# Toward a theory of observation

## Abstract

Quantum mechanics is usually formulated in terms of a single Hilbert space and observables are defined as operators on this space. Attempts to describe entire spacetimes and their resident matter in this way often encounter paradoxes. For example, it has been argued that an observer falling into a black hole may be able to witness deviations from unitary, violations of semi-classical quantum field theory, and the like. This thesis argues that the essential problem is the insistence on the use of a single, global Hilbert space, because in general it may be that a physical observer cannot causally probe all of the information described by this space due to the presence of horizons. Instead, one could try to define unitary quantum physics directly in terms of the information causally accessible to particular observers. This thesis makes steps toward a systematization of this idea. Given an observer on a timelike worldline, I construct coordinates which (in good cases) cover precisely the set of events to which she can send and then receive a signal. These coordinates have spatial sections parametrized by her proper time, and the metric manifestly encodes the equivalence principle in the sense that it is flat along her worldline. To describe the quantum theory of fields according to these observers, I define Hilbert spaces in terms of field configurations on these spatial sections and show how to implement unitary time-evolution along proper time. I explain how to compare the observations of a pair of observers, and how to obtain the description according to some particular observer given some a priori global description. In this sense, the program outlined here constructs a manifestly unitary description of the events which the observer can causally probe. I give a number of explicit examples of the coordinates, and show how the quantum theory works for a uniformly accelerated observer in flat spacetime and for an inertial (co-moving) observer in an inflating universe.