Considerations of particle definitions in the functional Schroedinger formalism

Abstract

It is often remarked in the literature that particles in QFT on curved spacetime are akin to coordinates in general relativity and hence are physically meaningless. This moral is given an explicit demonstration by giving the correspondence between the coordinates on phase space for a field theory and the particle number. Usually the ambiguity in particle definitions is only as varied as the possible sets of observers on the spacetime. However, there is a greater choice in coordinates on the phase space, especially for a field system with infinite degrees of freedom. Hence, for one set of coordinates on the spacetime (one class of comoving observers) there are many different coordinates to choose on the phase space. This demonstrates the true vacuousness of the concept of particles when defined as energy levels of the harmonic oscillator. In order to give a definition of particles we must specify the apparatus that detects them. The Unruh-Dewitt detector is one such apparatus, so we are not surprised to find that it gives a physically meaningful definition of particles. We give an explicit example on de Sitter spacetime and explain how the definition of particles as energy levels of the harmonic oscillator is meaningless even in simple cases. This is done first by comparing the response of an Unruh-Dewitt detector to the expectation of the number operator. Second, we demonstrate that by a choice of coordinates on phase space one can turn the Hamiltonian of a free Klein-Gordon field on FRW with flat spatial sections into a set of harmonic oscillators with time-independent mass and frequency. Also, we demonstrate a new method for determining the wave functional of known states such as the conformal vacuum.