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Quantum scalar field in quantum gravity: the vacuum in the spherically symmetric case

Rodolfo Gambini, Jorge Pullin, Saeed Rastgoo

TL;DR

This paper tackles the problem of formulating loop quantum gravity for a spherically symmetric spacetime with a scalar field by employing uniform discretization and a variational minimization of the master constraint. The authors construct a lattice-based, polymer-quantized framework, derive an effective Hamiltonian for the scalar field on a semiclassical quantum geometry, and implement Gaussian trial states to minimize the master constraint. They find that the minimum master constraint is nonzero, corresponding to a vacuum state that is effectively a product of a Fock vacuum for the scalar field and a Gaussian, semiclassical gravitational state, with no true continuum limit but good semiclassical behavior at finite lattice spacing. The work highlights a coordinated pathway to connect quantum gravity with quantum field theory in curved spacetime while underscoring the need to treat diffeomorphism and Hamiltonian constraints jointly to fully regulate matter in quantum geometry and to address the cosmological constant problem.

Abstract

We study gravity coupled to a scalar field in spherical symmetry using loop quantum gravity techniques. Since this model has local degrees of freedom, one has to face ``the problem of dynamics'', that is, diffeomorphism and Hamiltonian constraints that do not form a Lie algebra. We tackle the problem using the ``uniform discretization'' technique. We study the expectation value of the master constraint and argue that among the states that minimize the master constraint is one that incorporates the usual Fock vacuum for the matter content of the theory.

Quantum scalar field in quantum gravity: the vacuum in the spherically symmetric case

TL;DR

This paper tackles the problem of formulating loop quantum gravity for a spherically symmetric spacetime with a scalar field by employing uniform discretization and a variational minimization of the master constraint. The authors construct a lattice-based, polymer-quantized framework, derive an effective Hamiltonian for the scalar field on a semiclassical quantum geometry, and implement Gaussian trial states to minimize the master constraint. They find that the minimum master constraint is nonzero, corresponding to a vacuum state that is effectively a product of a Fock vacuum for the scalar field and a Gaussian, semiclassical gravitational state, with no true continuum limit but good semiclassical behavior at finite lattice spacing. The work highlights a coordinated pathway to connect quantum gravity with quantum field theory in curved spacetime while underscoring the need to treat diffeomorphism and Hamiltonian constraints jointly to fully regulate matter in quantum geometry and to address the cosmological constant problem.

Abstract

We study gravity coupled to a scalar field in spherical symmetry using loop quantum gravity techniques. Since this model has local degrees of freedom, one has to face ``the problem of dynamics'', that is, diffeomorphism and Hamiltonian constraints that do not form a Lie algebra. We tackle the problem using the ``uniform discretization'' technique. We study the expectation value of the master constraint and argue that among the states that minimize the master constraint is one that incorporates the usual Fock vacuum for the matter content of the theory.

Paper Structure

This paper contains 9 sections, 57 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Spherically symmetric gravity with a scalar field: the classical theory
  3. Quantization of the matter field on a fixed flat background
  4. Full quantization of the model
  5. Variational technique to study the expectation value of the master constraint
  6. The discrete master constraint
  7. Construction of the trial states
  8. Minimizing the master constraint
  9. Discussion

Figures (1)

  • Figure 1: The expectation value of the master constraint as a function of the lattice spacing. We see that the value of the master constraint is small unless one chooses lattice separations of order Planck length. The figure does not show it, but for separations of the order of $10^{-23}$cm the master constraint is very small, of the order of $10^{-20}$ (we are using units in which $\hbar$ is one and therefore the master constraint is dimensionless).