Simultaneous Quantization and Reduction of Constrained Systems
Jianhao M. Yang
TL;DR
The paper tackles the ambiguity in choosing whether to quantize before or after imposing constraints in constrained quantum systems by introducing an extended stationary action principle that adds an information metric $I_f$ to the classical action, forming a total action $A_t = A_c + \frac{\hbar}{2} I_f$. Constraints are incorporated directly into the Lagrangian via Lagrange multipliers, enabling quantization and constraint enforcement to occur simultaneously in a single variational step. The framework reproduces the Schrödinger equation via a five-step procedure, and demonstrates a nontrivial constrained quantization for a one-dimensional ensemble with vanishing local momentum, while remaining consistent with reduced and Dirac quantization for a translationally invariant bipartite system. An information-theoretic interpretation emerges, with the Bohm quantum potential arising from the extremization of $I_f$, and the approach offers a pathway to extend standard quantization to constraints not expressible as linear operators on the wave function, with potential applications to gauge theories and quantum gravity.$
Abstract
We present a novel framework for quantizing constrained quantum systems in which the processes of quantization and constraint enforcement are performed simultaneously. The approach is based on an extension of the stationary action principle, incorporating an information-theoretic term arising from vacuum fluctuations. Constraints are included directly in the Lagrangian via Lagrange multipliers, allowing the subsequent variational procedure to yield the quantum dynamics without ambiguity regarding the order of quantization and reduction. We demonstrate the method through two examples: (i) a one-dimensional system with vanishing local momentum, where the simultaneous approach produces the time-independent Schrödinger equation while conventional reduced and Dirac quantization yield only trivial states, and (ii) a bipartite system with global translational invariance, where all three methods agree. These results show that the proposed framework generalizes standard quantization schemes and provides a consistent treatment of systems with constraints that cannot be expressed as linear operators acting on the wave function. In addition to a unified variational principle for constrained quantum systems, the formalism also offers an information-theoretic perspective on quantum effects arising from vacuum fluctuations.