Feynman-Lagrangian Approach to Density Matrix/Partition Function for High Temperature

It is known one may use Feynman’s path integral approach to solve for a quantum propagator. Setting it=b where b=1/Tp (Tp=temperature), the approach may also be used to calculate the partition function (which is linked to a density matrix calculation). Furthermore, one may assume a form for the propagator and note that it is a solution of the time-dependent Schrodinger equation with id/dt replaced with -d/db. For example, setting P= Q(t) exp[i ( b(t)XX + g(t)YY + g1(t) XY] and equating coefficients for XX, XY and YY separately (and equating imaginary terms for Q(t)), yields differential equations for Q(t), b(t), g(t) and g1(t). In (1), this approach is used to find the partition function for an oscillator potential with an extra ax term. The differential equations are established and solved, then the solutions are expanded in powers of b=1/Tp i.e. for high temperature (tiny b) and a high-temperature partition function obtained. In (2), we calculated the propagator (and hence partition function) using Q(t) exp( i Action) where Action = Integral dt (0,T) L where L is the Lagrangian. In this approach: x(t)= XA(T-t)/A(T) + Y A(t-0)/A(T) (3) where X and Y are parameters, T=Tf and Ti=0 and T is associated with b=1/Tp. We argue in this note, that one may perform a first order expansion in b=1/Tp of A(T-t) and A(t-0) and then perform the integral dt L. This yields the high temperature result for the oscillator plus ax potential partition function without any need of solving differential equations and then taking high temperature limits