[Paper Review] The Classical Equations of Motion of Quantized Gauge Theories, Part I: General Relativity

David E. Kaplan, Tom Melia+1 more|arXiv (Cornell University)|May 2, 2023

Cosmology and Gravitation Theories4 citations

TL;DR

This paper demonstrates that quantum field theories with gauge symmetries—specifically general relativity—allow for a broader class of classical states than traditionally assumed, where initial conditions violate the Hamiltonian and momentum constraints. Despite this, gauge-invariant time evolution leads to effective dynamics indistinguishable from Einstein's equations with an auxiliary, covariantly conserved energy-momentum tensor, enabling novel gravitational phenomena such as cosmic bounces and wormholes without ghost instabilities.

ABSTRACT

In this and a companion paper, we show that quantum field theories with gauge symmetries permit a broader class of classical dynamics than typically assumed. In this article, we show that the dynamics extracted from the path integral or Hamiltonian formulation of general relativity allows for classical states that do not satisfy the full set of Einstein's equations. This amounts to loosening the Hamiltonian and momentum constraints that are imposed on the initial state. Nevertheless, the quantum theory permits gauge invariant time evolution of these states. The time evolution of these states is such that at the classical level the full set of Einstein's equations would appear to hold, with the physical effects of these states being attributable to an auxiliary, covariantly conserved energy-momentum tensor with no internal degrees of freedom. We derive the generalized Einstein equations for these states and show that a homogeneous and isotropic initial background state contributes to expansion identical to cold dark matter. The inhomogeneous components of this state could source curvature perturbations that grow linearly at linear order. This auxiliary contribution to Einstein's equations could have either sign and thus provide a trivial way to violate the null energy condition, enabling novel gravitational dynamics such as cosmic bounces and wormholes.

Motivation & Objective

• To explore whether quantum field theories with gauge symmetries permit classical states that do not satisfy the full set of Einstein’s equations.
• To investigate if such states, violating Hamiltonian and momentum constraints, still yield consistent, gauge-invariant time evolution.
• To derive the generalized Einstein equations governing these non-standard classical states in general relativity.
• To examine cosmological implications, particularly the role of homogeneous and inhomogeneous components in mimicking cold dark matter and sourcing curvature perturbations.
• To assess the potential for such states to violate the null energy condition without instabilities, enabling exotic dynamics like cosmic bounces and wormholes.

Proposed method

• Using the Schwinger-Dyson procedure and path integral formulation, the authors derive classical equations of motion from quantum dynamics without imposing constraints on initial states.
• Applying canonical quantization in minisuperspace, they construct coherent states that evolve unitarily under the Schrödinger equation, even when constraints are initially violated.
• The time evolution of these states is shown to generate an effective, covariantly conserved energy-momentum tensor in the classical limit, absent internal degrees of freedom.
• The generalized Einstein equations are derived by identifying the effective stress-energy contribution from the quantum state’s evolution.
• Cosmological solutions are analyzed, showing that the homogeneous part of the effective tensor mimics cold dark matter with either sign.
• The inhomogeneous components are studied in the linear regime, showing linear growth similar to dark matter and potential for supporting exotic spacetime structures.

Experimental results

Research questions

• RQ1 Can quantum field theories with gauge symmetries produce classical dynamics that violate the standard Einstein equations while remaining gauge-invariant?
• RQ2 What is the form of the effective stress-energy tensor that emerges from quantum states violating Hamiltonian and momentum constraints in general relativity?
• RQ3 How do the homogeneous and inhomogeneous components of such quantum states affect cosmological evolution, particularly in mimicking cold dark matter?
• RQ4 Can these states violate the null energy condition without introducing ghost-like instabilities, and what gravitational phenomena might result?
• RQ5 To what extent can such quantum initial states be erased by inflation, and what observational signatures might remain?

Key findings

• The time evolution of quantum states violating the Hamiltonian and momentum constraints in general relativity still yields gauge-invariant dynamics, with the classical limit appearing to satisfy Einstein’s equations.
• The effective dynamics are described by generalized Einstein equations with an auxiliary, covariantly conserved energy-momentum tensor that has no internal degrees of freedom.
• The homogeneous component of the effective stress tensor contributes to cosmic expansion identical to cold dark matter, regardless of its sign.
• The inhomogeneous components grow linearly at linear order in the perturbative regime, potentially sourcing curvature perturbations with cosmological significance.
• These states can violate the null energy condition without ghost instabilities, enabling novel gravitational phenomena such as cosmic bounces and wormholes.
• A period of inflation can dynamically erase the effects of such initial quantum states, making their observation a potential probe of pre-inflationary physics.