Trading drift and fluctuations in entropic dynamics: quantum dynamics as an emergent universality class

Abstract

Abstract. Entropic Dynamics (ED) is a framework that allows the formulation of dynamical theories as an application of entropic methods of inference. In the generic application of ED to derive the Schrödinger equation for N particles the dynamics is a non-dissipative diffusion in which the system follows a "Brownian" trajectory with fluctuations superposed on a smooth drift. We show that there is a family of ED models that differ at the "microscopic" or sub-quantum level in that one can enhance or suppress the fluctuations relative to the drift. Nevertheless, members of this family belong to the same universality class in that they all lead to the same emergent Schrödinger behavior at the "macroscopic" or quantum level. The model in which fluctuations are totally suppressed is of particular interest: the system evolves along the smooth lines of probability flow. Thus ED includes the Bohmian or causal form of quantum mechanics as a special limiting case. We briefly explore a different universality class -a nondissipative dynamics with microscopic fluctuations but no quantum potential. The Bohmian limit of these hybrid models is equivalent to classical mechanics. Finally we show that the Heisenberg uncertainty relation is unaffected either by enhancing or suppressing microscopic fluctuations or by switching off the quantum potential.

Entropic dynamics: reconstructing quantum field theory in curved space-time

“…The transition probability-Next we maximize (1) subject to (5) and normalization. As discussed in [56] the multiplier α ′ associated to the global constraint (5) turns out to have no influence on the dynamics: it can be absorbed into the drift potential α ′ φ → φ which means we can effectively set α ′ = 1.…”

Quantum Trajectories in Entropic Dynamics

“…(2.12) for some integer n. For n = 0, the particles follow Brownian trajectories, which in the limit of η → 0 and ξ → 0 recovers the smooth Bohmian trajectories [12].…”

“…The Lagrange multiplier α plays the role of controlling the relative strength of the fluctuations [12].…”

A covariant approach to entropic dynamics

“…As with any tool, its purpose is pragmatic in nature; that is, it is built to accomplish a particular 1 The Lagrange multiplier α ′ associated with the global constraint (3) turns out to be unimportant and can be chosen so that α ′ = 1. The interested reader can look to [19][20] for more information.…”

“…This choice of dynamics has the added feature in that it singles out a preferred choice of representation for implementing path independence. That is, the generators G Ax appearing in (19) can be represented as Hamiltonian generators H Ax , with the brackets [ ·, · ] being identified with Poisson brackets. Naturally, a covariant dynamics for ρ and Φ can then be achieved by choosing the Hamiltonian generators that satisfy the algebra (19).…”