renormalization procedure for the self-interacting 
scalar model in two-loop approximation of perturbation theory. In our calculations
we will use  the method of counterterms.

Let the Lagrangian of the self-interacting $\lambda\varphi^{4}$ model be
\bd
L={1 \over 2}\left( {\partial \varphi } \right)^2-{1 \over 2}
m_B^2\varphi ^2-{{\lambda _B} \over {4!}}\varphi ^4 \label{new1}
\ed
We may assume that (\ref{new1}) is
\bd
L=L_0+L_I \label{new2} 
\ed
where the Lagrangian 
\bd
L_0={1 \over 2}\left( {\partial \varphi } \right)^2-{1 \over 2}
m_R^2\varphi ^2 \label{new3} 
\ed
describes propagation of free particles,
and the Lagrangian 
\bd
L_I=-{1 \over 2}\delta m^2\varphi ^2-{{\lambda _B} \over {4!}}
\varphi ^4 \label{new4} 
\ed
describes interaction.

Let the coefficient  $\delta m^2$ be the difference of the form 
 \bd
\delta m^2=m_B^2-m_R^2 \nonumber
\ed
and constant $\lambda_B$ is expressed as
\bd
\lambda_B=\mu ^{4-n}(\lambda _R+\delta \lambda ) \label{new5}
\ed
We determine $m_B$ and $m_R$  as bare and renormalizable boson masses and
$\lambda_B$ and $\lambda_R$ as bare and renormalizable constants of interaction.

It is easy to see from (\ref{new3}) and (\ref{new4}) 
that the  free propagator of a  scalar field is

\begin{picture}(5,2.5)
\put(6,1.0){\line(1,0){1}}
\put(4,1.0){\makebox(0,0){$G=i/ (p^2-m_R^2)=$}}
\end{picture}

and its  vertex is 

\begin{picture}(5,2.5)
\put(6,1.0){\line(1,1){0.5}}
\put(6,1.0){\line(1,-1){0.5}}
\put(6,1.0){\circle*{.1}}
\put(6,1.0){\line(-1,1){0.5}}
\put(6,1.0){\line(-1,-1){0.5}}
\put(4,1.0){\makebox(0,0){$-i\lambda _R\mu ^{4-n}=$}}
\end{picture}

Feynman rules for this model have a standard form \cite{its1}.


The Feynman diagrams of the counterterms may be found from the  
Lagrangian of interaction 
(\ref{new4}).  

The  two point counterterm diagram is 

\begin{picture}(8,2.5)
\put(5,1){\makebox(0,0){$-i\delta m^2=$}}
\put(6,1){\line(1,0){1.5}}
\put(6.7,1){\makebox(0,0){$\times$}}
\end{picture}

and the  vertex counterterm is 

\begin{picture}(5,2.5)
\put(6,1.0){\line(1,1){0.5}}
\put(6,1.0){\line(1,-1){0.5}}
\put(6,1.0){\circle{0.2}}
\put(6,1.0){\line(-1,1){0.5}}
\put(6,1.0){\line(-1,-1){0.5}}
\put(4,1.0){\makebox(0,0){$-i\delta \lambda \mu ^{4-n}=$}}
\end{picture}

 Taking into account  these counterterms one   can construct
 contributions of the order $\lambda_B$ to Feynman
 propagator $G$

\begin{picture}(8,3)
\put(1.5,1){\line(1,0){1.5}}
\put(3.4,1){\makebox(0,0){+}}
\put(2.2,0.3){\makebox(0,0){a)}}
\put(4.,1){\line(1,0){1.5}}
\put(4.7,1.5){\circle{1}}
\put(4.7,1){\circle*{.1}}
\put(4.7,0.3){\makebox(0,0){b)}}
\put(6.2,1){\makebox(0,0){+}}
\put(7,1){\line(1,0){1.5}}
\put(7.7,1){\makebox(0,0){$\times$}}
\put(7.7,0.3){\makebox(0,0){c)}}
\end{picture}

Fig. II-1 One loop and counterterm contributions to the self energy of the boson.

Propagator Fig. 1a) has a standard form.  

The self-energy diagram of the first order to $\lambda_B$ for 
Fig. 1-b) may be constructed  from the  above mentioned  graphs  in the form 

\begin{picture}(8,3)
\put(4,1){\makebox(0,0){$Fig.1b~~=~~$}}
\put(5,1.){\line(1,0){1.5}}
\put(5.7,1.5){\circle{1}}
\put(5.7,1.){\circle*{.1}}
\put(7,1){\makebox(0,0){$~~=~~$}}
\end{picture}
\vspace{-5mm}
\begin{eqnarray}
={1 \over 2}{{\left( {-i\lambda _R\mu ^{4-n}} \right)} 
\over {\left( {2\pi } \right)^n}}\int {d^n}k{i \over {k^2-m_R^2}} \nonumber
\ed
\veb
\bd
={1 \over 2}{{\left( {\lambda _R\mu ^{4-n}} \right)}
 \over {\left( {2\pi } \right)^n}}\int {d^n}k{1 \over {k^2-m_R^2}} \label{new6}
\ed
The factor $(1/2)$ is the symmetry factor and $\mu$  is a parameter with dimension of
mass. This parameter is used to absorb the dimension of the coupling constant.

Using the expression \cite{key4} for the  $n$-dimensional integral 
\bd
\int {d^n}k{1 \over {\left( {k^2+2kq-m^2} \right)^\alpha }}
=\left( {-1} \right)^\alpha i\pi ^{{n \over 2}}{{\Gamma 
\left( {\alpha -{n \over 2}} \right)} \over {\Gamma 
\left( \alpha  \right)}}{1 \over {\left( {q^2+m^2} \right)^{\alpha 
-{n \over 2}}}} \label{new7}
\ed
we find

\begin{picture}(8,3)
\put(3,1){\makebox(0,0){$Fig.1b~~=~~$}}
\put(4,1.){\line(1,0){1.5}}
\put(4.7,1.5){\circle{1}}
\put(4.7,1.){\circle*{.1}}
\put(6,1){\makebox(0,0){$~~=~~$}}
\end{picture}
\vspace{-5mm}
\begin{eqnarray}
={{\lambda _R} \over 2}{{\mu ^{4-n}} \over {\left( {2\pi } 
\right)^n}}{{\left( {-i\pi ^{{n \over 2}}} \right)} 
\over {\left( {m^2} \right)^{1-{n \over 2}}}}\Gamma 
\left( {1-{n \over 2}} \right)= \nonumber
\ed
\veb
\bd
={{-i\lambda _R} \over 2}{{m^2} \over {16\pi ^2}}
\left( {{{m^2} \over {4\pi \mu ^2}}} \right)^{{n \over 2}
-1}\Gamma \left( {1-{n \over 2}} \right) \label{new8}
\ed
From the expression for gamma function 
\bd
\Gamma \left( {1-{n \over 2}} \right)={2 \over {n-4}}
+\gamma -1 \label{new9}
\ed
 finally get the divergent part of the self-energy diagram

\begin{picture}(8,3)
\put(3,1){\makebox(0,0){$Fig.1a~~=~~$}}
\put(4,1.){\line(1,0){1.5}}
\put(4.7,1.5){\circle{1}}
\put(4.7,1.){\circle*{.1}}
\put(6,1){\makebox(0,0){$~~=~~$}}
\end{picture}
\vspace{-5mm}
\begin{eqnarray}
=-{{i\lambda _R} \over {16\pi ^2}}{{m^2} \over {n-4}}
+\lambda _R\times finite\;term+O\left( {\lambda _R^2} \right) \label{new10}
\ed
The renormalization of the first order of $\lambda_R$ may be done with the equation
\bd
i\Gamma ^{(2)}=i\left[ {{{p^2-m_R^2} \over i}+(-i\delta m^2)-
{{i\lambda _Rm_R^2} \over {16\pi ^2}}{1 \over {n-4}}+...} \right] \label{new11}
\ed
Let us express term $\delta m^2$  in the form of series \cite{col1}
\bd
\delta m^2=m_R^2\sum\limits_{\nu =1}^\infty  
{\sum\limits_{j=\nu }^\infty  {{{b_{\nu j}\lambda _R^j} \over {(n-4)^\nu }}=}} \nonumber
\ed
\veb
\bd
=m_R^2\left\{ {{{b_{11}\lambda _R} \over {(n-4)}}+{{b_{12}\lambda _R^2} 
\over {(n-4)}}+{{b_{22}\lambda _R^2} \over {(n-4)^2}}+...} \right\} \label{new12}
\ed
where the coefficients $b_{\nu j}$  are the numbers.

In order for vertex $\Gamma ^{(2)}$ in one loop approximation to be finite 
\bd
i\Gamma ^{(2)}=p^2-m_R^2+\left( {\delta m^2+{{\lambda _Rm_R^2} 
\over {16\pi ^2}}{1 \over {n-4}}} \right)=finite\label{new13}
\ed
 assume
\bd
\delta m^2=-{{\lambda _Rm_R^2} \over {16\pi ^2}}{1 \over {n-4}} \label{new14}
\ed
then
\bd
b_{11}=-{1 \over {16\pi ^2}} \label{new15}
\ed
This result  completes the one loop calculations.

Now we will find the vertex corrections for $\Gamma^{(4)}$.

There are four  graphs of the order $\lambda^2_R$:

\vspace{5mm}
\begin{picture}(8,3.2)

\put(0,1.5){\line(1,1){0.5}}
\put(0,1.5){\line(1,-1){0.5}}
\put(0,1.5){\circle{.2}}
\put(0,1.5){\line(-1,1){0.5}}
\put(0,1.5){\line(-1,-1){0.5}}

\put(0,0){\makebox(0,0){a)}}

\put(1.3,1.3){\makebox(0.5,0.5){+}}

\put(3,1.5){\line(-1,1){0.5}}
\put(3,1.5){\line(-1,-1){0.5}}
\put(3,1.5){\circle*{.1}}
\put(3.5,1.5){\circle{1}}
\put(4,1.5){\circle*{.1}}
\put(4,1.5){\line(1,1){0.5}}
\put(4,1.5){\line(1,-1){0.5}}

\put(3.5,0){\makebox(0,0){b)}}
\put(5,1.3){\makebox(0.5,0.5){+}}

\put(6.5,2){\line(-1,1){0.5}}
\put(6.5,1){\line(-1,-1){0.5}}
\put(6.5,2){\circle*{.1}}
\put(6.5,1.5){\circle{1}}
\put(6.5,1){\circle*{.1}}
\put(6.5,2){\line(1,1){0.5}}
\put(6.5,1){\line(1,-1){0.5}}

\put(6.5,0){\makebox(0,0){c)}}
\put(8,1.3){\makebox(0.5,0.5){+}}

\put(9.5,2){\line(-1,1){0.5}}
\put(9.5,1){\line(-1,-1){0.5}}
\put(9.5,2){\circle*{.1}}
\put(9.5,1.5){\circle{1}}
\put(9.5,1){\circle*{.1}}
\put(9.5,2){\line(2,-1){1.5}}
\put(9.5,1){\line(2,1){1.5}}

\put(9.5,0){\makebox(0,0){d)}}

\end{picture}

\vspace*{0.5cm}

\vspace{5mm}
Fig. II-2  Feynman diagrams contributing to the vertex correction $\Gamma^{(4)}$.
 
The vertex counterterm Fig. 2a) was introduced before.

The loop contribution (Fig. 2b)) of the order $\lambda^2_R$   to the vertex function is

\begin{picture}(8,3)
\put(5,1){\line(-1,1){0.5}}
\put(5,1){\line(-1,-1){0.5}}
\put(5,1){\circle*{.1}}
\put(5.5,1){\circle{1}}
\put(6,1){\circle*{.1}}
\put(6,1){\line(1,1){0.5}}
\put(6,1){\line(1,-1){0.5}}
\put(7,1){\makebox(0,0){=}}
\end{picture}
\vspace{-3mm}
\begin{eqnarray}
={1 \over 2}{{\left( {-i\lambda _R\mu ^{4-n}} \right)^2} 
\over {(2\pi )^{2n}}}\int {d^nk{{(i)^2} \over {\left( {k^2-m^2} 
\right)\left[ {\left( {p+k} \right)^2-m^2} \right]}}}= \nonumber
\ed
\veb
\bd
={1 \over 2}{{\lambda _R^2\mu ^{8-n}} \over {(2\pi )^{2n}}}
\int {d^nk\int\limits_0^1 {dx{1 \over {\left[ {\left( {k^2-m^2}
 \right)\left( {1-x} \right)+\left[ {\left( {p+k} \right)^2-m^2} 
\right]x} \right]^2}}}}= \nonumber
\ed
\veb
\bd
={1 \over 2}{{\lambda _R^2\mu ^{8-n}} \over {(2\pi )^{2n}}}
\int\limits_0^1 {dx\int {d^nk{1 \over {\left[ {k^2+2pkx-\left( {m^2-p^2x}
 \right)} \right]^2}}}} \label{new16}
\ed
Taking into account (\ref{new7}) we get

\begin{picture}(8,3)
\put(5,1){\line(-1,1){0.5}}
\put(5,1){\line(-1,-1){0.5}}
\put(5,1){\circle*{.1}}
\put(5.5,1){\circle{1}}
\put(6,1){\circle*{.1}}
\put(6,1){\line(1,1){0.5}}
\put(6,1){\line(1,-1){0.5}}
\put(7,1){\makebox(0,0){=}}
\end{picture}
\vspace{-3mm}
\begin{eqnarray}
={{\lambda _R^2\mu ^{8-n}i\pi ^{{n \over 2}}}
 \over {2(2\pi )^{2n}}}\Gamma \left( {2-{n \over 2}} 
\right)\int\limits_0^1 {dx{1 \over {\left[ {m^2-p^2x(1-x)} 
\right]^{2-{n \over 2}}}}}= \nonumber
\ed
\veb
\bd
=-\frac{i\lambda ^2_R}{16 \pi ^2}\frac{1}{n-4}
+\lambda ^2_R \times finite~~term \label{new17}
\ed
Here we used the representation for $\Gamma \left( {2-{n / 2}} \right)$
 in the form
\bd
\Gamma \left( {2-{n \over 2}} \right)=-{2 \over {n-4}}-\gamma \label{new18} 
\ed
Two other loops have the same divergent contributions,
therefore $\Gamma^{(4)}$ vertex structure can be expressed in the form
\bd
\Gamma ^{\left( 4 \right)}=-i\lambda _R\mu ^{4-n}-{3 \over {16\pi ^2}}
{{\lambda _R^2\mu ^{4-n}} \over {\left( {n-4} \right)}}
-i\delta \lambda \mu ^{4-n}= \nonumber
\ed
\veb
\bd
\Gamma ^{\left( 4 \right)}=-i\lambda _R\mu ^{4-n}-i
\left[ {{3 \over {16\pi ^2}}{{\lambda _R^2\mu ^{4-n}}
 \over {\left( {n-4} \right)}}+\delta \lambda } \right]\mu ^{4-n}
=finite \label{new19}
\ed
From the equation (\ref{new19}) we find, that
\bd
\delta \lambda =-{3 \over {16\pi ^2}}\lambda _R^2 \label{new20}
\ed
Putting
\bd
\lambda _B=\mu ^{4-n}(\lambda _R+\delta \lambda ) \nonumber
\ed
\veb
\bd
=\mu ^{4-n}\left[ {\lambda _R^2+\sum\limits_{\nu =1}^\infty 
 {\sum\limits_{j=\nu }^\infty  {{{a_{\nu j}\lambda _R^j} 
\over {\left( {n-4} \right)^\nu }}}}} \right] \label{new21}
\ed
we find coefficient $a_{12}$:
\bd
a_{12}=-{3 \over {16\pi ^2}} \label{new22}
\ed
Therefore
\bd
\lambda _B=\mu ^{4-n}\left[ {\lambda _R-{3 \over {16}}{{\lambda _R^2} 
\over {(n-4)}}+O\left( {\lambda _R^3} \right)} \right] \label{new23}
\ed
Further we will consider two-loop contributions to two-point vertex $\Gamma^{(2)}$ 

\begin{picture}(8,4)
\put(0,1.){\line(1,0){1.5}}
\put(0.7,1.5){\circle{1}}
\put(0.7,1.){\circle*{.1}}
\put(0.7,2.){\makebox(0,0){$\times$}}
\put(0.7,0){\makebox(0,0){a)}}
\put(1.7,0.9){\makebox(0.5,0.5){+}}
\put(3.,1.){\line(1,0){1.5}}
\put(3.7,1.5){\circle{1}}
\put(3.7,1.){\circle{.2}}
\put(3.7,0){\makebox(0,0){b)}}
\put(5,0.9){\makebox(0.5,0.5){+}}
\put(6,1){\line(1,0){1.5}}
\put(6.9,1.5){\circle{1}}
\put(6.9,1.){\circle*{.1}}
\put(6.9,2.5){\circle{1}}
\put(6.9,2){\circle*{.1}}
\put(6.7,0){\makebox(0,0){c)}}
\put(8,0.9){\makebox(0.3,0.5){+}}
\put(8.5,1.){\line(1,0){1.6}}
\put(9.3,1.){\circle{1}}
\put(8.8,1.){\circle*{.1}}
\put(9.8,1.){\circle*{.1}}
\put(9.3,0){\makebox(0,0){d)}}
\end{picture}

\vspace{5mm}
Fig. II-3  Counterterms and the loop contributions of the order $\lambda^2_R$ 
to the self energy. 

To find counterterms to the order $O(\lambda^3_R)$, 
we will make loop calculation of all these contributions.

Diagram 3(a) gives

\begin{picture}(8,3)
\put(3,1){\makebox(0,0){$Fig.3a~~=~~$}}
\put(4,1.){\line(1,0){1.5}}
\put(4.7,1.5){\circle{1}}
\put(4.7,1.){\circle*{.1}}
\put(6,1){\makebox(0,0){$=$}}
\put(4.7,2){\makebox(0,0){$\times$}}
\end{picture}
\vspace{-5mm}
\begin{eqnarray}
={1 \over 2}{{\left( {-i\lambda _R\mu ^{4-n}} \right)}
 \over {\left( {2\pi } \right)^n}}\left( {-i\delta m^2} \right)\int {d^nk{{(i)^2}
 \over {\left( {k^2-m^2} \right)^2}}=} \nonumber
\ed
\veb
\bd
={1 \over 2}{{\lambda _R\mu ^{4-n}} \over {\left( {2\pi } \right)^n}}
\left( {\delta m^2} \right){{i\pi ^{{n \over 2}}\Gamma 
\left( {2-{n \over 2}} \right)} \over {\left( {m^2} \right)^{2-{n \over 2}}}}= \nonumber
\ed
\veb
\bd
=\frac{\lambda_R}{32 \pi^2}(\delta m^2)\Gamma\left(2-\frac{n}{2}\right)
\left(\frac{4\pi \mu ^2}{m^2}\right) ^{2-\frac{n}{2}} \label{new24}
\ed
Inserting  (\ref{new18}) into (\ref{new24}) we get 

\begin{picture}(8,3)
\put(3,1){\makebox(0,0){$Fig.3a~~=~~$}}
\put(4,1.){\line(1,0){1.5}}
\put(4.7,1.5){\circle{1}}
\put(4.7,1.){\circle*{.1}}
\put(4.7,1.){\circle*{.1}}
\put(4.7,2){\makebox(0,0){$\times$}}
\end{picture}
\vspace{-5mm}
\begin{eqnarray}
={{i\lambda _R} \over {(16\pi ^2)^2}}{{m_R^2} \over {(n-4)^2}}
+{{i\gamma \,\lambda _Rm_R^2} \over {2(16\pi ^2)^2}}{{1}
 \over {(n-4)}}+finite\;terms \label{new25}
\ed
Diagram 3(b) gives

\begin{picture}(8,2.8)
\put(3,1){\makebox(0,0){$Fig.3b~~=~~$}}
\put(4.,1.){\line(1,0){1.5}}
\put(4.7,1.5){\circle{1}}
\put(4.7,1.){\circle{.2}}
\put(6,1){\makebox(0,0){$~~=~~$}}
\end{picture}
\vspace{-5mm}
\begin{eqnarray}
={1 \over 2}{{\left( {-i\delta \lambda \mu ^{4-n}} \right)} 
\over {\left( {2\pi } \right)^n}}\int {d^nk{i \over {k^2-m^2}}=} \nonumber
\ed
\veb
\bd
={1 \over 2}{{\delta \lambda \mu ^{4-n}} \over {\left( {2\pi }
 \right)^n}}\left( -{i\pi ^{{n \over 2}}} \right)
\frac{\Gamma \left( {1-{n \over 2}} \right)} 
{\left( {m^2} \right)^{1-{n \over 2}}}\label{new26}
\ed
Inserting (\ref{new9}) into (\ref{new26}) we find

\begin{picture}(8,3)
\put(3,1){\makebox(0,0){$Fig.3b~~=~~$}}
\put(4.,1.){\line(1,0){1.5}}
\put(4.7,1.5){\circle{1}}
\put(4.7,1.){\circle{.2}}
\put(6,1){\makebox(0,0){$~~=~~$}}
\end{picture}
\vspace{-5mm}
\begin{eqnarray}
={{3i} \over {(16\pi ^2)^2}}{{\lambda _R^2} \over {(n-4)^2}}
+{{3i} \over {2(16\pi ^2)^2}}{{\lambda _R^2} \over {(n-4)}}
\left( {\gamma -1} \right) \label{new27}
\ed
Diagram 3(c) gives:

\begin{picture}(8,4)
\put(3,1){\makebox(0,0){$Fig.3c~~=~~$}}
\put(4,1){\line(1,0){1.5}}
\put(4.9,1.5){\circle{1}}
\put(4.9,1.){\circle*{.1}}
\put(4.9,2.5){\circle{1}}
\put(4.9,2){\circle*{.1}}
\put(6.2,1){\makebox(0,0){$~~=~~$}}
\end{picture}
\vspace{-5mm}
\begin{eqnarray}
={1 \over 4}{{\left( {-i\lambda _R\mu ^{4-n}} \right)^2} 
\over {\left( {2\pi } \right)^{2n}}}
\int {d^nk\int {d^nl{{\left( i \right)^3} \over {\left( {k^2-m^2}
 \right)^2\left( {l^2-m^2} \right)}}}}= \nonumber
\ed
\veb
\bd
={i \over 4}{{\lambda _R^2\mu ^{8-2n}} \over {\left( {2\pi }
 \right)^{2n}}}\int {d^nk{1 \over {\left( {k^2-m^2} \right)^2}}
\int {d^nl{1 \over {\left( {l^2-m^2} \right)}}}}= \nonumber
\ed
\veb
\bd
={i \over 4}{{\lambda _R^2\mu ^{8-2n}} \over {\left( {2\pi } 
\right)^{2n}}}\left\{ {\left( {i\pi ^{{n \over 2}}} \right){{\Gamma 
\left( {2-{n \over 2}} \right)} \over {\left( {m^2} \right)^{2-{n \over 2}}}}}
 \right\}\left\{ {\left( {-i\pi ^{{n \over 2}}} \right){{\Gamma 
\left( {1-{n \over 2}} \right)} \over {\left( {m^2} \right)^{1-{n \over 2}}}}} 
\right\}= \nonumber
\ed
\veb
\bd
={i \over 4}{{\lambda _R^2} \over {(16\pi ^2)^2}}\left( {{{4\pi \mu ^2} 
\over {m^2}}} \right)^{4-n}\Gamma \left( {2-{n \over 2}} \right)\Gamma 
\left( {1-{n \over 2}} \right)= \nonumber
\ed
\veb
\bd
={-{i\lambda _R^2} \over {(16\pi ^2)^2}}{{m_R^2} \over {\left( {n-4} \right)^2}}
-{{i\lambda _R^2} \over {(32\pi ^2)^2}}{{m_R^2} \over {\left( {n-4} \right)}}
\left[ {4\gamma -2} \right]+finite\;terms \label{new28}
\ed
Diagram 3(d) is

\begin{picture}(8,2.8)
\put(3,1){\makebox(0,0){$Fig.3d~~=~~$}}
\put(4.5,1.){\line(1,0){1.6}}
\put(5.3,1.){\circle{1}}
\put(4.8,1.){\circle*{.1}}
\put(5.8,1.){\circle*{.1}}
\put(7,1){\makebox(0,0){$~~=~~$}}
\end{picture}
\vspace{-5mm}
\begin{eqnarray}
={1 \over 6}{{\left( {-i\lambda _R\mu ^{4-n}} \right)^2}
 \over {\left( {2\pi } \right)^{2n}}}\int {d^nk
\int {d^nl{{\left( i \right)^3} \over {\left( {k^2-m^2} \right)
\left( {l^2-m^2} \right)\left[ {\left( {p+k+l} \right)-m^2} \right]}}}}= \nonumber
\ed
\veb
\bd
={{i\lambda _R^2\mu ^{8-2n}} \over {6\left( {2\pi } 
\right)^{2n}}}\int {d^nk\int {d^nl{1 \over {\left( {k^2-m^2} \right)
\left( {l^2-m^2} \right)\left[ {\left( {p+k+l} \right)-m^2} \right]}}}}= \nonumber
\ed
\veb
\bd
={{i\lambda _R^2\mu ^{8-2n}} \over {6\left( {2\pi } \right)^{2n}}}
\pi ^n\Gamma \left( {3-n} \right)\left\{ {-{6 \over {n-4}}(m^2)^{n-3}-{{p^2}
 \over 2}+3m^2} \right\} \label{new29}
\ed
The function $\Gamma(3-n)$  may be written as
\bd
\Gamma \left( {3-n} \right)={1 \over {n-4}}+\gamma -1 \label{new30}
\ed
As the result, the expression for 3(d) will be

\begin{picture}(8,2.8)
\put(3,1){\makebox(0,0){$Fig.3d~~=~~$}}
\put(4.5,1.){\line(1,0){1.6}}
\put(5.3,1.){\circle{1}}
\put(4.8,1.){\circle*{.1}}
\put(5.8,1.){\circle*{.1}}
\put(7,1){\makebox(0,0){$~~=~~$}}
\end{picture}
\vspace{-2mm}
\begin{eqnarray}
=-{{i\lambda _R^2m_R^2} \over {\left( {16\pi ^2} \right)^2}}
\left( {{{m^2} \over {4\pi \mu ^2}}} \right)^{n-4}{1 \over {(n-4)^2}}  \nonumber
\ed
\veb
\bd
-{{i\lambda _R^2} \over {\left( {16\pi ^2} \right)^2}}{1 \over {(n-4)}}
\left\{{ {{p^2} \over {12}}-{{m^2} \over 2}+(\gamma -1)m^2} \right\} \label{new31}
\ed
The complete two loop calculations give us the following
expression for $\Gamma^{(2)}$
\bd
\Gamma^{(2)}={{p^2-m^2} \over i}-i\delta m^2-{{i\lambda _Rm_R^2}
 \over {16\pi ^2(n-4)}}+ \nonumber
\ed
\veb
\bd
+{{3i} \over {(16\pi ^2)^2}}{{\lambda _R^2m_R^2} \over {(n-4)^2}}
+{{3i} \over {2(16\pi ^2)^2}}{{\lambda _R^2m_R^2} \over {(n-4)}}\left( {\gamma -1}
 \right) \nonumber
\ed
\veb
\bd
+{i \over {(16\pi ^2)^2}}{{\lambda _R^2m_R^2} \over {(n-4)^2}}
+{i \over {2(16\pi ^2)^2}}{{\lambda _R^2m_R^2} \over {(n-4)}}\gamma  \nonumber
\ed
\veb
\bd
-{{i\lambda _R^2} \over {(16\pi ^2)^2}}{{m_R^2}
 \over {\left( {n-4} \right)^2}}-{{i\lambda _R^2} 
\over {(32\pi ^2)^2}}{{m_R^2} \over {\left( {n-4} \right)}}
\left[ {4\gamma -2} \right] \nonumber
\ed
\veb
\bd
-{{i\lambda _R^2m_R^2} \over {\left( {16\pi ^2} \right)^2}}
\left( {{{m-R^2} \over {4\pi \mu ^2}}} \right)^{n-4}
{1 \over {(n-4)^2}} \nonumber
\ed
\veb
\bd
-{{i\lambda _R^2} \over {\left( {16\pi ^2} \right)^2}}
{1 \over {(n-4)}}\left\{ {{{p^2} \over {12}}-{{m-R^2} \over 2}
+(\gamma -1)m_R^2} \right\} \label{new32}
\ed
or
\bd
i\Gamma ^{(2)}=p^2-m^2+\delta m^2+{{\lambda _Rm_R^2} \over {16\pi ^2(n-4)}}- \nonumber
\ed
\veb
\bd
-{2 \over {(16\pi ^2)^2}}{{\lambda _R^2m_R^2} \over {(n-4)^2}}
+{1 \over {(16\pi ^2)^2}}{{\lambda _R^2} \over {(n-4)}}
\left[ {{{p^2} \over {12}}-{{m_R^2} \over 2}} \right] \label{new33}
\ed
To make   $\Gamma^{(2)}$ finite in the second order of 
 $\lambda^2_R$, we will put
\bd
\delta m^2=m_R^2\left\{ {{{\lambda _R^{}} \over {(n-4)}}b_{11}
+{{\lambda _R^2} \over {(n-4)}}b_{12}+{{\lambda _R^2} \over {(n-4)^2}}
b_{22}+O(\lambda _R^3)} \right\} \label{new34}
\ed
Then, combining terms in the proper way, we get
\bd
i\Gamma ^{(2)}={{m_R^2} \over {(n-4)}}\left\{ {b_{11}
+{1 \over {16\pi ^2}}} \right\}+ \nonumber
\ed
\veb
\bd
\left[ {1+{1 \over {12(16\pi ^2)^2}}{{\lambda _R^2}
 \over {(n-4)}}} \right]\times \left\{ {p^2-m_R^2
\left[ {1+{1 \over {12(16\pi ^2)^2}}{{\lambda _R^2}
 \over {(n-4)}}} \right]^{-1}\times } \right. \nonumber
\ed
\veb
\bd
\left. {\times \left\{ {1+{{\lambda _R^2} \over {(n-4)^2}}
\left[ {{1 \over {2(16\pi ^2)^2}}-b_{12}} \right]
+{{\lambda _R^2} \over {(n-4)}}\left[ {{2 \over {(16\pi ^2)^2}}-b_{22}} \right]}
 \right\}} \right\} \label{new35}
\ed
It follows from (\ref{new35}) that  coefficient 
 $b_{11}$ is exactly (\ref{new15}).

Two other coefficients 
are found from  the suggestion that $\Gamma^{(2)}$  is analytic at $n=4$.
In the result we get that 
\bd
b_{22}={1 \over {2(16\pi ^2)^2}} \label{new36}
\ed
and $b_{12}$ is the solution of the equation
\bd
{{\lambda _R^2} \over {(n-4)^2}}\left[ {b_{12}
-{5 \over {12(16\pi ^2)^2}}} \right]=0 \label{new37}
\ed
It gives
\bd
b_{12}={5 \over {12(16\pi ^2)^2}} \label{new38}
\ed
After these calculations we will have
\bd
i\Gamma ^{(2)}=\left[ {1+{1 \over {12(16\pi ^2)^2}}{{\lambda _R^2} 
\over {(n-4)}}} \right]\times \left( {p^2-m_R^2} \right) \label{new39}
\ed
or
\bd
\Gamma ^{(2)}=Z\Gamma ^{(2)}_{reg} \label{new40}
\ed
where (wave function) renormalization constant $Z$ is 
\bd
Z=1+{1 \over {12(16\pi ^2)^2}}{{\lambda _R^2} \over {(n-4)}} \label{new41}
\ed
From the calculations  of this section we found that the bare mass $m_B$ and 
coupling constant $\lambda_B$ for the  second order of perturbation theory  are
\bd
m_B^2=m_R^2\left\{ {1+{{\lambda _R} \over {(n-4)}}
\left[ {-{1 \over {16\pi ^2}}+{5 \over {12}}
{{\lambda _R} \over {(16\pi ^2)^2}}} \right]} \right.+ \nonumber
\ed
\veb
\bd
\left. {+{{2\lambda _R^2} \over {(16\pi ^2)^2}}
{1 \over {(n-4)^2}}+O\left( {\lambda _R^3} \right)} \right\}  \label{new42}
\ed
and
\bd
\lambda _B=\mu ^{4-n}\left\{ {\lambda _R-{3 \over {(16\pi ^2)}}
{{\lambda _R^2} \over {(n-4)}}+O\left( {\lambda _R^3} \right)} \right\} \label{new43}
\ed
The expressions (\ref{new42}) and  (\ref{new43}) connect non-renormalizable 
and renormalizable parameters of the model, and the model is renormalized 
in two loop approximation of the perturbative regime.


\chapter{GREEN'S FUNCTION OF  BOSON}
\centerline{\Large \bf IN FINITE TEMPERATURE REGIME }
\vs
	
The aim of this chapter is  to construct
renormalizable Green's function for a boson in a heat bath with a definite 
temperature. For this purpose we will use the  real time representation for the  
finite temperature propagator that will let  us obtain necessary results 
in a natural and elegant way.  

We will repeat  calculations for
the contributions in self energy of the boson in one and 
two loop approximations based on the scheme, developed in the previous chapter.

In contrast to chapter XIV  we will consider that all internal lines of Feynman 
graphs are the  finite temperature propagators of the form
\bd
D(k)=D_0(k)+D_\beta (k)={i \over {k^2-m^2}}
+{{2\pi \delta (k^2-m^2)} \over {e^{\beta |k^0|}-1}}, \label{n1}
\ed
where $\beta^{-1}=T$ is the temperature.

Finite temperature calculations
don't change Feynmann rules for loop calculations
 \cite{mor1}, \cite{key21}.

Fig 1b) gives

\begin{picture}(8,3)
\put(3,1){\makebox(0,0){$Fig.1b~~=~~$}}
\put(4,1.){\line(1,0){1.5}}
\put(4.7,1.5){\circle{1}}
\put(4.7,1.){\circle*{.1}}
\put(4.8,0.5){\makebox(0,0){$(T \neq 0)$}}
\put(6,1){\makebox(0,0){$~~=~~$}}
\end{picture}
\vspace{0.0mm}
\begin{eqnarray}
={1 \over 2}{{\left( {-i\lambda _R\mu ^{4-n}} \right)}
 \over {(2\pi )^n}}\int {d^nk}D(k)= \nonumber
\ed
\veb
\bd
={1 \over 2}{{\left( {-i\lambda _R\mu ^{4-n}} \right)}
 \over {(2\pi )^n}}\int {d^nk}D_0(k)+
{1 \over 2}{{\left( {-i\lambda _R\mu ^{4-n}} \right)} 
\over {(2\pi )^n}}\int {d^nk}D_\beta (k) \nonumber
\ed
or in a more compact form

\begin{picture}(8,3)
\put(3,1){\makebox(0,0){$Fig.1b~~=~~$}}
\put(4,1.){\line(1,0){1.5}}
\put(4.7,1.5){\circle{1}}
\put(4.7,1.){\circle*{.1}}
\put(4.8,0.5){\makebox(0,0){$(T = 0)$}}
\put(6,1){\makebox(0,0){$-$}}
\end{picture}
\vspace{0.0mm}
\begin{eqnarray}
-{1 \over 2}{{\left( {-i\lambda _R\mu ^{4-n}} \right)}
 \over {(2\pi )^n}}\int {d^nk}D_{\beta}(k). \label{n2}
\ed
\vspace{1cm}
The counterterm Fig. 3a) is

\begin{picture}(8,2.3)
\put(3,1){\makebox(0,0){$Fig.3a~~=~~$}}
\put(4,1.){\line(1,0){1.5}}
\put(4.7,1.5){\circle{1}}
\put(4.7,1){\circle*{.1}}
\put(4.7,2){\makebox(0,0){$\times$}}
\put(4.8,0.5){\makebox(0,0){$(T \neq 0)$}}
\put(6,1){\makebox(0,0){$~~=~~$}}
\end{picture}
\vspace{0.0mm}
\begin{eqnarray}
={1 \over 2}{{\left( {-i\lambda _R\mu ^{4-n}} \right)}
 \over {(2\pi )^n}}(-i\delta m^2)\int {d^nk}D_0^2(k)
-\lambda _R\delta m^2\int {{d^4k}\over(2\pi)^4}D_\beta (k)D_0(k), \nonumber
\ed
or

\begin{picture}(8,3)
\put(3,1){\makebox(0,0){$Fig.3a~~=~~$}}
\put(4,1.){\line(1,0){1.5}}
\put(4.7,1.5){\circle{1}}
\put(4.7,1){\circle*{.1}}
\put(4.7,2){\makebox(0,0){$\times$}}
\put(4.8,0.5){\makebox(0,0){$(T = 0)$}}
\put(6,1){\makebox(0,0){$~~-~~$}}
\end{picture}
\vspace{0.0mm}
\begin{eqnarray}
-\lambda _R\delta m^2\int {{d^4k\over(2\pi)^4}}D_\beta (k)D_0(k). \label{n3}
\ed
The contribution of the counterterm Fig. 3b) is

\begin{picture}(8,3)
\put(3,1){\makebox(0,0){$Fig.3b~~=~~$}}
\put(4.,1.){\line(1,0){1.5}}
\put(4.7,1.5){\circle{1}}
\put(4.7,1.){\circle{.2}}
\put(4.8,0.5){\makebox(0,0){$(T \neq 0)$}}
\put(6,1){\makebox(0,0){$~~=~~$}}
\end{picture}
\vspace{0.0mm}
\begin{eqnarray}
={1 \over 2}{{\left( {-i\delta\lambda \mu ^{4-n}} \right)}
 \over {(2\pi )^n}}\int {d^nk}D(k) \nonumber
\ed
\veb
\bd
={1 \over 2}{{\left( {-i\delta \lambda \mu ^{4-n}} \right)} 
\over {(2\pi )^n}}\int {d^nk}(D_0(k)+D_\beta (k)), \nonumber
\ed
and

\begin{picture}(8,3)
\put(3,1){\makebox(0,0){$Fig.3b~~=~~$}}
\put(4.,1.){\line(1,0){1.5}}
\put(4.7,1.5){\circle{1}}
\put(4.7,1.){\circle{.2}}
\put(4.8,0.5){\makebox(0,0){$(T=0)$}}
\put(6,1){\makebox(0,0){$~~-~~$}}
\end{picture}
\vspace{0.0mm}
\begin{eqnarray}
-{\left( {i\delta\lambda } \right)}\int {{d^nk}\over{(2\pi)^n}}D_{\beta}(k).
\ed \label{n4}
\vspace{1cm}
Two loop contribution Fig.  3c) may be written in the form

\begin{picture}(8,3.5)
\put(3,1){\makebox(0,0){$Fig.3c~~=~~$}}
\put(4,1){\line(1,0){1.5}}
\put(4.9,1.5){\circle{1}}
\put(4.9,1.){\circle*{.1}}
\put(4.9,2.5){\circle{1}}
\put(4.9,2){\circle*{.1}}
\put(5,0.5){\makebox(0,0){$(T \neq 0)$}}
\put(6.2,1){\makebox(0,0){$~~=~~$}}
\end{picture}
\vspace{0.0mm}
\begin{eqnarray}
={1 \over 4}{{(-i\lambda _R\mu ^{4-n})^2}
 \over {(2\pi )^{2n}}}\int {d^nk}\int {d^nq}(D_0(k)+D_\beta (k))^2
(D_0(q)+D_\beta (q)) \nonumber
\ed
\veb
\bd
={1 \over 4}{{(-i\lambda _R\mu ^{4-n})^2} 
\over {(2\pi )^{2n}}}\int {d^nk}\int {d^nq\left[ {D_0^2(k)D_0(q)}
 \right.+D_0^2(k)D_\beta (q)+} \nonumber
\ed
\veb
\bd
\left. {+2D_0(k)D_\beta (k)D_0(q)+2D_0(k)D_\beta (k)D_\beta (q)
+D_\beta ^2(k)D_0(q)+D_\beta ^2(k)D_\beta (q)} \right]. \nonumber
\ed
Then

\begin{picture}(8,3.8)
\put(3,1){\makebox(0,0){$Fig.3c~~=~~$}}
\put(4,1){\line(1,0){1.5}}
\put(4.9,1.5){\circle{1}}
\put(4.9,1.){\circle*{.1}}
\put(4.9,2.5){\circle{1}}
\put(4.9,2){\circle*{.1}}
\put(5,0.5){\makebox(0,0){$(T = 0)$}}
\put(6.2,1){\makebox(0,0){$~~-~~$}}
\end{picture}
\vspace{0.0mm}
\begin{eqnarray}
-{{\lambda _R^2} \over 4}\int {{{d^4q} \over {(2\pi )^4}}D_\beta (q)}
\left\{ {{{\mu ^{4-n}} \over {(2\pi )^n}}\int {d^nk}D_0^2(k)} \right\} \nonumber
\ed
\veb
\bd
-{{\lambda _R^2} \over 2}\left\{ {\int {{{d^4q} \over {(2\pi )^4}}
D_0(k)D_\beta (k)}} \right\}\left\{ {{{\mu ^{4-n}} \over {(2\pi )^n}}
\int {d^nq}D_0(q)} \right\} \nonumber
\ed
\veb
\bd
-{{\lambda _R^2} \over 2}\left\{ {\int {{{d^4q} \over {(2\pi )^4}}
D_\beta (q)}} \right\}\left\{ {\int {{{d^4k} \over {(2\pi )^4}}}.
D_0(k)D_\beta (k)} \right\} \label{n5}
\ed
Finally the contribution Fig.  3d) will be

\begin{picture}(8,2.8)
\put(3,1){\makebox(0,0){$Fig.3d~~=~~$}}
\put(4.5,1.){\line(1,0){1.6}}
\put(5.3,1.){\circle{1}}
\put(4.8,1.){\circle*{.1}}
\put(5.8,1.){\circle*{.1}}
\put(7,1){\makebox(0,0){$~~=~~$}}
\put(5.3,0){\makebox(0,0){$(T \neq 0)$}}
\end{picture}
\bd
{1 \over 6}{{(-i\lambda _R\mu ^{4-n})^2}
 \over {(2\pi )^{2n}}}\int {d^nk}D_\beta (k)\times \nonumber
\ed
\veb
\bd
\times\left[ {\int {d^nq}D_\beta (q)D_0(q-p-k)} \right.
+\int {d^nq}D_\beta (q)D_0(k-p-q) \nonumber
\ed
\veb
\bd
+{\int {d^nq}D_\beta (q)D_0(q+p+k)}+\int {d^nq}D_\beta (q)D_0(q-k+p) \nonumber
\ed
\veb
\bd
+\left. {\int {d^nqD_0(q)D_0(q-p-k)+\int {d^nqD_0(q)D_0(q+p+k)}}} \right] \nonumber
\ed
or

\begin{picture}(8,2.6)
\put(3,1){\makebox(0,0){$Fig.3d~~=~~$}}
\put(4.5,1.){\line(1,0){1.6}}
\put(5.3,1.){\circle{1}}
\put(4.8,1.){\circle*{.1}}
\put(5.8,1.){\circle*{.1}}
\put(7,1){\makebox(0,0){$~~+~~$}}
\put(5.3,0){\makebox(0,0){$(T = 0)$}}
\end{picture}
\bd
+{1 \over 2}{{(-i\lambda _R\mu ^{4-n})^2}
 \over {(2\pi )^{2n}}}\int {d^nk}D_\beta (k)\times \nonumber
\ed
\veb
\bd\times\left[ {\int {d^nq}D_\beta (q)D_0(q+p+k)} 
+\int {d^nq}D_0 (q)D_0(k+p+q)\right] \nonumber
\ed
So, we get

\begin{picture}(8,2.8)
\put(3,1){\makebox(0,0){$Fig.3d~~=~~$}}
\put(4.5,1.){\line(1,0){1.6}}
\put(5.3,1.){\circle{1}}
\put(4.8,1.){\circle*{.1}}
\put(5.8,1.){\circle*{.1}}
\put(7,1){\makebox(0,0){$~~-~~$}}
\put(5.3,0){\makebox(0,0){$(T = 0)$}}
\end{picture}
\bd
-{{\lambda _R^2} \over 2}\int {{{d^4k} \over {(2\pi )^4}}}
D_\beta (k)\left\{ {\int {{{d^4q} \over {(2\pi )^4}}}.
D_\beta (q)D_0(k+q+p)} \right\} \nonumber
\ed
\veb
\bd
-{{\lambda _R^2} \over 2}\int {{{d^4k} 
\over {(2\pi )^4}}}D_\beta (k)\left\{ {{{\mu ^{4-n}} 
\over {(2\pi )^n}}\int {d^nq}D_0(q)D_0(k+q+p)} \right\} \label{n6}
\ed
Now we can find counterterms. 

Assuming the sum of the finite temperature 
contributions of the equations (\ref{n3}) and (\ref{n5})  zero,
 \bd
\left\{ {\lambda _R\delta m^2+{{\lambda _R^2} \over 2}
{{\mu ^{4-n}} \over {(2\pi )^n}}\int {d^nq}D_0(q)} \right\}
\int {{{d^4k} \over {(2\pi )^4}}}D_0(k)D_\beta (k)=0. \nonumber
\ed
we find the expression for $\delta m^2$ in the form
\bd
{\delta m^2=-{{\lambda _R} \over 2}{{\mu ^{4-n}}
 \over {(2\pi )^n}}\int {d^nq}D_0(q)}. \label{n7}
\ed
The divergent part of this counterterm will be
\bd
{\delta m_{div}^2=-{{\lambda _R} \over {16\pi ^2}}{{m_R^2}
 \over {(n-4)}}}. \label{n8}
\ed
The following counterterm $\delta\lambda$  may be found by the summation of the 
finite temperature contributions of the equations (\ref{n4}), (\ref{n5}) 
and (\ref{n6}) 
\bd
\int {{d^4k} \over {(2\pi )^4}}D_\beta (k)
\left\{ i\delta \lambda 
+{{\lambda _R^2} \over 2}
\left[ {{\mu ^{n-4}} \over {(2\pi )^n}}
\int {d^nqD_0^2(q)} \right]\right. \nonumber
\ed
\veb
\bd
\left.+\lambda _R^2
\left[ {{\mu ^{n-4}} \over {(2\pi )^n}}
\int {d^nqD_0(q)D_0(k+q+p)} \right] \right\}=0. \nonumber
\ed
For zero external momentum $p$ the divergent part of the  $\delta\lambda$
will be determined by the  divergent part of the integral
\bd
{\delta \lambda ={3 \over 2}(i\lambda _R^2)
\left[ {{{\mu ^{n-4}} \over {(2\pi )^n}}\int {d^nqD_0^2(q)}} \right]} \label{n9}
\ed
and the divergent contribution will be
\bd
{\delta \lambda_{div} =-{{3\lambda _R^2} \over {16\pi ^2}}
{1\over {(n-4)}}}.\label{n10}
\ed
The  temperature counterterms (\ref{n8}) and (\ref{n10}) have the same structure as the 
counterterms which annihilate the divergent parts of zero temperature  loop 
contributions.

It is easy to see that at $T=0$ the sum of the loop contribution 
and the counterterm gives zero:

\begin{picture}(8,3)

\put(3,1.){\line(1,0){1.5}}
\put(3.7,1.5){\circle{1}}
\put(3.7,1.){\circle*{.1}}


\put(5.3,1){\makebox(0,0){+}}
\put(6.,1){\line(1,0){1.5}}
\put(6.7,1){\makebox(0,0){$\times$}}

\put(8,1){\makebox(0,0){$=$}}
\end{picture}
\vspace{-5mm}
\begin{eqnarray}
{={{(-i\lambda _R\mu ^{4-n})} \over {2(2\pi )^n}}
\int {d^nk}D_0(k)-\delta m^2=0}. \nonumber
\ed
It leads us to the equation (\ref{n7}).

Loop contributions and counterterm at $T=0$ in $\Gamma^{(4)}$ (Fig.II-2)
gives the equation
\bd
{{3 \over 2}{{\lambda _R^2\mu ^{4-n}} \over {(2\pi )^n}}
\int {d^nk}D_0(k)D_0(k+p)-i\delta \lambda =0}, \nonumber
\ed
which coincides with (\ref{n9}). 


From the  mentioned above analysis we can conclude that the following loops 
have the same divergent structure:

\begin{picture}(8,4)
\put(0,1.){\line(1,0){1.5}}
\put(0.7,1.5){\circle{1}}
\put(0.7,1.){\circle*{.1}}
\put(2.3,1){\makebox(0,0){+}}
\put(1.7,0.0){\makebox(0,0){$T=0$}}
\put(3.,1){\line(1,0){1.5}}
\put(3.7,1){\makebox(0,0){$\times$}}
\put(5.4,1){\makebox(0,0){$\leftrightarrow$}}
\put(6,1){\line(1,0){1.5}}
\put(6.9,1.5){\circle{1}}
\put(6.9,1.){\circle*{.1}}
\put(6.9,2.5){\circle{1}}
\put(6.9,2){\circle*{.1}}
\put(8,1){\makebox(0,0){+}}
\put(8,0){\makebox(0,0){$T \neq 0$}}
\put(8.5,1.){\line(1,0){1.6}}
\put(9.3,1.){\circle{1}}
\put(8.8,1.){\circle*{.1}}
\put(9.8,1.){\circle*{.1}}
\end{picture}
\vspace*{8mm}

and

\begin{picture}(8,4)
\put(0,1){\line(1,1){0.5}}
\put(0,1){\line(1,-1){0.5}}
\put(0,1){\circle{.2}}
\put(0,1){\line(-1,1){0.5}}
\put(0,1){\line(-1,-1){0.5}}
\put(1,1){\makebox(0,0){+}}
\put(1,0){\makebox(0,0){$T=0$}}
\put(1.4,1){\makebox(0,0){$3 \times$}}
\put(2,1){\line(-1,1){0.5}}
\put(2,1){\line(-1,-1){0.5}}
\put(2,1){\circle*{.1}}
\put(2.5,1){\circle{1}}
\put(3,1){\circle*{.1}}
\put(3,1){\line(1,1){0.5}}
\put(3,1){\line(1,-1){0.5}}
\put(3.7,1){\makebox(0,0){$\leftrightarrow$}}
\put(4.1,1){\line(1,0){1.2}}
\put(4.7,1.5){\circle{1}}
\put(4.7,1.){\circle{.2}}
\put(6,1){\makebox(0,0){+}}
\put(6.3,1){\line(1,0){1.2}}
\put(6.9,1.5){\circle{1}}
\put(6.9,1.){\circle*{.1}}
\put(6.9,2.5){\circle{1}}
\put(6.9,2){\circle*{.1}}
\put(8,1){\makebox(0,0){+}}
\put(7,0){\makebox(0,0){$T \neq 0$}}
\put(8.5,1.){\line(1,0){1.6}}
\put(9.3,1.){\circle{1}}
\put(8.8,1.){\circle*{.1}}
\put(9.8,1.){\circle*{.1}}
\end{picture}

\vspace{10mm}
The Green's function $D^{'}(p)$ of the boson is the object which takes into 
account virtual processes of  creation and  anihilation of the additional 
particles when  this boson  moves through the vacuum. 

The graph composing $D^{'}(p)$ may be divided into two distinct and unique 
classes of 
proper and improper graphs (Fig.II-4)\footnote{The proper graphs cannot be
 divided into two disjoint  parts by the removal of 
a single line, whereas the improper ones can be disjoint \cite{key6}}.

\begin{picture}(8,3)
\put(0,1){\makebox(0,0){$D^{'}(p)=$}}
\put(1,1){\circle{.2}}
\put(1,1){\line(1,0){0.5}}
\put(1.5,1){\circle{.2}}
\put(1.2,0.2){\makebox(0,0){$D(p)$}}
\put(2,1){\makebox(0,0){+}}
\put(2.5,1){\line(1,0){0.5}}
\put(2.7,0.2){\makebox(0,0){$D(p)$}}
\put(2.5,1){\circle{.2}}
\put(3.0,1){\circle{.2}}
\put(3.5,1){\oval(1,0.7)}
\put(4,1){\circle{.2}}
\put(4,1){\line(1,0){0.5}}
\put(4.3,0.2){\makebox(0,0){$D(p)$}}
\put(4.5,1){\circle{.2}}
\put(5,1){\makebox(0,0){+}}
\put(5.5,1){\circle{.2}}
\put(5.5,1){\line(1,0){0.5}}
\put(5.7,0.2){\makebox(0,0){$D(p)$}}
\put(6,1){\circle{.2}}
\put(6.5,1){\oval(1,0.7)}
\put(7,1){\circle{.2}}
\put(7.5,1){\circle{.2}}
\put(7,1){\line(1,0){0.5}}
\put(7.1,0.2){\makebox(0,0){$D(p)$}}
\put(7.8,0.2){\makebox(0,0){$D(p)$}}
\put(7.5,1){\line(1,0){0.5}}
\put(8,1){\circle{.2}}
\put(8.5,1){\oval(1,0.7)}
\put(9,1){\circle{.2}}
\put(9,1){\line(1,0){0.5}}
\put(9.2,0.2){\makebox(0,0){$D(p)$}}
\put(9.5,1){\circle{.2}}
\put(10,1){\makebox(0,0){+}}
\put(10.8,1){\makebox(0,0){$.~~.~~.$}}
\end{picture}

\vspace{1cm}
Fig.II-4  Green's function $D^{'}(p)$ of the boson as  sum of proper self-energy 
insertions.

In  accordance with Fig.II-4 the Green's function $D^{'}(p)$ is obtained by the 
summing  of the series:
\bd
D^{'}(p)=D(p)+D(p)\left( {{{\Sigma (p)} \over i}} \right)D(p) \nonumber
\ed
\veb
\bd
+D(p)\left( {{{\Sigma (p)} \over i}} \right)D(p)D(p)
\left( {{{\Sigma (p)} \over i}} \right)D(p)+... \nonumber
\ed
\veb
\bd
=D(p){1 \over {1+i\Sigma (p)D(p)}}={i \over {p^2-m^2-\Sigma (p)}} \label{n11}
\ed
In this equation  $\Sigma(p)$  is the sum of all two point improper
graphs. All divergent improper graphs and their counterterm graphs (Fig.II-1,II-3)
may be divided into two parts of the first and the second orders with respect 
to $\lambda_{R}$. We will define them as $\Sigma_1$ and $\Sigma_2$ self-energy 
graphs. 

One  can write finite contributions in $\Sigma_2$ in the form

\begin{picture}(8,4)
\put(2,1){\makebox(0,0){$(-i)\Sigma_2=$}}
\put(3.3,1){\line(1,0){1.2}}
\put(3.9,1.5){\circle{1}}
\put(3.9,1.){\circle*{.1}}
\put(3.9,2.5){\circle{1}}
\put(3.9,2){\circle*{.1}}
\put(5,1){\makebox(0,0){+}}
\put(5.5,1.){\line(1,0){1.6}}
\put(6.3,1.){\circle{1}}
\put(5.8,1.){\circle*{.1}}
\put(6.8,1.){\circle*{.1}}
\put(7.8,1){\makebox(0,0){$=$}}
\end{picture}
\vspace{0.0mm}
\begin{eqnarray}
=-{{\lambda _R^2} \over 2}\int {{{d^4q} \over {(2\pi )^4}}}D_\beta (q)
\left\{ {\int {d^4k}D_\beta (k)D_0(k+q+p)} \right\} \nonumber
\ed
\veb
\bd
-{{\lambda _R^2} \over 2}\int {{{d^4q} \over {(2\pi )^4}}}D_\beta (q)
\left\{ {{{\mu ^{4-n}} \over {(2\pi )^n}}
\int {d^4k}D_0(k)D_0(k+q+p)} \right. \nonumber
\ed
\veb
\bd
\left. {-{{\mu ^{4-n}} \over {(2\pi )^n}}
\int {d^4k}D_0(k)D_0(k+p)} \right\}_{p=0} \label{n12}
\ed
Let us introduce functions
\bd
F_\beta ={1 \over 2}\int {{{d^4q} \over {(2\pi )^4}}}D_\beta (q) \label{n13}
\ed
and
\bd
iI_\beta (p)=\int {{{d^4q} \over {(2\pi )^4}}}D_\beta (q)D_0(q+p). \label{n14}
\ed
Then we get 
\bd
\Sigma _2=\lambda _R^2F_\beta I_0(0)+{{\lambda _R^2} \over 2}G_\beta \label{n15} 
\ed
where
\bd
G_\beta =(F_\beta ,(I_\beta +I_0))_{finite} \label{n16}
\ed
The first equation in (\ref{n16}) is a scalar product of the form
\bd
(F_\beta ,I_{\beta})=\int {{{d^4k} 
\over {(2\pi )^4}}}D_\beta (k)\int {{{d^4q} 
\over {(2\pi )^4}}}D_0(q)D_\beta (k+q)  \label{n117}
\ed
and the following one is
\bd
(F_\beta ,I_0)_{finite}=\int {{{d^4k} 
\over {(2\pi )^4}}}D_\beta (k)\left\{\int {{{d^4q} 
\over {(2\pi )^4}}}{D_0(q)D_0 (k+q)}\right\}_{finite}  \label{n18}
\ed
The contribution of the first order in $\Sigma_1$ may 
be found from (\ref{n2}).

This contribution is
\bd
\Sigma _1={{\lambda _R} \over 2}\int {{{d^4q} \over {(2\pi )^4}}}
D_\beta (q)  
={\lambda _R}F_{\beta}.\label{n19}
\ed
The temperature contribution in the boson's mass\footnote{For finite temperature
quantum electrodynamics  the   non-equality of fermionic  masses 
$\delta m_\beta /m \sim\alpha(T/m)^2$  is described by a similar
equation  \cite{per1}} 
will have the following form
\bd
m^2(T)=m_R^2+\Sigma _1+\Sigma _2 \nonumber
\ed
\veb
\bd
=m_R^2+\lambda _{R}F_\beta+\lambda _R^2F_\beta I_\beta (0)
+{{\lambda _R^2} \over 2}\lambda _R^2G_\beta  \label{n20} 
\ed
As the result the Green's function will be
\bd
D^{'}(p)=\frac{i}{p^2-m^2(T)}  \label{n21} 
\ed
Thus we have  computed the  finite temperature Green's function of a  boson 
in two-loop approximation in the form of Feynman propagator with 
finite temperature dependent mass parameter (\ref{n20}). 

\chapter{THERMAL  PROPERTIES OF BOSON}
\vs
	
In chapter XV we showed that the model is renormalizable in each order 
of the  perturbative regime, and found the   finite temperature propagator of a 
 boson in 
a heat bath in two loop approximation.   We also got the expression for the  finite 
temperature mass of  a  boson. These results may help us to get an  effective Hamiltonian 
of the particle and  to study its finite temperature 
behavior in gravitational fields.

\section {Effective Hamiltonian of the boson}
\lum
\hspace{22mm}{\Large \bf   in non-relativistic approximation}
\vsse

After   renormalization   the  pole of boson propagator (\ref{n21})  
 may be written as 
\bd
E=\left[{\vec{p}}^{~2}
+m_R^2+\lambda _{R}F_\beta+\lambda _R^2F_\beta I_\beta (0)
+{{\lambda _R^2} \over 2}\lambda _R^2G_\beta\right]^{1\over2}  \label{n22} 
\ed
We can rewrite the  equation (\ref{n22}) in non-relativistic approximation in
the following form
\bd
E=m_R\left\{ {1+\lambda _Rf(\beta m_R)+o(\lambda _R^2)} \right\}^{{1 \over 2}}
\left[ {1+{{\vec{p}^{~2}} \over {m_R^2\left\{ {1+\lambda _Rf(\beta m_R)
+o(\lambda _R^2)} \right\}}}} \right]^{{1 \over 2}} \nonumber
\ed
\veb
\bd
=m_R+{1 \over 2}\lambda _Rm_Rf(\beta m_R)
+{\vec{p}^{~2}\over{2m_R}}
{\left( 1+{1 \over 2}\lambda _Rf(\beta m_R) \right)}^{-1}
+o(\lambda _R^2)  \label{n23}
\ed
Here the function $f(y)$ is connected with the function $F_\beta (y)$  (\ref{n13}) 
in the following way
\bd
F_\beta(\beta m_R) ={1 \over 2}\int {{{d^3k} \over {(2\pi )^3}}
{1 \over {\varepsilon \left( {e^{\beta \varepsilon }-1} \right)}}}
=m_R^2f(\beta m_R) \nonumber
\ed
The function $f(y)$ has an asymptotic form (for $(y\ll1)$) \cite{i15}:
\bd
f(y)={1 \over {(2\pi)^2}}\int\limits_1^\infty  {dx}
{{\sqrt {x^2-1}} \over {e^{xy}-1}} \nonumber
\ed
\veb
\bd
=\frac{1}{24y^2}
-{1 \over {8\pi y}}+O\left( {y^2\ln y^2} \right),  \label{n24}
\ed
which is very useful for the analysis of the  high temperature
behavior of the model.

\section{Inertial and gravitational masses of a boson}
\vsse

For our following calculations we will consider that the quantum system 
interacts with the gravitational field which is described by the metric
\bd
g_{\mu \nu }=\eta _{\mu \nu }+{\Phi\over2}{\delta_{0\mu}}\delta_{0\nu},  \label{n25}
\ed
where $\Phi$ is a gravitational potential.

The second term in (\ref{n25})
describes a small correction to the Minkowski metric which is connected with the  
presence of  the gravitational field \\
\cite{land2}, \cite{miz1}.

In order to write the  Hamiltonian of the boson in the presence of gravitational 
field one can consider that temperature $T$ changes according to 
Tolmen's law \cite{tol1}.
\bd
T={{T_0} \over {1+\Phi }},  \label{n26}
\ed
where $T_0$ is the temperature with $\Phi=0$.

The finite temperature Hamiltonian with precision to the first leading 
term of the series (\ref{n24}) will be
\bd
H={{\vec{p}^{~2}} \over 2}\left( {m_R+{\lambda_{R} \over {48}}
{{T_0^2} \over {(1+\Phi )^2m_R}}}
 \right)^{-1}+m_R+{\lambda_{R} \over {48}}{{T_0^2}
 \over {m_R(1+\Phi )^2}}+m_R\Phi +...  \label{n27}
\ed  
The last term of the equation (\ref{n27}) describes energy of boson's interaction  
with gravitational field.

Let us rewrite the Hamiltonian (\ref{n27}) as
\bd
H={{\vec{p}^{~2}} \over 2}\left( {m_R+{{\lambda _R} \over {48}}{{T_0^2} 
\over {(1+\Phi )^2m_R}}} \right)^{-1}
+\left( {m_R-{{\lambda _R} \over {24}}{{T_0^2} \over {m_R}}} \right)\Phi
 +...  \label{n28}
\ed
The acceleration of the boson in a gravitational field  may be found from
the quantum mechanical  relation
\bd
\vec{a}=-\left[ {H,\left[ H,{\vec{r}} \right]} \right]=
-\left( {1-{{\lambda _R} \over {12}}{{T_0^2} 
\over {m_R^2}}} \right)\nabla \Phi,  \label{n29} 
\ed
and the mass ratio  will be
\bd
{{m_g} \over {m_i}}=1-{{\lambda _R} \over {12}}{{T_0^2} \over {m_R^2}}  \label{n30}
\ed
so the  inertial and gravitational masses
of the boson in the heat bath are seen to be unequal. 

One can estimate the value of $\delta m_g/m_i$ for some gravitational source. 
Let the source of gravitational field be the Sun ( $1.989\times10^{30} kg$)
then the relation (\ref{n30}) for the combined boson (Cooper pair with 
mass $m_b=1Mev$) 
in the heat bath with temperature $300 K$  gives the following corrections for 
non-equality between masses
\bd
\frac{\delta m_g}{m_i}=\frac{\lambda _R}{12}\frac{T_0^2}{m_R^2}
\sim \lambda  \times 10^{-17} \label{n31}
\ed
or
\bd
10^{-21}< \frac{\delta m_g}{m_i} <10^{-17} \label{n32}
\ed
for the range of the coupling constant $10^{-3}<\lambda <10^{-2}$.

From the analysis we made in this chapter
one may conclude that thermal  interaction of the bosons
in a gravitational field causes  non-equality between inertial 
and gravitational masses. Non-thermal systems do not demonstrate such properties.

The calculations for non-equality between inertial and gravitational mass of electron
were made for thermal quantum electrodynamics by Donoghue\\
 \cite{don1}.
His  result for massive fermions has the same functional structure as 
the equation (\ref{n30}).







 \begin{center} 
\vspace*{8mm}{\LARGE \bf PART IV} 
\end{center}
\vspace{2mm}
\begin{center} 
{\LARGE\bf{TOPOLOGICALLY MASSIVE}} 
\end{center}
\begin{center} 
{\LARGE\bf{GAUGE THEORIES}} 
\end{center}

\vspace{2mm}
\begin{center} 
{\Large\bf  Introduction } 
\end{center}

\vspace{5mm}

3-D non-Abelian gauge models describing   systems of point particles carrying 
  non-Abelian charge   have been under investigation for over 
two decades. These models (so called topologically massive models)
 have a number of interesting features:

1) For vector fields, these models possess  single, parity-violating, massive, 
spin 1 excitations, in contrast to  single, massless, spin 0 exitations 
in the Maxwell  theory, and to a pair of spin 1 degrees
 of freedom in gauge non-invariant  models with a mass \\
\cite{int2} 

2) For second-rank tensor fields, describing gravity, the topological 
model leads to a single, parity-violating, spin 2 particle, whereas a 
conventional  (gauge non-invariant) mass term gives rise to a spin 2 doublet.
Furthermore, the topological term is of third-derivative order, yet the 
single propagating mode is governed by the Klein-Gordon equation.
 Einstein gravity, which is trivial and without propagation in three dimensions,
becomes a dynamical theory with  propagating particles.

3) Particles interacting via the Abelian Cherm-Simons term (CS-term)
acquire anomalous spin and fractional statistics.  They are
 called anyons \cite{int4}.
 Anyons  play a role in the fractional quantum Hall effect \cite{arov1},
[Lee \& Fisher 1989] and perhaps also in high temperature superconductivity
[Lauglin 1988].

This  is not a complete list of interesting properties of gauge theories 
in an odd number of dimensions.
An  important part of these models is the   Chern-Simons action (CS-term).  
Some interesting aspects of quantum field models 
arising from the topology of odd-dimensional manifolds are  discussed in
chapter XIX. The origin of the  CS-term of vector type that is induced  
by   gauge interaction  of 3-D fermions is studied in chapter XX. 
In this chapter  
the influence of topology and temperature effects are also considered.
The origin of the  induced gravitational CS-term at finite temperature 
is considered in chapter XXI.
   
\chapter{INTRODUCTION}
\centerline{\Large \bf TO  TOPOLOGICAL FIELD MODELS }
\vs

As an  introduction to odd dimensional topological field models 
let us consider  their origin and topological significance.

\mad
{\large \it Topological aspect of the model}
\mad

From gauge-invariant fields in even dimensions we may construct  
 gauge invariant 
Pontryagin densities:
\bd
 P_{(2)}=-(1/2 \pi )\epsilon ^{\mu \nu}{^*F}^{\mu \nu} \nonumber
\ed
\veb
\bd 
 P_{(4)}=-(1/16 \pi ^2) \mbox{tr} ^*F^{\mu \nu}F_{\mu \nu}\label{int1}
\ed
whose integrals over the even dimensional space are    invariants
that measure the topological content of the model. 

These gauge invariant objects can also be written  as total derivatives of gauge
invariant quantities
\bd 
P_n=\partial_\mu X^\mu _n \label{int2}
\ed
The two and four-dimensional expressions are 
\bd
X^\mu _2=(1/2\pi)\epsilon^{\mu \nu}A_\nu \nonumber
\ed
\veb
\bd 
X^\mu _4=(1/2\pi)\epsilon^{\mu \alpha \nu \beta}
\mbox{tr} (A_\alpha F_{\beta \gamma}-(2/3)A_\alpha A_\beta A_\gamma)  \label{int3}
\ed
The Chern-Simons (CS) secondary characteristic class is obtained by integrating one 
component of $X^ \mu _n$ over the $(n-1)$ dimensional space which does 
not include that component.

Therefore the  3-D action $S_{CS}$ is proportional to
\bd 
S_{CS} \sim \int dx^0dx^1dx^2 X^ 3 _4 \label{int4}
\ed 
A  topological massive  term (we will name it CS term) can be added  to the fundamental action for 
a gauge fields, but unlike the ways in which
gauge fields are usually given a mass, no gauge symmetry is broken
by its introduction.

\mad
{\large \it Quantum aspects of 3-D field theory}
\mad

Let us consider  non-Abelian quantum model with topological
mass term. The  Lagrangian of this model is
\bd 
L=L_0+L_{CS} +L_{gauge}  \label{int5}
\ed
where $L_0$ is the usual action for non-Abelian gauge field
\bd 
L_0=-(1/2)\mbox{tr}(F_{\mu \nu}F^{\mu \nu})  \label{int6} 
\ed
with
\bd 
F_{\mu \nu}=\partial_\mu A_\nu-\partial_\nu A_\mu +g[A_\mu, A_\nu] \label{int7}
\ed
$L_{CS}$ is CS term
\bd 
L_{CS}=-im\epsilon ^{\mu \nu \rho}
\mbox{tr} (A_\mu \partial_\nu A_\rho-(2/3)A_\mu A_\nu A_\rho)  \label{int8}
\ed
and $L_{gauge}$ includes the gauge-fixing term 
\bd 
L_{gauge}=
(\partial_\mu\bar{\eta}^a)(\partial^\mu \eta^a)
+gf_{abc}(\partial_\mu\bar{\eta}^a)D^\mu \eta  \label{int9}
\ed
We introduce  $SU(N)$ gauge group here with matrix notation:
$A_\mu=A^a_\mu\tau^a$,  where $\tau^a$ are anti-Hermitian 
matrices in the fundamental representation:
\bd 
[\tau^a, \tau^b]=f^{abc}\tau^c,~~~ \mbox{tr} (\tau^a \tau^b)
=-(1/2)\delta^{ab} \label{int10}
\ed
and  $f^{abc}$ are the structure constants of $SU(N)$.

The  theory is defined in three space-time dimensions with Euclidean signature 
$(+~+~+)$.
The coupling of the CS term is imaginary in Euclidean space-time and real 
in Minkowski space-time.

\mad
{\large \it Properties of the model}:
\mad

For an odd number of dimensions, the operation of parity, P, can be defined
 as reflection in all axes:
\bd 
x^\mu  \mathop{\to}\limits_{P}-x^\mu ,
~~~A_\mu  \mathop{\to}\limits_{P} -A_\mu \label{int11}
\ed
The usual gauge Lagrangian is even under parity,
\bd 
L_0 \mathop{\to}\limits_{P}+L_0 \label{int12}
\ed
but the  CS term is odd
\bd 
L_{CS} \mathop{\to}\limits_{P}-L_{CS} \label{int13}
\ed
Under gauge transformation 
\bd 
 A_\mu \to \Omega^{-1}\left\{(1/g)\partial_\mu +A_\mu\right\}\Omega \label{int14}
\ed
The  Lagrangian $L_0$ is invariant but $L_{CS}$ is not:
\bd
\int d^3x L_{CS} \to \int d^3x L_{CS} \nonumber
\ed
\veb
\bd 
+(im/g)\int d^3x \epsilon^{\mu \nu \rho}\partial_\mu 
\mbox{tr} [(\partial_\nu\Omega)\Omega^{-1}A_\rho] +8\pi^2 (m/g^2)i\omega \label{int15}
\ed
where:
\bd 
\omega=(1/24)\int d^3x \epsilon^{\mu \nu \rho}
 \mbox{tr}[\Omega^{-1}(\partial_\mu\Omega)\Omega^{-1}
(\partial_\nu\Omega)\Omega^{-1}(\partial_\rho\Omega)] \label{int16}
\ed
The set of gauge transformations is divided into global gauge rotations,
$\partial_\mu \Omega=0$, and all others, for which we assume that 
$\Omega (x) \to 1$ as $x^\mu \to \infty$. Integrating  over global 
gauge rotations requires the system to have a total color 
charge equal to zero. In this case, $A_\mu (x)$ falls off faster than $1/|x|$  
as $x^\mu \to \infty$, and the second term on the right-hand  side
of (\ref{int15}), which is a  surface integral, vanishes. 

The last term in (\ref{int15}) does not vanish in general.
The $\omega$ of (\ref{int16}) is a winding number, which labels 
the homotopy class of $\Omega (x)$ \cite{nak1} 

Changing variables $A \to A^U$, where $A^U$ is a gauge transformation  of $A$, 
implies that the vacuum average of the value $\hat{Q}$ is 
\bd 
<Q>=\exp |i8\pi^2 (m/g^2)\omega(U)|<Q> \label{int17}
\ed
This invariance gives us a quantization condition for the dimensionless ratio
\bd 
4\pi^2 (m/g^2)=n,~~~n=0,\pm 1,\pm 2,   \label{int18}
\ed
Therefore, for the theory to be invariant under certain large gauge 
transformations    (for a non-Abelian gauge group), 
which are not continiously deformable to the identity, 
the ratio of the CS mass $m$ 
and the gauge coupling $g^2$ must be quantized \cite{int6}.

Further we can study the problem of massive excitations.

For spinor electrodynamics in three dimensions we have
\bd 
L=L_g+L_f +L_{int}  \label{int19}
\ed
where
\bd 
L_g=-(1/4)F^{\mu \nu}F_{\mu \nu}+
(\mu/4)\epsilon ^{\mu \nu }F_{\mu \nu}A_\alpha, ~~
F_{\mu \nu}=\partial_\mu A_\nu-\partial_\nu A_\mu \label{int20}
\ed
\veb
\bd 
L_f=i\bar{\psi}\hat{\partial}\psi-m\bar{\psi}\psi \label{int21}
\ed
\veb
\bd 
L_{int}=-J^\mu A_\mu,~~ J^\mu=-e\bar{\psi}\gamma^\mu \psi \label{int22}
\ed
The coupling constant $e$ has dimension $(mass)^{-1/2}$.

The equations of motion will be
\bd 
\partial_\mu F^{\mu \nu}+
(\mu /4)\epsilon ^{\nu \mu \alpha}F_{\mu \alpha }=J^\mu \label{int23}
\ed
\veb
\bd 
(i\bar{\partial}+e\bar{A}-m) \psi =0\label{int24}
\ed
One can introduce the dual field strength tensor in 3-D space-time 
\bd 
^*F^\mu=(1/2)\epsilon ^{\mu \alpha \beta}F_{\alpha \beta}~~
F^{\alpha \beta}={\epsilon ^{ \alpha \beta \mu}}{^*}F_\mu \label{int25}
\ed
The  Bianchi identity follows from (\ref{int23}): 
\bd 
{\partial_\mu}{^*F^\mu}=0 \label{int26}
\ed
and the equation (\ref{int23}) may be written in a dual form
\bd 
{\partial_\alpha}{^*F_\beta}-{\partial_\beta}{^*F_\alpha}-
\mu F_{\alpha \beta}=-\epsilon _{\alpha \beta \mu}J^\mu \label{int27}
\ed
or 
\bd 
(\Box+ \mu^2){^*F^\mu}=\mu\left(\eta^{\mu \nu}
-\epsilon^{\mu \nu \alpha} \frac{\partial_\alpha}{\mu}\right)J_\nu \label{int28}
\ed
This equation demonstrates that the gauge excitations are massive.

\mad
{\large \it 3-D gravity. Connection with topology}
\mad

The results of the topological part of the introduction gives us a hint how to 
construct the  topological term for three dimensional gravity 
from a four-dimensional  $^*RR$ Pontryagin density.
\bd 
^*RR=(1/2)\epsilon^{\mu \nu \alpha \beta}R_{\mu \nu \rho \sigma}
{R_{\alpha \beta}}^{\rho \sigma}=\partial_\mu X^\mu \label{int29}
\ed 
Let us find $X^\mu$ from (\ref{int29}). To do this we will rewrite $^*RR$
in the following way
\bd
^*RR=(1/2)\epsilon^{\mu \nu \alpha \beta}R_{\mu \nu \rho \sigma}
{R_{\alpha \beta}}^{\rho \sigma}
=(1/2)\epsilon^{\mu \nu \alpha \beta}R_{\mu \nu ab}
{R_{\alpha \beta}}^{ab} \nonumber
\ed
\veb
\bd
=(1/2)\epsilon^{\mu \nu \alpha \beta}R_{\mu \nu ab}
\left\{
\partial_\alpha {\omega_\beta}^{ab}-\partial_\beta {\omega_\alpha}^{ab}
+{\omega_\alpha}^{ac}{\omega_{\beta c}}^{b}
-{\omega _\beta }^{ac}{\omega _{\alpha c}}^{b}
\right\} \nonumber
\ed
\veb
\bd
=\epsilon^{\mu \nu \alpha \beta}R_{\mu \nu ab}
\left\{
\partial_\alpha {\omega_\beta}^{ab}
+{\omega_\alpha}^{ac}{\omega_{\beta c}}^{b}
\right\} \label{int30}
\ed
In our calculations we used the following expression for the Ricci connection
\bd
R_{\mu \nu \rho \sigma}
=\partial_\alpha {\omega_{\beta a b}}-\partial_\beta {\omega_{\alpha a b}}
+{\omega_{\alpha a}}^{c}{\omega_{\beta c}}_{b}
-{\omega _{\beta a}}^{c}{\omega _{\alpha c}}_{b} \label{int31}
\ed
where $\omega_{\mu ab}$ is  3-D spin connection. 

Then the expression for Pontryagin density will be
\bd 
^*RR=\epsilon^{\mu \nu \alpha \beta}R_{\mu \nu ab} 
~\partial_\alpha {\omega_\beta}^{ab}
+\epsilon^{\mu \nu \alpha \beta}R_{\mu \nu ab}
~ {\omega_\alpha}^{ac}{\omega_{\beta c}}^{b} \label{int32}
\ed 
Let us find these two contributions separately.

The first contribution  to (\ref{int32}) gives
\bd
\epsilon^{\mu \nu \alpha \beta}R_{\mu \nu ab} 
~\partial_\alpha {\omega_\beta}^{ab}=
\epsilon^{\mu \nu \alpha \beta}\partial_\alpha (R_{\mu \nu ab} 
{\omega_\beta}^{ab})-
\epsilon^{\mu \nu \alpha \beta}
{\omega_\beta}^{ab}\partial_\alpha R_{\mu \nu ab} \nonumber
\ed
\veb
\bd
=\epsilon^{\mu \nu \alpha \beta}\partial_\alpha (R_{\mu \nu ab} 
{\omega_\beta}^{ab}) \nonumber
\ed
\veb
\bd 
-\epsilon^{\mu \nu \alpha \beta}
{\omega_\beta}^{ab}\partial_\alpha 
\left\{
\partial_\mu \omega_{\nu ab}-\partial_\nu \omega_{\mu ab}
+\omega_{\mu a}^c \omega_{\nu cb}\omega_{\nu a}^c \omega_{\mu cb}
\right\} \label{int33}
\ed 
\veb
\bd 
=\epsilon^{\mu \nu \alpha \beta}\partial_\alpha (R_{\mu \nu ab} 
{\omega_\beta}^{ab})-2\epsilon^{\mu \nu \alpha \beta}
{\omega_\beta}^{ab}\partial_\alpha 
\left\{
\omega_{\mu a}^c \omega_{\nu cb}
\right\} \label{int34}
\ed 
The second contribution to (\ref{int32}) will be
\bd
\epsilon^{\mu \nu \alpha \beta}R_{\mu \nu ab}
~ {\omega_\alpha}^{ac}{\omega_{\beta c}}^{b}= \nonumber
\ed
\veb
\bd
-\epsilon^{\mu \nu \alpha \beta}
 \left\{
\partial_\mu \omega_{\nu ab}-\partial_\nu \omega_{\mu ab}
+\omega_{\mu a}^c \omega_{\nu cb}-\omega_{\nu a}^c \omega_{\mu cb}
\right\}
{\omega_\alpha}^{ac}{\omega_{\beta c}}^{b} \nonumber
\ed
\veb
\bd 
=2\epsilon^{\mu \nu \alpha \beta}
\partial_\mu \omega_{\nu ab}
{\omega_\alpha}^{ac}{\omega_{\beta c}}^{b}
+2\epsilon^{\mu \nu \alpha \beta}\omega_{\mu a}^c \omega_{\nu cb}
{\omega_\alpha}^{ac}{\omega_{\beta c}}^{b} \label{int35}
\ed 
On the other hand the second one  to (\ref{int35}) is zero, then we have
\bd
^*RR=\epsilon^{\mu \nu \alpha \beta}\partial_\alpha (R_{\mu \nu ab} 
{\omega_\beta}^{ab}) \nonumber
\ed
\veb
\bd 
-2\epsilon^{\mu \nu \alpha \beta}
{\omega_\beta}^{ab}\partial_\alpha 
\left\{
\omega_{\mu a}^c \omega_{\nu cb}
\right\}+2\epsilon^{\mu \nu \alpha \beta}
\partial_\mu \omega_{\nu ab}
{\omega_\alpha}^{ac}{\omega_{\beta c}}^{b} \label{int36}
\ed 
One can find that 
\bd
-2\epsilon^{\mu \nu \alpha \beta}
{\omega_\beta}^{ab}\partial_\alpha 
\left\{
\omega_{\mu a}^c \omega_{\nu cb}
\right\}
+2\epsilon^{\mu \nu \alpha \beta}
\partial_\mu \omega_{\nu ab}
{\omega_\alpha}^{ac}{\omega_{\beta c}}^{b}\ \nonumber
\ed
\veb
\bd 
=2\epsilon^{\mu \nu \alpha \beta}
\partial_\mu \omega_{\nu a}^b
{\omega_\alpha b}^{c}{\omega_{\beta c}}^{a} \label{int37}
\ed 
and
\bd 
2\epsilon^{\mu \nu \alpha \beta}
\partial_\mu \omega_{\nu a}^b
{\omega_\alpha b}^{c}{\omega_{\beta c}}^{a}
=(2/3)\epsilon^{\mu \nu \alpha \beta}
\partial_\mu 
\left\{
\omega_{\nu a}^b
{\omega_\alpha b}^{c}{\omega_{\beta c}}^{a}
\right\}  \label{int38}
\ed 
In the result we will have
\bd 
^*RR=\partial_\mu \epsilon^{\mu \nu \alpha \beta}
\left\{
{\omega_\nu}^{ab}R_{\alpha \beta ab}+
(2/3)\omega_{\nu a}^b
{\omega_{\alpha b}}^{c}{\omega_{\beta c}}^{a}
\right\}  \label{int39}
\ed 
Therefore
\bd 
X^\mu=\epsilon^{\mu \nu \alpha \beta}
\left\{
{\omega_\nu}^{ab}R_{\alpha \beta ab}+
(2/3){\omega_{\nu a}}^b
{\omega_{\alpha b}}^{c}{\omega_{\beta c}}^{a}
\right\}  \label{int40}
\ed 
Let parameter $\mu$ be equal to zero, 
then determining  $\epsilon ^{0 \nu \alpha \beta}=\epsilon ^{\nu \alpha \beta}$
with $\nu, \alpha, \beta=1,2,3$ we get the  CS action in the form 
\bd 
S_{CS}\sim \int d^3x X^0=\int d^3x\epsilon^{\nu \alpha \beta}
\left\{
{\omega_\nu}^{ab}R_{\alpha \beta ab}+
(2/3){\omega_{\nu a}}^b
{\omega_{\alpha b}}^{c}{\omega_{\beta c}}^{a}
\right\}  \label{int41}
\ed 
As we can see from (\ref{int41}) the  CS term is of the third derivative order  
in contrast to the  first one as in the vector case (\ref{int3}).

\mad
{\large \it Nonlinear theory of gravity}
\mad

We can construct total gravitational action in the form 
\bd 
S_{tot}=(1/k^2) \int d^3x \sqrt{g}R+(1/k^2\mu)S_{CS}\label{int42}
\ed 
The sign of the Einstein part  is opposite to the conventional
one in four dimensions. The Einstein part of the action has 
coefficient $k^{-2}$ with the dimension of mass, while the topological part has 
a dimensionless coefficient ($\mu$ has a dimension of mass). 

Now we can find some interesting properties from this action.

Varying (\ref{int42}) with respect to the metric, we get field equations
\bd 
\Theta^{\mu \nu}\equiv G^{\mu \nu} +(1/\mu)C^{\mu \nu}=0 \label{int43}
\ed 
where the second rank  Weyl tensor $C^{\mu \nu}$ is  
\bd 
C^{\mu \nu}=(1/\sqrt{g})
\epsilon ^{\mu \alpha \beta} D_\alpha \tilde{R}^\nu _\beta \label{int44}
\ed 
and 
\bd 
\tilde{R}_{\alpha \beta} =R_{\alpha \beta}-(1/4)g_{\alpha \beta}R,
~~ R=R^\alpha_\alpha\label{int45}
\ed 
The components of the Einstein tensor $G^{\alpha \beta}$ are 
\bd 
G^{\alpha \beta} =R^{\alpha \beta}-(1/2)g^{\alpha \beta}R,
\label{int46}
\ed 
and the components of the Riemann tensor ${R^\alpha}_{\beta \gamma \delta}$ are
\bd 
{R^\alpha}_{\beta \gamma \delta}=
\partial _\delta{\Gamma^\alpha}_{\beta \gamma }
-\partial _\gamma {\Gamma^\alpha}_{\beta \delta}
+{\Gamma^\alpha}_{\mu \gamma }{\Gamma^\mu}_{\beta \delta}
-{\Gamma^\alpha}_{\mu \delta}{\Gamma^\mu}_{\beta \gamma} \label{int47}
\ed 
From (\ref{int43}) we get first order form for field equations
\bd 
{K_{\mu \nu}}^{\lambda \sigma} (\mu) R_{\lambda \sigma}=0 \label{int48}
\ed 
where  ${K_{\mu \nu}}^{\lambda \sigma}$ is operator of the form
\bd 
{K_{\mu \nu}}^{\lambda \sigma} =(\delta^\lambda _\mu \delta^\lambda _\nu 
-(1/2)g_{\mu \nu}g^{\lambda \sigma})+
\frac{1}{\mu \sqrt{g}}{\epsilon _\mu}^{\alpha \beta}
(\delta^\lambda _\beta  \delta^\sigma _\nu 
-(1/2)g^{\lambda \sigma}g^{\nu \beta})  \label{int49}
\ed 
Operator ${K_{\mu \nu}}^{\lambda \sigma}(\mu)$ may be multiplied by
 ${K_{\mu \nu}}^{\lambda \sigma}(- \mu)$
to yield a second-order equation for Ricci tensor.

From (\ref{int48}) we find
\bd 
{K_{\alpha \beta}}^{\mu \nu}(-\mu){K_{\mu \nu}}^{\lambda \sigma}(\mu)
R_{\lambda \sigma}=0  \label{int50}
\ed 
that is 
\bd 
(D_\alpha D^\alpha +\mu^2)R_{\mu \nu}=-g_{\mu \nu}R^{\alpha \beta}
R_{\alpha \beta}+
3R^\alpha _\beta R_\alpha ^\beta \label{int51}
\ed 
This exhibits the massive character of the excitations.

After this short review of topological field models we will consider 
3-D  fermionic models interacting with vector and with tensor fields.
These models may give effective induced topological action of the  CS type. 
\vspace*{8mm}
\begin{center}
{\Large \bf GRAPHICS}
\end{center}

\vspace*{10mm}
\begin{picture}(7,6)

\put(3,3){\line(1,0){3}}
\put(6,1){\line(0,1){3}}
\put(3,1){\vector(0,1){4}}
\put(3,1){\vector(1,0){4}}
\put(3.02,1){\vector(0,1){4}}
\put(3,1.02){\vector(1,0){4}}

\put(6,0.5){\makebox(0,0){$1$}}
\put(3,0.5){\makebox(0,0){$0$}}
\put(2.3,3){\makebox(0,0){$g_{3/2}(1)$}}
\put(3,5.5){\makebox(0,0){$g_{3/2}(z,R)$}}
\put(7,3.5){\makebox(0,0){$R<0$}}
\put(7,2.5){\makebox(0,0){$R>0$}}
\put(7,3){\makebox(0,0){$R=0$}}
\put(7.5,1){\makebox(0,0){$z(R)$}}

\put(2.3,3.5){\makebox(0,0){$g_{3/2}(1,R)$}}
\put(3,3.5){\line(1,0){3}}
\put(2.3,2.5){\makebox(0,0){$g_{3/2}(1,R)$}}
\put(3,2.5){\line(1,0){3}}


\end{picture}
\vspace*{5mm}
\begin{center}
Fig. I-1  Graphical expression of the function $g_{3/2}(z,R)$
\end{center}

\newpage
\vspace*{20mm}
\begin{picture}(9,7)

\put(3,2.3){\line(1,0){2.8}}
\put(6.5,1){\line(0,1){5}}
\put(3,1){\vector(0,1){5}}
\put(3,1){\vector(1,0){5}}
\put(3.02,1){\vector(0,1){5}}
\put(3,1.02){\vector(1,0){5}}

\put(4.8,1){\line(0,1){1.3}}
\put(5.3,1){\line(0,1){1.3}}
\put(5.8,1){\line(0,1){1.3}}
\put(3,5){\line(1,0){3.5}}
\put(2.9,6.5){\makebox(0,0){$g_{3/2}(z,R)+\lambda^3 n_0$}}
\put(2.3,5){\makebox(0,0){$2.612$}}

\put(1.7,5.5){\makebox(0,0){$g_{3/2}(1,R)+\lambda^3 n_0$}}
\put(3,5.5){\line(1,0){3.5}}
\put(1.7,4.5){\makebox(0,0){$g_{3/2}(1,R)+\lambda^3 n_0$}}
\put(3,4.5){\line(1,0){3.5}}

\put(6.6,0.5){\makebox(0,0){$1$}}
\put(3,0.5){\makebox(0,0){$0$}}
\put(8.5,1){\makebox(0,0){$z(R)$}}
\put(4.8,0.5){\makebox(0,0){$z_1$}}
\put(5.3,0.5){\makebox(0,0){$z_0$}}
\put(5.8,0.5){\makebox(0,0){$z_2$}}

\put(7,5.5){\makebox(0,0){$R<0$}}
\put(7,4.5){\makebox(0,0){$R>0$}}
\put(7,5){\makebox(0,0){$R=0$}}

\put(6.5,5.5){\line(-1,0){0.1}}
\put(6.5,4.5){\line(-1,0){0.1}}

\put(2.3,2.3){\makebox(0,0){$\lambda^3 n$}}

\end{picture}
\vspace*{5mm}
\begin{center}
Fig.I-2  Graphical solution for bosons.\\
Solution of the equation (\ref{bos19}) for different curvatures 
and fixed temperature 
and density $z_1$ for $R<0$,
$z_0$ for $R=0$ and $z_2$ for $R>0$
\end{center}



\newpage
\vspace*{30mm}
\begin{picture}(8,6)



\put(2.5,1){\vector(0,1){4}}
\put(2.5,3){\vector(1,0){6}}
\put(2.52,1){\vector(0,1){4}}
\put(2.5,3.02){\vector(1,0){6}}


\put(5,3){\line(0,-1){0.15}}
\put(6,3){\line(0,-1){0.15}}
\put(7,3){\line(0,-1){0.15}}

\put(2.2,3){\makebox(0,0){$0$}}

\put(9.5,3){\makebox(0,0){$(\lambda^3 n)^{-1}$}}

\put(2.5,5.5){\makebox(0,0){$\mu_{eff}(R)$}}
\put(5,3.5){\makebox(0,0){$T_c^{'}$}}
\put(6,3.5){\makebox(0,0){$T_c$}}
\put(7,3.5){\makebox(0,0){$T_c^{''}$}}

\put(5.8,1.3){\makebox(0,0){$R<0$}}
\put(7,1.3){\makebox(0,0){$R=0$}}
\put(8,1.3){\makebox(0,0){$R>0$}}



\end{picture}

\vspace*{5mm}

\begin{center}
Fig.I- 3 Chemical potential $\mu _{eff}(R)$ as a 
functional of a curvature of space-time.
\end{center}

\newpage
\vspace*{20mm}
\begin{picture}(7,6)



\put(3,1){\vector(0,1){4}}
\put(3,1){\vector(1,0){5}}
\put(3.02,1){\vector(0,1){4}}
\put(3,1.02){\vector(1,0){5}}

\put(4,1){\line(0,1){2}}
\put(5,1){\line(0,1){2}}
\put(6,1){\line(0,1){2}}
\put(3,3){\line(1,0){3}}

\put(3,0.5){\makebox(0,0){$0$}}
\put(8.5,1){\makebox(0,0){$z(R)$}}
\put(4,0.5){\makebox(0,0){$z_1$}}
\put(5,0.5){\makebox(0,0){$z_0$}}
\put(6,0.5){\makebox(0,0){$z_2$}}

\put(3,5.5){\makebox(0,0){$f_{3/2}(z,R)$}}
\put(4.5,1.6){\makebox(0,0){$R<0$}}
\put(5.5,1.6){\makebox(0,0){$R=0$}}
\put(6.5,1.6){\makebox(0,0){$R>0$}}
\put(2.2,3){\makebox(0,0){$(n/s)\lambda^3(T)$}}

\end{picture}

\vspace*{5mm}
\begin{center}
Fig.I-4  Graphical solution for fermions.\\
Solution of the equation (\ref{tf11})
for different curvatures: $z_1$ for $R>0$,
$z_0$ for $R=0$ and $z_2$ for $R<0$
\end{center}


\newpage
\vspace*{20mm}
\hspace{2.8cm}$R_1\times R_1\times S_1$\hspace{2cm}$R_1\times Mobius~strip$

\begin{picture}(6,5.5)
\put(2,1){\line(1,0){2}}
\put(2,1){\line(0,1){4}}
\put(2,5){\line(1,0){2}}
\put(4,1){\line(0,1){4}}
\put(2,2){\vector(1,0){2}}
\put(4,2){\vector(-1,0){2}}
\put(2,4){\vector(1,0){2}}
\put(4,4){\vector(-1,0){2}}


\put(3,0.5){\makebox(0,0){$L$}}

\put(6,1){\line(1,0){2}}
\put(6,1){\line(0,1){4}}
\put(6,5){\line(1,0){2}}
\put(8,1){\line(0,1){4}}
\put(6,1){\vector(1,2){2}}
\put(6,5){\vector(1,-2){2}}
\put(6,3){\vector(1,0){2}}
\put(7,0.5){\makebox(0,0){$L$}}
\end{picture}

\vspace*{5mm}
Fig. III-1 Topologies of cylinder and Mobius strip in $(y,z)$
spaces.\\
 Identification of the points
$\psi(x,y,0)=\psi(x,y,L)$ and $\psi(x,y,0)=-\psi(x,y,L)$
\vspace{1cm}

\newpage
\vspace*{20mm}
\hspace{2.5cm}$R_1\times R_1\times S_1$\hspace{2.5cm}$R_1\times Mobius~strip$

\begin{picture}(6,5.5)
\put(2,1){\line(1,0){2}}
\put(2,1){\line(0,1){4}}
\put(2,5){\line(1,0){2}}
\put(4,1){\line(0,1){4}}
\put(2,2){\vector(1,0){2}}
\put(4,2){\vector(-1,0){2}}
\put(2,4){\vector(1,0){2}}
\put(4,4){\vector(-1,0){2}}
\put(3,1){\vector(0,1){4}}
\put(3,5){\vector(0,-1){4}}
\put(3,0.5){\makebox(0,0){$L^{'}$}}
\put(1.5,3){\makebox(0,0){$L$}}
\put(6,1){\line(1,0){2}}
\put(6,1){\line(0,1){4}}
\put(6,5){\line(1,0){2}}
\put(8,1){\line(0,1){4}}
\put(6,1){\vector(1,2){2}}
\put(6,5){\vector(1,-2){2}}
\put(6,3){\vector(1,0){2}}
\put(7,1){\vector(0,1){4}}
\put(7,5){\vector(0,-1){4}}
\put(5.5,3){\makebox(0,0){$L$}}
\put(7,0.5){\makebox(0,0){$L^{'}$}}
\end{picture}

\vspace*{5mm}

Fig. III-2 Topologies of torus and Klein bottle in $(y,z)$
spaces.\\
 Identification of the points
$\psi(x,,0)=\psi(x,L,L')$ and $\psi(x,0,0)=-\psi(x,L,L')$




\begin{center}

{\Large \bf TEMPERATURE, TOPOLOGY} \\

{\Large \bf AND QUANTUM FIELDS}

{A Thesis Presented to The Faculty of the Division of Graduate Studies by} \\
 

{\Large  Igor Konstantinovich Kulikov}

{ In Partial Fulfillment of the Reqirements for the Degree Doctor of Philosophy in
Physics at the Georgia Institute of Technology, May, 1996} \\

{Copyright c 1996 by Igor K. Kulikov}

\vspace*{0.5cm}

{\Large \it  Dedicated to My Parents} \\
\end{center}

\vspace*{0.5cm}

This thesis uses Path Integrals and Green's Functions to study \\ 
Gravity, Quantum Field Theory and Statistical Mechanics, \\
particularly with respect to:  \\ 
finite temperature  quantum systems of different spin in  gravitational fields; \\ 
finite temperature interacting quantum systems in perturbative regime; \\   
self-interacting fermi models in non-trivial space-time of different dimensions; \\ 
non-linear quantum models at finite temperatures in a background curved space-time; \\ 
3-D topological field models in non-trivial space-time and at finite temperatures; \\ 
thermal quantum systems in a background curved space-time. \\
Results include:  NON-EQUIVALENCE of INERTIAL and GRAVITATIONAL Mass.  
\vspace*{5mm}
\begin{center}
{\Large \bf CONTENTS} \\

\vspace*{5mm}

{page numbers may be approximate due to latex processing}
\end{center}

\hspace{13.5cm}Page
\begin{tabbing}
ACKNOWLEDGEMENTS     \` 8 \\

LIST OF FIGURES \`9 \\

CONVENTIONS AND ABBREVIATIONS \`  10  \\

SUMMARY \`   11  \\

PART I. LOCAL QUANTUM STATISTICS\\

AND THERMODYNAMICS  IN CURVED SPACE-TIME\\
 
Introduction \`   17 \\

CHAPTER I. QUANTUM FIELD METHODS\\

\hspace{25mm} IN STATISTICAL PHYSICS\\

 I.1 Equilibrium statistical mechanics \`   23  \\

 I.2. Statistical mechanics of  simple systems\\

 \hspace{7mm}  Formalism of second quantization \`   28  \\

CHAPTER II. PATH INTEGRALS\\

\hspace{27mm} IN STATISTICAL MECHANICS \`  34   \\

 II.1. Partition function  in path integral formalism \`   34  \\

 II.2. Partition function for bosons \`  38   \\

 II.3. Green's  function for boson  field \`   43  \\

 II.4. Notation \`   47  \\

 II.5. Partition function for fermions \`   49  \\

 II.6. Green's  function for fermi field  \`   53  \\

CHAPTER III. THERMODYNAMICS OF QUANTUM GASES\\

\hspace{28mm} AND   GREEN'S FUNCTIONS\\

 III.1. Thermal bosonic fields \`   58  \\ 

 III.2. Bosonic finite temperature\\ 

Green's function in the  Schwinger representation \`  63   \\

 III.3. Thermal fermionic fields \`   66  \\

 III.4. Fermionic finite temperature\\ 

Green's function in Schwinger representation \`   68  \\

CHAPTER IV. FINITE TEMPERATURE GAUGE FIELDS\\

 IV.1. Gauge theories: Pure Yang-Mills theory \`   73  \\

 IV.2. Ghost fields \`   77  \\

 IV.3. Effective action \`   80  \\

 IV.4. Propagator for vector field \`   82  \\

 IV.5 Partition function for gauge fields \`   84  \\

CHAPTER V. QUANTUM FIELDS IN CURVED SPACE-TIME\\

 V.1. Lorentz group and quantum fields \`   88  \\

 V.2. Fields in curved space-time \`  90   \\

 V.3. Spinors in general relativity \`   95  \\

CHAPTER VI. BOSE FIELD IN CURVED SPACE-TIME\\ 

 VI.1. Momentum-space representation of\\ 

the  bosonic Green's function\`   102  \\ 

 VI.2. The  Green's function and the  Schwinger-DeWitt method \`  111   \\

 VI.3. Connection between the two methods \`   115  \\

CHAPTER VII. FINITE TEMPERATURE\\ 

BOSONS IN CURVED SPACE-TIME \`  117   \\

CHAPTER VIII. FERMI FIELDS IN CURVED SPACE-TIME\\

 VIII.1. Momentum space representation\\ 

for the  Green's function of a fermion \`   124  \\ 

 VIII.2. The bi-spinor function in the  Schwinger-DeWitt 
representation \` 128  \\

CHAPTER IX. FINITE TEMPERATURE FERMIONS\\ 

IN CURVED SPACE-TIME\\

 IX.1.The  Helmholtz free energy of\\ 

 a fermi gas in curved space-time \`   130  \\

CHAPTER X. THERMODYNAMICS OF VECTOR BOSONS\\

 X.1 The Green's function of photons  \`   136  \\

 X.2. The thermodynamic  potential  of a  photon gas \`  141   \\

 X.3. Internal energy 

 and heat capacity of photon gas \`   143  \\

CHAPTER XI. RENORMALIZATIONS IN  LOCAL\\ 

  STATISTICAL MECHANICS\\

 XI.1. Divergencies of    finite temperature field models \`   145  \\

CHAPTER XII. LOCAL QUANTUM STATISTICS\\ 

  AND  THERMODYNAMICS  OF BOSE GAS\\

 XII.1. Density of Grand thermodynamical potential \`  151   \\

 XII.2. Statistics and  thermodynamics of bose gas \` 154    \\

 XII.3.  Bose-Einstein condensation \`   157  \\

CHAPTER XIII. LOCAL STATISTICS  AND THERMODYNAMICS\\ 

  OF FERMI GAS\\

 XIII.1. Grand thermodynamical potential and low\\ 

   temperature properties  of fermi gases \`  160   \\ 

PART II. INTERACTING FIELDS AT FINITE TEMPERATURE\\ 

 Introduction \`   167  \\

CHAPTER XIV. TWO LOOP RENORMALIZATIONS\\

 IN  $\lambda\phi^4$ MODEL \`  169   \\

CHAPTER XV. GREEN'S FUNCTION OF A  BOSON IN\\ 

    FINITE TEMPERATURE REGIME \`  184   \\

CHAPTER XVI. TEMPERATURE PROPERTIES OF A  BOSON\\

 XVI.1. Effective Hamiltonian of the  boson\\ 

 in non-relativistic approximation \`  197   \\

 XVI.2. Internal and gravitational masses of a  boson \`   198  \\

PART III. NON-LINEAR MODELS\\ 

IN TOPOLOGY NON-TRIVIAL SPACE-TIME\\

 Introduction \`   202  \\

CHAPTER XVII. NON-PERTURBATIVE EFFECTS\\ 

  IN GROSS-NEVEU MODEL\\

 XVII.1. Trivial case. Euclidean space-time \`   207  \\

 XVII.2. Non-trivial topology of space-time \`   209  \\

CHAPTER XVIII. $(\bar{\psi}\psi)^2$ NON-LINEAR  SPINOR MODEL\\

 XVIII.1. Dynamical mass and symmetry breaking  \`   212  \\

 XVIII.2. Model  with topologies $R_1 \times R_1\times S_1$\\ 

    and $R_1 \times Mobius~strip$ \`  215   \\

 XVIII.3. Torus topology $R_1\times R_1\times S_1$\\ 

    and topology  $R_1 \times Klein~bottle$ \`   219   \\

 XVIII.4. Non-linear spinor $(\bar{\psi}\psi)^2$ model\\

 in  Riemann space-time at finite temperature \`  220   \\

PART IV. TOPOLOGICALLY MASSIVE  GAUGE THEORIES\\

(non-trivial space-time and finite temperatures)\\

 Introduction \`   224  \\  
 
CHAPTER XIX. INTRODUCTION  TO\\ 

  TOPOLOGICAL FIELD MODELS \`   226  \\

CHAPTER XX. INDUCED CHERN-SIMONS  MASS TERM\\ 

  IN NON-TRIVIAL TOPOLOGICAL SPACE-TIME \\

 XX.1. Euclidean space-time. Trivial topology \`   240  \\

 XX.2. Non-trivial topology \`  245   \\

CHAPTER XXI. GRAVITATIONAL CHERN-SIMONS\\ 

  MASS TERM AT FINITE TEMPERATURE\\  

 XXI.1. Induced gravitational Chern-Simons mass term   \`  249   \\

 XXI.2. Induced gravitational Chern-Simons mass term\\ 

   at finite temperature \`   254  \\ 

APPENDIX\\

 I. Integral representations of modified Bessel functions I. \`  256  
\\

 II. Integral representations of modified Bessel functions II. \`  
260  \\

 III. Graphics \`  264   \\

BIBLIOGRAPHY	 \`  270   \\


\end{tabbing}


\newpage
\vspace*{5mm}
\begin{center}
{\Large \bf ACKNOWLEDGEMENTS}
\end{center}


I acknowledge and thank many people for their help in my work.
In particular I thank Professor David Finkelstein for  
encouraging me in my research, for his help and  great  contribution in my
professional growth. 
I thank my first graduate teachers Dr. Petr I. Pronin, Professor D.D. Ivanenko,
Professor V.N. Ponomariev and  Dr. G.A. Sardanashvilli,
who taught, supported and encouraged me during
my research work in Moscow State University.

Especially I am greatful to
Dr. Pronin whose physical intuition and deep knowledge of gravitation, field theory
 and quantum statistics helped to develop the subject of
local statistics and thermodynamics in curved space-time.
I thank him for his support during all the period of our collabaration.

I thank graduate students Frank (Tony) Smith, Jeffrey Hasty, Sarah Flynn, William 
Kallfelz, Zhong Tang and  Bereket Berhane for reading, correcting and discussing 
parts of the text. 
  
I thank  my family for patience and support, especially
my wife Yelena who encouraged me in my work  and helped to  correct the text.

I am thankful to  the  School of Physics of the Georgia Institute of Technology
for giving me the opportunity to complete this research.  



\newpage
\vspace*{5mm}
\begin{center}
{\Large \bf LIST OF FIGURES}
\end{center}

Figure \hspace{12cm} page
\begin{tabbing}
I-1 Graphical expression of the function $g_{3/2}(z,R)$ \`  239 \\

I-2  Graphical solution for bosons  \` 240  \\

I-3 Chemical potential $\mu _{eff} (R)$ as a functional\\ 

of a  space-time curvature  \`  241 \\

I-4 Graphical solution for fermions \` 242  \\

II-1 One loop and counterterm contributions to the self\\

energy of the boson \`  150 \\

II-2 Feynman diagrams contributing to the vertex correction $\Gamma^{(4)}$ \`  153 \\

II-3 Counterterms and loop contributions\\ 

of the order $(\lambda ^2_R)$ to the self energy \`  156 \\

II-4 Green's function $D^{'}(p)$ of the boson\\ 

as a sum of proper self-energy insertions \`  172 \\

III-1 Topologies of cylinder and Mobius strip in $(x,y)$ space \` 243  \\

III-2 Topologies of torus  and Klein bottle  in $(x,y)$ space \`  244 \\
\end{tabbing}


\newpage   
\begin{center}
{\Large \bf CONVENTIONS AND ABBREVIATIONS}
\end{center}

The sign conventions of the metric of the Part  I. is $sign(+2)=diag(-,+,+,+)$ 
with Riemann tensor 
${R^\alpha}_{\beta \gamma \delta}=
\partial_\delta {\Gamma ^\alpha}_{\beta \gamma}-\partial_  \gamma{\Gamma
^\alpha}_{\beta\delta }+{\Gamma^\alpha }_{\xi \gamma }{\Gamma^\xi }_{\beta
\delta} -{\Gamma^\alpha }_{\xi\delta  }{\Gamma^\xi }_{\beta \gamma }$   
and Ricci  tensor  as contraction of the form 
$R_{\mu \nu} ={R^\alpha}_{\beta \gamma \alpha}$

In Part II.    metric with signature $sign(-2)=diag(+,-,-,-)$ is used   
in order to preserve standard formulation theory in real time formalism.

The units $\hbar=c=1$ and Boltzmann constant  $k=1$ are  used in the thesis. 

The following special symbols are used throughout:

$*$    complex conjugate

$+$    Hermitian  conjugate

$-$    Dirac  conjugate

$\partial_\alpha =\frac{\partial}{\partial x^\alpha}$ partial derivative

$\nabla_\alpha $ or $;$  covariant derivative 

${\Gamma ^\delta}_{\beta \gamma}$   Christoffel symbols

${\omega^\alpha}_{\beta \gamma}$ spin connection

$\mbox{tr}$ or $\mbox{Tr}$  are traces

$[a,b]=ab-ba$ commutator

$\{a,b\}=ab+ba$ anticommutator

$\sim$ order of magnitude estimate

$\simeq$ approximately equal

$\equiv$ defined to be equal to

\newpage
\begin{center}
{\Large \bf SUMMARY}
\end{center}

The   problems which are studied 
in this work  belong to  three different fields of theoretical 
physics:  gravity, quantum field theory and statistical mechanics.
Two well known methods of  modern quantum field theory, path integrals  and
Green's functions, allow one to connect these different branches of physics.
 It is possible to get a number of interesting physical
predictions by  combining these two  methods.  
This combination of path integral and Green's function methods is applied to:  
finite temperature  quantum systems of different spin in  gravitational fields;
finite temperature interacting quantum systems in perturbative regime;   self
interacting fermi models in non-trivial space-time of different dimensions; 
non-linear quantum models at finite temperatures in a background curved space-time;
 3-D topological  field models in non-trivial space-time and at finite
temperatures; and 
construction of the statistics and thermodynamics of thermal quantum systems in
a background curved space-time.\\
The thesis is divided into four parts.

The core of the thesis is Part I.
 It is concerned with the development of ideas of local quantum statistics 
and thermodynamics of ideal thermal quantum systems.
The goal is to apply the  methods of quantum field theory 
to thermal quantum systems, and so
to extend and to improve them as  convenient  mathematical tools for
describing the thermal behavior of quantum systems of different 
spins in external gravitational field. Particular attention is devoted to
the development of the  connection between Green's functions 
of the models  and their distribution  functions.
For the description of thermal quantum systems in an external curved space-time, 
Schwinger-Dewitt and momentum-space representations of Green's functions 
are introduced. The problem of introducing   temperature into field models in
curved space-time for these two methods is discussed.
The equivalence and difference between these two formalisms are studied.  
The aim of introducing   these different representations 
is to show how to introduce chemical potential in the 
momentum-space formulation of partition functions  of quantum systems
and to study their  thermal behavior in external gravitational 
fields. 
The concept of local thermodynamics is introduced, and densities of 
Helmholtz free energies  and  grand 
thermodynamical potentials for bose and fermi gases are computed.
It is shown how  average occupation numbers of  bosonic and fermionic
quantum system depend on   external gravitational fields.
Low temperature behavior of bose and fermi gases is explored, and 
the phenomenon of Bose condensation in thermal bose systems in a background
curved space-time is studied.     
Thermodynamical properties of a photon gas in a background 
gravitational field are considered.

In part II finite temperature  interacting quantum 
fields are explored from the  point of view of thermofield dynamics.
Two loop renormalizations of a  self-interacting $\lambda \varphi ^4$
scalar field  at zero and finite temperatures are discussed and analyzed.
On this basis, the  two loop renormalized finite temperature Green's function of
a boson and non-relativistic Hamiltonian of a boson in a heat bath 
in an external gravitational field are computed. 
The interesting problem of non-equality between inertial 
and gravitational masses of a boson is studied in this part.

In part III non-linear spinor models in 3-D and 4-D space-time are considered.
The Gross-Neveu model  and the 3-D Heisenberg model in   non-trivial space-times
with different topologies are studied. The problem of the dynamical mass 
of   composite fermions is discussed.
The 4-D Heisenberg model in a background curved space-time at finite temperature 
is explored. The influence of  curvature 
on the  dynamical mass generation of fermions is considered.

Part IV is devoted to the analysis of topological field models in 
 non-Euclidean 3-D space-time and finite temperature effects.
The properties of vector and tensor field models with Chern-Simons mass term 
are considered. 
The effective Chern-Simons terms of vector and tensor fields
appearing as the result of interaction of 3-D fermions  with external 
gauge vector  and tensor fields are discussed.
The ideas and methods developed  in part I are applied  to the analysis of  
topological models with non-trivial space-time structure.     
The  influences of the topology of 3-D space-time    
and of thermal behavior of interacting fermions on generation of
the effective Chern-Simons terms are studied and analyzed. 

\centerline{\Large New results contained in  thesis }

Finite temperature field theory in flat space-time has been studied 
in a  number of   journal publications (references in the text of part I)  
and  books 
[ Bonch-Bruevich et al 1962, Abrikosov et al. 1963, Popov 1987,  Kapusta 1989 et
al.]. Quantum field theory in curved space-time has been developed by many authors
(journal references in the text of part I)
and is  well described in the books [De-Witt 1965,
 Birrell \& Devies 1982,  Fulling 1989
et al.] Finite temperature field theory in curved space-time 
in the Schwinger-DeWitt  formalism [De-Witt 1965,1975] is considered
in many  journal publications  (c.f. references in the text of part I). 
Most important  are
the works of Dowker, Kennedy, Denardo et al.   
 [Dowker \& Kritchley 1977, Dowker \& Kennedy 1978, Denardo \& Spalucci 1983].
The momentum-space method in applications to a field theory in curved
space-time was discussed  in the works of Bunch, Parker, Panangaden  
[Bunch \& Parker 1979,  Panangaden  1981]. This method was used in  the study 
of the $\lambda \varphi^4$-model and the quantum  
electrodynamics in perturbative regimes in curved
space-time. 
In   Part I, on the basis of the  above mentioned  works, the author develops the 
theory of  local thermodynamics  and a statistics in curved space-time.
The author shows that the thermodynamical potentials 
(free energy, grand thermodynamical potential) of thermal quantum systems
may be computed  directly through the thermal Green's functions of corresponding 
quantum fields  in  chapter III (III.24) and (III.52); chapter VII (VII.22) and 
chapter IX (IX.15).
In chapters VII and IX the author shows that the densities of
free energy of thermal bose and  and fermi systems in curved space-time 
can be derived  with the Green's function in the  Schwinger-DeWitt formalism. 
The author shows the equivalence between the  Schwinger-DeWitt and momentum 
space methods
in applications to thermal quantum systems in  chapter VII (VII.25) and chapter IX
(IX.18). The author analyzes the difficulties of the computation of the 
grand thermodynamical potentials for bose and fermi systems
using the  Schwinger-DeWitt formalism (chapter IX) and computes 
the  grand thermodynamical potentials of ideal bose and fermi 
systems in gravitational fields  with momentum-space method in
chapter  XII (XII.1 and XII.2) and chapter XIII. 
The author  considers low temperature properties of these systems 
in chapters XII and XIII, and  studies the behavior of the chemical potentials 
of the bose and fermi systems  with respect to  space-time curvature
 (XII.19) and (XIII.20).   
The author analyzes the  phenomenon of Bose condensation  (section XII.3) and finds
the  variation of the  critical temperature of condensation with respect to
the  variation of the curvature (XII.25). 
The author studies the  thermodynamical properties of a photon gas in curved
space-time with momentum space-methods in chapter X. The author computes
Bose-Einstein  and Fermi-Dirac distributions  in curved space-time in
chapters XII (XII.14) and XIII (XIII.10)

The temperature properties of self-interacting  and gauge fields are studied in
imaginary and in real time formalisms in the  
works of Kislinger , Kapusta,  Donoghue et al., from the  point of view of their
renormalizability [Kislinger et al., 1976] and  finite temperature behavior of the
constant of the interaction   [Kapusta 1979, Fujimoto et al., 1987]. In the work of
Donoghue  the gravitational field  was  also introduced in the electrodynamics with
the Tolman's shift of temperature, and the  non-equality between inertial and
gravitational masses of fermion was also discussed [Donoghue et al.,1984].
In part II of this work
  the author computes  the  finite temperature effective mass and  renormalized  
Green's function of boson in a  heat bath  in two loop approximation
with a real time formalism in   chapter XV (XV.21), finds the  
non-relativistic finite temperature Hamiltonian of bosons in a  weak
gravitational field (section XVI.1),  and gets the ratio between the  inertial and 
the gravitational masses of a boson (section XVI.2) (XVI.9). 

Twisted fields were introduced   by Isham in 1978 and field
models in space-time with a non-Minkowskian topology were studied in the works
of Ford, Toms et al.,  [Ford 1980, Toms 1980]. Non-linear 3-D field models
with $\gamma_5$   symmetry   and  a large  number of flavors
and $(\bar{\psi}\psi)^2$ non-linear models in 3-D and 4-D space-time 
were studied in works  Gross \& Neveu 1974,
 Bender 1977,  Tamvakis,  Kawati \& Miyata  1981 et al.   
In part III the author applies the idea of twisted and untwisted spinor fields
to obtain the  dependence of the dynamical fermionic mass with respect to  
a  non-trivial topology of space-time in chapter  XVIII (section XVIII.2).
The author analizes the behavior of the dynamic fermionic mass with  respect to 
the  curvature of space-time and the  thermal behavior of interacting fields (section
XVIII.4).

The idea of computing the  induced Chern-Simons action for vector and tensor fields 
as the result of the  interaction of 3-D fermi fields with external vector and tensor
fields was studied  in the  works of Redlich,  Das, Ojima  et al.
Das  proposed the method of the  computation of finite temperature induced
Chern-Simons action of vector type  with the method of derivative
expansion [Babu \& Das 1987]. Ojima computed the Chern-Simons action of vector 
and tensor fields using the  path integral method [Ojima
1989].    In Part IV the author, using the  above mentioned works computes the 
induced Chern-Simons mass term of  a vector field in   space-times with  
topologies $R^2\times S^1$ and  $R^2\times mobius~strip$ in section (XX.2).     
In chapter XXI  the author computes the finite temperature  gravitational induced
Chern-Simons mass  term using  the  momentum space method developed in part I.

In the list of references the author includes only the publications most important
for undestanding the text.




\documentstyle[12pt]{report}
\includeonly{ikpm,ikpt1,ikin,ikchap1a,ikchap2b,ikchap2c,ikchap3d,ikchap4e,ikcvtg,ikcur1h,ikfin1i,ikfmioncur,ikfrm1j,ikveck,ikrenl,ikBEin,ikltemfer,ikinterectn,iktopol,ikintrcs,ikveccs,ikGCS,ikapp1m,ikpic1,ikbibnew}
\setlength{\unitlength}{.5in}
\setlength{\rightmargin}{.5in}
\setlength{\topmargin}{0in}
\setlength{\textwidth}{6in}
\setlength{\textheight}{8.625in}
\renewcommand{\baselinestretch}{2}

\newcommand{\ed}{\end{eqnarray} \vspace{-6mm} \newline}

%begin eq. with dist.15mm

\newcommand{\bd}{\newline \vspace{-6mm} \begin{eqnarray}}

%end eq. with dist.15mm

\newcommand{\veb}{\vspace{-17mm}}

%distance 15mm between equations 

\newcommand{\vs}{\vspace{24pt}}

%interval between chapter and text 15mm.

\newcommand{\vsse}{\vspace{1mm}}

%interval between section and text 15mm.

\newcommand{\mad}{\vspace{3mm}}

% interval between head in Large and text 

\newcommand{\lum}{\vspace{-4mm}}
  
% space inside section


\begin{document}
\include{ikpm}
\include{ikpt1}
\include{ikin}
\include{ikchap1a}
\include{ikchap2b}
\include{ikchap2c}
\include{ikchap3d}
\include{ikchap4e}
\include{ikcvtg}
\include{ikcur1h}
\include{ikfin1i}
\include{ikfmioncur}
\include{ikfrm1j}
\include{ikveck}
\include{ikrenl}
\include{ikBEin}
\include{ikltemfer}
\include{ikinterectn}
\include{iktopol}
\include{ikintrcs}
\include{ikveccs}
\include{ikGCS}
\include{ikapp1m}
\include{ikpic1}
\include{ikbibnew}
\end{document}
 \begin{center} 
\vspace*{8mm}{\LARGE \bf PART III} 
\end{center}
\vspace{2mm}
\begin{center} 
{\LARGE\bf{NON-LINEAR MODELS IN}} 
\end{center}
\begin{center} 
{\LARGE\bf{TOPOLOGY NON-TRIVIAL }} 
\end{center}
\begin{center} 
{\LARGE\bf{ SPACE-TIME}} 
\end{center}
\vspace{2mm}
\begin{center} 
{\Large\bf  Introduction } 
\end{center}

\vspace{5mm}

The usual method of generating spontaneous symmetry breaking in quantum field 
theory is to introduce multiplets of vector  or scalar  fields which develop 
a nonvanishing vacuum expectation values \cite {top1}. 
This mechanism is not necessary.
The general features of spontaneous symmetry breaking 
are independent of whether the Goldstone  or Higgs particle is associated with an 
elementary field or with a composite field. 
However it is   possible to develop a dynamical theory of 
elementary particles in which the origin of the  spontaneous symmetry 
breaking is  dynamical \cite {top3}.
The finite mass appears in analogy with the  phenomenon of superconductivity. 
This part III deals with the research of  \cite {top4}, \cite{top5}, \cite{km1}
in the field of dynamical symmetry breaking and phase transitions.
Models with dynamical mass generation are interesting for the following reasons:

First: Initially massless fermi fields expose $\gamma_5$ invariance, which is 
violated by the dynamical mass generation. 
Thus the initial symmetry will be broken, 
and this  broken symmetry may appear in experimental measurements \cite {top6}.

Second: Dynamical mass of particles in such models is a function of
 a constant of interaction of primary fields. Perhaps this will indicate how 
to solve the puzzle of the origin of particle mass. 

Third: Higgs mechanism introduces into the  theory additional parameters which are
connected 
with Higgs-Goldstone fields. These are  additional difficulties of the model.
\cite{kaku3}.

Distinctive and important characteristics of the  models with dynamic symmetry 
violation  are the introduction of   bound states of particles, and 
calculations in non-perturbative regime. Estimations of the  condensate 
may be obtained from the sum rule \cite{top9}, or from Monte-Carlo method \cite{top7},
 or for the semiclassical 
instanton solutions of the Yang-Mills equations \cite{top8}. 

To develop further the theory one should take into account also 
the topology of space-time \cite{toms1}, \cite{isham1}, \cite{ford1}. 
One should consider 3-D and 4-D spinor models with non-linearity 
of the type $(\bar{\psi}\psi)^2$.
 The non-linear Gross-Neveu model in 3-D space-time in non-trivial topology is studied 
in chapter XVII. 3-D Heisenberg-Ivanenko  model in non-Euclidean space-time
and 4-D  Heisenberg-Ivanenko  model in Riemann space-time are studied in
chapter XVIII. The influence of topology  and curvature on the generation of 
dynamical mass   
of fermions and the effects of violation of symmetry 
in these models are also  studied in  chapter XVIII.   

\chapter{NON-PERTURBATIVE EFFECTS}
\centerline{\Large \bf IN GROSS-NEVEU MODEL }
\vs
	
	Quantum Field Theory may be essentially symplified in the limit of high 
interanal symmetries such as O(N), SU(N) and so on.  
In some appropriate cases field models can be solved strictly 
in the limit of large flavor numbers N \cite{top10}.  Asymptotically free 
Gross-Neveu model without dimensional parameters in the Lagrangian 
is one of such models. This model is renormalizable in 3-D dimensions.  
The solution of this model shows that the phenomenon of dimensional 
transmutation \cite{top11} has place and there is a gap in mass spectrum 
of the model. 
Studying this model one will find the connection between topological
characteristics of space time and the behavior of the solution of the
mass gap equation and also the behavior of dinamical mass of the model.
N-flavor Gross-Neveu model is described by  the Lagrangian
\bd
L=\bar \psi _ii\hat \partial \psi _i+{{g^2} \over 2}
\left( {\bar \psi _i\psi _i} \right)^2 \label{f1}
\ed
This Lagrangian is invariant under the discrete transformations:
\bd
\psi \to \gamma _5\psi,~~\bar{\psi} \to -\bar{\psi}\gamma _5 \label{f2} 
\ed
The generating functional of the model
\bd
Z[\eta,\bar{\eta}]=\int D\psi D\bar{\psi}
\exp \left[i\int d^nx\left(i\bar{\psi}\partial \psi
+(1/2) g^2(\bar{\psi}\psi)^2
+\bar{\eta}\psi+\bar{\psi}\eta\right)\right]    \nonumber
\ed
may be rewritten in the form:
\bd
Z[\eta,\bar{\eta},\sigma]=\int D\psi D\bar{\psi}D\sigma   \nonumber
\ed
\veb
\bd
\times \exp \left[i\int d^nx\left(i\bar{\psi}\partial \psi
- g(\bar{\psi}\psi)\sigma-(1/2)\sigma^2
+\bar{\eta}\psi+\bar{\psi}\eta\right)\right] \label{f3} 
\ed
Here we introduced the new field $\sigma$ and used a useful relation
\bd
\int D\sigma
\exp \left[-i(1/2)\left( \sigma, \sigma \right)
-i\left(g\bar{\psi}\psi,\sigma\right)\right]
\propto \exp \left[(i/2)g^2 \left (\bar{\psi}\psi\right) ^2 \right] \label{f4} 
\ed
Since 
\bd
m\bar{\psi}\psi \to m\left(-\bar{\psi}{\gamma_5}^2\psi \right)
=-m\bar{\psi}\psi \label{f5} 
\ed
we must  put $m=0$ for symmetry of the model.

Therefore a new Lagrangian may be written in the form
\bd
L_\sigma =\bar \psi _ii\hat \partial \psi _i-\sigma 
\left( {\bar \psi _i\psi _i} \right)-{1 \over {2g^2}}\sigma ^2 \label{f6}
\ed
and the symmetry of (\ref{f6}) is
\bd
\psi \to \gamma _5\psi,~~\bar{\psi} \to-\gamma _5\bar{\psi},
~~\sigma \to -\sigma    \label{f7}  
\ed
The generating functional of the model after integration over the matter 
fields will be 
\bd
Z[0]=\int {D\bar \psi D\psi D\sigma }\exp \,i\int {L_\sigma \left( x \right)}dx   \nonumber
\ed
\veb
\bd
=\int {D\sigma \cdot }Det(i\hat \partial -\sigma )
\exp [{{-i} \over {2g^2}}(\sigma ,\sigma )] \label{f8}
\ed
Then effective action is written as
\bd
\Gamma[\sigma ]=-i\ln \,Det(i\hat \partial -\sigma )
-{1 \over {2g^2}}(\sigma ,\sigma ) \label{f9}
\ed
and the effective potential is
\bd
V_{eff}={{iN} \over 2}(\mbox{Tr} \hat 1)\int {{{d^2k}
 \over {\left( {2\pi } \right)^2}}}\ln \left( {k^2-\sigma ^2} \right)
+{1 \over {2g^2}}\sigma ^2 \label{f10}
\ed
The conditions for the energy to be minimal are
\bd
\left({{\delta V_{eff}} \over {\delta \sigma }}\right)_{|\sigma 
=\sigma _c}=0,\quad \left({{\delta ^2V_{eff}} 
\over {\delta \sigma ^2}}\right)_{|\sigma =\sigma _c}>0 \label{f11}
\ed
From these  conditions one can get the gap equation for definition of $\sigma _c$
  ($\sigma _c$ defines the minimum of the effective potential) in the form
\bd
{1 \over \lambda }=\mbox{Tr} \hat 1\int {{{d^2\bar k}
 \over {\left( {2\pi } \right)^2}}}{1 \over {\bar k^2+\sigma _c^2}} \label{f12}
\ed
where constant  $\lambda =g^2N$.

\section  {Trivial case. Euclidean space time.}
\vsse

Let us find the solution of the gap equation (\ref{f12}).

Ultraviolet cut-off of the integral gives:
\bd
\int_{-\Lambda}^{\Lambda} {{{d^2\bar k}
\over {\left( {2\pi } \right)^2}}}{1 \over {\bar k^2+\sigma _c^2}}   \nonumber
\ed
\veb
\bd
=(1/2\pi)\int_{0}^{\Lambda} \frac{2\pi dk^2}{\bar k^2+\sigma _c^2} =(1/4\pi)
\ln \frac{\Lambda^2}{\sigma^2} \label{f13}
\ed
Then, after regularization of (\ref{f13}) we get the equation 
\bd
{1 \over {\lambda \left( \Lambda  \right)}}
={1 \over {2\pi }}\ln {{\Lambda ^2} \over {\sigma ^2}} \label{f14}
\ed
Let the subtraction point be $\mu =\sigma $,
then the renormalized coupling constant  may be written as
\bd
{1 \over {\lambda \left( \mu  \right)}}
={1 \over {2\pi }}\ln {{\mu ^2} \over {\sigma ^2}}\label{f15}
\ed
To eliminate the parameter $\mu$ one can use the methods 
of the renormalization group.
The $\beta$-function in one loop approximation is
\bd
\beta \left( {\lambda _R(\mu )} \right)=-{1 \over \pi }
\left( {\lambda _R(\mu )} \right)^2 \label{f16}
\ed
then Gell-Mann Low equation 
\bd
{{d\lambda _R} \over {\lambda _R^2}}=-{1 \over \pi }{{d\xi } \over \xi } \label{f17}
\ed
with initial condition $\lambda \left( {\xi _0} \right)=\lambda _0$
determines the behavior of the coupling constant with respect to 
scaling of the momentum:
\bd
\lambda _R( t )=\frac{\lambda _0}{1+(\lambda _0 /\pi )t} \label{f18}
\ed
where $t=\ln ( \xi /\xi _0)$

The dynamical mass can be found from (\ref{f11}) in the form
\bd
\sigma _c(triv.)=\mu \exp (-{\pi  \over {\lambda _R\left( \mu  \right)}})
=\mu \exp \left( {-{\pi  \over {\lambda _0}}} \right)=const.\label{f19}
\ed

\section{ Non-trivial topology of space time.}
\vsse

In this case the constant of interaction can be written as
\bd
{1 \over {\lambda \left( {\mu ,L} \right)}}
={1 \over {2\pi }}\left[\ln {{\mu ^2} \over {\sigma _c^2}}+f(L,\sigma _c)
\right] \label{f20}
\ed
where $f(\sigma _c,L)$ is some function which depends on topological parameter L.

The  $\beta$ function is
\bd
\beta \left( {\lambda _R(\mu ,L)} \right)
=-\frac{1}{\pi }\left( {\lambda _R(\mu ,L)} \right)^2 \label{f21}
\ed
and the solution of Gell-Mann-Low equation with 
$\lambda _R\left( {\xi _0,L} \right)=\lambda _0(L)$
is expressed by the equation:
\bd
\lambda _R\left( {t,L} \right)=\frac{\lambda _0(L)}
{1+(\lambda _0(L) /\pi )t} \label{f22}
\ed
The dynamical mass is governed by the equation
\bd
\sigma _c=\mu \exp \left( {-{1 \over {\lambda _0(\mu ,L)}}
-{{f(\sigma _c,L)} \over 2}} \right) \label{f23}
\ed
or
\bd
\sigma _c=\sigma _c(triv)\exp \left( {-{{f(\sigma _c,L)} \over 2}}
 \right) \label{f24}
\ed
One can see that the dynamical mass depends on function $f(\sigma _c,L)$.
 
The explicit expression of the function $f(\sigma _c,L)$ is 
\bd
f_{(\pm )}(L,\sigma _c)=\pm {1 \over {\pi ^2}}\int\limits_0^\infty  
{dx{1 \over {\sqrt {x^2+(L\sigma _c)^2}}}}
\left( {\exp \sqrt {x^2+(L\sigma _c)^2}-1} \right)^{-1} \label{f25}
\ed
for  topologies of the cylinder $(+)$ and the  Mobius strip $(-)$.
Therefore non-Euclidean structure of space time leads 
to redefinition of the gap equation (\ref{f12}) for the Gross-Neveu model. 
That gives us the possibility to define the dependence of the 
fermionic mass on topology of the space time.
Dynamical violation of the $\gamma_5$ symmetry occurs when $\sigma$
is not equal zero.
The method developed above is very useful when the  number of "flavors" 
of the fundamental fields is big ($N \to \infty$). 
This method does not work for a small "flavor" number $N$. 
In Quantum  Field Theory  
there is another method, based on an analogy with superconductivity. This is  
a  Mean Field Method \cite{km1},
\cite{key14}, which works very well for any number 
of "flavors" of the particles. The idea of the method is based on 
effective potential\footnote{Effective potential is the generating functional 
for  (1PI) Green's functions \cite{jak1,rs1}}
 calculations,  that allows us to take  into consideration the 
effects of topology \cite{toms1} and curvature for self-interacting 
and gauge models \cite{ish1}.

 In this work the Mean Field Method is used in dynamical modeling 
of the behavior
of  elementary  particles  and is  based on the idea that the masses of  
compound particles (e.g. nucleons) are  generated by the  self-interaction of 
some fundamental  fermion fields  through the same mechanism 
as  superconductivity. Here the combined particles are treated 
as the quasi-particles excitations. 
The  Mean Field Method also leads to the  mass gap equation, and the solution gives 
the dynamical mass of the particle. In the following section we will treat 
the problem of non-Euclidean space-time structure in 
 models with a dynamical mass.      

\chapter{ $(\bar{\psi}\psi)^2$ NON-LINEAR SPINOR MODELS}
\vs

\section{Dynamical mass and}
\lum 
\hspace{33mm}{\Large \bf  symmetry breaking}
\vsse

In the construction of the unified  models of the elementary
particles one
can admit the possibility that the mass of combined particles 
appears as the result of self-interaction of certain fundamental fields, 
for instance, quarks and leptons from preons, or  compound 
fermions in technicolor models \cite{ro1,ta1}, \cite{ito1},
\cite{sd1}. 
 Following this idea we can get the gap equation. 
Its solution   can predict  the dynamical 
mass of the compound particles.  As in the previous case of 
Gross-Neveu model, we will study here the effects of 
non-trivial topology and, also, geometry of background space-time.

In this section we will consider  the phenomenon of the dynamical 
generation of mass of particles  in the application 
to the models in non-trivial  space-time.


Let us consider Heisenberg-Ivanenko \cite{top12} non-linear spinor model
with Lagrangian
\bd
L=\bar \psi i\hat \partial \psi +{{g_0^2}
 \over {2\mu _0^2}}\left( {\bar \psi \psi } \right)^2 \label{f26}
\ed
where $g_0^2$ is a massless parameter and  parameter $\mu_0^2$  has 
 dimension which is connected with the dimension of space time.
The symmetry of the Lagrangian (\ref{f26}) is
\bd
\psi \to \gamma _5\psi, ~~\bar \psi \to -\bar \psi \gamma _5 \label{f27}
\ed
We can rewrite this Lagrangian in the new form
\bd
L_\sigma =\bar \psi i\hat \partial \psi -g_0\sigma 
\left( {\bar \psi \psi } \right)-{{\mu _0^2} \over 2}\sigma ^2 \label{f28}
\ed
The symmetry of the last one is (\ref{f7}).

The equation of motion for the   $\sigma$ field is
\bd
\sigma =-{{g_0} \over {\mu _0^2}}\left( {\bar \psi \psi } \right)  \label{f29}
\ed
thus we may assume that the  field $\sigma$ is a collective field.  

Let us  consider that   $\sigma$  is $\sigma \to \sigma +\tilde \sigma $,
where 
\bd
\sigma =(g_0 /{\mu _0}^2)<\psi \bar \psi >\ne 0   \nonumber
\ed
is the background field and $\tilde \sigma $  is quantum 
fluctuations of the $\sigma $-field.

Then 
\bd
L_\sigma =\bar \psi (i\hat \partial -g_0\sigma )\psi 
-g_0\left( {\bar\psi  \tilde{\sigma} \psi } \right)
-{{\mu _0^2} \over 2}\sigma ^2
-{{\mu _0^2} \over 2}{\tilde{\sigma }}^2-{\mu_0}^2\sigma\tilde \sigma \label{f30}
\ed
The last term of (\ref{f30}) may be eliminated because of the 
redefinition of the sources
of the quantum field $\tilde \sigma$, and the  Feynman graphs will be

\begin{picture}(8,3.5)
\put(2,1){\line(-1,1){1}}
\put(2,1){\line(-1,-1){1}}
\put(2,0.99){\line(1,0){1}}
\put(2,1.01){\line(1,0){1}}
\put(1,1.5){\makebox(0,0){$\psi$}}
\put(1,0.5){\makebox(0,0){$\bar \psi$}}
\put(3.5,1){\makebox(0,0){$\tilde \sigma$}}
\put(1.5,1){\makebox(0,0){$g_0$}}
\put(2,1){\circle*{.1}}
\put(2,0){\makebox(0,0){$a)$}}
\put(5,1.01){\line(1,0){1.5}}
\put(5.8,1.3){\makebox(0,0){$i\hat k-g_0\sigma_0$}}
\put(5.8,0){\makebox(0,0){$b)$}}
\put(8,0.99){\line(1,0){1}}
\put(8,1.01){\line(1,0){1}}
\put(8.5,1.3){\makebox(0,0){$(i/\mu_0^2)$}}
\put(8.5,0){\makebox(0,0){$c)$}}
\end{picture}

\vspace{10mm}

Fig. III-1  Feynman graphs including a  collective field.

Graph $1 a)$ describes the interaction of fermi field with collective field,
$1 b)$ is the  propagator of fermi field, and $1 c)$ is the  
propagator of collective field.

In tree approximation with respect to  $\sigma$-field we can write 
an effective action:
\bd
\Gamma_{eff} [\sigma]=(-i/2)\ln Det(i\hat k-g_0\sigma)
-\frac{\mu_0^2}{2}(\sigma,\sigma) \label{f31}
\ed
The effective potential of this model will be
\bd
V_{eff}={i \over 2}(\mbox{Tr} \hat 1)\int {{{d^nk} 
\over {\left( {2\pi } \right)^n}}}\ln 
\left( {k^2-(g_0\sigma )^2} \right)+{{\mu _0^2} \over 2}\sigma ^2 \label{f32}
\ed
Minimum  $V_{eff}$ gives the gap equation 
\bd
\sigma _c=(g_0 / \mu _0)^2\sigma _cI\,(g_0\sigma _c) \label{f33}
\ed
or, in another form,
\bd
m=s\lambda _0mI\left( m \right) \label{f34}
\ed
where $m$ is the dynamical mass of fermionic field $m=g_0\sigma _c$,
$s$ is the dimension of $\gamma$- matrices 
and $\lambda _0=(g_0^2/ \mu _0)^2$

Now one can solve the equation (\ref{f34}) for different space time topologies.

\section{ Model  with topologies} 
 \lum 
\hspace{33mm} {\Large \bf $R_1 \times R_1 \times S_1$ 
and $R_1 \times  Mobius~strip$}
\vsse

The solution of this gap equation is connected with 
the calculation of the function  I(m) of  the equation (\ref{f34}).

Let us consider two types of topologies:

1)  $R_1\times R_1\times S_1$ with $\psi(x,y,0)=\psi(x,y,L)$\\ 
and 

2) $R_1\times Mobius~strip$ with $\psi(x,y,0)=-\psi(x,y,L)$ (Fig.III- 2)

We can find for 3-D space time that 
\bd
I(m)=\Lambda -F_{(\pm )}(L,m) \label{f35}
\ed
where
\bd
F_{(\pm )}(L,m)={1 \over {\pi L}}\ln (1\;_+^-\;e^{-Lm}) \label{f36}
\ed
	with (+) for $R_1\times R_1\times S_1$
 topology and (-) for $R_1\times Mobius\;strip$  topology.

The gap equation will be
\bd
m=m\lambda _0\Lambda \left( {1-{1 \over {\pi L\Lambda }}
\ln (1\;_+^-\;e^{-Lm})} \right) \label{f37}
\ed
The analysis of this expression can be made by the theory 
of bifurcations\\
 \cite{hk1}. 
For this purpose let us write the gap equation (\ref{f37}) in the form 
\bd
m=f_{(\pm )}(m,\bar \lambda ) \label{f38}
\ed
where the functions $f_{(\pm )}(m,\bar \lambda )$ 
for topologies $(+)$and $(-)$  are 
\bd
f_{(\pm )}(m,\bar \lambda )=m\bar \lambda 
\left( {1-{1 \over {\pi L\Lambda }}\ln (1\;_+^-\;e^{-Lm})} \right) \label{f39}
\ed
The  equation (\ref{f38}) has stable m=0 solutions, if $f_m(0,\bar \lambda )<1$.

For $f_m(0,\bar \lambda )>1$ the equation (\ref{f38}) has no stable 
trivial solutions.  

One can see that there is a stable m=0 solution for 
(-) topology with the critical parameter 
\bd
L_c={{\ln 2} \over \pi }{{\lambda _0} \over {\lambda _0\Lambda -1}}
\approx {{\ln 2} \over {\pi \Lambda }} \label{f40}
\ed
for $\lambda _0\Lambda =\bar \lambda >1$

The gap equation for topology (+) has no stable trivial solutions.  

The dynamical mass in this case is a smooth function with respect to parameter L:
\bd
m_{(+)}=-{1 \over L}\ln \left( {1-f(\Lambda )
\exp \left( {-{{\pi L} \over {\lambda _0}}} \right)} \right) \label{f41}
\ed
for  $\lambda _0\Lambda =\bar \lambda <1$

The solution for topology $R_1\times Mobius\;strip$  is
\bd
m_{(-)}=-{1 \over L}\ln \left( {\exp \left( {{L \over {L_c}}\ln 2} \right)-1}
 \right)\label{f42}
\ed
The restoration of the symmetry takes place when $L=L_c$.
In this case the condensate function of the bound state equals zero
\bd
\sigma _c={{g_0} \over {\mu _0^2}}<\psi \bar \psi >=0 \label{f43}
\ed
Now we can consider  the models with  the  more complicated 
space-time structures of Klein bottle and Torus topologies.

\newpage

\section{Torus topology $M_{3)}=R_1 \times S_1 \times S_1$} 
\lum 
\hspace{35mm}{\Large \bf and topology $M_{3)}=R_1\times Klein\;bottle$}
\vsse

We can introduce  topologies $R_1\times S_1\times S_1$
and $R_1\times Mobius~strip$ as an identification of space points
for wave functions Fig. III-3:
\bd
\psi(x,0,0)=\psi(x,L,L)~and~\psi(x,0,0)=-\psi(x,L,L)
\ed 
For simplicity  consider that these topologies have only one parameter
 $L=L^{'}$.
The gap equations for the topologies will be
\bd
m=m\bar {\lambda} s\left\{1+\sqrt {m /(\Lambda ^2L)}
( \pi)^{-3 /2}
[\sum\limits_{n=1}^\infty  {1 \over {\sqrt {2n}}}K_{-1/2}(2Lmn)\right.   \nonumber
\ed
\veb
\bd
\left.+\sum\limits_{n,k=1}^\infty  K_{-1/2}
(2Lm\sqrt {n^2+k^2})\left( {n^2+k^2} \right)^{-1/4}]\right\} \label{f44}
\ed
and
\bd
m=m\bar {\lambda} s\left\{1+\sqrt {m /(\Lambda ^2L)}
( \pi)^{-3 /2}
[\sum\limits_{n=1}^\infty  {1 \over {\sqrt {2n}}}K_{-1/2}(2Lmn)\right.   \nonumber
\ed
\veb
\bd
\left.+\sum\limits_{n,k=1}^\infty (-1)^k K_{-1/2}
(2Lm\sqrt {n^2+k^2})\left( {n^2+k^2} \right)^{-1/4}]\right\} \label{f45}
\ed
The solutions of (\ref{f44}) and (\ref{f45}) give dynamical masses for 
these topologies\\
 \cite{top13}.

The results of this paragraph show that the application of the  Mean Field Method
to non-linear models  gives non-renormalized solutions, 
though we can obtain some information about  the influence 
of topology on dynamical mass behavior.
The evaporation of condensate and restoration of chiral symmentry 
proceed in different ways and are ruled by topology. There are topologies 
in which these phenomena do not take place. We believe that these results
are  important in the bag models, because the energy of bag is dependent 
on non-perturbative effects and boundary conditions\\
 \cite{guid1}. 

\section{Non-linear spinor  $(\bar \psi \psi)^2$ model in}
\lum  
\hspace{23mm}{\Large \bf  Riemann space-time at finite temperature.}
\vsse

In this section we will  treat the problem of dynamical mass generation 
of the  non-linear spinor model in 4-D Riemann space-time at finite temperature.
We will get the  finite temperature effective potential  
and find out information about the   influence of curvature of 
the background gravitational field and temperature 
on the value of  dynamical fermionic mass.

Let the total Lagrangian of the model be
\bd
L=L_g+L_m \label{f46}
\ed
where the  gravitational Lagrangian is\footnote{For clear understanding 
of the problem, we study the fermionic system in a weak gravitational field.} 
\bd
L_g={1 \over {16\pi G_0}}(R-2\tilde \Lambda _0) \label{f47}
\ed
and the  Lagrangian of matter field $L_m$ is
\bd
L_m=\bar \psi i\bar \gamma ^\mu \nabla _\mu \psi +{{g_0^2} 
\over {2\mu _0^2}}\left( {\bar \psi \psi } \right)^2 \label{f48}
\ed
$\nabla _\mu $ is a covariant derivative.

The first  term of (\ref{f48}) describes 
kinetics of the fermi field and its interaction with the  gravitational field,
 and the second one describes the interaction of the fields.

The effective potential of the model is written from the action
\bd
\Gamma_{eff}[\sigma]=-{i \over 2}\ln Det[\nabla ^2
+{1 \over 4}R-(g_0\sigma )^2]-{{\mu _0^2} \over 2}(\sigma ,\sigma ) \label{f49}
\ed
After making ultraviolet regularization and renormalizations of the model we get
\bd
L_{tot}=L_g+L_m=\left( {L_g+L_m(\infty )} \right)
+L_m(\beta )   \nonumber
\ed
\veb
\bd
=L_g^{ren}+L_m(\beta ) \label{f50}
\ed
The renormalized gravitational constant  $G_R$ 
and the   $\tilde{\Lambda} _R$-term are:
\bd
{1 \over {8\pi G_R}}\tilde \Lambda _R
={1 \over {8\pi G_0}}\tilde \Lambda _0
+{1 \over {16\pi ^2}}\Lambda ^2-{1 \over {16\pi ^2}}m^2\ln \Lambda ^2   \nonumber
\ed
\veb
\bd
{1 \over {16\pi G_R}}={1 \over {16\pi G_0}}
+{1 \over {16\pi ^2}}\ln \Lambda ^2 \label{f51}
\ed
where $\Lambda$ is a cut-off parameter, and $m$ is a dynamical mass.

In the  calculations (\ref{f50}) and (\ref{f51}) there was used the cut-off 
method  of regularization of divergent integrals  \cite{key7}. 


The effective potential can be written from (\ref{f49}) in the form
\bd
V_{eff}[\sigma ]=\sum\limits_{j=1}^2 {\hat 
\alpha _j(R)f^j(\beta g_0\sigma )+{{\mu _0^2} \over 2}}\sigma ^2 \label{f52}
\ed
where
\bd
f^0(\beta g_0\sigma )={{2m^2} \over {\pi ^2\beta ^2}}
\sum\limits_{n=1}^\infty  {{{\left( {-1} \right)^n}
 \over {n^2}}}K_{-2}(\beta g_0\sigma )   \nonumber
\ed
\veb
\bd
=-{4 \over \beta }
\int {{{d^3k} \over {\left( {2\pi } \right)^3}}
\ln \left( {1+e^{-\beta \varepsilon }} \right)} \label{f53}
\ed
and
\bd
f^j(\beta g_0\sigma_0 )={1 \over {4g_0}}
\left( {-{\partial  \over {\partial \sigma ^2}}}
 \right)^jf^0(\beta g_0\sigma ) \label{f54}
\ed
The  coefficients $\hat \alpha _j(R)$ are 
\bd
\hat \alpha _0=1,\quad \hat \alpha _0={1 \over {12}}R,\quad ....\label{f55}
\ed
The solution of the gap equation
\bd
\frac{\partial}{\partial \sigma } 
 V_{eff}[\sigma ]_{|\sigma=\sigma_c}
=0 \label{f56}
\ed
gives, in high temperature approximation (\ref{bb15}), the expression 
for dynamical mass without redefinition of 
the coupling constant and temperature
\bd
m^2(R,T)=(g_0 \sigma_c)^2 = A/\lambda + b\cdot T^2 + C\cdot R+...\label{f57}
\ed
where the  constants 
\bd
A=32\pi^2/3.84,~~
B=A/24,~~
C=0.7/12,~~
\lambda=(g_0/\mu_0)^2   \nonumber
\ed
are numerical positive coefficients.
As we can see from the equation (\ref{f57}), the effective dynamical mass
of fernion is a positive function for any temperature and curvature.


