Perennials and the group-theoretical quantization of a parametrized scalar field on a curved background

Abstract

The perennial formalism is applied to the real, massive Klein–Gordon field on a globally-hyperbolic background space–time with compact Cauchy hypersurfaces. The parametrized form of this system is taken over from the accompanying paper. Two different algebras 𝒮can and 𝒮loc of elementary perennials are constructed. The elements of 𝒮can correspond to the usual creation and annihilation operators for particle modes of the quantum field theory, whereas those of 𝒮loc are the smeared fields. Both are shown to have the structure of a Heisenberg algebra, and the corresponding Heisenberg groups are described. Time evolution is constructed using transversal surfaces and time shifts in the phase space. Important roles are played by the transversal surfaces associated with embeddings of the Cauchy hypersurface in the space–time, and by the time shifts that are generated by space–time isometries. The automorphisms of the algebras generated by this particular type of time shift are calculated explicitly. The construction of the quantum theory using the perennial formalism is shown to be equivalent to the Segal quantization of a Weyl system if the time shift automorphisms of the algebra 𝒮can are used. In this way, the absence of any timelike Killing vector field in the background space–time leads naturally to the ‘‘problem of time’’ for quantum field theory on a background space–time. Within the perennial formalism, this problem is formally identical to the problem of time for any parametrized system, including general relativity itself. Two existing strategies—the ‘‘scattering’’ approach, and the ‘‘algebraic’’ approach—for dealing with this problem in quantum field theory on a background space–time are translated into the language of the perennial formalism in the hope that this may give some insight into how the general problem can be solved. The non-unitary time evolution typical of the Hawking effect is shown to be due to global properties of the corresponding phase space: specifically, the time shifts map a global transversal surface to a non-global one. Thus, the existence of this effect is closely related to the global time problem.