Functional density matrix formulation of quantum statistics

We present a unified formulation for quantum statistical physics based on the representation of the density matrix as a functional integral. For quantum statistical (thermal) field theory, the stochastic variable of the statistical theory is a boundary field configuration. We explore the properties of an effective theory for such boundary configurations and apply it to the computation of the partition function of an interacting one-dimensional quantum-mechanical system at finite temperature. Plots of free energy and specific heat show excellent agreement with more involved semiclassical results. The method of calculation provides an alternative to the usual sum over periodic trajectories: it sums over paths with coincident end points and includes nonvanishing boundary terms. An appropriately modified expansion into modified Matsubara modes is presented.

Figure 1

(Color online) Schematic plot in $\left[-\beta ,\beta \right]$ of series (18) of a configuration $x$ defined originally in $\left[0,\beta \right]$.

Figure 2

(Color online) Plot of ${\Theta }_{\text{min}}$ as a function of the dimensionless coupling constant $g$. Each approximation is applicable for temperatures larger than the associated ${\Theta }_{\text{min}}$. The inset is the same plot for smaller values of $g$.

Figure 3

(Color online) Free energy of the anharmonic oscillator as a function of the dimensionless temperature for $g=0.4$ in the high-temperature region (see text for details).

Figure 4

(Color online) Free energy of the anharmonic oscillator as a function of the dimensionless temperature for $g=0.4$ in the low-temperature region (see text for details).

Figure 5

(Color online) Specific heat of the anharmonic oscillator as a function of the dimensionless temperature for $g$ (see text for details).