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Open Quantum Systems from Dynamical Constraints

Yu Su, Yao Wang

Abstract

Open quantum systems are traditionally described by decomposing the total Hilbert space into a system and an external environment, linked by an explicit interaction Hamiltonian. We propose an alternative framework in which the environment is not introduced as an independent sector a priori, but instead emerges from the dynamical activation of constraints in an initially constrained quantum system. Within Dirac quantization, the physical degrees of freedom define the system, whereas the constraint sector, once promoted to carry its own dynamics, functions as an environment. In this picture, the system-environment coupling is not added through a separate interaction term, but is encoded directly in the constraint structure. As an example, we study a quantum particle coupled to a Brownian-oscillator environment and show how the resulting environmental influence can be formulated in this constraint-based setting. Our results provide a new perspective on the origin of quantum environments and point toward a general framework for open quantum systems rooted in constrained quantization.

Open Quantum Systems from Dynamical Constraints

Abstract

Open quantum systems are traditionally described by decomposing the total Hilbert space into a system and an external environment, linked by an explicit interaction Hamiltonian. We propose an alternative framework in which the environment is not introduced as an independent sector a priori, but instead emerges from the dynamical activation of constraints in an initially constrained quantum system. Within Dirac quantization, the physical degrees of freedom define the system, whereas the constraint sector, once promoted to carry its own dynamics, functions as an environment. In this picture, the system-environment coupling is not added through a separate interaction term, but is encoded directly in the constraint structure. As an example, we study a quantum particle coupled to a Brownian-oscillator environment and show how the resulting environmental influence can be formulated in this constraint-based setting. Our results provide a new perspective on the origin of quantum environments and point toward a general framework for open quantum systems rooted in constrained quantization.

Paper Structure

This paper contains 6 sections, 27 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Dynamical constraint
  3. Dirac--Bergmann alogrithm
  4. Dirac quantization
  5. Auxiliary stochastic fields
  6. Summary and perspective

Figures (2)

  • Figure 1: Comparison between the conventional construction of an open quantum system and the constraint-based construction proposed in this work. In the conventional picture, the system and the environment are introduced as distinct sectors from the outset and are coupled through an explicit interaction (left panel). In the present framework, the same system is constrained to move on a ring, while the constraint sector acquires its own dynamics and thereby plays the role of an environment. The multiple ring radii indicate the breathing motion of the constraint, namely its dynamical activation. Open-system behavior then arises through coupling mediated by the dynamical constraint, rather than by a separately prescribed interaction Hamiltonian (right panel).
  • Figure 2: The interaction between the system and environment is mediated with the constraint-induced stochastic fields.