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.
Highlights
The Basic PictureThis paper discusses two subjects: quantum mechanics and gravity, especially in the context of cosmology
We need to fix a reference frame when we describe a system with gravity quantum mechanically—this is the real meaning of the phrase: “physics must be described from the viewpoint of a single observer” in Ref. [14]
In the rest of the paper, we study the structure of the Hilbert space describing the entire quantum universe, starting with the well-known discussion on quantum mechanics of black holes
Summary
This paper discusses two subjects: quantum mechanics and gravity, especially in the context of cosmology. The fact that we observe an ordered, classical world can be explained by a combination of spacetime locality and the fact that the multiverse evolves into a Minkowski (or singularity) world, which has an infinite-dimensional Hilbert space This results in irreversibility of quantum measurement, despite the fact that the evolution of the multiverse state is unitary. To make actual predictions in the context of the multiverse, e.g. of the value of a physical parameter we observe, we still need to know the explicit form of the time evolution operator as well as the initial condition for the multiverse state (except for a few special cases, including that for calculating the distribution of the cosmological constant [28]). The picture of the multiverse from a local viewpoint, which arises here as a consequence of quantum mechanics, has been promoted in the context of geometric cutoff measures; see Refs. [33,34,35] for example
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