Path Integral Theory

Bruce L. Nelson

We present a well-defined path integral for scattering theory. This path integral is particularly well suited for evaluating the semiclassical expansion for the cross section. As an example, we show how the semiclassical cross section for rainbow scattering can be obtained in a systematic fashion. We indicate how the path-integral approach to semiclassical scattering theory may be more general and more reliable than other approaches.

Leon W. Cohen

Seth Lloyd, Olaf Dreyer

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.

Piers Coleman

Jonathan Dimock

Aleksandar Bogojević

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.