\chapter{Feynman's path integrals in quantum mechanics}
Path integrals is a formulation of quantum mechanics where particles
move simultaneously along all possible paths with certain probability
amplitude.

\section{Time evolution in Schr\"odinger's wave-mechanics}
The time evolution of a quantum system with Hamiltonian $H$ is
described by the time dependent Schr\"odinger equation,
	\begin{equation}
i\hbar\frac{\partial\psi}{\partial t}=H\psi\:,
	\end{equation}
with a formal solution
	\begin{equation}
\psi=e^{-\tfrac{i}{\hbar}Ht}\psi_0\:,
	\end{equation}
where $\psi_0=\psi|_{t=0}$ and
	\begin{equation}
e^{-\tfrac{i}{\hbar}Ht}\equiv U
	\end{equation}
is the
time-evolution operator which propagates the system from time zero to
time $t$. The operator can be explicitly calculated if all eigenvalues
and eigenvectors of the Hamiltonian $H$ are known.

For example, it can be calculated in the case of a free particle with
mass $m$ in one dimension, with the free Hamiltonian
	\begin{equation}
H_0=-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}\:.
	\end{equation}
The matrix elements of the free time-evolution operator
$U_0=e^{-\tfrac{i}{\hbar}H_0t}$ are given in momentum space as
	\begin{equation}
\langle k^\prime|U_0|k\rangle =\delta_{k^\prime k}e^{-i\frac{\hbar
k^2}{2m}t}\:,
	\end{equation}
and in coordinate space as
	\begin{eqnarray}\label{pi:eq:xux}
\langle x^\prime|U_0|x\rangle =\int\frac{dk}{2\pi}e^{-i\frac{\hbar
k^2}{2m}t}e^{ik(x-x^\prime)} \nonumber\\
=
\sqrt{\frac{m}{2\pi i\hbar t}}
e^{\frac{i}{\hbar}\frac{m}{2t}(x^\prime-x)^2}\:.
	\end{eqnarray}
The latter matrix element is the probability amplitude for the
particle to propagate (move) from point $x$ at time zero to
$x^\prime$ at time $t$.

\section{Time evolution in path integral formulation}
In path integral formulation a particle can propagate from an initial
position $x$ to the final position $x^\prime$ simultaneously along all
possible paths. The amplitude of the probability for the particle to
start at $x$ and end up at $x^\prime$ is given as the ``path
integral'',
	\begin{equation}\label{pi:eq:ps}
\langle x^\prime|U|x\rangle=
\sum_{p~\in~\mathrm{paths}}w(p)~e^{\frac{i}{\hbar}S(p)}\:,
	\end{equation}
where $w(p)$ is a certain weight factor of path $p$ and
	\begin{equation}
S(p)=\int_{p} L dt
	\end{equation}
is the action integral along the path $p$, and $L$ is the Lagrangian of
the particle.

Apparently, when $S\gg \hbar$ only the stationary classical path, $\delta
S(p)=0$, gives a non-vanishing contribution to the path integral, which
is the obvious limit of classical physics.

The weight factors $w(p)$ depend upon the class of the paths and cannot
be written in a general form. The polygonal paths is the usual class of
paths considered in path integral formulations of quantum mechanics.

\subsection{Line segment propagator}
Any path can be approximated by a sequence of connected line segments
forming a broken line path, also called piecewise linear path, or
polygonal path.

The weight factor for the line segment propagator can be found e.g. by
considering the motion of a free particle along a straight line and
comparing the result with (\ref{pi:eq:xux}).

A free one-dimensional particle with mass $m$, coordinate
$x$, and velocity $\dot{x}$ has the Lagrangian
	\begin{equation}
 L(x,\dot{x})=\frac{m\dot{x}^{2}}{2}\:.
	\end{equation}

Assuming the particle moves from $x$ to $x^\prime$ along a straight line
with constant velocity $\dot{x}=(x^\prime-x)/t$, the action integral
along the path in~(\ref{pi:eq:ps}) is given as
	\begin{equation}
e^{\frac{i}{\hbar}S}=
e^{\frac{i}{\hbar}\frac{m}{2}\left(\frac{x^\prime-x}{t}\right)^2t}
=e^{\frac{i}{\hbar}\frac{m}{2t}(x^\prime-x)^2}\:.
	\end{equation}
The exponent is identical to that of the Schr\"odinger propagator
(\ref{pi:eq:xux}). If now the weight factor is chosen as
	\begin{equation}
w=\sqrt{\frac{m}{2\pi i\hbar t}}\:,
	\end{equation}
the line segment propagator for the free particle becomes identical
to (\ref{pi:eq:xux}).

Generally, if the particle is not free but moves in a potential $V$
with the Lagrangian
	\begin{equation}
 L(x,\dot{x})=\frac{m\dot{x}^{2}}{2}-V(x)\:,
	\end{equation}
the short line propagator can be approximated as
	\begin{equation}
we^{\frac{i}{\hbar}S}= \sqrt{\frac{m}{2\pi
i\hbar t}} e^{\frac{i}{\hbar}\left(
\frac{m}{2t}(x^\prime-x)^2-V(\frac{x+x^\prime}{2})t \right)} \:.
	\end{equation}

\subsection{Polygonal path formulation}
In the polygonal path formulation the time is discretized into
$N+1$ equidistant points $t_{n}=n\Delta{}t$ where $n=0\dots N$ and
$\Delta{}t=t/N$; and the particle is assumed to move from $x=a$ to $x=b$
along a polygon $\left\{(t_0,a),(t_1,x_1),\dots,(t_N,b)\right\}$.

Using the obvious composition rule
	\begin{equation}
\langle b|U|a\rangle=\int dx
\langle b|U|x\rangle
\langle x|U|a\rangle\:,
	\end{equation}
the path integral along this polygon can be written as
\begin{eqnarray}
\lefteqn{\langle b|U|a\rangle=}\\
&&\int dx_1
\sqrt{\frac{m}{2\pi i \hbar \Delta t}}e^{\frac{i}{\hbar}S_{1}}
\int dx_2
\sqrt{\frac{m}{2\pi i \hbar \Delta t}}e^{\frac{i}{\hbar}S_{2}}
\dots\nonumber\\
&&\int dx_{N-1}
\sqrt{\frac{m}{2\pi i \hbar \Delta t}}e^{\frac{i}{\hbar}S_{N-1}}
\sqrt{\frac{m}{2\pi i \hbar \Delta t}}e^{\frac{i}{\hbar}S_{N}}
\nonumber
\end{eqnarray}
where the short time action is defined using the midpoint rule,
\begin{equation}
S_{n} =  \Delta{}t
L\left(\frac{x_{n}+x_{n-1}}{2},\frac{x_{n}-x_{n-1}}{\Delta t}\right)
\end{equation}

\subsection{Space discretization}
If the space is also discretized into $M$ equidistant points
$x_{n}=x_\mathrm{min}+n\Delta x$, where $n=0\dots M-1$ and $\Delta
x=(x_\mathrm{max}-x_\mathrm{min})/(M-1)$, the short-time propagator
$U(\Delta t)$ becomes a complex matrix with matrix elements
	\begin{eqnarray}
\langle x_n|U(\Delta t)|x_{n'}\rangle \equiv
U(\Delta{}t)_{nn^\prime}\nonumber\\
=\sqrt{\frac{m}{2\pi i\hbar\Delta t}}\;
e^{\frac{i\Delta{}t}{\hbar} L\left(
\frac{x_{n}+x_{n'}}{2},\frac{x_{n}-x_{n'}}{\Delta t}
\right)}
	\end{eqnarray}
and the finite time propagator $U(t)$ becomes a product of short-time
propagators,
	\begin{equation}
U(t)=\Delta{}x^{N-1}U(\Delta{}t)^{N}.
	\end{equation}

\subsection{Real time propagation and the spectrum of the system}
The energy levels can be extracted by a
Fourier transform of the trace of the propagator:
	\begin{equation}
\mathrm{trace}\big(U(t)\big) =
\sum_{\nu} \langle \nu | e^{-\frac{i}{\hbar}Ht} | \nu \rangle
= \sum_{\nu} e^{-\frac{i}{\hbar}E_{\nu} t}\:,
	\end{equation}
where $E_{\nu}$ and $|\nu\rangle$ are the eigenvalues and
eigen-functions of the system's Hamiltonian,
	\begin{equation}
H|\nu\rangle = E_{\nu}|\nu\rangle\:.
	\end{equation}

\subsection{Imaginary time propagation and eigen-function}
Propagation of a random state $|\psi\rangle$ in imaginary time
$t=-i\hbar\tau$,
	\begin{equation}
U(-i\hbar\tau)=e^{-\tau H}\:.
	\end{equation}
apparently reduces the contributions of excited states
and converges to the ground state $|0\rangle$,
	\begin{equation}
e^{-\tau H}|\psi\rangle
\stackrel{\tau\to\infty}{\longrightarrow}
|0\rangle\:.
	\end{equation}

The trace of the imaginary time propagator is apparently the partition
function, with the temperature $T=1/\tau$,
	\begin{equation}
\mathrm{trace}\left(U(-i\hbar\tau)\right)=\sum_\nu e^{-\tau{}E_\nu}\:,
	\end{equation}
For small temperatures the system cools down to the ground state
$|0\rangle$.

% \p{Problems}
% \begin{itemize}
% \item Consider (the lowest states of) a one-dimensional oscillator with time
% and space discretized path integral method with real- and complex-time
% propagation.
% \end{itemize}
