We begin our discussion of the Feynman path integral with the canonical ensemble. The expressions for the partition function and expectation value of an observable A are, respectively
It is clear that we need to be able to evaluate traces of the type appearing in these expressions. We have already derived expressions for these in the basis of eigenvectors of H. However, since the trace is basis independent, let us explore carrying out these traces in the coordinate basis. We will begin with the partition function and treat expectation values later.
Consider the ensemble of a one-particle system. The partition function evaluated as a trace in the coordinate basis is
We see that the trace involves the diagonal density matrix element
. Let us solve the more general problem of
any density matrix element 
.
If the Hamiltonian takes the form
then we cannot evaluate the operator 
explicitly because
the operators for kinetic (T) and potential energies (U) do not commute with
each other, being, respectively, functions of momentum and position, i.e.,
In this instance, we will make use of the Trotter theorem, which states that
given two operators A and B, such that 
, then for any number
,
Thus, for the Boltzmann operator,
and the partition function becomes
Define the operator in brackets to be 
:
Then,
In between each of the P factors of , the coordinate
space identity operator
is inserted. Since there are P factors, there will be P-1 such insertions.
the integration variables will be labeled 
. Thus, the
expression for the matrix element becomes
The next step clearly involves evaluating the matrix elementx
Note that in the above expression, the operators involving the potential U(X) act on their eigenvectors and can thus be replaced by the corresponding eigenvalues:
In order to evaluate the remaining matrix element, we introduce the momentum space identity operator
Letting 
, the matrix remaining matrix element becomes
Using the fact that
it follows that
The remaining integral over p can be performed by completing the square, leading to the result
Collecting the pieces together, and introducing the 
limit,
we have for the density matrix
The partition function is obtained by setting x=x', which is equivalent to
setting 
and integrating over x, or equivalently 
.
Thus, the expression for 
becomes
where we have introduced a ``frequency''
When expressed in this way, the partition function, for a finite value of P, is isomorphic to a classical configuration integral for a P-particle system, that is a cyclic chain of particles, with harmonic nearest neighbor interactions and interacting with an external potential U(x)/P. That is, the partition function becomes
where
Thus, for finite (if large) P the partition function in the discretized
path integral representation can be treated as any ordinary classical
configuration integral. Consider the integrand of in the limit
that all P points on the cyclic chain are at the same location x. Then
the harmonic nearest neighbor coupling (which is due to the quantum kinetic
energy) vanishes and 
, and the integrand
becomes
which is just the true classical canonical position space distribution function. Therefore, the greater the spatial spread in the cyclic chain, the more ``quantum'' the system is, since this indicates a greater contribution from the quantum kinetic energy. The spatially localized it is, the more the system behaves like a classical system.
It remains formally to take the limit
that . There we will see an elegant formulation for
the density matrix and partition function emerges.