Partition Function and Density Matrix Path Integral Representation

Density Matrix Path Integral Representation

For now, let us proceed in the evaluation of the density matrix via the functional integration in a similar manner as we did for the transition amplitude.

Density Matrix Operator Average Value in Phase Space

First we write the density matrix operator average value in phase space:

Here,

This gives the basis for path integral application in non-equilibrium statistical physics as well as the kinetic theory. We consider a statistical ensemble of a quantum system at the absolute temperature T and described by the Hamiltonian of a stationary system:

For brevity, the unnormalized density matrix can be written as:

and the normalized equilibrium density matrix as:

Here, the statistical partition function Z(/l) is:

From the above, equation 15.4 can equally be obtained from the solution of the following Cauchy problem:

which can be rewritten in the operator form referred to as the Bloch equation [1,20] for the density matrix of a canonical ensemble:

and the Bloch equation in the coordinate ^-representation:

The second equation in 15.9 is the boundary condition at/? = 0 and expresses the completeness condition on the eigenfunctions.

In the formulation of statistical physics, we pass from the real time path integral formulation to the Euclidean time formulation where the pure imaginary time is introduced by the analytic continuation of the time interval to the negative imaginary value:

where r is a real number [2] on the negative imaginary axis (Figure 15.1):

and

So, similarly, considering the Cauchy problems 4.7 and 4.9 then the solution of 4.8 imitates 4.11:

Here T is the operator of the chronological-ordered product that orders the times chronologically with the latest time to the left and the ordering parameter г has the dimension of inverse energy.

We can now find the matrix element of the equilibrium statistical density matrix operator in the path integral representation:

Here,

is the dimensionless and real quantum-statistical action functional (Euclidean action) and r, the Euclidean time.

We can now rewrite the statistical partition function in 6: or

This involves path integral in the sub-space of generalized coordinates. We find how 15.14 relates 15.17 by calculating the matrix element of the normalized equilibrium statistical density matrix defined in 5:

Generalized Gaussian Functional Path Integral in Phase Space

For brevity, we consider harmonic oscillation with the Lagrangian:

The boundary condition for our problem is, respectively, the starting and ending coordinates of the path:

We write the Euler-Lagrange equation described by the relation in 3.30: with solution being the classical path qd.

From quasi-classical approximation, we select a fluctuating path q'(r) interpreted as the quantum fluctuations around the classical path qd(f'):

The classical action is obtained:

So,

where the reduced propagator is:

We apply this to the case of a charged particle in a uniform magnetic field in the z-direction where the action functional is:

and the reduced propagator:

The partition function can now be written as: where V is the volume.

 
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