Time-Slicing Path-integral in Curved Space

Mingnan Ding1 and Xiangjun Xing1,2,3

1Wilczek Quantum Center, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, 200240 China
2T.D. Lee Institute, Shanghai Jiao Tong University, Shanghai, 200240 China
3Shanghai Research Center for Quantum Sciences, Shanghai 201315 China

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Path integrals constitute powerful representations for both quantum and stochastic dynamics. Yet despite many decades of intensive studies, there is no consensus on how to formulate them for dynamics in curved space, or how to make them covariant with respect to nonlinear transform of variables (NTV). In this work, we construct a rigorous and covariant formulation of time-slicing path integrals for dynamics in curved space. We first establish a rigorous criterion for equivalence of $\textit{time-slice Green's function}$ (TSGF) in the continuum limit (Lemma 1). This implies the existence of infinitely many equivalent representations for time-slicing path integral. We then show that, for any dynamics with second order generator, all time-slice actions are equivalent to a Gaussian (Lemma 2). We further construct a continuous family of equivalent path-integral actions parameterized by an interpolation parameter $\alpha \in [0,1]$ (Lemma 3). The action generically contains term linear in $\Delta \boldsymbol x$, whose concrete form depends on $\alpha$. Finally we also establish the covariance of our path-integral formalism, by demonstrating how the action transforms under NTV. The $\alpha = 0$ representation of time-slice action is particularly convenient because it is Gaussian and transforms as a scalar, as long as $\Delta \boldsymbol x$ transforms according to $\textit{Ito's formula}$.

According to Feynman, the propagator of a quantum system can be expressed as integral over all possibles paths, with each path carries a weight $e^{i S[\mathbf{x}(t)]/\hbar}$, where $S[\mathbf{x}(t)]$ is the classical action. It has been known for long that such a prescription breaks down if the space is curved, or if curvilinear coordinates are used. Even though there have been a very large body of past researches on the correct path integral representation of quantum mechanics in curved space, no consensus has been achieved.

Following Feynman's time-slicing procedure, one only needs to calculate the quantum propagator for an infinitesimal time interval $\Delta t$. When calculating the action for this interval, however, there comes the issue exactly where in this interval the potential energy $V(\mathbf{x})$ should be evaluated. Luckily, the choice of this point does not matter in the limit $\Delta t \rightarrow 0$, as one can show via power counting. Now for quantum mechanics in curved space, or using curvilinear coordinates, there is also an issue where the metric tensor $g_{ij}(\mathbf{x})$ should be evaluated, and the choice turns out to matter significantly. One can, of course, compensate the change of the evaluation point by changing the action accordingly. Such an operation is remarkably similar to renormalization group transformation, where one tunes the cutoff and coupling parameters simultaneously such that the long scale physics remains invariant.

Built on this crucial observation, which has been missed miraculously for so long time, we systematically construct a continuous family of time-slicing path integral representations for quantum mechanics in curved space. It turns out that typical quantum trajectories (which makes dominant contribution to the path integral) behaves as biased random walk, a phenomenon which we call $\textit{quantum spurious drift}$. We also show that, when carrying out nonlinear transformation of variables in path integral, usual rules of calculus, such as Leibniz rules, must be replaced by more sophisticated rules in stochastic calculus. Our theory also solves the problem of time-slicing path integral presentation for classical stochastic processes with multiplicative noises, and shed new light on the origin of various anomalies in quantum field theories.

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[1] Mingnan Ding, Fei Liu, and Xiangjun Xing, "Unified theory of thermodynamics and stochastic thermodynamics for nonlinear Langevin systems driven by non-conservative forces", Physical Review Research 4 4, 043125 (2022).

[2] Mingnan Ding and Xiangjun Xing, "Covariant nonequilibrium thermodynamics from Ito-Langevin dynamics", Physical Review Research 4 3, 033247 (2022).

[3] Thibaut Arnoulx de Pirey, Leticia F. Cugliandolo, Vivien Lecomte, and Frédéric van Wijland, "Path integrals and stochastic calculus", Advances in Physics 71 1-2, 1 (2022).

[4] Cécile Monthus, "Large deviations at level 2.5 and for trajectories observables of diffusion processes: the missing parts with respect to their random-walks counterparts", Journal of Physics A Mathematical General 57 9, 095002 (2024).

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