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\lhead{\scriptsize\textbf{PHYSICAL REVIEW}}
\chead{\scriptsize\textbf{VOLUME 97, NUMBER 3}}
\rhead{\scriptsize\textbf{FEBRUARY 1, 1955}}
\title{\large\fontseries{bx}\selectfont{Slow Electrons in a Polar Crystal}\vspace{-3ex}}
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\author{
    R. P. Feynman\\
    \textit{California Institue of Technology, Pasdena, California}\\
    (Received October 19, 1954)\\
  }
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       A variational principle is developed for the lowest energy of a system described by a path integral. It is applied to the problem of the interaction of an electron with a polarizable lattice, as idealized by Frohlich. The motnion of the electron, after the phonons of the lattice field are eliminated, is described as a path integral. The variational method applied to this gives an energy for all values of the coupling onstant. It is at least as accurate as previosly known results. The effective mass of the electron is also calculated, but the accurtact here is difficult to judge.
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\lettrine[findent=2pt]{\textbf{A}}{ }N electron in an iconic crystal polarizes the lattice in its neighborhood. This interaction changes the energy of the electron. Furthermore, when the electron moves the polarization state must move with it. An electron moving with its accompanying distortion of the lattice has sometimes been called a polaron. It has an effective mass higher than that of the electron. We wish to compute the energy and effective mass of such an electron. A summary giving the present state of this problem has been given by Frohlich.\footnote{H. Frohlich,Advances in Physics 3,325 (1954). Refrences to other work is given here.} He makes simplifying assumptions, such that the crystal lattice acts much like a dielectric medium, and that all the important phonon waves have the same frequency. Whe will not discuss the validity of these assumptions here, but will consider the problem described by Frohlich as simply a mathematical problem. Aside from its intrinsic interest, the problem is much simplified analog of those which occur in the conventional meson theory when perturbation theory is inadequate. The method we shall use to solve the polaron problem is new, but the pseudoscalar symmetric meson field problems involve so many further complications that it cannot be directly applied there without further development
\par We shall show how the variational technique which is so succesful in ordinary quantum mechanic can be extended to integrals over trajectories.
\begin{center}
\textbf{\small{STATEMENT OF THE PROBLEM}}
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\par With Frohlich's assumptions, the problem is reduced to that of finding the properties of the following Hamiltonian
\begin{equation*}
\scriptsize H=\frac{1}{2}P^2+\sum_{k}\alpha_{K}+\alpha_{K}+i\sqrt{2}_{pi\alpha}/V)^{\frac{1}{2}}\sum_{K}\mathlarger{\frac{1}{K}}
\end{equation*}
%\begin{flushright}
% EQUATION WITH NUMBER (1)
\begin{equation}\label{eq:1}
 \mathlarger{\times}[\alpha_{K}+\exp(-i\mathlarger{\boldsymbol{K}}\cdot\mathlarger{\boldsymbol{X}})-\alpha_{K}exp(i\mathlarger{\boldsymbol{K}}\cdot\mathlarger{\boldsymbol{X}})].
\end{equation}
%\end{flushright}
Here \textbf{X} is the vector position of the electron, \textbf{P} its conjugate momentum, $\alpha_{K}^+$,$\alpha_{K}$, the creation and annihilation operators of a phonon (of momentum \textbf{K}). Our units are such that $\hbar$, this freuency and the electron mass are unity. The quantity $\alpha$ acts as a coupling constant, which may be large or small. In conventional units it is given by
\begin{center}
$\alpha=\frac{1}{2}(\frac{1}{\epsilon_{\infty}}-\frac{1}{\epsilon})\frac{e^2}{\hbar\omega}(\frac{2m\omega}{\hbar})^{\frac{1}{2}}$,
\end{center}
where $\epsilon$, $\epsilon_{\infty}$ are the static and high frequency dielectric constant, respectively. In a typical case, such as NaCl, $\alpha$\ may be about 5. The wave function of the system satisfies ($\hbar = 1$) \\ \\
\begin{equation}\label{eq:2}
i\delta \psi / \delta t = H\psi
\end{equation}

so that if $\varphi_{n}$ and $E_{n}$ are the eigenfunctions and eigenvalues of $H$,
\begin{equation}\label{eq:3}
 H\varphi_{n}=E_{n}\varphi_{n}
\end{equation}
then any solutions of (\ref{eq:2}) is of the form
\begin{equation*}
\psi=\sum\nolimits_nC_{n}\varphi_{n}e^{-iE_{n}t}
\end{equation*}
\par Now we can cast (\ref{eq:1}) and (\ref{eq:2}) int the Lagrangian form of quatum mechanics and then eliminate the field oscillators (specializing to the case that all phonons are virtual). Doing this in exact analogy to quantum electrodynamics,\footnote{R.P.Feynman, Phys. Rev. 80, 440 (1950)} we find that we must study the sum over all trajectories \textbf{X}(t) of exp($iS'$), where
%LEFT ALIGNED EQUATION
\begin{fleqn}
\begin{align*}
S'=\frac{1}{2}\int\bigg(\frac{d\boldsymbol{X}}{dt}\bigg)^{2}dt
\end{align*}
\end{fleqn}
%RIGHT ALIGNED EQUATION
\begin{flalign}\label{eq:4}
+2^{-\frac{3}{2}}\alpha i \int\int |\boldsymbol{X_{t}} - \boldsymbol{X_{8}}|^{-1}e^{-i|t-8|}dtds.
\end{flalign}
This sum will depend on the initial and final conditions and on the time interval T. Since it is a solution of the Schrodinger Eq. (\ref{eq:2}), considered as a function of T it will contain frequencies $E_{n}$, the lowest of which we seek. It is difficult to isolate the lowest frequency, however. \par For that reason, consider the mathematical problem of solving
\begin{equation}\label{eq:5}
\scriptstyle \delta\psi / \delta t = -H\psi
\end{equation}
without question as to the meaning of t. This has the same eigenvalues and eigenfunctions as (\ref{eq:3}), but a
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