Algebraic Generalization

Abstract

Abstract The path integral method discussed up to the preceding section is based on Trotter’s formula (1.10) and its accuracy is of the order of Δt for the time slice Δt. This is sufficient for discussing quantum systems qualitatively. Higher accuracy of the Quantum Monte method is required for studying quantum systems numerically. In the present chapter, we explain the algebraic generalization of the path integral method. By applying the integral representations given in chapters 1−4 to such generalized product formulae, some new kinds of path integral of higher accuracy can be obtained. This is crucial when the essence of a phenomenon can be found only by increasing the accuracy of calculations. It should be emphasized that such higher order decomposition formulae to be explained in the present chapter can also be applied not only to the path integral method but to many other problems such as non-linear dynamics in Hamiltonian systems.