[Paper Review] Global Dynamics for Newton and Planck

Alexander Vikman|arXiv (Cornell University)|Jul 20, 2021

Cosmology and Gravitation Theories23 references4 citations

TL;DR

This paper proposes a framework in which the Newton constant and Planck constant are global dynamical degrees of freedom, emerging from scale-free Einstein equations that lose trace information. By extending the Henneaux-Teitelboim covariant action for unimodular gravity, the authors derive Heisenberg uncertainty relations for these constants and show that quantum fluctuations of $G_N$ and $ ar{\hbar} $ are bounded, with implications for quantum gravity and the cosmological constant problem.

ABSTRACT

We discuss recently introduced scale-free Einstein equations, where the information from their trace part is lost. These equations are classically equivalent to General Relativity, yet the Newton constant becomes a constant of integration or a global dynamical degree of freedom. Thus, from the point of view of standard quantization, this effective Newton constant is susceptible to quantum fluctuations. This is similar to what happens to the cosmological constant in the unimodular gravity where the trace part of the Einstein equations is lost in a different way. Using analogy with the Henneaux-Teitelboim covariant action for the unimodular gravity, we consider different general-covariant actions resulting in these dynamics. This setup allows one to formulate the Heisenberg uncertainty relations for the Newton constant and canonically conjugated quantities. Unexpectedly, one of such theories also promotes the Planck's quantum constant to a global degree of freedom, which is subject to quantum fluctuations. Following analogy with the unimodular gravity, we discuss non-covariant "unimatter" and "unicurvature" gravities describing the scale-free Einstein equations. Finally, we show that in some limit of the Yang-Mills gauge theory a "frozen" axion-like field can emulate the gravitational Newton constant or even of the quantum Planck constant.

Motivation & Objective

• To reformulate gravity such that the Newton constant $G_N$ becomes a global dynamical degree of freedom, not a fixed parameter.
• To extend this framework to treat the Planck constant $\hbar$ as a global dynamical variable subject to quantum fluctuations.
• To derive quantum uncertainty relations for $G_N$ and $\hbar$ using canonical quantization of global degrees of freedom.
• To embed these dynamics into axion-like fields in Yang-Mills theories, providing a field-theoretic realization of dynamical constants.
• To explore the implications for quantum cosmology, singularities, and the cosmological constant problem through non-covariant 'unimatter' and 'unicurvature' gravity models.

Proposed method

• Formulate scale-free Einstein equations (e.g., $G_{\mu\nu}/G = T_{\mu\nu}/T$) that are classically equivalent to GR but lose trace information, making $G_N$ a constant of integration.
• Adopt the Henneaux-Teitelboim covariant action for unimodular gravity as a template to construct actions where $\Lambda$ or $G_N$ become global degrees of freedom.
• Introduce frozen axion-like fields $\nu$ and $\eta$ that couple non-minimally to gravity and matter, with $\nu^2$ determining the effective Newton constant $G_N = 1/(8\pi\nu^2)$.
• Construct a Yang-Mills-type action with a confined gauge field and a frozen axion $\alpha$, where the coupling $\alpha/f_\alpha$ generates a cosmological constant and $\alpha$ acts as a dynamical $G_N$.
• Derive canonical conjugate pairs for $\varepsilon_\Lambda$ and cosmic time $\tau$, leading to the uncertainty relation $\delta\varepsilon_\Lambda \cdot \delta\tau \geq \hbar/2$, which generalizes to $\delta\Lambda \cdot \delta\Omega \geq 4\pi \ell_{Pl}^2$.
• Extend the formalism to include both $G_N$ and $\hbar$ as dynamical, with $\bar{\hbar} = \hbar M_m^2 / \eta^2$, and derive bounds on quantum fluctuations of $\bar{\hbar}$ via uncertainty relations.

Experimental results

Research questions

• RQ1 Can the Newton constant $G_N$ be promoted to a global dynamical degree of freedom in a generally covariant theory?
• RQ2 What are the quantum uncertainty bounds on the effective Newton constant $G_N$ when it is a dynamical global variable?
• RQ3 Can the Planck constant $\hbar$ also be treated as a global dynamical degree of freedom in a consistent quantum gravity framework?
• RQ4 How can such dynamical constants emerge from a fundamental field-theoretic construction, such as in Yang-Mills theories with axion-like fields?
• RQ5 What are the implications of these global degrees of freedom for quantum cosmology and singularities in general relativity?

Key findings

• The effective Newton constant $G_N = 1/(8\pi\nu^2)$ becomes a global dynamical degree of freedom when the trace of the Einstein equations is discarded, leading to scale-free dynamics.
• A lower bound on quantum fluctuations of $G_N$ is derived as $\delta G_N \gtrsim \frac{1}{8\pi} \frac{\hbar}{\nu^2 \delta \Omega}$, where $\delta \Omega$ is the uncertainty in spacetime volume.
• The Planck constant $\hbar$ can also be made dynamical via a second axion field $\eta$, with $\bar{\hbar} = \hbar M_m^2 / \eta^2$, leading to a new uncertainty relation $\delta \bar{\hbar} \cdot \delta \eta \gtrsim \frac{\hbar M_m^2}{2\eta^2}$.
• The uncertainty relation $\delta \Lambda \cdot \delta \int d^4x \sqrt{-g} \geq 4\pi \ell_{Pl}^2$ is derived, showing unavoidable quantum fluctuations of the cosmological constant in this framework.
• A Yang-Mills theory with a frozen axion $\alpha$ can emulate the Newton constant via $G_N \propto \alpha^{-1}$, and even the Planck constant via $\bar{\hbar} \propto \alpha^{-1}$, in the limit of strong coupling.
• The construction leads to a natural emergence of a cosmological constant and a non-minimal coupling to gravity, breaking shift symmetry and stabilizing the vacuum energy.