[Paper Review] Path integrals for classical-quantum dynamics

Jonathan Oppenheim, Zachary Weller-Davies|arXiv (Cornell University)|Jan 11, 2023

Quantum Mechanics and Applications4 citations

TL;DR

This paper derives a general path integral formulation for classical-quantum (CQ) dynamics that ensures complete positivity and trace preservation—key requirements for physical consistency. By generalizing Feynman’s quantum path integral and stochastic path integrals, it constructs a hybrid path integral using a classical-quantum action, enabling a covariant formulation of CQ dynamics, especially when the Hamiltonian is quadratic in momenta, yielding a configuration-space path integral mapping to CPTP master equations.

ABSTRACT

Consistent dynamics which couples classical and quantum degrees of freedom exists. This dynamics is linear in the hybrid state, completely positive and trace preserving. Starting from completely positive classical-quantum master equations, we derive a general path integral representation for such dynamics in terms of a classical-quantum action, which includes the necessary and sufficient conditions for complete positivity and trace preservation. The path integral we study is a generalization of the Feynman path integral for quantum systems, and the stochastic path integral used to study classical stochastic processes, allowing for interaction between the classical and quantum systems. When the classical-quantum Hamiltonian is at most quadratic in the momenta we are able to derive a configuration space path integral, providing a map between master equations and covariant classical-quantum path integrals.

Motivation & Objective

• To develop a consistent path integral formulation for classical-quantum systems that respects the complete positivity and trace preservation (CPTP) conditions required for physical dynamics.
• To unify the Feynman path integral for quantum systems and the stochastic path integral for classical processes into a single hybrid formalism for CQ dynamics.
• To establish a direct correspondence between CPTP master equations and covariant classical-quantum path integrals, particularly in the quadratic-in-momenta case.
• To enable the application of path integral techniques to quantum control, effective field theories, and semi-classical gravity, where space-time and gauge symmetries must be preserved.
• To provide a framework for studying non-Markovian and constrained CQ dynamics, including gravity, through a manifestly CPTP path integral approach.

Proposed method

• Derives a classical-quantum path integral from the most general form of CPTP classical-quantum master equations, as introduced in Oppenheim (2018) and Oppenheim et al. (2022b).
• Introduces a hybrid action functional that generalizes both the quantum Feynman action and the classical Onsager-Machlup action, incorporating interaction terms between classical and quantum degrees of freedom.
• Imposes the necessary and sufficient conditions for complete positivity and trace preservation directly on the path integral through the structure of the diffusion kernels in the action.
• For systems with quadratic Hamiltonians in momenta, derives a configuration-space path integral, enabling a direct map between master equations and path integrals.
• Uses the path integral to study semi-classical gravity, where the CQ interaction term enforces decoherence of quantum superpositions into mass eigenstates, correlated with classical gravitational fields.
• Analyzes the constraints and symmetry structure of the path integral, showing that CQ evolution leads to quartic-in-momenta actions, altering the standard constraint structure of general relativity.

Experimental results

Research questions

• RQ1 How can a path integral formulation be consistently constructed for classical-quantum dynamics while preserving complete positivity and trace preservation?
• RQ2 What is the general form of the classical-quantum action that unifies quantum and classical stochastic path integrals in a hybrid framework?
• RQ3 Under what conditions does the path integral reduce to a configuration-space formulation, and how does this relate to the master equation formalism?
• RQ4 How does the CQ path integral enforce decoherence of quantum superpositions in a way that correlates with classical gravitational fields, avoiding the pathologies of standard semi-classical gravity?
• RQ5 Can the path integral formulation be extended to non-Markovian or diffeomorphism-breaking theories, and what are the implications for effective field theories?

Key findings

• The paper derives a general path integral representation for classical-quantum dynamics that is manifestly CPTP, with the conditions for complete positivity and trace preservation encoded directly in the structure of the action’s diffusion kernels.
• For classical-quantum systems with a Hamiltonian quadratic in momenta, the path integral reduces to a configuration-space formulation, establishing a direct and covariant map between CPTP master equations and path integrals.
• The path integral formulation correctly enforces decoherence of quantum superpositions into mass eigenstates, with classical paths correlated to the source of the gravitational field, avoiding the failure of standard semi-classical Einstein equations.
• The action for gravity in the path integral includes a CQ interaction term that correlates quantum state decoherence with classical metric evolution, leading to a healthier semi-classical dynamics than the standard $ G_{\mu\nu} = 8\pi G \langle T_{\mu\nu} \rangle $ equation.
• The path integral formulation reveals that CQ dynamics leads to a quartic-in-momenta action, fundamentally altering the constraint structure of general relativity and complicating the transition to the momentum representation.
• The framework allows for the study of non-Markovian CQ dynamics and foliation-preserving diffeomorphism invariance, suggesting a path toward effective quantum gravity theories with controlled symmetry breaking.