% File: qft_gen.tex
% Section: QFT
% Title: QFT: General formulas
% Last modified: 27.12.2006 
%
\documentclass[a4paper,12pt]{article}
\usepackage{amsmath,amssymb}
\textwidth 16cm
\textheight 25cm
\oddsidemargin 0cm
\topmargin -1.5cm
\pagestyle{empty}
\begin{document}

\begin{center}
\large\textbf{QFT: GENERAL FORMULAS}
\end{center}

\vspace{.0cm}

\begin{center}
\large\textbf{Minkowski vs Euclidean}
\end{center}

\vspace{.1cm}

In the flat Minkowski 4-space the Lorentz indices are handled as follows:
\begin{gather}
a_\mu b_\mu = a^\mu b_\mu = a_\mu b^\mu = a^\mu b^\mu
= a_0 b_0 - \vec{a}\vec{b} = a_0 b_0 - a_1 b_1 - a_2 b_2 - a_3 b_3 \\
g_{\mu\nu} = \text{diag}\,(1, -1, -1, -1)\,, \ \
g_{\mu\nu}=g_{\nu\mu}\,, \ \
g_{\mu\nu}a_\mu = a_\nu\,, \ \ \ g_{\mu\mu} = 4\,, \ \ \
\partial_\mu x_\nu = g_{\mu\nu} \\
\partial_\mu = (\partial_0\,, \vec{\partial})
= (\tfrac{\partial}{\partial x_0}\,, -\tfrac{\partial}{\partial \vec x})\,,
\ \ \ a_\mu\partial_\mu = a_0\partial_0 - \vec{a}\vec{\partial}
= a_0\tfrac{\partial}{\partial x_0}
+ \vec{a}\tfrac{\partial}{\partial\vec x}\,.
\end{gather}
In the corresponding Euclidean space, 4th components replace 0th in a
universal way, except for derivatives,
\begin{equation}
\label{}
a_0 = -i a_4\,, \ \ \ a_\mu b_\mu = -a_i b_i\,, \ \ \
\partial_0 = i\partial_4 = i\tfrac{\partial}{\partial x_4}\,, \ \ \ \
\nabla_i = (\partial_4\,, -\vec{\partial})
= (\tfrac{\partial}{\partial x_4}\,, \tfrac{\partial}{\partial \vec x})\,,
\end{equation}
Kronecker's $\delta$-symbol replaces $g_{\mu\nu}$,
\begin{equation}
\label{}
\delta_{ij} = \text{diag}\,(1, 1, 1, 1)\,, \ \
\delta_{ij}a_i = a_j\,, \ \ \nabla_i x_j = \delta_{ij}\,,
\end{equation}
while d'Alembertian in both cases means the same:
\begin{equation}
\label{}
\Box = - \partial_\mu\partial_\mu
= -\partial_0\partial_0 + \vec{\partial}\vec{\partial}
= \nabla_i\nabla_i
= \tfrac{\partial}{\partial x_i}\tfrac{\partial}{\partial x_i}\,.
\end{equation}
Relations between evolutionary operators as well as between the action
functionals are obvious,
\begin{gather}
e^{-i t H} \leftrightarrow e^{-\tau H} \ \ (t=x_0\,,\tau=x_4)\,, \ \ \ \
\int\!d^4x_{\scriptscriptstyle M}
= -i\!\!\int\!d^4x_{\scriptscriptstyle E}\,, \\
i S_M = i\!\!\int\!dx_{\scriptscriptstyle M}\,
(\tfrac{1}{2}\partial_\mu\varphi\partial_\mu\varphi-\tfrac{m^2}{2}\varphi^2)
\ \Leftrightarrow \ -\!\int\!dx_{\scriptscriptstyle E}\,
(\tfrac{1}{2}\nabla_i\varphi\nabla_i\varphi+\tfrac{m^2}{2}\varphi^2) = -S_E
\end{gather}
whereas a sign before Euclidean momentum integrals stems
from the `Wick rotation' being in agreement with causality (position of the
poles of propagators in the Minkowski momentum space)\,:
\begin{equation}
\label{}
p_0 = -i p_4\,, \ \ \ p_\mu q_\mu = -p_i q_i\,, \ \ \
\int\!d^4p_{\scriptscriptstyle M} = i\!\!\int\!d^4p_{\scriptscriptstyle E}\,.
\end{equation}
Here are the typical Fourier transform formulas in the $n$-dimensional
Minkowski
\begin{gather}
\frac{1}{(2 \pi)^n}\!\int\frac{d^n\!p\,e^{-ipx}}{(p^2+i0)^\Delta}
= \frac{i}{4^\Delta(-\pi)^{n/2}}\frac{\Gamma(\frac{n}{2}-\Delta)}
{\Gamma(\Delta)}\frac{1}{(x^2-i0)^{\frac{n}{2}-\Delta}} \\
\int\frac{d^n\!x\,e^{ipx}}{(x^2-i0)^\Delta}
= \frac{-i(2\pi)^n}{4^\Delta(-\pi)^{n/2}}\frac{\Gamma(\frac{n}{2}-\Delta)}
{\Gamma(\Delta)}\frac{1}{(k^2+i0)^{\frac{n}{2}-\Delta}}
\end{gather}
and Euclidean
\begin{gather}
\frac{1}{(2 \pi)^n}\!\int\frac{d^n\!p\,e^{ipx}}{(p^2)^\Delta}
= \frac{1}{4^\Delta\pi^{n/2}}\frac{\Gamma(\frac{n}{2}-\Delta)}
{\Gamma(\Delta)}\frac{1}{(x^2)^{\frac{n}{2}-\Delta}} \\
\int\frac{d^n\!x\,e^{-ipx}}{(x^2)^\Delta}
= \frac{(2\pi)^n}{4^\Delta\pi^{n/2}}\frac{\Gamma(\frac{n}{2}-\Delta)}
{\Gamma(\Delta)}\frac{1}{(k^2)^{\frac{n}{2}-\Delta}}
\end{gather}
cases (note that $p^2_{\scriptscriptstyle M}
= -p^2_{\scriptscriptstyle E}$, etc.)\,.

\newpage

\begin{center}
\large\textbf{Gaussian continual integrals}
\end{center}

\vspace{.1cm}

In Minkowski space, general representation of the generating functional
in terms of continual integration is
\begin{equation}
\label{z(j)}
Z(J) = \int\!\mathcal{D}\varphi\,e^{i\int dx(\mathcal{L}+\varphi J)}\,.
\end{equation}
In what follows, we omit the (space-time) integration symbol in the
exponents. A generic Gaussian integration formula for real-like fields is
\begin{equation}
\label{real}
\int\!\mathcal{D}\varphi\,e^{i(\varphi\frac{K}{2}\varphi+\varphi J)}
= (\det K)^{-1/2}\,e^{-\frac{i}{2}JK^{-1}J}
\end{equation}
whereas in complex-like case (with independent variables $z,\bar z$) one has
\begin{equation}
\label{compl}
\int\!\mathcal{D}z\mathcal{D}\bar{z}\,
 e^{i(\bar{z}Kz+\bar{z}\xi+\bar{\eta}z)}
= (\det K)^{\mp 1}\,e^{-i\bar{\eta}K^{-1}\xi}
\end{equation}
( - sign for bosons, + for fermions). Continual $\delta$-function:
\begin{gather}
\delta(\varphi) \doteq \int\!\mathcal{D}\lambda\,e^{i\lambda\varphi}\,, \ \
F(\varphi') = \int\!\mathcal{D}\varphi\,\delta(\varphi-\varphi')F(\varphi) \\
\int\!\mathcal{D}\varphi\,\delta(A\varphi) = (\det A)^{-1}
= \int\!\mathcal{D}\lambda\,\mathcal{D}\varphi\,e^{i\lambda A\varphi}\,.
\end{gather}
For $K=\Box-m^2$ (scalar field) or $K=i\hat{\partial}-m$ (spinors)
a free propagator is
\begin{gather}
<T\varphi(x)\varphi(0)>_0\, = iK^{-1} \doteq -iD^c(x)
= \frac{i}{(2\pi)^n}\!\int\frac{d^n\!p\,e^{-ipx}}{p^2-m^2+i0} \\
<T\psi(x)\psi(0)>_0\, = iK^{-1} \doteq -iS^c(x)
= \frac{i}{(2\pi)^n}\!\int\frac{d^n\!p\,e^{-ipx}(\hat{p}+m)}{p^2-m^2+i0}
\end{gather}

The corresponding formulas in Euclidean case are:
\begin{gather}
Z(J) = \int\!\mathcal{D}\varphi\,e^{-\int dx(H+\varphi J)} \\
\int\!\mathcal{D}\varphi\,e^{-(\varphi\frac{K}{2}\varphi+\varphi J)}
= (\det K)^{-1/2}\,e^{\frac{1}{2}JK^{-1}J} \\
\int\!\mathcal{D}z\mathcal{D}\bar{z}\,
 e^{-(\bar{z}Kz+\bar{z}\xi+\bar{\eta}z)}
= (\det K)^{\mp 1}\,e^{\bar{\eta}K^{-1}\xi} \\
\delta(\varphi) \doteq \int\!\mathcal{D}\lambda\,e^{i\lambda\varphi}, \ \ \
F(\varphi') = \int\!\mathcal{D}\varphi\,\delta(\varphi-\varphi')F(\varphi) \\
<\varphi(x)\varphi(0)>_0\, = K^{-1} = (-\Box+m^2)^{-1} \doteq S_2(x)
= \frac{1}{(2\pi)^n}\!\int\frac{d^n\!p\,e^{ipx}}{p^2+m^2}
\end{gather}

\newpage

\begin{center}
\large\textbf{Generating functionals}
\end{center}

\vspace{.1cm}

Main definitions (in Minkowski space)\,:
\begin{gather}
Z(J) \doteq \int\!\mathcal{D}\varphi\,
\exp\Bigl[\,i\!\!\int dx(\mathcal{L}(\varphi)+\varphi(x)J(x))\Bigr]
\equiv \int\!\mathcal{D}\varphi\,e^{i(\mathcal{L}+\varphi J)}
\doteq e^{iW(J)}\,, \\
Z(J) = \sum_{n}\frac{1}{n!}\!\int\!dx_1\ldots dx_n\,
G_n(x_1\ldots x_n)\,J(x_1)\ldots J(x_n) \ \ \ \ \ \ \
\text{all diagrams} \\
W(J) = \sum_{n}\frac{1}{n!}\!\int\!dx_1\ldots dx_n\,
W_n(x_1\ldots x_n)\,J(x_1)\ldots J(x_n) \ \ \ \ \ \ \
\text{connected diagrams} \\
<\!F(\varphi)\!>_J \
=\int\!\mathcal{D}\varphi\,F(\varphi)\,e^{i(\mathcal{L}(\varphi)+\varphi J)}
= F(-i\frac{\delta}{\delta J})Z(J) \\
\Phi(y) \doteq \frac{\delta W}{\delta J(y)}
= \frac{1}{Z(J)}<\!\varphi(y)\!>_J \ \ \ \ \ \ \
\text{functional argument alternative to $J$}
\end{gather}
We choose either $\Phi = \Phi\,(J)$ or $J = J(\Phi)$ in accordance with
the situation, and define
\begin{gather}
\Gamma(\Phi) \doteq W-J\Phi \equiv
W(J(\Phi)) - \int\!dx\,J(\Phi(x))\,\Phi(x) \\
\Gamma(\Phi) = \sum_{n}\frac{1}{n!}\!\int\!dx_1\ldots dx_n\,
\Gamma_n(x_1\ldots x_n)\,\Phi(x_1)\ldots \Phi(x_n) \ \ \ \ \ \
\text{1PI diagrams} \\
\frac{\delta\Gamma}{\delta\Phi}
=\frac{\delta W}{\delta J}\,\frac{\delta J}{\delta\Phi}
 - \frac{\delta J}{\delta\Phi}\,\Phi - J = -J(\Phi)
\end{gather}
In brief, \ \ $\Phi(J) = W'(J)\,, \ \ J(\Phi) = -\Gamma'(\Phi)$\,.
The functional derivatives are represented as follows,
\begin{equation}
\label{}
\frac{\delta}{\delta J}
= \int\!dx\,\frac{\delta\Phi(x)}{\delta J}\,\frac{\delta}{\delta\Phi(x)}
= W''\frac{\delta}{\delta\Phi}\,, \ \ \
\frac{\delta}{\delta\Phi} = -\Gamma''\frac{\delta}{\delta J}\,,
\end{equation}
therefore,
\begin{equation}
\label{}
\delta(x-y) = \frac{\delta\Phi(x)}{\delta\Phi(y)}
= -\!\int\!dz\,\Gamma''(z,y)\,\frac{\delta\Phi(x)}{\delta J(z)}
= -\!\int\!dz\,W''(x,z)\,\Gamma''(z,y)\,,
\end{equation}
i.e., \ $W''\,\Gamma''=-1$\,,\, and we find
\begin{align}
W''(x,y) &= W_2(x,y) \doteq D(x,y) \ \ \ \ \ \ \
&&\text{full (connected) propagator} \\
\Gamma''(x,y) &= \Gamma_2(x,y) = -D^{-1}(x,y) \ \ \ \ \ \
&&\text{inverse propagator}
\end{align}

\end{document}