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The direct integration of the harmonic oscillator path integral obscures thefundamental structure of its discrete, imaginary time propagator (densitymatrix). This work, by first proving an operator identity for contracting twofree propagators into one in the presence of interaction, derives the discretepropagator by simple algebra without doing any integration. This discretepropagator is $universal$, having the same two hyperbolic coefficient functionsfor all short-time propagators. Individual short-time propagator only modifiesthe coefficient function's argument, its $portal$ parameter, whose convergentorder is the same as the thermodynamic energy. Moreover, the thermodynamicenergy can be given in a closed form for any short-time propagator. Since theportal parameter can be systematically optimized by matching the expansion ofthe product of the two coefficients, any short-time propagator can be optimized$sequentially$, order by order, by matching the product coefficient's expansionalone, without computing the energy. Previous empirical findings on theconvergence of fourth and sixth-order propagators can now be understoodanalytically. An eight-order convergent short-time propagator is also derived.

journal of chemical physics online/the journal of chemical physics/journal of chemical physics

Anatomy of Path Integral Monte Carlo: algebraic derivation of the  harmonic oscillator's universal discrete imaginary-time propagator and its  sequential optimization