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Quantum Mechanics, Spacetime Locality, and Gravity

Yasunori Nomura

TL;DR

The paper proposes a unified framework linking quantum measurement with cosmology by treating probabilities in the multiverse as dictated by quantum mechanics. It introduces a locality-based Hilbert-space structure for dynamical spacetime with gravity, restricting states to regions on and inside stretched apparent horizons, which leads to horizon complementarity and observer-dependent spacetime. An infinite-dimensional Minkowski/singularity sector is essential to realize irreversible classical histories while preserving unitary evolution, thereby reconciling many-worlds with eternal inflation through an extended Born rule. The resulting picture explains why our world appears classical and ordered, and provides a concrete, frame-dependent formalism to compute probabilities across scales from laboratory experiments to the multiverse.

Abstract

Quantum mechanics introduces the concept of probability at the fundamental level, yielding the measurement problem. On the other hand, recent progress in cosmology has led to the "multiverse" picture, in which our observed universe is only one of the many, bringing an apparent arbitrariness in defining probabilities, called the measure problem. In this paper, we discuss how these two problems are intimately related with each other, developing a complete picture for quantum measurement and cosmological histories in the quantum mechanical universe. On one hand, quantum mechanics eliminates the arbitrariness of defining probabilities in the multiverse, as discussed in arXiv:1104.2324. On the other hand, the multiverse allows for understanding why we observe an ordered world obeying consistent laws of physics, by providing an infinite-dimensional Hilbert space. This results in the irreversibility of quantum measurement, despite the fact that the evolution of the multiverse state is unitary. In order to describe the cosmological dynamics correctly, we need to identify the structure of the Hilbert space for a system with gravity. We argue that in order to keep spacetime locality, the Hilbert space for dynamical spacetime must be defined only in restricted spacetime regions: in and on the (stretched) apparent horizon as viewed from a fixed reference frame. This requirement arises from eliminating all the redundancies and overcountings in a general relativistic, global spacetime description of nature. It is responsible for horizon complementarity as well as the "observer dependence" of horizons/spacetime---these phenomena arise to represent changes of the reference frame in the relevant Hilbert space. This can be viewed as an extension of the Poincare transformation in the quantum gravitational context, as the Lorentz transformation is viewed as an extension of the Galilean transformation.

Quantum Mechanics, Spacetime Locality, and Gravity

TL;DR

The paper proposes a unified framework linking quantum measurement with cosmology by treating probabilities in the multiverse as dictated by quantum mechanics. It introduces a locality-based Hilbert-space structure for dynamical spacetime with gravity, restricting states to regions on and inside stretched apparent horizons, which leads to horizon complementarity and observer-dependent spacetime. An infinite-dimensional Minkowski/singularity sector is essential to realize irreversible classical histories while preserving unitary evolution, thereby reconciling many-worlds with eternal inflation through an extended Born rule. The resulting picture explains why our world appears classical and ordered, and provides a concrete, frame-dependent formalism to compute probabilities across scales from laboratory experiments to the multiverse.

Abstract

Quantum mechanics introduces the concept of probability at the fundamental level, yielding the measurement problem. On the other hand, recent progress in cosmology has led to the "multiverse" picture, in which our observed universe is only one of the many, bringing an apparent arbitrariness in defining probabilities, called the measure problem. In this paper, we discuss how these two problems are intimately related with each other, developing a complete picture for quantum measurement and cosmological histories in the quantum mechanical universe. On one hand, quantum mechanics eliminates the arbitrariness of defining probabilities in the multiverse, as discussed in arXiv:1104.2324. On the other hand, the multiverse allows for understanding why we observe an ordered world obeying consistent laws of physics, by providing an infinite-dimensional Hilbert space. This results in the irreversibility of quantum measurement, despite the fact that the evolution of the multiverse state is unitary. In order to describe the cosmological dynamics correctly, we need to identify the structure of the Hilbert space for a system with gravity. We argue that in order to keep spacetime locality, the Hilbert space for dynamical spacetime must be defined only in restricted spacetime regions: in and on the (stretched) apparent horizon as viewed from a fixed reference frame. This requirement arises from eliminating all the redundancies and overcountings in a general relativistic, global spacetime description of nature. It is responsible for horizon complementarity as well as the "observer dependence" of horizons/spacetime---these phenomena arise to represent changes of the reference frame in the relevant Hilbert space. This can be viewed as an extension of the Poincare transformation in the quantum gravitational context, as the Lorentz transformation is viewed as an extension of the Galilean transformation.

Paper Structure

This paper contains 27 sections, 100 equations, 3 figures.

Table of Contents

  1. Introduction---The Basic Picture
  2. Probabilistic Interpretation of Quantum Mechanics
  3. Quantum measurement a la von Neumann
  4. The problem of the preferred basis
  5. Physical Predictions and Spacetime Locality
  6. Physical information is in matrix elements
  7. Spacetime locality and the basis in Hilbert space
  8. What physical questions may one ask?
  9. Classical Reality in Quantum Mechanical Systems
  10. A double-slit experiment in a large system
  11. Dynamical selection of a measurement basis---spacetime locality
  12. Ordered observations require infinitely large Hilbert space
  13. Spacetime Locality in Theories with Gravity
  14. Black hole complementarity
  15. Quantum states are defined in restricted spacetime regions
  16. ...and 12 more sections

Figures (3)

  • Figure 1: The double-slit experiment with an electron. The electron at the two slits are represented by $\left| 1 \right>$ and $\left| 2 \right>$, respectively, while that at the screen at position $x$ by $\left| x \right>$.
  • Figure 2: Entropy fluctuations in a closed quantum system can provide "observations," indicated by (both solid and dashed) arrows, with each fluctuation corresponding to two "regular worlds" obeying the second law of thermodynamics (as well as the standard rules of Copenhagen quantum mechanics). However, the vast majority of these fluctuations correspond to random, irregular observations (dashed arrows), as opposed to ordinary, ordered observations, which compose only a tiny fraction of the fluctuations (solid arrows).
  • Figure 3: In the situation where a Minkowski bubble forms in a meta-stable de Sitter vacuum, there are two possible ways to retrieve information carried away from the horizon of $p$ by some traveler: from Hawking radiation and from a direct signal. The conditions for successful information retrieval from these sources, however, are mutually incompatible, prohibiting faithful duplication of quantum information.