Abstract
BRST quantization is an elegant and powerful method to quantize theories with local symmetries. In this article we study the Hamiltonian BRST quantization of cosmological perturbations in a universe dominated by a scalar field, along with the closely related quantization method of Dirac. We describe how both formalisms apply to perturbations in a time-dependent background, and how expectation values of gauge-invariant operators can be calculated in the in-in formalism. Our analysis focuses mostly on the free theory. By appropriate canonical transformations we simplify and diagonalize the free Hamiltonian. BRST quantization in derivative gauges allows us to dramatically simplify the structure of the propagators, whereas Dirac quantization, which amounts to quantization in synchronous gauge, dispenses with the need to introduce ghosts and preserves the locality of the gauge-fixed action.
Highlights
In order to explain the properties of the primordial fluctuations, in many models of the origin of structure one needs to quantize general relativity coupled to a scalar field
The main idea behind BRST quantization is the replacement of the local symmetry under gauge transformations by a nilpotent global symmetry generated by what is known as the BRST charge
There are basically two approaches to quantize a gauge theory: One can quantize a complete set of gauge-invariant variables in phase space, a method known as reduced phase space quantization, or one can fix the gauge by appropriately modifying the action of the theory
Summary
In order to explain the properties of the primordial fluctuations, in many models of the origin of structure one needs to quantize general relativity coupled to a scalar field. The quantization of gauge theories is somewhat subtle, but becomes relatively straightforward if one is interested in tree-level calculations alone: One can either fix the gauge and break the local symmetry, clearing the way, say, to canonical quantization, or one can work with the essentially equivalent method of quantizing an appropriate set of gaugeinvariant variables, a procedure known as reduced phase space quantization. The most common method of quantizing a gauge theory is that of Faddeev and Popov (and DeWitt.) This method is inherently linked to the choice of gauge-fixing “conditions.” In the case of cosmological perturbations, these can be taken to be four functions f μ of the inflaton and metric perturbations δφ, δhμν that are not invariant under the four independent diffeomorphisms with infinitesimal parameters ξν Once these conditions have been chosen, the action of the theory needs to be supplemented with appropriate ghost terms, Sghost =. Both share the property that a pair of additional fields is introduced for each of the four constraints of the theory, but whereas BRST-invariance guarantees that the ghosts do not change the gauge-invariant content of the theory, the dust fields of [18] do seem to survive as additional degrees of freedom in the system
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