Entropic dynamics: The Schrödinger equation and its Bohmian limit

Abstract

Abstract. In the Entropic Dynamics (ED) derivation of the Schrödinger equation the physical input is introduced through constraints that are implemented using Lagrange multipliers. There is one constraint involving a "drift" potential that correlates the motions of different particles and is ultimately responsible for entanglement. The purpose of this work is to deepen our understanding of the corresponding multiplier α . We find that α must take integer values. Its main effect is to control the strength of the drift relative to the fluctuations. We show that ED exhibits a symmetry: models with different values of α can lead to the same Schrödinger equation; different "microscopic" or sub-quantum models lead to the same "macroscopic" or quantum behavior. In the limit of large α the drift prevails over the fluctuations and the particles tend to move along the smooth probability flow lines. Thus ED includes the causal or Bohmian form of quantum mechanics as a special limiting case.

Entropic Dynamics

“…In the ED approach a particle has a definite position, but its velocity-the tangent to the trajectory-is completely undefined. We can also see that the effect of α is to enhance or suppress the magnitude of the drift relative to the fluctuations -a subject that is discussed in detail in [12]. However, for our current purposes we can absorb α into the so far unspecified drift potential, α φ → φ, which amounts to setting α = 1.…”

“…From Equation (43) we see that m AB I AB is a contribution to the energy such that those states that are more smoothly spread out tend to have lower energy. The case ξ < 0 leads to instabilities and is therefore excluded; the case ξ = 0 leads to a qualitatively different theory and will be discussed elsewhere [12]. With this choice of F [ρ] the generalized Hamilton-Jacobi Equation (32) becomes…”

“…These include the quantum measurement problem [5,7,8]; momentum, angular momentum, their uncertainty relations, and spin [9,10]; relativistic scalar fields [11]; the Bohmian limit [12]; and the extension to curved spaces [13].…”

Relational entropic dynamics of particles

“…Continuity is achieved imposing that α n → ∞. As discussed in [10] the multiplier α ′ can be absorbed into φ, which amounts to setting α ′ = 1 without changing the dynamics.…”

The classical limit of entropic quantum dynamics

“…The form of the "ensemble" HamiltonianH is chosen so that the first equation reproduces the FP equation (10). Then, the second equation in (14) becomes a Hamilton-Jacobi equation.…”