Finite-Time Path Integrals and a Geometric Quantum of Action

N. L. Diaz, J. M. Matera, R. Rossignoli

Soon Hee@Bungsuh Yeo

Abstract Feynman's path integral formulation describes quantum propagation as a sum over all possible paths, weighted by a phase factor e^{iS/ℏ}. The classical path emerges via constructive interference of paths with stationary action. We reinterpret this mechanism within the Generalized Mass as Twisted Time (GMaTT) framework. The phase factor is identified as the accumulated phase difference Δφ of a G₂ root vector rotating around the √α axis. The condition for constructive interference is shown to be equivalent to the TUT (TwistUntwist Threshold) coherence condition |δφ| < √α. We clarify that the TUT window is not a dynamic laboratory filter but a static geometric boundary condition of the phase space. Classical paths are not "chosen" by the particle; they are the trajectories that maintain phase coherence within the TUT threshold during propagation. This provides a geometric origin for the path integral and the principle of least action, grounding Feynman's sum in the pregeometric phase geometry of GMaTT.

R. J. Rivers

Let us recapitulate. Although the form (4.32) for the path integral was motivated by analogy with finite-dimensional integrals, in no way is it a definition. Rather, as we have stressed repeatedly, it is an algorithm for handling the Feynman perturbation series. Strong coupling apart, all the manipulations of the path integral that we have performed (integration by parts, the introduction of auxiliary fields, the inclusion of background fields) stand in one-to-one correspondence to rearrangements of the perturbation series. In fact, path integrals are unnecessary in the derivation of expressions like (4.50). See Fried (1972), for example. The path integrals are as rigorous – or non-rigorous – as the series, irrespective of whether they can be given proper mathematical definition or not. At the same time, the analogy with finite-dimensional integrals suggests approaches that go beyond this purely operational stance. In particular, because of the dominant role played by the classical action in the path integral formalism we would like to use it to establish the relation of quantum dynamics to classical dynamics. This is the most compelling reason for introducing the path integral formalism, requiring a distinct relaxation in our treatment of it. By formal analogy with finite-dimensional saddle-point and stationary-phase methods, elevating the classical action to the phase of the integrand promotes series expansions in ħ as the natural way to proceed.

Michael C. Ogilvie

Matouš Ringel, Vladimir Gritsev

We develop a dynamical symmetry approach to path integrals for general interacting quantum spin systems. The time-ordered exponential obtained after the Hubbard-Stratonovich transformation can be disentangled into the product of a finite number of the usual exponentials. This procedure leads to a set of stochastic differential equations on the group manifold, which can be further formulated in terms of the supersymmetric effective action. This action has the form of the Witten topological field theory in the continuum limit. As a consequence, we show how it can be used to obtain the exact results for a specific quantum many-body system which can be otherwise solved only by the Bethe ansatz. This represents an example of a many-body system treated exactly using the path-integral formulation. Moreover, our method can deal with time-dependent parameters, which we demonstrate explicitly.