Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Local Wheeler-DeWitt Measure for the String Landscape

Bjoern Hassfeld, Arthur Hebecker, Manfred Salmhofer, Jonah Cedric Strauss, Johannes Walcher

TL;DR

The paper introduces a local Wheeler-DeWitt measure to confront the measure problem in the string landscape and eternal inflation. It posits a direct-sum Hilbert space comprising finite-dimensional static patches for each de Sitter vacuum and infinite-dimensional terminal sectors, with an inhomogeneous WDW equation encoding creation of new dS vacua and current flow to terminals. Time is not global but emerges locally through correlations within each horizon sector, yielding standard quantum evolution for observed subsystems while the universal state remains stationary. The authors derive a phenomenological rate-equation framework linking microscopic inputs (creation amplitudes, tunneling rates) to vacuum probabilities and discuss implications for anthropic predictions, comparing with other measure proposals and outlining future avenues for embedding realistic landscape data.

Abstract

According to the `Cosmological Central Dogma', de Sitter space can be viewed as a quantum mechanical system with a finite number of degrees of freedom, set by the horizon area. We use this assumption together with the Wheeler-DeWitt (WDW) equation to approach the measure problem of eternal inflation. Thus, our goal is to find a time-independent wave function of the universe on a total Hilbert space defined as the direct sum of a variety of subspaces: A finite-dimensional subspace for each de Sitter vacuum and an infinite-dimensional subspace for each terminal Minkowski or AdS vaccuum. We argue that, to be consistent with semiclassical intuition, such a solution requires the presence of sources. These are implemented as an inhomogenous term in the WDW equation, induced by the Hartle-Hawking no-boundary or the Linde/Vilenkin tunneling proposal. Taken together, these steps unambiguously lead to what we would like to think of as a `Local WDW measure,' where `local' refers to the fact that the dS part of the resulting wave function describes a superposition of static patches. The global 3-sphere spatial section of the entire multiverse makes no appearance.

A Local Wheeler-DeWitt Measure for the String Landscape

TL;DR

The paper introduces a local Wheeler-DeWitt measure to confront the measure problem in the string landscape and eternal inflation. It posits a direct-sum Hilbert space comprising finite-dimensional static patches for each de Sitter vacuum and infinite-dimensional terminal sectors, with an inhomogeneous WDW equation encoding creation of new dS vacua and current flow to terminals. Time is not global but emerges locally through correlations within each horizon sector, yielding standard quantum evolution for observed subsystems while the universal state remains stationary. The authors derive a phenomenological rate-equation framework linking microscopic inputs (creation amplitudes, tunneling rates) to vacuum probabilities and discuss implications for anthropic predictions, comparing with other measure proposals and outlining future avenues for embedding realistic landscape data.

Abstract

According to the `Cosmological Central Dogma', de Sitter space can be viewed as a quantum mechanical system with a finite number of degrees of freedom, set by the horizon area. We use this assumption together with the Wheeler-DeWitt (WDW) equation to approach the measure problem of eternal inflation. Thus, our goal is to find a time-independent wave function of the universe on a total Hilbert space defined as the direct sum of a variety of subspaces: A finite-dimensional subspace for each de Sitter vacuum and an infinite-dimensional subspace for each terminal Minkowski or AdS vaccuum. We argue that, to be consistent with semiclassical intuition, such a solution requires the presence of sources. These are implemented as an inhomogenous term in the WDW equation, induced by the Hartle-Hawking no-boundary or the Linde/Vilenkin tunneling proposal. Taken together, these steps unambiguously lead to what we would like to think of as a `Local WDW measure,' where `local' refers to the fact that the dS part of the resulting wave function describes a superposition of static patches. The global 3-sphere spatial section of the entire multiverse makes no appearance.
Paper Structure (23 sections, 89 equations, 1 figure)

This paper contains 23 sections, 89 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The naive, quantum-mechanical perspective
  3. Density of states
  4. Tunneling in the quantum mechanical toy model
  5. Ergodicity and distribution of eigenvectors
  6. Wheeler-DeWitt and 'time behind the horizon'
  7. From particle worldlines to mini-superspace and landscape dynamics
  8. The time variable as part of the horizon degrees of freedom
  9. The perspective of a 'WDW purist'
  10. Emergent time in its simplest form
  11. Including unobserved degrees of freedom
  12. The complete picture of a universe without time
  13. Terminal vacua
  14. Terminal vacua as infinite-N limits
  15. The creation of dS vacua and probability currents
  16. ...and 8 more sections

Figures (1)

  • Figure 1: Energy spectra of the two dS vacua. Most eigenvalues of the false vacuum (left) lie inside the dense region of eigenvalues of the true vacuum (right).