# qft-e (1).tex

qft-e (1).tex — TeX document, 111Kb

## Съдържание на файла

%% Lectures in QFT %% by E. Horozov. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%% LaTeX2e, uses amsfonts.sty and latexsym.sty %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass[11pt]{article} \usepackage{amsfonts,latexsym} \usepackage{amsxtra} \usepackage{amsmath, amscd} \usepackage{tikz} \input xypic \usepackage[all, knot]{xy} \xyoption{arc} %\usepackage{feynmf} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\setspacing}[1]{\renewcommand{\baselinestretch}{#1}\large\normalsize} \newcommand{\doublespacing}{\setspacing{1.6}} \newcommand{\normalspacing}{\setspacing{1}} \setlength{\paperheight}{11truein} \setlength{\paperwidth}{8.5truein} % changed for title page with hsize \setlength{\topmargin}{-0.275truein} \setlength{\headheight}{.25truein} \setlength{\headsep}{.125truein} \setlength{\textheight}{8.8truein} \setlength{\footskip}{.25truein} \setlength{\oddsidemargin}{.6truein} \setlength{\evensidemargin}{.1truein} \setlength{\textwidth}{5.8truein} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%% Equation counting %%%%%%%%%%%%%%%%%%%%%%%%%%%%% \renewcommand{\theequation}{\arabic{section}.\arabic{equation}} \newcommand{\cleqn}{\setcounter{equation}{0}} \newcommand{\clth}{\setcounter{theorem}{0}} \newcommand {\sectionnew}[1]{\section{#1}\cleqn\clth} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\beq}{\begin{equation}} \newcommand{\eeq}{\end{equation}} \newcommand{\beqa}{\begin{eqnarray}} \newcommand{\eeqa}{\end{eqnarray}} \newcommand{\beaa}{\begin{eqnarray*}} \newcommand{\ben}{\begin{eqnarray*}} \newcommand{\eaa}{\end{eqnarray*}} \newcommand{\een}{\end{eqnarray*}} \newcommand{\Nset}{\hfill\nonumber} \newcommand \nc {\newcommand} \nc \proof {\noindent {\em{Proof.\/ }}} \nc \qed {$\Box$\hfill} %%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{question}[theorem]{Question} %%%%%%%%%% \nc \bth[1] {\begin{theorem}\label{t#1} } \nc \ble[1] {\begin{lemma}\label{l#1} } \nc \bpr[1] {\begin{proposition}\label{p#1} } \nc \bco[1] {\begin{corollary}\label{c#1} } \nc \bde[1] {\begin{definition}\label{d#1}\rm } \nc \bex[1] {\begin{example}\label{e#1}\rm } \nc \bre[1] {\begin{remark}\label{r#1}\rm } \nc \bcon[1] {\begin{conjecture}\label{con#1}\rm } \nc \bque[1] {\begin{question}\label{que#1}\rm } %%%%%%%%%% \nc {\eth} { \end{theorem} } \nc {\ele} { \end{lemma} } \nc {\epr}{ \end{proposition} } \nc {\eco} { \end{corollary} } \nc {\ede} {\end{definition} } \nc {\eex} { \end{example} } \nc {\ere} {\end{remark} } \nc {\econ} { \end{conjecture} } \nc {\eque} {\end{question} } %%%%%%%%%% \nc \thref[1]{Theorem \ref{t#1}} \nc \leref[1]{Lemma \ref{l#1}} \nc \prref[1]{Proposition \ref{p#1}} \nc \coref[1]{Corollary \ref{c#1}} \nc \deref[1]{Definition \ref{d#1}} \nc \exref[1]{Example \ref{e#1}} \nc \reref[1]{Remark \ref{r#1}} \nc \conref[1]{Conjecture\ref{con#1}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nc \bth[1] { \begin{theorem}\label{t#1} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand {\normprod}[1]{ {\textrm{:}}{#1}{\textrm{:}} } %%%normal product \def \W {W_{1+\infty}} \def \WN {\W(N)} %\def \A {{\mathcal A}} \def \M {{\mathcal M}} \def \L {{\mathcal L}} \def \O {{\mathcal O}} \def \R {{\mathcal R}} \def \D {{\mathcal D}} \def \Dir{\partial \!\!\!/} \def \Dmom{p \!\!\!/} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def \g {\gamma} \def \G {\Gamma} \def \B {{\mathcal B}} \def \bb {b} \newcommand{\nn}{\hfill\nonumber} \def\dd{{\mathrm{\, d}}} \def \K {{\mathcal K}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\a{\alpha} \def\b{\beta} \def\de{\delta} \def\De{\Delta} \def \e{\epsilon} \def \ep{\varepsilon} \def\l{\lambda} \def\la{\lambda} \def\La{\Lambda} \def\ka{\varkappa} \def\om{\omega} \def\Om{\Omega} \def\ph{\varphi} \def\si{\sigma} \def\Si{\Sigma} \def\th{\theta} \def\Th{\Theta} \def\ze{\zeta} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def \d {{\partial}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def \Rset {{\mathbb R}} \def \Cset {{\mathbb C}} \def \Zset {{\mathbb Z}} \def \Nset {{\mathbb N}} \def \Vset {{\mathbb V}} %%%%%%%%%%%%%%%%%%%%%%%%%%% \def \A {{\mathbb A}} \def \F {{\mathbb F}} \def \N {{\mathbb N}} \def \Z {{\mathbb Z}} \def \Q {{\mathbb Q}} \def \R {{\mathbb R}} \def \C {{\mathbb C}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def \Hom{ {\mathrm{Hom}}} \def \Aut{ {\mathrm{Aut}}} \def \End{ {\mathrm{End}}} \def \tr{ {\mathrm{Tr}}} \def \coker { {\mathrm{Coker}} } \def \ord { {\mathrm{ord}} } \def \rank { {\mathrm{rank}} } \def \span { {\mathrm{span}} } \def \const { {\mathrm{const}} } \def \mod { {\mathrm{mod}} } \def \spec { {\mathrm{Spec}} } \def \diag { {\mathrm{diag}} } \def \deg { {\mathrm{deg}} } \def \mult { {\mathrm{mult}} } \def \res { {\mathrm{Res}} } \def \ad { {\mathrm{ad}} } \def \Ad { {\mathrm{Ad}} } \def \wt { {\mathrm{wt}} } \def \psd { {\mathrm{Psd}} } \def \Im { {\mathrm{Im}} } \def \Re { {\mathrm{Re}} } \def \p { {\partial}} \renewcommand \ker { {\mathrm{Ker}} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def \vect {\overrightarrow } %%%%%%%%%%%%% Grassmannians %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \nc \Wr {Wr} \nc \GRN { \Gr^{(N)} } \nc \GRA[1] { \Gr_A^{(#1)} } %% Gr_A \nc \GRAN { \GRA{N} } \nc \GrA[1] { \Gr_A(#1) }\nc \GrAa { \GrA{\alpha} } \nc \GRB[1] { \Gr_B^{(#1)} } %% Gr_B \nc \GRBN { \GRB{N} } \nc \GrB[1] { \Gr_B(#1) } \nc \GrBb { \GrB{\beta} } \nc \GRMB[1] { \Gr_{MB}^{(#1)} } %% Gr_{MB} \nc \GRMBN { \GRMB{N} } \nc \GrMB[1] { \Gr_{MB}(#1) } \nc \GrMBb { \GrMB{\beta} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\dfrac#1#2{{\displaystyle\frac{#1}{#2}}} \usepackage[hypertex]{hyperref} \headsep 10mm \oddsidemargin 0in \evensidemargin 0in \begin{document} \vspace{-8ex} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% Title %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \title{{\textsf {QUANTUM FIELD THEORY}} \\for mathematicians} \author{ E.~Horozov \thanks{E-mail: horozov@fmi.uni-sofia.bg and horozov@math.bas.bg} \\ \hfill\\ \normalsize \textit{Faculty of Mathematics and Informatics},\\ \normalsize \textit{Sofia University "St. Kliment Okhridski"}, \\ \hfill\\ and \\ \hfill\\ \normalsize \textit {Institute of Mathematics and Informatics, }\\ \normalsize \textit{ Bulg. Acad. of Sci., Acad. G. Bonchev Str., Block 8, 1113 Sofia, Bulgaria } } \date{} \maketitle \break %%%%%%%%%%%%%%%%%%%% Introduction %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \tableofcontents \setcounter{section}{-1} \break \sectionnew{Introduction} From the times of Newton Physics and Mathematics have lived most of the time in symbiosis. Physics has supplied Mathematics with deep problems. On its hand Mathematics has developed a language for writing the physical problems and laws and tools for solving the problems posed by physicists. Only for some part of 20-th century there was the fashion of "pure mathematics", which means the neglect of any motivation of mathematical research but the intrinsic mathematical problems. For some time this has been maybe useful for the advance of abstract algebraic geometry, number theory, etc. But after some really fruitful years for Mathematics there came the new marriage -- Mathematics and Physics once again became very close. Now there is a new ingredient in their relations -- Physics started to supply not only ideas but also intuition and tools for {\it posing and solving mathematical problems}. Let us mention here the problem of computing the intersection numbers of Chern classes on moduli spaces of Riemann surfaces \cite{Kon1, Wit1}, knot invariants, mirror symmetry, etc. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Phenomenology} \vspace{2cm} \begin{flushright} \begin{minipage}[l][6cm][r]{6cm}{\scriptsize} Thirty one years ago Dick Feynman told me about his "sum over histories" version of quantum mechanics. "The electron does anything it likes", he said. It goes in any direction with any speed, forward or backward in time, however it likes, and then you add up the amplitudes and it gives you the wave function." I said to him "You are crazy." But he wasn't. \vspace{0.5cm} \begin{flushright} --F. J. Dyson \end{flushright} \end{minipage} \end{flushright} The title of this subsection is a little bit misleading. Here we present only one simple (but very important) experiment whose goal is to justify the introduction of path integrals in physics. It is taken from R. Feynman's numerous popular lectures (see e.g. \cite{Fey1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Mathematical view on QFT} Before presenting a more complete account on the path integral method we would like to explain in few words some of the ideas on which it rests. The Feynman path (or more general -- functional) integrals are integrals depending on parameters and the "integration" is carried on infinite-dimensional spaces. First we are going to study integrals of the Feynman type but on finite-dimensional spaces. They should not be considered only as toy models for the real QFTs. They are a main ingredient in their study. We are going to introduce the famous Feynman graphs that help to express in a simple way their asymptotic expansions. With their help we are going {\it to define} the Feynman integrals in the cases relevant for QFT. The 0-dimensional QFT being a powerful mathematical tool has a lot of beautiful applications to areas far from QFT, e.g. -- topology of moduli spaces of Riemann surfaces. From mathematical point of view QFT studies "integrals" that are defined as follows. Let $\Sigma$ and $N$ be manifolds with Riemannian or pseudo-Riemannian metric. We shall denote by $Map(\Sigma, N)$ the set of all smooth (= infinitely differentiable) maps from $\Sigma$ to $N$. Let us also have an action function (or rather functional) $S(\phi)$ on $\phi \in Map(\Sigma, N)$. Let $\-h$ be a small constant (Planck's constant). We will be interested in the following object (including giving sense to it): \beq \int_{Map(\Sigma, N)} V(\phi) \exp(\frac{-S(\phi)}{\hbar}) D[\phi] \label{2.1} \eeq Here $V(\phi)$ is "an insertion function" in physicist's language. This is a smooth function on $Map(\Sigma, N)$, whose meaning will be explained later. The function $\exp(\frac{-S(\phi)}{\hbar})$ has the meaning of probability amplitude of the map $\phi \in Map(\Sigma, N)$ to the integral. The set of the objects $\big(\Sigma, N,S(\phi), \phi \in Map(\Sigma, N)\big)$ is called by physicists "theory". In the case when $V \equiv1$ the above integral \eqref{2.1} is called {\it partition function} of the theory and is denoted by \beq Z^E=\int_{Map(\Sigma, N)} \exp(\frac{-S(\phi)}{\hbar}) D[\phi] \label{2.2} \eeq The superscript "E" means that the theory is {\it "euclidean"}, e.i. the manifold $\Sigma$ is Riemannian. When $\Sigma$ is pseudoeucledian manifold with Lorentzian metric (of signature (-,+,+,+)) we call the theory {\it relativistic QFT}. The first coordinate is reserved for the time. In that case we replace the sign $(-)$ with the imaginary unit $i$: \beq Z^M=\int_{Map(\Sigma, N)} \exp(\frac{iS(\phi)}{\hbar}) D[\phi] \label{2.3} \eeq In this case the theory is {\it Minkovskian QFT}, the letter $M$ designating this fact. We are going to start with $0$-dimensional theory. Of course here there is no time or spacial coordinates. Let us start with the case when $\Sigma$ is one point and $N$ is the real line. Now the set $Map(\Sigma, N)$ consists of all real constants, e.i. it is the real line. The partition function becomes \beq \int_{-\infty}^{\infty} e^{-S(x)/{\hbar}}dx \label{2.4} \eeq This integral is studied by {\it the method of the steepest descent}, which will be explained in one of the next sections. The physicists are in particular interested in the integral \eqref{2.3} when the manifold $\Sigma$ has dimension $\geq 3$. As a rule its value cannot be computed explicitly. So they (Feynman) have invented an algorithm to "find" its {\it asymptotic expansion} in $\hbar$. From mathematical point of view even the definition needs clarification. One reasonable definition is to define the integral \eqref{2.3} just by a series (asymptotic, not convergent!), whose members are indexed by different types of graphs. (This is the reason why we put "find" in quotation marks.) But this is not the end. The problem is that the members of the series are defined via integrals that are not well defined. Then there comes a painful procedure to put meaning in each of this integrals ({\it renormalization}). The methods of renormalization are found by physical intuition but further the predictions are confirmed by experiments. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \sectionnew{QFT in dimension $0$ -- first Feynman graphs} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% In this chapter we are going to consider Feynman integrals for $0$-dimensional manifolds $\Sigma$, i.e. when $\Sigma$ consists of finite number (say $d$) of point. This is reason for the title of the chapter. In this case the set of all mappings from $\Sigma$ to $\Rset$ is just the vector space $V=\Rset^d$ and the integrals become \beq \int_V e^{-S(x)/{\hbar}}dx. \label{1.0} \eeq \subsection{Free theory} In quantum field theory the case when $S$ contains only quadratic terms is called {\it free theory}. We are going to consider the theory as a perturbation of the free theory Then the function $S$ is given in the form $S=B(x,x) + \sum_{m\geq 3}g_mB_m(x,x,\ldots ,x)$ where $B_m(x,x,\ldots ,x)$ are {\it the interactions} and $g_m$ are formal parameters. %%%%%%%%%%%%%%%%%%%%%%%%%Da se preraboti-11.03.09%%%%%%%%%%%%%%%%%%%%%%%%%!!!!! The simplest example of an integral of the form \eqref{2.4} is the Gaussian integral: \ben \int_{-\infty}^{\infty}e^{-\frac{1}{2}ax^2}dx = \sqrt{\frac{2\pi}{a}}. \een Its multidimensional generalization is defined via a symmetric quadratic positive-definite form $B(x,x)$ defined on a $d$-dimensional space $V$ in terms of a positive-definite symmetric matrix $B(x,y)= (Bx,y)$. Then the integral we want to study is \beq \int_Ve^{-\frac{1}{2}(Bx,x)}dx. \label{1.1} \eeq By a change of variables $ x\rightarrow Sx$ where $S \in SO(d)$ we can diagonalize the matrix $B$. Then the integral \eqref{1.1} is easily computed: \beq \int_Ve^{-\frac{1}{2}(Bx,x)}dx = \sqrt{\frac{(2\pi)^d}{\det B}}. \label{1.2} \eeq Sometimes it is useful in to consider slightly more general integrals, namely -- with a linear term in the exponent: \beq Z_b = \int_V e^{-\frac{1}{2}(Bx,x) + (b,x)}dx \label{1.2}. \eeq It is easy to check that \beq Z_b= (2 \pi)^{d/2}(\det B)^{-1/2} e^{\frac{1}{2}(b,B^{-1}b)} = Z_0e^{\frac{1}{2}(b,B^{-1}b}) \label{2.12} \eeq We also will be interested in integrals with insertions \beq \int_V P(x) e^{-\frac{1}{2}(Bx,x)} dx. \label{1.3} \eeq To compute the above integral it is enough to consider the case when the polynomial $P(x)$ is homogeneous and moreover that it is a product of linear forms $l_1(x)l_2(x)\ldots l_{2m}(x)$. Such an integral is a major object in quantum physics (and not only there!) and is called {\it correlation function} or {\it correlator}. It is denoted as follows \beq <l_1,l_2,\ldots,l_m> = \frac{1}{Z_0}\int_V l_1(x)l_2(x)\ldots l_{m}(x)e^{-\frac{1}{2}B(x,x)}dx, \label{1.4} \eeq The correlators can be computed by using the integral $Z_b$ as follows. First notice that \ben \frac{\partial}{\partial b_j}Z_b= \int_V e^{-\frac{1}{2}(Bx,x) + (b,x)}x_jdx \een Then for any product of coordinate functions $x^{i_1}x^{i_2}\ldots x^{i_k}$, not necessarily different, we obtain: \ben \frac{\partial}{\partial b_{i_1}}\ldots\frac{\partial}{\partial b_{i_k}} Z_b= \int_V e^{-\frac{1}{2}(Bx,x) + (b,x)}x^{i_1}x^{i_2}\ldots x^{i_k}dx. \een From this formula we obtain the correlator \beq <x^{i_1},x^{i_2},\ldots,x^{i_m}> = \frac{1}{Z_0} \frac{\partial}{\partial b_{i_1}}\ldots\frac{\partial}{\partial b_{i_k}} Z_b|_{b=0} = \frac{\partial}{\partial b_{i_1}}\ldots\frac{\partial}{\partial b_{i_k}}e^{\frac{1}{2}(b,B^{-1}b)}_{|b=0} \label{3a.3} \eeq In particular the two-point correlation functions are given by {\it the matrix elements of $B^{-1}$}: \ben <x^i,x^j> = (B^{-1})_{|ij}. \een We can use these results for more general functions and even for formal power series $f_1,\ldots,f_m$ to obtain \bpr{4.1} The correlator $<f_1,\ldots,f_m>$ is given by the formula \ben <f_1,f_2,\ldots,f_m> = f_1( \frac{\partial}{\partial b})\ldots f_m(\frac{\partial}{\partial b} ) \big(e^{\frac{1}{2}(b,B^{-1}b)}\big)_{|b=0}. \een \epr \proof The formula for the monomials is \eqref{3a.3}. The general formula is obtained as linear combination of the monomial formulas. \qed From this we obtain a simple but very important combinatorial theorem, known to physicists as {\it Wick's Lemma}. Before formulating it we will introduce some combinatorics. Consider the set $\{1,2, \dots ,2m\}$. A pairing of this set is a partition $\sigma$ of the set into $m$ disjoint pairs. Let us denote the set of all pairings of the above set by $\Pi_m$. It is known that $|\Pi_m|= \frac{(2m)\,!}{2^m \, m \, !}$. Any $\sigma \in \Pi_m$ can be considered as a permutation of $\{1,2, \dots ,2m\}$ without fixed points and such that $\sigma^2 = 1$. Any pair consists of an element $i$ and its image $\sigma (i)$. Now we are ready to formulate the Wick's lemma. \bth{3.1} (Wick's lemma) \beq < l_1\ldots l_{m}>= \begin{cases} \sum_{\sigma \in \Pi_m} \prod_{i \in \{1,\ldots, m\}/\sigma} <l_i,l_{\sigma(i)}> & \textrm {if $m$ is even}\\ 0 & \textrm {if $m$ is odd} \end{cases}. \label{3a.4} \eeq \eth \proof As before we shall first prove the theorem for coordinate functions. In this case our formula takes the form \ben <x^{i_1},\ldots, x^{i_m}>= \begin{cases} \sum_{\sigma \in \Pi_m} \prod_{i \in \{1,\ldots, m\}/\sigma} <x^{i}, x^{\sigma(i)}> & \text{if $m$ is even}\\ 0 & \textrm{if $m$ is odd} \end{cases}. \een By \eqref{3a.3} we need to compute the derivatives: \ben \frac{\partial}{\partial b_{i_1}}\ldots\frac{\partial}{\partial b_{i_k}}e^{\frac{1}{2}(b,B^{-1}b)}. \een Let us do this computation by induction. We have \ben \partial_{i} e^{\frac{1}{2}(b,B^{-1}b)} = \sum_j (B^{-1})_{ij}b_je^{\frac{1}{2}(b,B^{-1}b)}. \een It is clear that applying the next derivative $\partial_{k}$ produces by Leibnitz' rule: \ben (B^{-1})_{ik}e^{\frac{1}{2}(b,B^{-1}b)}+ Q_{ik}(b)e^{\frac{1}{2}(b,B^{-1}b)}= P_{ik}(b)e^{\frac{1}{2}(b,B^{-1}b)}, \een where $Q$ is some homogenous polynomial of degree two and $P$ has only even power terms, the free term $(B^{-1})_{ik}$ giving the result in this case. In general we proceed in the same way. Denote by $P_{i_1\ldots i_s}$ the corresponding polynomial in $b$ (but when it is clear we will drop the indeces): $$\frac{\partial}{\partial b_{i_1}}\ldots\frac{\partial} {\partial b_{i_s}}e^{\frac{1}{2}(b,B^{-1}b)} = P(b)e^{\frac{1}{2}(b,B^{-1}b)}.$$ Each new application of a derivative $\partial_j$ has the following effect on $P$ \ben \partial_j\big(P(b)e^{\frac{1}{2}(b,B^{-1}b)}\big) = \big(\partial_j P(b)\big) e^{\frac{1}{2}(b,B^{-1}b)}+ P(b)\big(\partial_je^{\frac{1}{2}(b,B^{-1}b)}\big), \een i.e. $P \rightarrow \big(\partial_j + \sum_m (B^{-1})_{jm}b_m\big) P(b)$. Notice that the function $P_{i_1\ldots i_s}$ is either even or odd depending on the number $s$. This proves the formula for odd $m$. At the same time we obtained a formula for $P_{i_1\ldots i_s}$: \ben P_{i_1\ldots i_s}= \big(\partial_{i_1} + \sum_m (B^{-1})_{{i_1}m}b_m\big)\ldots \big(\partial_{i_s} + \sum_m (B^{-1})_{{i_s}m}b_m\big)\cdot 1. \een From this formula we see that the free term consists of sum of products of the type $(B^{-1})_{{l_1}{m_1}}\ldots (B^{-1})_{{l_p}{m_p}}$, where $2p=s$ and ${l_j}{m_j}$ is a pair from the set of indices $i_1,\ldots i_s $. Moreover, each pair is present exactly once. The general case can be obtained using linearity as above. \qed Notice that each summand in the formula can be represented by simple graphs. For each $\sigma$ and each pair $(ij) \in \sigma$ draw an unoriented subgraph with two vertices -- $i$ and $j$ and a wedge connecting them. The disconnected union of these subgraphs is the desired graph $\Gamma_{\sigma}$, corresponding to the partition $\sigma$. Then our sum \eqref{3a.4} becomes {\it sum over the graphs $\Gamma_{\sigma}$}. \vspace{2cm} $\quad \quad \quad \quad \quad \quad $ \xymatrix{1 \ar[r]& 2 & 1\ar[d]& 2\ar[d] &1\ar[rd] &2 \ar[ld]\\ 3 \ar[r]& 4 & 3 & 4 & 3 & 4 \\ } \bigskip \begin{center} Figure 1. $\quad $ \end{center} \vspace{2cm} Although this is just change of notation we are going to use widely it in the computations involving general action, i.e. when $S$ is the perturbed function \ben S=\frac{B(x,x)}{2} + \sum_{r\geq 3} \frac{U_r(x,\ldots, x)}{r!} \een \subsection{Steepest descent and the stationary phase methods} The method of steepest descent gives the asymptotics of integral of the type \eqref{2.4}. \bth{2.1} Let the functions $f(x)$ and $g(x)$ are smooth functions defined on an interval $[a,b] \in \Rset $. Assume that the function $f(x)$ has a unique global minimum at a point $c\in [a,b]$ and $f^{''}(c) > 0$. Then the integral \beq \int_a^b g(x)e^{-f(x)/\hbar} dx \nn \eeq has the following asymptotic expansion: \beq \int_a^b g(x)e^{-f(x)/\hbar} dx = \hbar^{1/2} e^{-f(c)/\hbar} I(\hbar), \label{2.6} \eeq where $I(\hbar)$ is continuous function on $(0,\infty)$, which extends in $0$ as \beq \lim_{\hbar \rightarrow 0} I(\hbar) = \sqrt{\pi} \frac{g(c)}{\sqrt{{f^{''}(c)}}}. \eeq \eth \proof To simplify slightly notations we can consider that $c=0$. Then we are going to cut the singular point from the integration region, i.e. we define the integral over a small neighborhood of $0$. This we will do as follows. Take a small real number $\varepsilon$ satisfying $1/2 > \varepsilon > 0$ and define $I_1(\hbar)$ by the equation \ben \hbar^{1/2}e^{-f(0)/\hbar} I_1 = \int_{-\hbar ^{\frac{1}{2} - \ep}} ^{\hbar ^{\frac{1}{2} - \ep}} g(x)e^{-f(x)/\hbar} dx \een Then it is clear that the difference $|I(\hbar) - I_1(\hbar)|$ decays faster than $\hbar^N$ for any $N$. So it suffices to show that $I_1(\hbar)$ has the asymptotics \eqref{2.6}. Let us introduce new variable $y$ by $x=y\hbar$. Then the function $I_1$ can be written as \beq I_1 = \int_{-\hbar^{\ep}}^{\hbar^{\ep}} g(y\sqrt{\hbar})e^{(f(0)-f(y\sqrt{\hbar}))/\hbar} dy \nn \eeq Now it is clear that the integrand is a smooth function in $\sqrt{\hbar}$. Then we can replace $I_1(\hbar)$ by $I_2(\hbar)$ which is the Taylor expansion of $I_1(\hbar)$ modulo $N$. Then $|I_1(\hbar)-I_2(\hbar)|\leq C\hbar^N$. At the end we replace $I_2(\hbar)$ by $I_3(\hbar)$ which is the same integral but with limits from $-\infty$ to $\infty$. Then the difference $I_2(\hbar)- I_3(\hbar)$ is rapidly decaying. Hence it is enough to show that the $I_3(\hbar)$ has Taylor expansion in $\hbar^{1/2}$. In fact $I_3(\hbar)$ is a polynomial in $\hbar^{1/2}$. Also the odd powers of $\hbar^{1/2}$ vanish as the corresponding coefficients are integrals of odd functions. So we find that the Taylor expansion exists. Let us compute the value of $I_3(\hbar)$. We have \beq I_3(0) = g(0)\int_{-\infty}^{\infty} e ^{-\frac{f^{''}(0)y^2}{2}}dy, \nn \eeq Using the value of the Gaussian integral \eqref{1.1} we get the desired result. \qed \bex {2.1} Consider the integral $$ \int_{-\infty}^{\infty}e^{-\frac{x^2+x^4}{2\hbar} }dx = \sqrt{2\pi}\hbar^{1/2} I(\hbar) $$ Then the function $I(\hbar)$ is given by $$ I(\hbar) = \frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-\frac{y^2+\hbar \,y^4}{2} }dy $$ The integral has the asymptotic expansion $$ I(\hbar) = \sum_{n=0}^{\infty}a_n\hbar^n, $$ where $$ a_n = \frac{(-1)^n}{\sqrt{2\pi}} \int_{-\infty}^{\infty}e^{-\frac{y^2}{2} }\frac{y^{4n}}{2^{2n}} n \,!dy $$ \eex The method of {\it stationary phase} is slightly more complicated and uses the Fresnel integral \beq \int_{-\infty}^{\infty} e^{ix^2/2} dx = \sqrt{ 2\pi } e^{\pi i /4} \nn \eeq instead of Gaussian integral. We are going only to formulate the result. \bth{2.2} Assume that $f$ has unique critical point in $c \in (a,b)$ with $f^{''}(c) \neq 0$ and $g$ vanishes with all its derivatives at the ends $a$ and $b$. Then \beq \int_a^b g(x)e^{if(x)/\hbar} dx = \hbar^{1/2} e^{if(c)/\hbar} I(\hbar), \label{2.7} \eeq where $I(\hbar)$ extends to a smooth function on $[0,\infty)$, such that $$I(0)= \sqrt{2\pi} e^{ sign(f(c)) i\pi/4} \frac{g(c)}{|\sqrt{f^{''}}|}$$. \eth The methods of steepest descent and the stationary phase easily extend to the multidimensional case. We introduce the following notation. By $V$ we denote a real vector space of dimension $d$ and by $B$ -- a closed $d$-dimensional box in it. We assume that the functions $f(x)$ and $g(x)$ are defined on $B$ and smooth. \bth{2.4a} Let the function $f$ have a unique global minimum at a point $c \in B$ and the form $D^2 f(c)$ be positive-definite. Then \beq \int_Bg(x)e^{-f(x)/\hbar} = \hbar^{d/2} e^{-f(c)/\hbar}I(\hbar), \label{2.8} \eeq where $I(\hbar)$ extends as smooth function on $[0,\infty)$, such that \beq I(0)= (2\pi)^{d/2}\frac{g(c)}{\sqrt{\det D^2 f(c)}}. \nn \eeq \eth In a similar manner we formulate the stationary phase method. \bth{2.4} Let the function $f$ have a unique global minimum at a point $c \in B$ and let the form $D^2 f(c)$ be non-degenerate. Then \beq \int_Bg(x)e^{if(x)/\hbar} = \hbar^{d/2} e^{if(c)/\hbar}I(\hbar), \label{2.9} \eeq where $I(\hbar)$ extends as smooth function on $[0,\infty)$, such that \ben I(0)= (2\pi)^{d/2}e^{{\pi i \sigma}/4}\frac{g(c)}{\sqrt{\det D^2f(c)}}. \nn \een Here $\sigma$ is the signature of the symmetric bilinear form $D^2f(c)$. \eth Notice that the multidimensional Gaussian and Fresnel integrals become respectively \beq \int_V e^{-B(x,x)}= (2 \pi)^{d/2}(\det B)^{-1/2} \label{2.10} \eeq for positive-definite form B and \beq \int_V e^{-B(x,x)}= (2 \pi)^{d/2}e^{\pi i \sigma}|\det B|^{-1/2} \eeq for nondegenerate form B. We leave the details of the proofs for the reader. \subsection{Definitions of Feynman graphs } We aim to compute the entire asymptotic expansion of integrals of the form: \ben \int_V l_1\ldots l_N e^{-S(x)/\hbar} dx. \een in terms of some combinatorics. The result will be useful as it gives the model to define Feynman integrals in physically meaningful theories. Here \ben S= \frac{1}{2}(Bx,x) + \sum \frac{g_jU_j(x)}{n!} \een The functions $U_j(x)$ are homogeneous polynomials of degree $j$q i.e. symmetric $j$-tensors. The integral is a formal power series in $\hbar$ and $g_m$ in a form that will be explained below. For simplicity assume that $c=0$ and $S(0)=0$. The expansion will be done in terms of {\it Feynman diagrams}, which are a major object in quantum field theory. Also we make the change of the variables $x/\sqrt{\hbar} \rightarrow x$. We use the same variables. The correlator above becomes: \ben \hbar^{(N+d)/2} \int_V l_1\ldots l_N e^{-(Bx,x) + \sum \hbar^{j/2}\frac{g_jU_j(x)}{n!}} dx. \een In what follows we are going to drop the factor $\hbar^{(N+d)/2}$. We expand the exponential function above as follows: \ben e^{-\frac{1}{2}(Bx,x) + \sum \frac{\hbar^{j/2}g_jU_j(x)}{j!} }= e^{-\frac{1}{2}(Bx,x)}\Big(1 + \frac{1}{1!} \sum \hbar^{j/2}\frac{g_jU_j(x)}{j!} + \\ \frac{1}{2!} \big(\sum \hbar^{j/2}\frac{g_jU_j(x)}{j!} \big)^2 + \ldots + \frac{1}{k!}\big(\sum \hbar^{j/2}\frac{g_jU_j(x)}{j!} \big)^k + \ldots \Big) \een The correlator becomes \ben \int_V l_1\ldots l_Ne^{-\frac{1}{2}(Bx,x)}\Big(1 + \frac{1}{1!} \sum_j \hbar^{j/2}\frac{g_jU_j(x)}{j!} + \frac{1}{2!} \big(\sum \hbar^{j/2}\frac{g_jU_j(x)}{j!} \big)^2 + \ldots \Big)dx=\\ \sum_{n=0}^{\infty}\int_V l_1\ldots l_Ne^{-\frac{1}{2}(Bx,x)} \frac{1}{n!} \big(\sum_{j=3}^{\infty} \sum \hbar^{j/2}\frac{g_jU_j(x)}{j!} \big)^n + \ldots \Big)dx. \een Also we expand the $n$-th power of the infinite sum $\big(\sum_{j=3}^{\infty} \sum \hbar^{j/2}\frac{g_jU_j(x)}{j!} \big)^n$. This is the formal series we are interested in. In the case when there are no functionals $l_j$ the corresponding function is called {\it partition function}. Explicitly it is \ben Z_U = \int e^{-\frac{1}{2}(Bx,x) + \hbar^{1/2} U(x)}dx \een We have the obvious \bpr{3.2} \ben Z_U=Z_0e^{\hbar^{1/2} U(\frac{\partial}{\partial b})} e^{\frac{1}{2} (b, B^{-1}b)}|_{b=0}. \een \epr Next we define the correlation function of $f_1,\ldots, f_m$ with respect to the above perturbed action: \ben <f_1,\ldots,f_m>_U= \frac{1}{Z_U}\int f_1\ldots f_me^{-\frac{1}{2}(Bx,x) + \hbar^{1/2} U(x)} dx \een And again we have \bpr{3.5} \ben <f_1,\ldots,f_m>_U= \frac{Z_0}{Z_U}e^{\hbar^{1/2} U(\frac{\partial}{\partial b})}f_1(\frac{\partial}{\partial b})\ldots f_m(\frac{\partial}{\partial b}) e^{\frac{1}{2} (b,B^{-1}b)}|_{b=0}. \een \epr We want to express the correlator in terms of Feynman's graphs, which we define below. We denote by $G_{\geq 3}(N)$ the set of isomorphism classes of graphs with $N$ $1$-valent external vertices, labeled by $1,\ldots , N$ and finite number of unlabeled internal vertices of valency $\geq 3$. For each graph $\Gamma$ we define a Feynman amplitude of $\Gamma$ by the following rules: \begin{enumerate} \item Put the covector $l_j$ at the the $j$-th external vertex. \item Put the tensor $-g_mU_m$ at each $m$-valent vertex. \item Take the contraction of the tensors along the edges of $\Gamma$, using the bilinear form $B^{-1}$. The result will be a number denoted by $F_{\Gamma}$. This is the {\it Feynman amplitude}. \end{enumerate} \subsection{Feynman's theorem} \bth{3.12} (Feynman) The correlation function $<l_1\ldots l_N>$ is given by the asymptotic series: \beq <l_1\ldots l_N> = Z_0\sum_{\Gamma\in G_{\geq 3(N)}} \frac{\hbar^b(\Gamma)}{|Aut(\Gamma)|}F_{\Gamma}(l_1,\ldots,l_N) \label{3.13} \eeq \eth We will give another version of this theorem, easier to prove. Before that let us introduce some notation. Let $\bf{n} = (n_0,n_1,\ldots)$ be a sequence of nonnegative integers, only a finite number of which are nonzero. Let $G(\bf{n})$ be the set of isomorphism classes of graphs with $n_0$ $0$-valent vertices, $n_1$ $1$-valent vertices, etc. The version of Feynman's theorem that we have in mind goes as follows. \bth{3.13} The partition function has the following asymptotic expansion \ben Z= Z_0{\sqrt{\det{B}}}\sum_{\bf{n}} \big(\prod_i g_i^{n_i}\big) \sum_{\Gamma \in G(\bf{n})} \frac{\hbar^b(\Gamma)}{|Aut(\Gamma)|}F_{\Gamma}(l_1,\ldots,l_N) \label{3.14} \een \eth \proof First expand the exponential function in Taylor series. The partition function becomes \ben Z=\sum_{\bf{n}}Z_{\bf{n}}, \een where \beq Z_{\bf{n}}=\int_Ve^{-\frac{1}{2}B(x,x) }\prod_i \frac{g_i^{n_i}}{(i!)^{n_i}n_i!} \big( -\hbar^{i/2 -1} U_i(x,\ldots ,x)^{n_i}dx\big) \label{3.14a} \eeq We can write the terms $U_i$ as sums of products of linear functions. Then we can apply Wick's lemma. It gives that each $Z_{\bf{n}}$ can be computed as follows. \begin{enumerate} \item Define {\it a flower} -- a graph with one vertex and $i$ outgoing edges (see fig. 1). Attach it to the tensor$U_i$. \vspace{1cm} \begin{tikzpicture}\filldraw [black] (0,0) circle (2pt); \draw (0,0)..controls (1,0) ..(1.4,0) ; \draw (0,0)..controls (0.7,0.3) ..(1.4,0.6) ; \draw (0,0)..controls (0.7,-0.3) ..(1.4,-0.6) ;\draw (1.45,0) circle (2pt);\draw (1.5,0.6) circle (2pt);\draw (1.5,-0.6) circle (2pt); \end{tikzpicture} \vspace{1cm} \begin{flushleft} \bf{Figure 2.} \end{flushleft} \item Consider the set $T$ of these outgoing edges (see fig.) and for any pairing of this set, consider the corresponding contraction of the tensor $-U_i$ using the form $B^{-1}$. This will give the a number $F_{\sigma}$ corresponding to this pairing. \end{enumerate} We can visualize a pairing $\sigma$ by drawing its elements as points and connecting the points in each pair them by an edge (see fig .). In this way we obtain an unoriented graph $\Gamma=\Gamma_{\sigma}$. The number $F_{\sigma}$ is called {\it an amplitude} of the graph $\Gamma$. \vspace{1cm} \begin{tikzpicture}\filldraw [black] (0,0) circle (2pt); \draw (0,0)..controls (1,0) ..(1.4,0) ; \draw (1.4,0) circle (2pt); \filldraw [black] (0,2) circle (2pt); \draw (0,2)..controls (1,2) ..(1.4,2) ; \draw (0,2)..controls (0.7,2.3) ..(1.4,2.6) ; \draw (0,2)..controls (0.7,1.7) ..(1.4,1.4) ;\draw (1.4,2) circle (2pt);\draw (1.4,1.4) circle (2pt);\draw (1.4,2.6) circle (2pt); \filldraw [black] (6,0) circle (2pt); \draw (4.4,0)..controls (5,0) ..(6,0) ; \draw (4.4,0.6)..controls (5.1,0.3) ..(6,0) ; \draw (4.4,-0.6)..controls (5.1,-0.3) ..(6,0) ;\draw (4.4,0) circle (2pt);\draw (4.4,0.6) circle (2pt);\draw (4.4,-0.6) circle (2pt); \filldraw [black] (6,2) circle (2pt); \draw (4.4,2)..controls (5,2) ..(6,2) ; \draw[step= 0.2, black, dashed](1.4,0)..controls (2,-0.3) .. (4.4,-0.6); \draw[step= 0.2, black, dashed](1.4,1.4)..controls (3.6,1) .. (4.4,0); \draw[step= 0.2, black, dashed](1,2)..controls (3.6,1.8) .. (4.4,0.6); \draw[step= 0.2, black, dashed](4.4,2)..controls (3.6,2.1) .. (1.4,2.6); \end{tikzpicture} \vspace{2cm} \begin{flushleft} \bf{Figure 3.} \end{flushleft} \vspace{1cm} \begin{tikzpicture}\filldraw [black] (0,0) circle (2pt); \draw (0,0)..controls (1,0) ..(1.4,0) ; \draw (0,0)..controls (0.7,0.3) ..(1.4,0.6) ; \draw (0,0)..controls (0.7,-0.3) ..(1.4,-0.6) ;\draw (1.45,0) circle (2pt);\draw (1.5,0.6) circle (2pt);\draw (1.5,-0.6) circle (2pt); \filldraw [black] (0,2) circle (2pt); \draw (0,2)..controls (1,2) ..(1.4,2) ; \draw (0,2)..controls (0.7,2.3) ..(1.4,2.6) ; \draw (0,2)..controls (0.7,1.7) ..(1.4,1.4) ;\draw (1.4,2) circle (2pt);\draw (1.4,1.4) circle (2pt);\draw (1.4,2.6) circle (2pt); \filldraw [black] (6,0) circle (2pt); \draw (4.4,0)..controls (5,0) ..(6,0) ; \draw (4.4,0.6)..controls (5.1,0.3) ..(6,0) ; \draw (4.4,-0.6)..controls (5.1,-0.3) ..(6,0) ;\draw (4.4,0) circle (2pt);\draw (4.4,0.6) circle (2pt);\draw (4.4,-0.6) circle (2pt); \filldraw [black] (6,2) circle (2pt); \draw (4.4,2)..controls (5,2) ..(6,2) ; \draw (4.4,2.6)..controls (5.1,2.3) ..(6,2) ; \draw (4.4,1.4)..controls (5.1,1.7) ..(6,2) ;\draw (4.4,2) circle (2pt);\draw (4.4,1.4) circle (2pt);\draw (4.4,2.6) circle (2pt);\node at (0,0) {$a$}; \draw[step= 0.2, black, dashed](1.4,0)..controls (2,0) .. (4.4,0); \draw[step= 0.2, black, dashed](1.4,-0.6)..controls (2,-0.4) .. (4.4,-.6); \draw[step= 0.2, black, dashed](1.4,2)..controls (1.6,1.8) .. (1.4,1.4); \draw[step= 0.2, black, dashed](4.4,2.6)..controls (4,2.2) .. (4.4,2); \draw[step= 0.2, black, dashed](4.4,1.4)..controls (4,.8) .. (1.4,0.6); \draw[step= 0.2, black, dashed](4.4,0.6)..controls (4,.8) .. (1.4,2.6); \end{tikzpicture} \vspace{2cm} \begin{flushleft} \bf{Figure 3.} \end{flushleft} \vspace{1cm} \begin{tikzpicture}\filldraw [black] (0,0) circle (2pt); \draw (0,0) ..controls (0.3,0)..(1,0); \draw (1.7,0) circle (20pt); \draw (-0.55,0) circle (15pt); \filldraw [black] (1,0) circle (2pt); \filldraw [black] (2.4,0) circle (2pt); \draw (2.4,0) ..controls (3,0)..(3.3,0);\filldraw [black] (3.3,0) circle (2pt); \draw (3.8,0) circle (15pt); \end{tikzpicture} \vspace{.5cm} \begin{flushleft} \bf{Figure 4.} \end{flushleft} It is easy to see that each graph with $n_i$ $i$-valent vertices can be obtained in this way. But it can be obtain many times and need to count this number. This means that we need to count how many $\sigma$-s can produce a fixed graph $\Gamma$. For this we need to find the group $G$ of permutations which preserve the "flowers". It consists of the following elements: (1) Permutations which permute the set of flowers of fixed valency; (2) Permutations which permute the edges of a fixed flower. \bigskip We see that the group $G$ is a semi-direct product $(\prod_i S_{n_i}) \rhd \!\!\!\! < (\prod_i S_i^{n_i})$ where $S_j$ is the permutation group of $j$ elements. Its cardinality $|G|$ is $\prod_i(i!)^{n_i}n_i!$. This is exactly the product of the integers at the denominators in \eqref{3.14a}. The group $G$ acts on the set of all pairings of $T$. The action is transitive on the set $P_{\Gamma}$ of the pairings which produce a fixed graph $\Gamma$. On the other hand the stabilizer of a fixed pairing is $Aut(\Gamma)$. Thus the number of the pairings producing $\Gamma$ is \ben \frac{\prod_i (i!)^{n_i}n_i! }{|Aut(\Gamma)|} \een In this way we obtain a formula connecting the sum of the numbers $F_{\sigma}$ and the sum of the amplitudes with weights: \ben \sum_{\sigma} F_{\sigma}= \sum_{\Gamma} \frac{\prod_i (i!)^{n_i}n_i! }{|Aut(\Gamma)|} F_{\Gamma} \een At the end we will compute the powers of $\hbar$ at the amplitudes. We note that the power of $\hbar$ is given by the number of edges of $\Gamma$ minus the number of vertices, i.e. $b(\Gamma)$. This gives exactly $i/2-1$. This proves the theorem.\qed Now we are going to extract Feynman's theorem. \proof of \thref{3.12}. As in Wick's lemma we can use the symmetry of the correlation function with respect to $l_j$. So it is enough to consider the case of $l_1=l_2=\ldots=l_N=l$. The corresponding correlation function is denoted by $<l^N>$ and is also called expectation value of $l^N$. Let us compute the expectation value $<e^tl>$. Obviously this is the generating function of the expectation values $<l^N>\frac{1}{N!}$. If we put in \thref{3.13} $g_i=1, \,\,i \geq 3$, $g_0=g_2=0$, $g_1=-\hbar t$ and $B_1=l,\,\,B_0=B_2=0$ we get the result.\qed \subsubsection{Sums over connected graphs} Here we are going to show that the reduce the computation of the correlator to a sum only over connected graphs. This is very useful in studies of Feynman's integrals in real physics. We denote the set of connected graphs in $G(\bf{n})$ by $G_c(\bf{n})$. \bth{3.14} The logarithm of the partition function $\ln(Z_U)$ has the following asymptotic expansion: \beq \ln(Z_U) = \sum_{\bf{n}} \prod_i g_i^{n_i} \sum_{\Gamma \in G_c(\bf{n})} \frac{\hbar^{b(\gamma)}}{|Aut(\Gamma)|}F_{\Gamma} \label{3.17} \eeq \eth \proof Denote by $\Gamma_1\Gamma_2$ the disjoint union of two graphs $\Gamma_1$ and $\Gamma_2$. Following this notation we use $\Gamma^n$ for the disjoint union of $n$ copies of $\Gamma$. Thus any graph can be written as $\Gamma_1^{k_1}\ldots\Gamma_l^{k_l}$ with some connected graphs $\Gamma_j$. Then we have $F_{\Gamma_1\Gamma_2}=F_{\Gamma_1}F_{\Gamma_2}$, $b_{\Gamma_1\Gamma_2}=b_{\Gamma_1}+b_{\Gamma_2}$ and $|Aut(\Gamma_1^{k_1}\Gamma_2^{k_2})| = |Aut(\Gamma_1)|^{k_1} k_1! |Aut(\Gamma_2)|^{k_2} k_2!$ After exponentiating \eqref{3.17} and expanding the r.h.s. in Taylor series we find the expression of the partition function, given by \thref{3.13}. \qed \subsection{Computations with Feynman's graphs} \subsubsection{Loop expansions} Note that the number $b({\Gamma})$ in \thref{3.14} is the number of the loops of $\Gamma $ minus $1$. For this reason this expansion is referred to as "loop expansion". Denote by $G^{(j)}_{\bf n}$ the set of graphs from $G_c({\bf n})$ with $j$ loops. Also denote the $j$-loop term of $\ln (Z)$ by \ben \big(\ln(Z)\big)_j = \sum_{\bf{n}} \prod_i g_i^{n_i} \sum_{\Gamma \in G^(j)(\bf{n})} \frac{\hbar^{b(\gamma)}}{|Aut(\Gamma)|}F_{\Gamma} \label{3.18} \een We are especially interested in the $0$-th and the first terms, i.e. in {\it the tree expansion} and {\it the one-loop expansion}. \bth{3.15} (i) The tree expansion of $\ln(Z)$ is given by the value of the action $S$ with minus sign at the singular point: \beq \big(\ln(Z)\big)_0= -S(x_0). \label{3.19} \eeq (ii) The value of $\big(\ln(Z)\big)_1$ is: \beq \big(\ln(Z)\big)_1 = \frac{1}{2}\ln \frac{\det(B)}{\det D^2S(x_0)}. \label{3.20} \eeq \eth \proof It is enough to study the case when $S$ is a polynomial $U=\sum^m_jg_jU_j/j!$. Also assume that the numbers $g_j$ are small enough and that the integration takes place on small box $B$ around $x_0$. Then the function $S$ has a global maximum at $x_0$ and we can apply the method of steepest descent. It gives \ben Z Z_0= \hbar^{d/2}e^{-S(x_0)/\hbar} I(\hbar). \een where \ben I(\hbar) = (2\pi)^{d/2} \sqrt {\frac{1}{\det D^2S(x_0) }} (1+a_1\hbar^{d/2} + \ldots)\,\,\, \textrm{(asymptotically)} \een Using the value of $Z_0 = (2\pi)^{d/2}\hbar^{d/2}(\det B)^{-1/2}$ we find: \ben Z = e^{-S(x_0)/\hbar} \sqrt {\frac{\det B}{\det D^2S(x_0) }}. \een After taking a logarithm this yields that \ben \ln (Z) = -S(x_0)/\hbar +\frac{1}{2} \ln( {\frac{\det B}{\det D^2S(x_0) }})+ O(\hbar) \een which are exactly the desired equalities. \subsubsection{1-particle irreducible diagrams} A powerful method widely used by physicists to compute the partition function is to find a new action $ {S_{{\texttt{eff}}}}_{|0}$, called {\it effective action} such that \ben \ln(S_{\texttt{eff}})_0 = \ln(S) \een Then using the simple formula \eqref{3.19} for $ \ln(S_{\texttt{eff}})_0$ we can find the partition function for the initial action. Before that we need some definitions. \bde{2.4} An edge of a connected graph is called {\it a bridge} if when removed the graph becomes disconnected. A connected graph without bridges is called {\it 1-particle irreducible} (1PI). \ede \begin{tikzpicture}\filldraw [black] (1,0) circle (2pt); \draw (1,0) ..controls (2,0)..(2.4,0); \draw (1.7,0) circle (20pt); \filldraw [black] (1,0) circle (2pt); \filldraw [black] (2.4,0) circle (2pt); \end{tikzpicture} \vspace{.5cm} \begin{flushleft} \bf{Figure 6.} \end{flushleft} The graph on Fig.4 obviously isn't 1-particle irreducible. The graph on Fig.6 is an example of 1PI graph. Note that the 1PI graphs are what in mathematics are known as "2-connected". We are ready to describe the rules for computing of the effective action. We will consider graphs with at least one internal and one external vertices. Such a graph is called 1PI if the graph obtained by removing the external vertices is 1PI. Denote by $G_{1PI}({\bf n},N)$ the set of isomorphism classes of 1PI graphs with $N$ external vertices and $n_i$ $i$-valent vertices. Here the isomorphisms are taken to keep the external vertices fixed. \bth{3.17} The effective action is given by the formula \ben S_{\texttt{eff}}= \frac{(Bx,x)}{2} - \sum_{i\geq 0} \frac{\mathcal{U}_i}{i!}, \een where \ben \mathcal{U}_i(x,x,\ldots,x) = \sum_{\bf{n}}\big(\prod_ig_i\big) \sum_{\Gamma \in G_{1PI}(\bf{n},N)} \frac{\hbar^{b(\Gamma)+1}}{|Aut \Gamma|}F_{\Gamma} (x_*,\ldots, x_*). \een and the functional $x_*\in V^*$ is defined as $x_*(y):=B(x,y)$ \eth Before giving the proof let us make few comments. Write $ S_{\texttt{eff}}$ as a power series: \ben S_{\texttt{eff}} = S + \hbar S_1 + \hbar^2 S_2 + \ldots \een The expression $\hbar^j S_j$ is called $j$-{\it loop correction to the effective action.} The theorem formulated above shows that we can work only with 1PI diagrams. Physicists rarely use other diagrams, see e.g. the cited textbooks. Notice that the 1PI diagrams are considerably less than all the diagrams. \proof of \thref{3.13} We will make use of the following theorem from graph theory (see e.g. \cite{Bol}). \bpr{3.4} Any connected graph can be uniquely represented as a tree (called skeleton), whose vertices are 1PI subgraphs (with external edges) and the edges of the tree are bridges of $\Gamma$. \epr \proof of the proposition. \qed \vspace{1cm} Graph \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad Skeleton \begin{tikzpicture} \draw (-.30,0) ..controls (0.3,0)..(2.7,0);\draw (1,.30) ..controls (1,0.5)..(1,1.3); \draw (0,0) circle (8pt); \draw (1,0) circle (8pt); \draw (2.4,0) circle (8pt); \draw [black] (1,1) circle (8pt); %%%%%%%%%%%%%%%%%%%%%%%%%%%%% \filldraw [black] (5,0) circle (2pt); \draw (5,0) ..controls (5.3,0)..(7.4,0);\draw (6,0) ..controls (6,0.5)..(6,1); \filldraw [black] (6,0) circle (2pt); \filldraw [black] (7.4,0) circle (2pt); \filldraw [black] (6,1) circle (2pt); \end{tikzpicture} \vspace{.5cm} \begin{flushleft} {\bf Figure 7.} The skeleton of a graph.\end{flushleft} \vspace{2cm} \subsubsection{Legendre Transform} In this section we are going to express the effective action in terms of the Legendre transform of the logarithm of the partition function. Consider an action $S$ and perturb it with a linear term: \ben S(b,x)=S(x)- (b,x) \een Consider the corresponding partition function \ben Z_U(b) = \frac{ \int_V e^\frac{-S(x)+(b,x)}{\hbar}dx} {\int_V e^{-(Bx,x)/2}dx} \een Using \thref{3.15} we have \ben \ln(Z_U(b))= -S_{\texttt{eff}}(0,b) \een Let us find the perturbed effective action \ben S_{\texttt{eff}}(x,b). \een \thref{3.14} tells us that $S_{\texttt{eff}}(x,b)$ is given by the expansion in 1PI graphs. One of these graphs is the graph having a single wedge with vertices -- tensors $(b,x)$. The only one of them is a wedge connecting two vertices labeled by $(b,x)$. \sectionnew{Quantum mechanics} \subsection{Preliminaries} There is a dictionary that translates the objects from classical mechanics into the corresponding objects from quantum mechanics. Naturally we start with the phase space $M$. Its analog in quantum mechanics is a Hilbert space $\mathcal{H}$. This Hilbert space here will be the space $L^2(M)$ of functions on the configuration space with integrable square. {\it The observables}, i.e. functions of positions and momenta become self-adjoint operators in this Hilbert space. The eigenvalues and the eigenvectors are interprest as follows. An eigenvalue $a$ of a self-adjoint operator $A$ is the probability to measure an observable $A$ at the eigenstate $|a>$ (= normed eigenvector). In particular, the position $q_j$ translates into the operator $\hat{q_j}$ of multiplication by $q_j$ and the momentum $p_j$ translates into the operator of differentiation $i\hbar \partial_j$. Then we see that the Hamiltonian translates into the Schr\"odinger operator: \beq \hat{H}= \frac{-\hbar^2}{2m}\sum_j \partial^2_{q_j} + V(q) \label{3.1} \eeq The function $V(q)$ is again called potential and obviously $\hat{H}= -\frac{-\hbar^2}{2m}\Delta + V(q)$. The constant $\hbar$ is called Planck's constant. The rule (na\"ive) to write the Schr\"odinger operator is obvious: we put $i\hbar\partial_{x_j}$ instead of $p_j$. Much more important is the analog and the interpretation of the Hamiltonian equations. They read \beq i\hbar\frac{\partial\psi}{\partial t} = \hat{H}\psi \label{3.2} \eeq This is the famous Schr\"odinger equation. It describes a particle (or more particles) under the action of a potential $V$. The unknown function $\psi(x,t)$ is called wave function. Its physical interpretation is that $|\psi(x,t)|^2$ is a probability density, i.e. the probability to find a particle, described by the equation \eqref{3.2} in an infinitesimally small volume $d^3x$ at the point $x$ and the time $t$, is $|\psi(x,t)|^2d^3 x$. The standard method for solving Schr\"odinger equation is by the method of separation of variables. We seek solution of the form $\psi(x,t)= \psi(x) e^{-iEt/\hbar}$ where the constant $E$ means energy. Then the spacial part $\psi(x)$ of the wave function satisfies the time-independent Schr\"odinger equation \beq \hat{H}\psi = E\psi \label{3.3} \eeq The main problem of quantum mechanics is to solve the eigenvalue problem \eqref{3.3}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \vspace{1cm} Unfortunately most of the operators needed in quantum mechanics have no eigenfunctions in $\mathcal{H}$. E.g., the operator $-\frac{i \partial}{\partial_{q}}$, acting in $L^2(\mathbb{R})$, which is basic for quantum physics has no eigenfunction in that space. On the other hand na\"ively one can say that any function of the form $e^{ipq}$ is an eigenfunction with an eigenvalue $p$ in some bigger space. The operator $q$ is even worse; it has no eigenfunction in the class of functions but only in the class of distributions. One standard way to get out of this situation (but not the only one) is to consider the sequence $\mathcal{S}\subset \mathcal{H} \subset \mathcal{S}^*$, where $\mathcal{S}$ is the space of $\mathbb{C}^{\infty}$-functions that decay faster than any polynomial and $\mathcal{S}^*$ is the space of its functionals. \bex{3.4} {\it (Fourier transform.)} Consider the operator $-i\frac{d}{dq}$. Its eigenfunctions are $e^{-ipq}$ with any fixed $p$. The linear functional \beq f(q)\rightarrow \hat{f}(p)\,=\, \int f(q) e^{-ipq}dq, \label{3.4} \eeq where $f$ is a test function, belongs to $\mathcal{S}^*$. The inverse Fourier transform \ben f(q)= \frac{1}{2\pi}\, \int \hat{f}(q) e^{ipq}dq, \een gives the expansion of $f$ in the eigenfunctions of the operator $-i\frac{d}{dq}$. Of course, they do not belong to the Hilbert space. \eex Each physical state is represented by a vector, i.e. a $L^2$-function. We are going use Dirac's "ket" and "bra" notation. By the "ket" $|\psi>$ we (following Dirac) are going to denote the states (vectors in $\mathcal{H}$. Here $\psi$ could, be e.g. an eigenvalue, a vacuum state or any letter denoting some physical object. In a similar way we denote by "bra" $<\phi|$ the dual vector. The scalar product $(\phi,A\psi)$ will be denoted by $<\phi|A|\psi>$ and called {\it matrix element} of $A$. The name comes from the situation when $|\phi>$ and $|\psi>$ are both members of an orthogonal basis of $\mathcal{H}$. In that case $<\phi|A|\psi>$ is really an element of the matrix of $A$ in that basis. Let $\{a\}$ be a complete orthonormal set of eigenvectors of a self-adjoint operator $A$ in $\mathcal{H}$. One can expand any vector $\psi$ as \ben |\psi> = \sum_a|a> <a|\psi> \een i.e. - in Fourier series. This equality will be used quite frequently and referred to as {\it insertion of a complete set of states}. In a general form it reads: \ben \sum_a|a> <a| = \bf{1}, \een where by $\bf 1$ we denote the identity operator in $\mathcal{H}$. Here is an example. \bex{3.2} Let $|\psi>$ be a state. We want to find the average value of the measurements of the observable $A$ at the state $|\psi>$. We have \ben \sum_a a|<a|\psi>|^2 = \sum_a a|<\psi|a><a|\psi>=\\ \sum_a <\psi|A|a><a|\psi>= <\psi|A|\psi> \een \eex The most important observables are the coordinates $q_j$ and the momenta $p_j$. Using their definition \ben \hat{\,q}_j f(q) := q_jf(q), \,\,\, \hat{p}_jf(q) := -i\hbar\frac{ \,d}{d q_j} \een we find that they satisfy the following identities \ben [q_i,q_j]=0,\,\, [p_i,p_j]=0,\,\, [q_i,p_j]=i\delta_{ij}. \een Another important observable is the {\it energy} given by the Hamiltonian $\hat{H}$. Further we are going to skip the hat denoting quantization. We can consider Schr\"odinger equation \eqref{3.1} as a dynamical system in the Hilbert space $\mathcal{H}$. Then we can solve it by the formula: \beq \psi(t)= e^{-itH}|\psi(0)>. \label{3.5} \eeq The evolution is one parameter family of unitary operators $e^{-itH}$. \subsection{Heisenberg picture} Up to now the main role in our discussion was played by the Schr\"odinger equation \eqref{3.1}. This setting is referred to as {\it Schr\"odinger picture}. There is an equivalent quantum mechanical picture, called {\it Heisenberg picture}. The states $|\psi>$ at time $t$ is mapped to $e^{iHt}|\psi>$, and the operators $A$ are mapped to $e^{iHt}A e^{-iHt}$. The operator $e^{iHt}$ is unitary and hence it preserves the scalar products. Notice that all measurable quantities are given by matrix elements, i.e. by scalar products. This shows that we do not change the physical picture. In Schr\"odinger picture the observables do not change and the states change with time. In Heisenberg picture the situation is the opposite -- the observables change by the law \beq \frac{dA}{dt}= -i[A,H], \label{3.6} \eeq (this is obtained by differentiation) but the states stay constant. \subsection{The Harmonic oscillator} In classical mechanics the simplest but very important system is the harmonic oscillator. The importance lies in the fact that roughly speaking all other systems can be considered as sets of connected oscillators. The situation in quantum mechanics and quantum field theory is the same. The classical harmonic oscillator is governed by the Hamiltonian \beq H= \frac{p^2}{2m} + \frac{kx^2}{2} = \frac{p^2}{2m} + \frac{m\omega^2x^2}{2} \label{3.7} \eeq "Quantizing" it gives for the Schr\"odinger operator \beq H=\frac{-\partial_x^2}{2m} + \frac{m\omega^2x^2}{2} \label{3.8} \eeq Here we assume that the Planck constant $\hbar=1$. Our Hilbert space will be $L^2(\Rset)$. The above operator is essentially the Hermite operator whose eigenfunctions are expressed in terms of the Hermite polynomials. This is well known fact but we will derive it here below. In what follows we are going to use simple arguments from representation theory. Instead of using the operators $x$ and $p$ we are going to present $H$ in terms of the following two operators: \beqa a\,=\, x \sqrt{\frac{m\omega}{2}}\, + \, ip \sqrt{\frac{1}{2m\omega} }\\ a^{\dag}\,=\, x \sqrt{\frac{m\omega}{2}}\, - \, ip \sqrt{\frac{1}{2m\omega}} \eeqa Notice that the operators $a$ and $a^{\dag}$ satisfy {\it the canonical commutation relation} $[a,a^{\dag}]= 1$, which plays a crucial role below. Obviously the Hamiltonian can be written in the form \beq H = \frac{\omega}{2}(a^{\dag}a +aa^{\dag})= \omega(N+\frac{1}{2}), \eeq where $N=aa^{\dag}$. The Hermitian operator $N$ satisfies the relations \beq [N,a^{\dag}]=a^{\dag}\,\,\, \textrm{and}\,\,\, [N,a]=-a. \label{3.9} \eeq The above operators define an algebra, called {\it Heisenberg algebra}. We are going to study the representations of this algebra in order to obtain the spectrum of $N$. Let $|n>$ be a normalized eigenvector of $N$, i.e. $N|n>=n|n>$ and $<n|n>=1$. Consider the vectors $a^{\dag}|n>$ and $a|n>$. If we apply to them $N$ and use the the commutation relations \eqref{3.9} we obtain \beqa Na^{\dag}|n>=(a^{\dag}N +a^{\dag})|n> = a^{\dag}(N+1)|n>= (n+1)a^{\dag}|n>\\ Na|n> = (aN-a)|n> = a(N-1)|n>=(n-1)a|n> \label{3.10} \eeqa The equations \eqref{3.10} explain the names of the operators $a^{\dag}$ and $a$ -- operators of creation and annihilation. The last equations show that we can build new eigenstates from old ones. In particular we can obtain eigenstates with arbitrary negative eigenvalues. Below we are going to show that this cannot be true. The operator \eqref{3.7} $H$ is a sum of squares of Hermitian operators. This shows that it cannot have negative eigenvalues. This shows that from some positive $k$ further the vectors $a^k|n>$ are zero and we do not produce new eigenvectors from it. Let us denote by $|0>$ the last non-zero vector of the sequence $|n>, a|n>,\ldots, a^k|n>,\ldots$. The vector $|0>$ is called {\it vacuum}. (Notice that here we have denoted {\bf a non-zero state} by $|0>$! This is the vacuum and not the zero vector.) The uniqueness of the vacuum is also easy to prove, see below. We have $a|0>=0$. On the other hand all eigenvectors $|0>, a^{\dag}|0>,\ldots, a^{\dag k}|0>,\ldots$ are non-zero. Let us show this. Take a normalized eigenvector $|n>$ as above. The squared norm of $a^{\dag}n$ can be computed as follows. \ben <a^{\dag}n|a^{\dag}|n>= <n|aa^{\dag}|n> =(a^{\dag}a +1)|n>=(n+1)|n+1>. \een (Why the first equality is true?) One can easily show that the eigenspaces of $N$, corresponding to the eigenvalues $n$ are one-dimensional. Let us start with the vacuum $|0>$. It satisfies ordinary differential equation of order one, $a|n>=0$. Hence the statement is true. Assume that we have proved the statement for an eigenvalue $n-1$. If for the eigenvalue $n$ we have at least two independent eigenvectors $|n>$ and $|n^{'}>$ we can act upon them by $a$, Then the we obtain \ben Na|n> = (n-1)a|n>, \,\,\, Na|n^{'}> = (n-1)a|n^{'}>. \een This shows that $a|n-n^{'}> = 0$ and hence $|n-n^{'}>$ is the vacuum. On the other hand $N|n-n^{'}>=n|n-n^{'}>$ contradicting to the fact that the vacuum has zero eigenvalue. Finally the fact that the eigenvectors of $N$ form a complete orthogonal system in $L^2(\mathbb{R})$ is well known fact, e.g. from the theory of Hermite polynomials. In this way we obtained an orthogonal basis of $L^2(\mathbb{R})$ formed by the eigenvectors of $H$ with eigenvalues $\omega(n+1/2)$. %%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \sectionnew{Path integral formulation of quantum mechanics} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Definitions} We are going define the path integrals for quantum mechanics by the same expansion \eqref{3.13} we used in $0$-dimensional QFT. For this we need to define the Feynman amplitudes, which means we have to define the function $S$, the quadratic form $B$, to find its inverse -- $B^{-1}$ and finally to define the covectors $l_j$. Let us consider a classical particle with action functional \ben S(q)=\int L(q_j,\dot{q}_j) dt \een Then we need to define the Feynman integral, having the meaning of correlation function. \beq \mathcal{G}(t_1,\ldots , t_N) = <q(t_1),\ldots, q(t_N)>:=\\ \frac{ \int q(t_1)\ldots q(t_N) e^{iS(q)/\hbar} Dq}{\int e^{iS(q)/\hbar} Dq} \label{4.1} \eeq \bre{4.1} An obvious but important remark is that $q(t_j)$ has the meaning of a functional. Here $t_j$ is fixed and $q$ is the variable. \ere The notation $\mathcal{G}_n(t_1,\ldots,t_n)$ refers to another name of the correlator -- {\it Green's function}. Of course we consider first the Euclidian picture. For this we need to make Wick's rotation, i.e. to rotate the time in the complex domain. We are going to consider only Lagrangians of the form $L(q,\dot{q})= \dot(q)^2/2 - U(q)$ Then our action will become: \ben S= \int\big( - (\dot{q})^2/2 - U(q)\big)i dt \een and the Green's function will be given by the formula \beq \mathcal{G}^E(t_1,\ldots , t_N) = <q(t_1),\ldots, q(t_N)>:=\\ \frac{ \int q(t_1)\ldots q(t_N) e^{-S_E(q)/\hbar} Dq}{\int e^{-S_E(q)/\hbar} Dq} \label{4.1e} \eeq with $S_E=\int \big((\dot{q})^2/2 + U(q)\big)dt$. We may assume for simplicity that the particle moves in one-dimensional space. The general case is not much harder. The potential $U$ will be taken a power series of the form $U=\sum_{j=2}^{\infty}U_j$, i.e. without constant and linear terms. For further use introduce the notation $U_j=u_jx^j/j!$. Then in analogy with $0$-dimensional case we take the quadratic form $B$ to be \ben B=\int (\dot{q}^2 + m^2q^2 )dt. \een Here $m^2q^2=2U_2$. The coefficient $m$ has the meaning of mass. Integrating by parts we obtain \ben B= <Aq|q> , \een where $A= -d^2/dt^2+m^2$. This will help us define the inverse $B^{-1}$; namely we put $B^{-1}(f,f)=<A^{-1}f|f>$. The operator $A^{-1}$ is defined as in differential equations: if $Aq=f$, then the solution of this equation is given by $q=A^{-1}f$. In differential equations this is the integral operator with kernel {\it the Green function} $G(x,y)$: \ben q(x)= \int G(x,y)f(y)dy. \een It is well known that in our case the Green's function is given explicitly by the formula: \beq G(x,y) = \frac{e^{-m|x-y|}}{2m}. \label{4.2} \eeq We see that our Hilbert space $\mathcal{H}$ has to be the space of quadratically integrable functions $L^2$. But we are going to work with Schwartz spaces $S(\mathbb{R}^n)$ and $S^*(\mathbb{R}^n)$ as explained in {\bf Section 2.} Now we are ready to give definition of the Feynman integral \eqref{4.1} (Euclidean version). Introduce some numeration of the internal vertices. The formula below does not depend on the choice. \bde{4.1} The correlation (Green's) function \eqref{4.1} is given by the asymptotic series \beq \mathcal{G}(t_1,\ldots,t_N ) = \sum_{\Gamma\in G^*_{\geq 3}(N)} \frac{\hbar^b(\Gamma)}{|Aut(\Gamma)|}F_{\Gamma}(t_1,\ldots,t_N) \label{4.3} \eeq To define the numbers $F_{\Gamma}$ we fix the graph $\Gamma$. Then the following rules hold: \vspace{1cm} \begin{enumerate} \item Put the variable $t_j$ (the functional $q(t_j)$ at the $j$-th external vertex of $\Gamma$. \item Put the variable $s_k$ at the internal vertex $k$. \item For each edge connecting $u$ and $v$ write the Green's function $G(\alpha,\beta)$. \item The number $F_{\Gamma}$ is defined by the formula \beq F_{\Gamma} = \prod_j(-u_{v(j)})\int G({\bf t}, {\bf s})d{\bf s}, \label{4.3a} \eeq where $v(j)$ is the valency of the $j$-th vertex of $\Gamma$. \end{enumerate} \ede \vspace{1cm} \bex{4.1} (Wick's Lemma.) Let us examine in detail the free theory: \ben S=\int (-\frac{\dot{q}^2}{2} - \frac{m^2q^2}{2})dt. \een In this case each graph is disconnected union of subgraphs with two vertices and edge connecting them. The above formula gives us that \ben \mathcal{G}(t_1,\ldots , t_{2k}) = \hbar^k\sum_{\sigma \in \Pi_m} \prod_{i \in \{1,\ldots, 2m\}/\sigma} G(t_i- t_{\sigma(i)}). \een \eex \bex{4.2} $\phi^3$-theory. Consider action with Lagrangian $L=\dot{\phi}^2/2 - m^2\phi^2/2 + \phi^3$. Let us compute the two-point correlation function up to some order. \vspace{2cm} \begin{tikzpicture}\filldraw [black] (0,0) circle (2pt); \draw (0,0) ..controls (0.3,0)..(3.4,0); \draw (1.7,0) circle (20pt); \filldraw [black] (1,0) circle (2pt); \filldraw [black] (2.4,0) circle (2pt); \filldraw [black] (3.4,0) circle (2pt); \end{tikzpicture} \vspace{.5cm} \begin{flushleft} \bf{Figure 8.} \end{flushleft} \eex \subsection{Computations} \subsubsection{The partition function} Let us consider the partition function with slight modification -- {\it partition function with external current} $J$: \beq Z(J):={\int e^{iS_E(q)/\hbar}+<J|q> Dq}. \label{4.2a} \eeq Here the arbitrary function $J\in S$ (the space of fast decaying functions $S$. Then we have the equality (only formally!): \ben \frac{Z(J)}{Z(0)}= \sum_n\frac{\hbar^{-n}}{n!}\int_{\mathbb{R}^n} \mathcal{G}_n(t_1,\ldots,t_n)J(t_1)\ldots J(t_n)dt_1\ldots dt_n. \een This will be our definition for $\frac{Z(J)}{Z(0)}$. We see that this the generating function of all the Green's functions $ \mathcal{G}_n(t_1,\ldots,t_n)$. As in the $0$-dimensional QFT here we have \bpr{4.1a} The following formula holds: \ben W(J):=\ln \frac{Z(J)}{Z(0)}= \sum_n\frac{\hbar^{-n}}{n!}\int_{\mathbb{R}^n} \mathcal{G}^c_n(t_1,\ldots,t_n)J(t_1)\ldots J(t_n)dt_1\ldots dt_n. \een \epr The \proof is the same as in the $0$-dimensional QFT. In this way we have generating function of all the Green's functions $ \mathcal{G}^c_n(t_1,\ldots,t_n)$. Also as in $0$-dimensional QFT we have $j$-loops expansion: \ben W(J)= \hbar^{-1}W_0(J) +W_1(J) +\ldots +\hbar^{j-1}W_j(J) +\ldots, \een where $W_0$ is the sum over trees, $W_1$ is the $1$-loop contribution, etc. Furthermore, \bpr{4.2} (0) The tree approximation is given by \ben W_0(J)= -S_E(q_J) + <q_J,J>, \een where $q_J$ is the extremal of the functional $S^J_{E}:=S_E(q_J) - <q_J,J>$; (1) The one-loop contribution is given by \ben W_1(J)= -\frac{1}{2}\ln\det L_J, \een where $L_J$ is the linear operator on $\mathcal{H}$ such that $d^2S^J_{E}(q_j)(f_1,f_2)=d^2 S^0J_{E}(0)(Lf_1,f_2)$. In a similar vein we can write explicitly a generation function for the on-particle irreducible Green functions $\mathcal{G}^{1PI}_n(t_1,\ldots,t_n)$, i.e. the Green functions that are defined only over the 1PI-graphs. \epr \subsection{Motivation} \subsubsection{"Derivation" of Feynman's formula } \subsubsection{Feynman-Kac formula} \subsection{Example - the Harmonic Oscillator} \subsection{Example - $\phi^3$ Theory} \subsection{Momentum space formulation} The computations in the position variables are quite heavy. In particular the Feynman amplitude is given by an integral over a space of dimension equal to the number of internal vertices, which can be enormous even for trees. Instead one can pass to {\it momentum representation} by applying Fourier transform. Let us start with the classical equation: \ben (-\frac{\partial^2}{\partial t^2} + m^2)G(x)=\delta \een Applying Fourier transform to it (with the variable $p$ instead of $\xi$) we obtain: \ben (p^2 +m^2 )\hat{G}= 1 \een This gives \ben \hat{G}(E) = \frac{1}{E^2 +m^2} \een Of course \ben G(t-s)=\int \frac{e^{ip(t-s)}dp}{2\pi(p^2 +m^2)} \een Below we introduce the following notation. We denote by $p_j$ the edges of a fixed graph $\Gamma$ and by $\alpha(p_j), \beta(p_j)$ the vertices adjacent to $p_j$. Both $\alpha(p_j)$ and $\beta(p_j)$ denote either $t$ or $s$. We plug in the above expression for $G$ with corresponding variables into the formula for the amplitude $F_{\Gamma}(t_1,\ldots,t_N)$. We also can perform Fourier transform on $F_{\Gamma}(t_1,\ldots,t_N)$ (with respect to $(t_1,\ldots,t_N)$). Denote the dual variables by $E_1,\ldots, E_N$. Then we get $E_j=p_{k_j}$. All the exponentials will disappear. The integrations with respect to $p$-s will remain but with some connections between the $p$-s. We obtain %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5 \ben \hat{F}({\bf E}) = \prod_k \int_{t_k \in \mathbb{R}} \Big[\int_{\bf s} \Big(\prod_j \int_{p_j \in \mathbb{R}} \frac {e^{ip_j(\alpha(p_j)-\beta(p_j))}dp_j}{2\pi(p_j^2 +m^2)}\Big)d{\bf s}e^{iE_k(\alpha(p_k)-t_k)}\Big]dt_k \een We can change the order of integration; first integrate with respect to ${\bf s}$ and ${\bf t}$ and then with respect to ${\bf p}$. \ben \hat{F}({\bf E}) = \prod_j \int_{p_j \in \mathbb{R}} \frac {1}{2\pi(p_j^2 +m^2)} \int_{\bf t} \int_{\bf s} e^{ip_j(\alpha(p_j)-\beta(p_j))}e^{iE_k(\alpha(p_k)-t_k)} d{\bf s} d{\bf t} dp_j \een The integration with respect to ${\bf s}$ and ${\bf t}$ will produce some delta-functions involving ${\bf p}$ and $E$. In more details each fixed $s_j$ gives $\delta$-s with all the edges (i.e. the variables $p$) connecting $s_j$ with the other vertex of $p$. Then using the meaning of $\delta(p_{i_1} + \epsilon_j p_{i_j} \ldots,) \,\, \epsilon_j =\pm 1$ we obtain a linear relation between $p$-s with coefficients $\pm 1$. Consider as an example the diagram on Fig. 8. We can first write the propagator as inverse Fourier transform: \ben G(t-s) = \int \frac{ e^{ip(t-s)}dp}{2\pi (p^2+m^2)} \een Then we plug it in the formula for the amplitude: \ben F(t)= \int \Big( \prod \int \frac{e^{i\sum p_j(t_1-s_1)} e^{i\sum p_j(t_2-s_2)} d{\bf p}}{2\pi (p_j^2+m^2)} \Big)d{\bf s}. \een Next perform Fourier transform with respect to $t$ where the dual variable is denoted by $E$. Also perform the integration with respect to $s$ This gives: \ben \hat{F}(E) = \int \frac{ \delta{(E_1- \sum_jp_j)} \delta{(E_2- \sum_jp_j)}} {\prod_{j=1}^3 2\pi(p_j^2+m^2)} d{\bf p}. \een This gives \ben \hat{F}(E) = \int \frac{ 1} {\prod_{j=1}^3 2\pi(p_j^2+m^2)} d{\bf p}, \een where the variables $E_1=E_2$ and $\sum p_j=E_1$. Below we give the rules defining Feynman's amplitude in momentum variables. Recall that the dual variables to $t$ are denoted by $E$. The dual variables to $s$ will be denoted by $Q$. The rules include checking what are the the signs. In fact we can choose quite arbitrary these signs but still there is some rules. \bde{5.1} (Feynman's rules for the amplitudes in momenta variables.) The Fourier transform of an amplitude $F_{\Gamma}$ are as follows: \begin{enumerate} \item Put a variable $E_j$ at each external edge and a variable $Q_j$ at each internal one; \item Assign a propagator $\frac{1}{p^2 +m^2}$ to each edge and substitute $p$ by $E_j$ for the external edges and by $Q_j$ by the internal ones. Multiply all the propagators and denote the result by $\Phi_{\Gamma}(E,Q)$; \item Orient the external edges inward; \item Orient the internal edges arbitrarily; \item For each internal vertex write "the Kirchhoff law": the sum of the incoming variables is equal to the sum of the outgoing ones. This will produce relations among the variables $Q$ and $E$. One of them is $\sum_j^N E_j=0$. The rest define a linear subspace $Y(E)$ of the space of the $Q$-s; \item Define the momentum-space amplitude of $\Gamma$ by \ben \hat{F}_{\Gamma}(E)= \prod_l (-a_{v(l)})\int_{Y(E)} \Phi(E,Q)dQ. \een \item The measure $dQ$ on $Y(E)$ is defined to be in such a way that the volume of $Y(E)/Y_Z(0) = 1$, where $Y_Z(0)$ is the set of integer points on $Y(0)$. \end{enumerate} \ede \bex{5.1} Consider the Feynman graph given on fig.5.1. \eex %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5 \sectionnew{Symmetries} Symmetries are everywhere around us. Quite often we attribute beauty to some visible symmetry. In science they are less visible but no less important. Even in classical mechanics the symmetries are responsible for the integrability of mechanical equations. Some of the corresponding symmetries can be seen easily, e.g. -- the rotational symmetry yields the conservation of the momentum. But other, as the symmetries in rigid body equations are not at all obvious. The adequate mathematical tool describing symmetries is group theory. In this section we assume some knowledge of groups and present some of the theory needed in the course. On the other hand we are going to consider simple enough examples that presumably would help the reader to get more insight even without preliminary acquaintance with groups. As the definition of a group is simple let's recall it. \bde{4a.1} A group is a set $G$ with the following properties: \begin{enumerate} \item Multiplication. For any ordered pair of elements $g_1,g_2 \in G$ there exits an element $g_1\cdot g_2 \in G$; \item Inversion. For any element $g \in G$ there exits an element $g^{-1} \in G$ \item Unit. There exist an element ${\bf 1}\in G$ such that ${\bf 1}\cdot g = g$ for any $g \in G$; \item Associativity. For any three elements $g_1,g_2,g_3 \in G$ the associativity equation holds: \ben g_1\cdot (g_2\cdot g_3)= (g_1\cdot g_2)\cdot g_3. \een \end{enumerate} If the order of multiplication is irrelevant, i.e. $g_1\cdot g_2= g_2\cdot g_1$ we say that the group is {\it commutative} or {\it Abelian.} \ede In the subsections that follow we study few examples all important for QFT. \subsection{The Group SO(2)} This is the group of rotations of the circle. (Show that it is a group.) We can identify its elements with the $2\times 2$ matrices of the form: \ben A=\left( \begin{array}{cc} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \\ \end{array} \right)\, \een Obviously such a matrix describes a rotation of angle $\theta$. In this example we meet one of the most important notions in mathematics and physics -- the notion of {\it representation}. In simple terms we described above our group as subgroup of a matrix group. The description of the representations of groups is a major goal of mathematics. Soon we will see its importance for quantum theory. First we will give a precise definition. \bde{4a.2} Let $V$ be a vector space (it could be infinite-dimensional). Denote by $Inv(V)$ the group of invertible linear operators in $V$. A homomorphism of a group $G$ into a subgroup of $Inv(V)$ is called representation. \ede \subsection{The Groups SO(3) and SU(2)} \subsection{The Lorentz and Poincar\'e groups} \subsection{Clifford algebras} Let $V$ be a complex space with scalar product. \bde{4a.3} Clifford algebra is an algebra spanned by the elements of $V$ and the complex numbers $\Cset$, satisfying the relation \ben \xi\eta +\eta \xi = 2( \xi,\eta), \,\, \xi,\eta \in V. \een \ede \sectionnew{Classical fields} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Multidimensional Variational Problems} Here we going to generalize the variational approach in mechanics to some other physical problems, where the configuration space is infinite-dimensional. \subsection{Klein-Gordon equation} The simplest non-trivial Poincar\'e-invariant Lagrangian is: \ben \mathcal{L}= \frac{1}{2}\partial_{\mu}\phi \partial^{\mu}\phi - \frac{1}{2}m^2\phi^2 \een Klein-Gordon equation is the corresponding Euler-Lagrange equation for the action $S=\int \mathcal{L}(\phi,\nabla \phi)d^4x$: \beq (\Box + m^2)\varphi = 0 \label{5.1} \eeq \subsection{Electromagnetic field} The electromagnetic field is governed by the Maxwell equations. We recall them. Let ${\bf E(x,t)} =(E^1,E^2,E^3)$ and ${\bf B(x,t)} =(B^1,B^2,B^3)$ be the electric and magnetic fields in a three-dimensional space correspondingly. Denote also by $j$ and by $\rho$ the current density and the charge density. Then the Maxwell's equations are: \beqa a)\,\,\, \nabla \cdot{\bf B}= 0,\qquad \qquad b)\,\, \nabla \times\, {\bf E} + \frac{\partial {\bf B}}{\partial t}=0\\ c)\,\,\, \nabla\cdot{\bf E}= \rho,\qquad \qquad d)\,\, \nabla \times\, {\bf B} - \frac{\partial {\bf E}}{\partial t}=j \eeqa The meaning of the equations is the following. Equation a) means that there are no magnetic charges. Next comes Faraday's law b) of induction: if the magnetic field is changing, then an electric field appears. Equation c) is nothing but Gauss's (Stokes, Green, Ostrogradsky, etc.) theorem in differential form. Finally equation d) is the Amp\`ere's circuital law, with the Maxwell correction. By Helmholtz's theorem, $\bf B$ can be written in terms of a vector field $\bf A$, called the {\it magnetic potential}: \ben {\bf B} = \nabla \times {\bf A}. \een Differentiating and using Faraday's law we find \ben \nabla \times ({\bf E} + \frac{\partial {\bf A}}{\partial t})=0. \een This shows, again by Helmholtz's theorem, that there exists a function $\varphi$ such that ${\bf E} + \frac{\partial {\bf A}}{\partial t}=\varphi$. Denote by $A^{\mu} = (\varphi, {\bf A})$. It is called {\it 4-potential}. We are going to write the Maxwell equations in terms of the 4-potential. Introduce the {\it electromagnetic tensor} $F^{\mu \nu}$ by the equalities: \beq F^{\mu \nu} = \partial^{\mu}A^{\nu}- \partial^{\nu}A^{\mu} = - F^{\nu \mu} \label{5.2}. \eeq Component-wise it reads: \beq F=\left( \begin{array}{cccc} 0 &-E^1 &-E^2 & -E^3 \\ E^1 & 0 &-B^3 & B^2\\ E^2 & B^3 &0 & -B^1\\ E^3 & -B^2 &B^1 &0 \label{5.3} \end{array} \right). \eeq It is quite obvious that the electromagnetic tensor is invariant under the transformation \ben A^{\mu}\rightarrow A^{\mu} + \partial^{\mu}\chi \een \subsection{Dirac Equation} Introduce the {\it Dirac matrices} for four-dimensional Minkowski space. They are: \beq \gamma^0=\left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right), \quad \gamma^i= \left( \begin{array}{cc} 0 & \sigma^i \\ -\sigma^i & 0 \label{5.4} \end{array} \right), \eeq where $\sigma^i$ are the Pauli matrices \begin{footnote} {Warning! This notation is not the only one used in literature.} \end{footnote}. This representation is called {\it Weyl} or {\it chiral representation} In terms of Dirac matrices we can write Dirac equation as: \ben (i\gamma^{\mu}\partial_{\mu} - m)\psi(x)=0, \een or in Dirac's notation $(i\partial \!\!\!/ - m)\psi(x)=0$ \bpr{5.1} (i) Dirac equation is Lorentz invariant. (ii) Klein-Gordon operator \ben \partial^2 +m^2 = (-i\gamma^{\mu}\partial_{\mu} - m) (i\gamma^{\mu}\partial_{\mu} - m), \een i.e. Klein-Gordon equation follows from Dirac equation. \epr \proof is elementary computation and is left for the reader. The Lagrangian for Dirac theory is: \beq L_{Dirac} = \bar{\Psi} (i\partial \!\!\!/ - m)\Psi, \eeq where $\bar{\Psi}=\psi^{\dag}\gamma^0$. Dirac propagator, i.e. the fundamental solution of Dirac equation is: %%%%%%%%%%%%%%%%555%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \sectionnew{Quantum fields} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% We start with one scalar field $\phi$ on Minkowski space. This means we have a vector space $V$ with a signature $-1,\ldots,1$. We also have an action $ S=\int L dx$ with Lagrangian $L(x,\partial_x)$. In QFT there is an operator of {\it time ordering} $T$ acting on fields as follows. If $(x-y)_M^2\geq0$. $(x-y)_M^2\geq0$ then $T(\phi(x) \psi(y) )= \phi(x) \psi(y)$. Otherwise $T(\phi(x) \psi(y) )= \psi(y)\phi(x)$. We want to put sense in the expressions of the form: \beq \mathcal{G}(x^1,\ldots , x^N) = <\phi(x^1),\ldots, \phi(x^N)>:=\\ \frac{ \int T( \phi(x^1)\ldots \phi(x^N)) e^{iS(\phi)/\hbar} D\phi}{\int e^{iS(\phi)/\hbar} D\phi}. \label{6.1} \eeq Of course after that we need to learn how to compute them. As explained earlier we are going to study first the Euclidian theory. In Euclidian theory we have to find the correlator \ben \mathcal{G}(x^1,\ldots , x^N) = <\phi(x^1),\ldots, \phi(x^N)>:=\\ \frac{ \int T( \phi(x^1)\ldots \phi(x^N)) e^{-S(\phi)/\hbar} D\phi}{\int e^{-S(\phi)/\hbar} D\phi}, \een We will proceed as in quantum mechanics. \subsection{$\phi^4$ Theory} Our first example will be Klein-Gordon's Lagrangian: \ben \mathcal{L}_{KG} =\frac{1}{2} (\partial_0\phi)^2 - \frac{1}{2}\sum_{j=1}^{m-1} (\partial_j\phi)^2 -m^2\phi^2 \een perturbed by $\sum_j U_j\phi^j$, i.e. \beq \mathcal{L}= \mathcal{L}_{KG}+\sum_j U_j\phi^j. \label{6.2} \eeq After Wick's rotation it becomes \ben \mathcal{L}_{KG}^E= - \frac{1}{2}\big((\nabla \phi)^2 + m^2\phi\big)dx. \een Denote the Green's function of the Euclidian Klein-Gordon equation by $G_{KG}^E(x-y)$. In other words we look for solution of the equation \ben (-\Delta+m^2) G_{KG}^E(x-y)=\delta(x-y) \een Performing Fourier transform on both sides gives: \ben (k^2+m^2)\hat{G}_{KG}(k)=1 \een Then the Green's function is given by the inverse Fourier transform: \ben G_{KG}^E(x-y)= \frac{1}{(2\pi)^d} \int e^{-i(x-y)k}\frac{dk}{k^2+m^2}. \een As in quantum mechanics we see the advantages of Wick's rotation -- the integrand has no poles in the real domain. We can define the Euclidian correlation functions as follows. \bde{6.1} The correlation function, corresponding to the Lagrangian \eqref{6.2} and to functionals $\phi(x^j)$ is given by the following rules: \begin{enumerate} \item Put the variable $y^j$ at the $j$-th external vertex. \item Put the variable $w^k$ at each internal vertex. \item Put the Green's function $G_{KG}(w^j-w^k)$ at each edge connecting two internal vertices $w^j,w^k$. \item Put the Green's function $G_{KG}(y^j-w^k)$ at each edge connecting the external vertex $y^j$ with the internal vertex $w^k$. \item Put the Green's function $G_{KG}(y^j-y^k)$ at each edge connecting two external vertices $w^j,w^k$. \item The number $F_{\Gamma}$ is defined by the formula \ben F_{\Gamma} = \prod_j (-U_v(j)) \int G({\bf{y}};{\bf{w}}) d{\bf w}; \een \end{enumerate} \ede %%%%%%%%%%%%%%%%555%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%% \sectionnew{Fermions} Elementary particles are divided into two types: bosons and fermions. Examples of the former are photons, $W$, and $Z$ particles. The electrons, protons, neutrons are examples of fermions. The bosons are characterized by the fact that several bosons can occupy the same quantum state, while fermions cannot. Mathematically this difference is expressed by the corresponding Hilbert spaces. If the Hilbert space for a single particle is $\mathcal{H}$ then for $k$ bosons it is $S^k\mathcal{H}$ (the $k$-th symmetric power of $ \mathcal{H}$), while for $k$ fermions it is $\Lambda^k \mathcal{H}$ (the $k$-th exterior power of $\mathcal{H}$). The quantum theory we have developed up to now describes mostly bosons -- the fields commute. Now we need to develop field theory of anti-commuting fields. The relevant mathematical tool is the notion of supermanifolds. \subsection{Linear Superspaces} Of course we start with the relevant linear algebra. \bde{7.1} A supervector space (or superspace) $V$ is $\Zset_2$-graded vector space -- $V= V_0\oplus V_1$ with the following additional structure. We define tensor product $v\otimes u$ of two vectors, where $v\in V_i, u \in V_j,\quad i,j \in \{0,1\}$ satisfying the rule $v\otimes u = (-1)^{ij} u\otimes v$. Let us define the operation of changing parity $\Pi$, by $\Pi V_i= V_{1-i},\quad i,j \in \{0,1\}$. With this notation we can define the following extension of the notions of symmetric and exterior powers: \ben S^m V= \Pi(\Lambda^m (\Pi V), \quad \Lambda^m V = \Pi(S \Pi (V). \een When $V_0= \Rset^n,\quad V_1=\Rset^m$ we denote $V$ by $\Rset^{n|m}$. In general we say that $V$ has dimension ${n|m}$, where ${n,m}$ are the dimensions of $V_0,V_1$, correspondingly. \ede The elements of $V_0$ are called {\it even} and the elements of $V_1$ are called {\it odd}. We define the algebra of polynomial functions $\mathcal{O}(V)$ on a superspace $V$ as $SV^*$, where $S$ acts as defined above on the superalgebra $V^*$. In more details if $x_1,\ldots, x_n$ are linear coordinates on $V_0$,called {\it even variables} and $\xi_1,\ldots, \xi_m$, called {\it odd variables} are linear coordinates $V_1$ then $\mathcal{O}(V)$ is $\Rset[x_1,\ldots, x_n,\xi_1,\ldots, \xi_m]$ with the relations \ben x_ix_j = x_jx_i, \quad \xi_i\xi_j = - \xi_j\xi_i, \quad x_i\xi_j=\xi_jx_i. \een The algebra spanned only by the odd variables $\xi_1,\ldots, \xi_m$ is called {\it Grassmann or exterior algebra}. Using the standard notation of {\it anticommutator} -- $\{a,b\} = ab+ba$ we can write the defining relations $\{a,b\}=0$ for any $a,b$ of the Grassmann algebra. It is a finite-dimensional space, while $\mathcal{O}(V)$ is (in general) an infinite-dimensional supervector space. \subsection{Supermanifolds} More generally we can define the algebra of smooth functions $C^{\infty}(V)$ on $V$ as $C^{\infty}(V_0) \otimes \Lambda V_1^*$. We can look on the smooth functions on a supermanifold $V$ as functions of the form: \ben F(x,\xi) = \sum f_i(x)\xi_1^{\alpha_1}\ldots \xi_m^{\alpha_m}, \een where $\alpha_i = 0\,\, \textrm{or} \,\,1$. \bde{7.2} A supermanifold $M$ is an ordinary manyfold $M_0$ but instead of the standard sheaf of smooth functions we consider a sheaf of smooth functions $C^{\infty}(V)$ on a superspace. This means that that the structure sheaf is locally isomorphic to $C^{\infty}_{M_0}\otimes \Lambda(\xi_1,\ldots , \xi_m)$. \ede \subsection{Calculus on Supermanifolds} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Let us define the notion of integral of Grassmann functions. It will have the properties: \ben \int 1 d\xi=0,\,\,\, \int \xi d\xi=1,\,\,\,\\ \int \xi_2 \Big( \int\xi_1 d\xi_1\Big)d\xi_2 = \int \int \xi_2\xi_1 d\xi_1d\xi_2 =1 \een %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Next we define an integral for functions in both even and odd variables. Consider functions $f(x,\xi)$ that in even variables are compactly supported, i.e. \ben f(x,\xi) = \sum_{\alpha} f_{\alpha}(x)\xi^{\alpha}, \een and the functions $f_{\alpha}$ are with compact support. It is enough to define the integral for the summands: $ f_{\alpha}(x)\xi^{\alpha}$. The integral will be \ben \int_{V} f_{\alpha}(x)\xi^{\alpha}dx d\xi = \int_{V_0}f_{\alpha}(x)dx \int_{V_1}\xi^{\alpha}d \xi. \een The general case is defined by linearity. We need to learn how to make changes of variables. Consider the case when there are only odd variables. To get an idea about the natural formulas we start with 2-dimensional $V=V_1$. The linear change of variables $F$ has the form: \ben \xi_1 = f_{11}\eta_1 + f_{12}\eta_2,\quad \xi_2 = f_{21}\eta_1 + f_{22}\eta_2. \een Then the function $\xi_1\xi_2$ transforms into \ben (f_{11}\eta_1 + f_{12}\eta_2)(f_{21}\eta_1 + f_{22}\eta_2)=\\ (f_{11}f_{22}-f_{12}f_{21})\eta_1\eta_2 = \det{(F)}\eta_1\eta_2. \een We want to keep the value of the integral $\int\xi_1\xi_2 d\xi_2 d\xi_1= 1$. This yields that the change of the variables must be: \ben \xi_1\xi_2d\xi_2 d\xi_1 = \det{(F)}^{-1}\eta_1\eta_2 d\eta_2d\eta_1. \een Obviously the same formula has to be applied to odd spaces in any dimension. To guess the formula for change the variables in general, i.e. when we have integral $f(x)\xi_1\ldots\xi_m$ we can apply the above arguments. Again take two odd variables. The linear map will be \begin{multline} \\ x = Ay +b_{11} \eta_1 +b_{12}\eta_2,\\ \xi_1 = c_1 y +d_{11}\eta_1 +d_{12}\eta_2.\\ \xi_2 = c_2 y +d_{21}\eta_1 +d_{22}\eta_2. \\ \label{7.1} \end{multline} Here $A,D$ have even elements while $B,C$ have odd elements. The matrix $B$ is $m\times 2$ and the matrix $C$ is $2\times m$. The change \eqref{7.1} gives \ben f(x)\xi_1\xi_2dx d\xi_2 d\xi_1 = f(x)\eta_1\eta_2( A dy +b_1d \eta_1 +b_2d \eta_2)\\ ( c_1 dy +d_{11}d \eta_1 +d_{12}d \eta_2) ( c_2 dy +d_{21}d \eta_1 +d_{22}d \eta_2). \een Assume that $\det D \neq 0$. After some manipulations we obtain \ben \xi_1 \xi_2 dx d\xi_1 d\xi_2 =\\ \eta_1 \eta_2 (\det A \det D) dy d\eta_1 d\eta_2 -\\ \eta_1 \eta_2 \det( B D^{-1}C )\det D dy d\eta_1 d\eta_2 =\\ \eta_1 \eta_2 \det\big( A -B D^{-1}C \big) \det D dy d\eta_1 d\eta_2 \een Having in mind that the integral of $\xi_1\xi_2 d \xi_1 d\xi_2$ must be $1$ we finally find that the formula for the change of the variables is given by \ben \int f(x)dx d\xi = \int f \textrm{Ber}(F)dy d\xi \een where {\it the Berezinian} of $F$ is the number \ben \textrm{Ber}(F) = \frac{\det(A - B D^{-1}C)}{\det D}. \een We also need to learn how to differentiate functions of anticommuting variables. Here we are going to distinguish between {\it left derivative} $\frac{ \partial^L}{\partial_{\xi}}$ and {\it right derivative} -- $ \frac{\partial^R}{\partial_{\xi}}$. It is enough to define them for the function $\xi_1\xi_2$. We have \ben \frac{\partial^L}{\partial_{\xi_i}}\xi_1\xi_2 =\delta_{i1}\xi_2 - \delta_{i2}\xi_1, \quad \frac{\partial^R}{\partial_{\xi_i}}\xi_1\xi_2 =\delta_{i2}\xi_1 - \delta_{i1}\xi_2. \een \subsection{ Fermionic Quantum Mechanics } The simplest fermionic Lagrangian is \ben \mathcal{L}= \psi \dot{\psi}. \een This is quantum-mechanical Lagrangian of a single massless fermion. \subsection{Path Integrals for Free Fermionic Fields} We already know some fermionic Lagrangians -- Weyl, Majorana, Yukava... .................. ................................ The Dirac Lagrangian is \ben \mathcal{L}_D = \psi^{\dag}_L\sigma \partial \psi_L + \psi^{\dag}_R\bar{\sigma} \partial \psi_R + i m\big(\psi^{\dag}_R \psi_L + \psi^{\dag}_L \psi_R \big). \een It describes a pair particle-antiparticle, for example electron and positron. Unlike Majorana's Lagrangian here the antiparticle is different from the particle. Using the four-component Dirac's spinor \ben \Psi_D= \left(\begin{array}{cc} \psi_L\\ \psi_R \end{array} \right) \een we can express Dirac's Lagrangian in a compact form: \ben \mathcal{L}_D = \bar{\Psi}_D (i\Dir - m)\Psi_D. \een Let us write the Feynman rules for free theories. We simply add a source coupling to get the generating functional: \ben Z =\frac{1}{N}\int e^{i\int [ \mathcal{L}_D + i\bar{\Psi}_D \zeta + i\bar{\zeta}\Psi_D]dx}D\Psi_D D\bar{\Psi}_D. \een Denote by $\hat{\Psi}_D({\xi})$ the Fourier transform of $\Psi_D$. Then the equation for the propagator in momenta variables reads: \ben ( -\Dmom -m)\hat{\Psi}_D({\xi})=1. \een %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \sectionnew{Renormalization} %%%%%%%%%%%%%%%%555%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%% Unfortunately many of the diagrams have divergent amplitudes. Let's consider the following example. In momenta variables the Klein-Gordon propagator is (after Wick rotation) \ben \hat{G} = \frac{1}{k^2 +m^2} \een Let us study $\phi^4$ theory. Consider the four-point function. It contains diagrams like the one on Fig. 8. Then by Feynman rules the amplitude for this graph is \ben F_{\Gamma}(x^{(1)}) = \int_{\Rset^d} \frac{dk}{(k^2 +m^2)((k-x^{(1)})^2 +m^2)} \een In the case of $d\geq 4$, which we need in QFT, the integral is divergent at $\infty$. This is the so called ultra-violet (UV) divergence. The physicists have invented a number of ways to overcome this difficulty. The objective of this section is to give idea of some of these methods. \subsection{Renormalizability of Field Theories} \subsection{Dimensional Regularization} %%%%%%%%%%%%%%%%555%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \sectionnew{Quantum electrodynamics} %%%%%%%%%%%%%%%%555%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \sectionnew{Gauge theories} %%%%%%%%%%%%%%%%555%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Chern-Simons theory} \subsection{Yang-Mills Theories} %%%%%%%%%%%%%%%%555%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \sectionnew{Appendix} %%%%%%%%%%%%%%%%555%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Linear and Multilinear Algebra} Here we give some definitions. For a more detailed treatment of the topics see e.g. \cite{Gel, Gr}. Let $U, V$ and $E$ be a vector spaces. \bde{9.1} We say that the mapping $\phi$ from $U\bigoplus V$ to $E$ is bilinear if it is linear in each argument when the other is fixed. \ede Exactly in the same manner we define a multilinear mapping from $\bigoplus_jV_j$ to $E$. \bde{9.2} Let the mapping $\phi$ from $V\bigoplus U$ to $E$ is bilinear. We say that $E$ is a tensor product of $U$ and $V$ if the image of $\phi$ is $E$ and if $\psi$ is a map from $U\bigoplus V$ to some vector space $F$ then there exists a linear mapping $\chi$ from $E$ to $F$ such that $\psi = \chi \circ \phi$. In other words the following diagram is commutative: \vspace{0.5cm} \xymatrix{V\bigoplus U \ar[r]^{\phi}\ar[rd]^{\psi} & E \ar[d]^{\chi}\\ & F\\ } \ede \subsection{Differential Geometry} The main object of differential geometry is the notion of {\it connection}. This notion makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. \subsection{Classical Mechanics} Here we give a brief account on some notions of classical mechanics. Our exposition follows \cite{Arn} and for more thorough course the same book is excellent. Of course there are many other books that could serve the purpose. We start with the notion of a functional. Roughly speaking this is a function whose arguments are functions. We will be interested in functionals defined as follows in a particular but very important case, describing classical mechanics. Let $\mathcal{L}(r,q)$ be a function defined on an open set $R\times U$ of $ \Rset^{2d}$ . Let $q(t)$ be a smooth path with $\dot{q},q$ taking values in $U$. {\it Action} is the functional (= function in which the variable is the path $q(t))$: \ben S(q) = \int_{t_0}^{t_1}\mathcal{L}(\dot{q}(t), q(t))dt \een The function $\mathcal{L}$ is called {\it Lagrangian}. Most of the time we will consider Lagrangians of the form \ben \mathcal{L}=T-U(q) = \frac{||\dot{q}||^2}{2} -U(q). \een The quadratic form $T$ is called {\it kinetic energy} and the function $U$ is called {\it potential (energy)}. One can define {\it variational derivative} of $S$ with respect to the path $q$ as usually. Let $\delta q(t) $ is a small change of the path $q(t)$. The difference: \ben \delta S(q)= S(q + \delta q(t)) - S(q) \een is small and can be written as \ben \delta S(q) = F(q)\delta q(t) + \mathcal{O}(|\delta q|^2). \een The function $F(q)$ is called variational derivative of $S$ and is denoted by \ben \frac{\delta S}{\delta q}. \een The paths for which the variational derivative becomes zero satisfy the {\em Euler-Lagrange equations}: \beq \frac{d}{dt}\frac{\partial \mathcal{L}}{\partial \dot{q}_j} - \frac{\partial L}{\partial q_j}=0, \quad j=1,\ldots,d. \label{12.1} \eeq We will need also the equivalent formulation of classical mechanics (and field theory) in Hamiltonian form. Let us first recall the notion of Legendre transform. Consider a convex (or concave) function $f(x)$ and define the function in $p$ and $x$ \ben F(x,p)= (p,x)-f(x). \een For a fixed $p \in V^*$ find the unique extremum of $F(x,p)$ as a function in $x$. This yields the equation: \ben p-Df(x)=0. \een Due to the convexity of $f$ this equation has a unique solution $x=x_0 \in V$. {\it The Legendre transform of $f$} is the function $g(p)$ defined by \ben L(f)(p)= (p,x_0(p))- f(x_0(p)). \een Using Legendre transform we can arrive at Hamiltonian formulation of classical mechanics in the following manner. Let $\mathcal{L}(\dot{q},q)$ be a Lagrangian. Fix the variable $q$ and perform Legendre transform with respect to the variable $\dot{q}$. We obtain a new function $H(p,q)$, which is called {\it Hamiltonan}. Then the Euler-Lagrange equations \eqref{12.1} are equivalent to the Hamiltonian system of equations \beq \dot{q}_j=\frac{\partial H}{\partial p_j},\quad \dot{p}_j=-\frac{\partial H}{\partial q_j},\quad j=1,\dots,d. \label{12.2} \eeq We will often use the following terminology. The variables $q$ will be called {\it position variables} this implying that $\dot{q}$ are velocities. The variables $p$ are {\it momenta}. The set where the position variables are defined is called {\it configuration space.} The entire space where the Hamiltonian is defined is called {\it phase space.} \subsection{Functional Analysis and Differential Equations} We will need quite often Hilbert spaces. Here is the definition. \bde{12.1} Hilbert space $\mathcal{H}$ is a linear space over $\Rset$ (or $\Cset$) with a vector product $(x,y)$ (hermitian vector product $(x,\bar{y}$) which is complete with respect to the norm $||x||=\sqrt{(x,x)}$. \ede \bex{12.1} Let $X$ be a set with a Lebesgue measure $d\mu$. Denote by $L^2(X)$ the space of complex functions with integrable square $\int_X |f|^2d\mu$. Define a scalar product by $\int_X f\bar{g}d\mu$ where $f,g \in L^2(X)$. Then by the theory of Lebesgue integral $L^2(X)$ is a Hilbert space. This is the most important example. \eex The continuous operators are exactly those which satisfy $||Ax|| \leq c||x||$ with some constant $c$. They are also called {\it bounded}. However we will need operators that are unbounded as well as operators that are defined only on subspace of $\mathcal{H}$. For example the operators $\hat{x}_j$ in $L^2(\Rset^n)$ (multiplication by ${x}_j$ is neither defined everywhere, nor bounded. The same is true for the Schr\"dinger operator $-\delta + U(x)$. We will need to find the spectrum of such operators. In fact this problem is in the center of quantum mechanics. More generally we will need to find solutions of partial differential equations. Even when they have "good solutions" (which is not so often) it is very convenient to have broader spaces of "functions" to operate with. The corresponding spaces are different spaces of {\it distributions} (= {\it generalized functions}). We are going to work with the space of tempered distributions, which we define below. First define the Schwartz space $\mathcal{S}$ of all infinitely differentiable functions on $\Rset^n$ which decay at infinity faster than any power of $x_j$. We define topology on this space by the semi-norms \ben p_{\alpha, \beta}(\phi) = sup_{x\in \Rset^n}|| x^{\alpha} D^{\beta}\phi|| . \een The space of continuous functionals on this space is called the {\it space of tempered distributions}. It is denoted by $\mathcal{S}^*$. A very important example of a tempered distribution is {\it Dirac's delta-function}. It is defined as \ben \delta(f) = f(0). \een {\it Fourier transform} of a function $f\in \mathcal{S}(\mathbb{R}^n)$ is defined by the formula: \ben \hat{f}(\xi) = \int_{\mathbb{R}^n}f(x) e^{-i(x,\xi)}dx \een The inverse transform is given by \ben f(x) = \frac{1}{(2\pi)^n} \int_{\mathbb{R}^n} \hat{f}(\xi) e^{i(x,\xi)}d\xi \een We can define {\it Fourier transform of tempered distribution} by the formula: \ben \hat{F}(\phi) := F (\frac{1}{(2\pi)^n}\hat{\phi}), \een where $F\in \mathcal{S}^*$ and $\phi \in \mathcal{S}$ for any test function $\phi \in \mathcal{S}$. Let us compute the Fourier transform of $\delta$. We have \ben \hat{\delta}(\phi) = \delta (\frac{1}{(2\pi)^n}\hat{\phi})= \frac{1}{(2\pi)^n} \hat{\phi}(0) = \frac{1}{(2\pi)^n} \int_{\Rset^n} \phi(x)dx. \een This yields that $\hat{\delta}=\frac{1}{(2\pi)^n}$. In physics and mathematics we often need $\delta$-functions supported at more complicated sets than one point. Hermitian operator $A$ is an operator, satisfying the equality $(Ax,y) = (x,Ay)$ for all vectors from the definition domain of $A$. Differential equations Spectral theorem Representation theory \subsection{Relativistic Notations} {\it Minkowski space} is a space $\Rset^n$ with Minkowskian metric, i.e. a metric with signature $(-1,1\ldots,1)$. Minkowski inner product is defined by $(x,y)_M:=x_0y_0 -x_1y_1-\ldots - x_{n-1}y_{n-1}$. In Minkowski space we define the {\it light cone} by the equation $x^2_0 -x^2_1-\ldots - x^2_{n-1}=0$. A point with coordinates $(x_0,x_1\ldots,x_{n-1})$ is said to be {\it space-like} if $(x,y)_M< 0$. If $(x,y)_M> 0$ the point is said to be {\it time-like}. We would like to introduce {\it time ordering}. If $x,y$ are points we say that $x$ chronologically precedes $y$ if $(x-y)^2>0$. \subsection{Miscellaneous Notations} \ben \left.\begin{aligned} & \nabla \cdot {\bf A } = \partial_{1} A^1 + \partial_{2} A^2 +\partial_{3} A^3 \,\, --\,\, \textrm{ called\,\it nabla of} \, {\bf A}\\ & \nabla \times{\bf A} = (\partial_{2}A^3-\partial_{3}A^2, \partial_{3}A^1-\partial_{1}A^3, \partial_{1}A^2-\partial_{2}A^1) \textrm{ called\,{\it rotor}, or \it curl of} \, \, {\bf A} \end{aligned}\right. \een %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% References %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{small} \begin{thebibliography}{AFM} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bibitem{Arn} V. I. Arnol'd, {\it Mathematical methods of classical mechanics}. Moscow, \bibitem{Ber} A. Berezin, {\em The Method of Second Quantization}, Academic Press, (1966). \bibitem{Bol} B. Bolobas \bibitem{Princ} Pierre Deligne, Pavel Etingof, Daniel S. Freed, Lisa C. Jeffrey, David Kazhdan, John W. Morgan, David R. Morrison, and Edward Witten, editors. {\em Quantum fields and strings: a course for mathematicians.} Vol. 1, 2. American Mathematical Society, Providence, RI, 1999. Material from the Special Year on Quantum Field Theory held at the Institute for Advanced Study, Princeton, NJ, 1996�1997. (also lecture notes available online) \bibitem{DfMS} Ph. Di Francesco, P. Mathieu, David S\'en\'echal, {\em Conformal Field Theory}. Springer, New York, 1997. \bibitem{Dol} I. Dolgachev, {\it Introduction to string theory}, preprint - Ann Arbor., lecture notes available online: http://www.math.lsa.umich.edu/ ~idolga/lecturenotes.html. \bibitem{DNF} B. Dubrovin, S. P. Novikov, A. Fomenko, {em Modern Geometry} Part 1 and Part 2, Springer, 1992. \bibitem{Et} P. Etingof, {\em Mathematical ideas and notions of quantum field theory}. preprint - MIT lecture notes available online: http://math.mit.edu/~etingof/. \bibitem{FY} L. D. Faddeev, O. A. Yakubovsky, {\em Lectures in quantum mechanics for students in mathematics}, Leningradskii universitet, 1980. (in Russian). English translation: Lectures on Quantum Mechanics for Mathematics Students - L. D. Faddeev, Steklov Mathematical Institute, and O. A. Yakubovskii(, St. Petersburg University - with an appendix by Leon Takhtajan - AMS, 2009 \bibitem{Fey1} R. P. Feynman, {\em The character of physical laws}. Cox and Wyman Ltd., London, 1965. \bibitem{FLS} R. P. Feynman, R. B. Leighton and M. Sands, {\em The Feynman Lectures on Physics}., (Addison-Wesley, 1963), Vol III, Chapter 1. \bibitem{Gel} G I. M. Gel'fand, {\em Lectures in linear algebra} \bibitem{Gr} W. H. Greub, {\em Multilinear algebra}, Springer, 1967 \bibitem{IZ} C. Itzykson, and J. B. Zuber, {\em Quantum Field Theory}, McGraw-Hill, 1980. \bibitem {Ka} M. Kaku, {\em Quantum Field Theory, A Modern Introduction}, Oxford University Press, 1993. \bibitem {Kon1} M. Kontsevich, Intersection theory on the moduli spaces of curves and the matrix. Airy function, Comm.Math.Phys.,vol.147(1992),1-23. \bibitem {Kon2} M. Kontsevich, Vassiliev's knot invariants, Adv.Soviet Math.,vol.16,Part 2(1993),137-150. \bibitem {PS} M. E. Peskin, D.V. Schr\"oder, {\em An introduction in quantum field theory}. Perseus Books, Reading Massachusetts, 1995. \bibitem{Pol} M. Polyak, {\em Feynman diagrams for pedestrians and mathematicians}, in: Graphs and Patterns in Mathematics and Theoretical Physics Edited by: Mikhail Lyubich, and Leon Takhtajan, Proceedings of Symposia in Pure Mathematics, AMS, 2005. \bibitem{Rab} J. Rabin, {\em Introduction to QFT for mathematicians}. in: Freed, D. and Uhlenbeck, K., eds., Geometry and Quantum Field Theory, American Mathematical Society, 1995. \bibitem{Ram} Ramond P. {\em Field theory: A modern primer} (2ed., Westview, 2001). \bibitem{R} L. Ryder, {\em Quantum Field Theory}. Cambridge University Press. \bibitem{ Schw} L. Schwartz, {\em Cours d'analyse} (French Edition), 1981. \bibitem{Tay} M.E. Taylor, {\em Partial differential equations 1. Basic theory}, AMS115, Springer, 1996. \bibitem{Tic} R. Ticciati, {\em Quantum field theory for mathematicians} (CUP, 1999). \bibitem{Wit1} E. Witten, "Quantum field theory and the Jones polynomials", CMP, (1988). \bibitem{Wit2} E. Witten, \bibitem{Wo} H Woolf, ed., {\em Some strangeness in proportion}, Addison-Wesley, 1980. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \end{thebibliography} \end{small} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \end{document}