Polchinski - String Theory, Volume 1: An Introduction to the Bosonic String

Path integrals are a powerful means of representing quantum theories, especially on surfaces of nontrivial topology. The introduction we present here is similar to that which the reader will find in any modern field theory text. We include it in order to emphasize certain ideas that we will need, such as the relation between the path integral and Hilbert space formalisms and the use of operator equations inside the path integral.

Consider first a quantum mechanics problem, one degree of freedom with Hamiltonian , where

Throughout the appendix operators are indicated by hats. A basic quantity of interest is the amplitude to evolve from one q eigenstate to another in a time T,

In field theory it is generally convenient to use the Heisenberg representation, where operators have the time dependence

The state |q, t is an eigenstate of (t),

In terms of the t = 0 Schrdinger eigenstates this is

In this notation the transition amplitude is qf, T \qi, 0.

By inserting a complete set of states, we can write the transition amplitude as a coherent sum over all states q through which the system might pass at some intermediate time t :

Divide the time interval further into N steps as in ,

Fig. A.1. Transition amplitude broken down into time steps. The dashed line suggests a piecewise linear path, which is one of many ways to define path integration.

and corresponding to each intermediate time insert a complete set of states

Translate back to Schrdinger formalism and introduce an integral over intermediate momenta,

By commuting, we can always write with all to the left and all to the right, so that

Use this to evaluate the matrix element in left of ; drop these. Then to this order becomes

Thus

In the last line we have taken a formal 0 limit, so the integral runs over all paths p(t), q(t) with given q(T) and q(0). This is the Hamiltonian path integral (or functional integral), the sum over phase space paths weighted by exp(iS/) with S the action in Hamiltonian form.

Let us integrate over p(t), making first the approximation that the integral is dominated by the point of stationary phase,

Solving for p in terms of q and and recalling that L = pH, the stationary phase approximation gives

Often the momentum integral is Gaussian, , so that the stationary phase approximation is exact up to a normalization that can be absorbed in the measure [dq]. In fact, we can do this generally: because the relation is the Lagrangian path integral, the integral over all paths in configuration space weighted by exp(iS/), where now S is the Lagrangian action.

We have kept so as to discuss the classical limit. As