Abstract
We investigate path integral quantization of two versions of unimodular gravity. First a fully diffeomorphism-invariant theory is analyzed, which does not include a unimodular condition on the metric, while still being equivalent to other unimodular gravity theories at the classical level. The path integral has the same form as in general relativity (GR), except that the cosmological constant is an unspecified value of a variable, and it thus is unrelated to any coupling constant. When the state of the universe is a superposition of vacuum states, the path integral is extended to include an integral over the cosmological constant. Second, we analyze the standard unimodular theory of gravity, where the metric determinant is fixed by a constraint. Its path integral differs from the one of GR in two ways: the metric of spacetime satisfies the unimodular condition only in average over space, and both the Hamiltonian constraint and the associated gauge condition have zero average over space. Finally, the canonical relation between the given unimodular theories of gravity is established.
Highlights
The idea of unimodular gravity is nearly as old as general relativity (GR) itself
We have studied path integral quantization of two versions of unimodular gravity
In the fully diffeomorphism-invariant theory defined by the action (2.13), the path integral has the same form as the one of GR with a cosmological constant Λ (3.82), except that the value of Λ is not set by the action
Summary
The idea of unimodular gravity is nearly as old as general relativity (GR) itself. Originally, Einstein considered the unimodular condition [1],. A path integral quantization of the Henneaux–Teitelboim version of unimodular gravity has been considered previously in [14] (see [20]), where the unimodular condition was shown to be imposed locally in the quantum theory. Two approaches regarding the interpretation of the cosmological constant are considered: either (i) the effective value of the cosmological constant is fixed by the physical boundary conditions of the path integral, or (ii) the state of the universe is taken as a superposition of states with different values of Λ, and the path integral includes an integral over Λ In the latter approach, we derive the path integral in the form originally proposed in [12] (see [13,14]).
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