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AIP conference proceedings

The Use of Path Integral Ideals: Deriving the Euler Summation Formula for Path Integrals

Aleksandar Bogojević

Published Jan 1, 2006• 0 citations
Publisher

Abstract

We present and comment on a new quantity that we have recently introduced: the path integral ideal. The new quantity governs the flow of a discrete quantum theory to its continuum limit. Path integral ideals satisfy a unique integral equation — the distinction between different quantum theories being in the boundary conditions. An asymptotic expansion of this equation has led to the derivation of a generalization of Euler’s summation formula for path integrals. The new analytical method has brought about a systematic improvement of the convergence of path integrals. Applied to numerical procedures, the new analytical input has resulted in the speedup of numerical simulations by many orders of magnitude. On the analytical side, the integral equation for ideals may turn out to be a useful setting for extending the obtained results to a wider setting — e.g. to p‐adic valued theories and theories on non‐commuting space‐times.

DOI
10.1063/1.2193134

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We present and comment on a new quantity that we have recently introduced: the path integral ideal. The new quantity governs the flow of a discrete quantum theory to its continuum limit. Path integral ideals satisfy a unique integral equation — the distinction between different quantum theories being in the boundary conditions. An asymptotic expansion of this equation has led to the derivation of a generalization of Euler’s summation formula for path integrals. The new analytical method has brought about a systematic improvement of the convergence of path integrals. Applied to numerical procedures, the new analytical input has resulted in the speedup of numerical simulations by many orders of magnitude. On the analytical side, the integral equation for ideals may turn out to be a useful setting for extending the obtained results to a wider setting — e.g. to p‐adic valued theories and theories on non‐commuting space‐times.
The definition of a path integral is proposed. The method suggested is analogous to Lagrange’s formulation of a path integral used in ordinary quantum mechanics. The notation of linear order on the set of p-adic number, p-adic segment, p-adic Lagrangian, integral of p-adic function of one variable and classical action are introduced. It is proven that if the action is stationary at some trajectory then the Euler–Lagrange equations are satisfied on this trajectory. A finite approximation of a path integral is constructed and the kernel of the operator of evolution is calculated for the case of p-adic harmonic oscillator.
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.
Abstract The focus of Chapter 12 is the Feynman path integral formulations of quantum mechanics and quantum statistical mechanics. The connection between real-time quantum propagator and the density matrix is established via the Wick rotation, and a Trotter expansion is used to derive discrete path integral expressions for both. The continuous limit of the discrete path integral is derived to yield the functional integral formulation of the path integral. The construction of estimators and their averages for computing quantum observables is discussed. Next, the path integral is extended to incorporate Boltzmann, Fermi-Dirac, and Bose-Einstein statistics, and a useful recursive formula for the partition function is derived that accelerates convergence by building up N-body partition functions from those of smaller systems. Quantum free energies are discussed, and algorithms for the evaluation of path integrals using molecular dynamics or Monte Carlo are presented. The chapter concludes with a discussion of higher-order factorizations and ring-polymer contraction methods to reduce the computational overhead of path-integral calculations
Path integrals represent a powerful route to quantization: they calculate probabilities by summing over classical configurations of variables such as fields, assigning each configuration a phase equal to the action of that configuration. This paper defines a universal path integral, which sums over all computable structures. This path integral contains as sub-integrals all possible computable path integrals, including those of field theory, the standard model of elementary particles, discrete models of quantum gravity, string theory, etc. The universal path integral possesses a well-defined measure that guarantees its finiteness, together with a method for extracting probabilities for observable quantities. The universal path integral supports a quantum theory of the universe in which the world that we see around us arises out of the interference between all computable structures.
A recently developed method systematically improved the convergence of generic path integrals for transition amplitudes, partition functions, expectation values and energy spectra. This was achieved by analytically constructing a hierarchy of discretized effective actions indexed by a level number p and converging to the continuum limit as 1/N p . Here we apply the above general method to numerical calculations using Metropolis Monte Carlo simulations of energy expectation values and energy spectra. We analyze and compare the ensuing increase in efficiency of several orders of magnitude.
A systematic classification of Feynman path integrals in quantum mechanics is presented and a table of solvable path integrals is given which reflects the progress made during the last 15 years, including, of course, the main contributions since the invention of the path integral by Feynman in 1942. An outline of the general theory is given which will serve as a quick reference for solving path integrals. Explicit formulas for the so-called basic path integrals are presented on which the general scheme to classify and calculate path integrals in quantum mechanics is based.