Lattice Approximation of Quantum Statistical Traces at a Complex Temperature

Ever since the introduction of the path-integrals by R. Feynman, the path-integral formulas with a complex \temperature" have presented an elusive mathematical problem|for instance, the Wiener measure, which is so useful for positive temperature statistics, cannot be extended to the complex case. In fact, since real time processes and real temperature statistics can be identi ed with each other via analytic continuation, the present numerical non-perturbative methods in quantum eld theories rely mostly on computations with lattice approximations to the imaginary time, i.e. real temperature, path-space measures. There have already been proposals for numerical evaluation of the complex pathintegrals, but only now can we establish rigorously the assumed relations between the complex lattice integrals and the continuum traces. Also, the result of Theorem III.5 has been used as an assumption in the derivation of a lattice approximation to the quantum microcanonical ensemble and the theorem now yields a wide range of potentials for which this approximation is valid. For an introduction to the mathematics of operators on a Hilbert space see e.g. the second part of the comprehensive series by Reed and Simon. The path-space measures and their applications are discussed in detail in another book by Simon and the proofs of the statements made in the following section can be found from Section 6 of that book.