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I have a question regarding Physics. The theme concerns some history behind the development of the physics we know and the questions it might evoke. In quantum physics, there is a notorious effort to assemble its different parts: gravitation, eletrodynamics, weak and strong forces. Its development bases on prior formalismus which, among others, Statistical Physics and Path integrals are within. My current master advisor developed an optimization method based on the Feynman path integral (Link). With this brief introduction, my question concerns gravitational quantum physics:

A brief description of the path summation (discrete version for the path integral, as far as I know, the statistical physics formalismus) develops a sequence of data points with the local evaluation of a positive function (namely, the Lagrangian). The expected value position for the particle corresponds to a gaussian function. The particle path follows, on average, the gradient of the function. Is there some comprehensible work (graduate or undergraduate level) about the same path summation approach, but in the sense one look for a computable way to obtain the approximate path from one point to another?

Formal statement: Given points A and B, in hand of Feynman path Integral, obtain systematic computable approach to obtain the path corresponding from A to a vicinity of B.

I hope, the question is comprehensible.

Best regards, Bruno Peixoto

Bruno Lobo
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1 Answers1

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In an analytic approach the paths that dominate the path integral (contribute most to the sum over trajectories) are those closest to the classical path - this can be seen by making a stationary phase (or laplace approximation in euclidean space) expansion of the action, S, about the classical solution. Indeed, in the classical limit ($\hbar \rightarrow 0$) the contribution comes entirely from the classical solution to the equations of motion, on which $\delta S =0$, and the principle of of least action is recovered.

With numerical methods the usual approach is a Monte Carlo simulation in which trajectories are sampled so as to favour those that minimise the (euclidean) action. A good sampling would provide a convergence towards solutions that explore close to the classical path. This was explored with complementary techniques (based on "worldline numerics") in Annals Phys. 411 (2019) 167966, https://doi.org/10.1016/j.aop.2019.167966 [arXiv:1903.00536] where the classical limit was analysed for some simple systems.

lux
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