Spacetime quanta?: the discrete spectrum of a quantum spacetime four-volume operator in unimodular loop quantum cosmology

Abstract

Abstract. This study considers the operatorT corresponding to the classical spacetime four-volumeT (on-shell) of a finite patch of spacetime in the context of Unimodular Loop Quantum Cosmology for the homogeneous and isotropic model with flat spatial sections and without matter sources. Since the spacetime four-volume is canonically conjugate to the cosmological "constant", the operatorT is constructed by solving its canonical commutation relation withΛ -the operator corresponding to the classical cosmological constant on-shellΛ. This conjugacy, along with the action ofT on definite volume states reducing toT , allows us to interpret thatT is indeed a quantum spacetime four-volume operator. The discrete spectrum ofT is calculated by considering the set of all τ 's where the eigenvalue equation has a solution Φ τ in the domain ofT . It turns out that, upon assigning the maximal domain D(T ) tô T , we have Φ τ ∈ D(T ) for all τ ∈ C so that the spectrum ofT is purely discrete and is the entire complex plane. A family of operatorsT (b0,φ0) was also considered as possible self-adjoint versions ofT . They represent the restrictions ofT on their respective domains D(T (b0,φ0) ) which are just the maximal domain with additional quasi-periodic conditions. Their possible self-adjointness is motivated by their discrete spectra only containing real and discrete numbers τ m for m = 0, ±1, ±2, ....

Canonical analysis of covariant unimodular gravity and an extension of the Kodama state

“…Note that the symmetry-reduced models of covariant unimodular gravity in the context of loop quantum cosmology has been studied in Refs. [24][25][26].…”

Time as a stochastic process

“…In the case of unimodular gravity with a cosmological constant (see Ref. [21]) after reduction to a cosmological model one is able to construct a 4-volume operator similar to Relativistic Time of Arrival Operator. This operator can be considered as a cosmological time operator.…”

“…According to Ref. [21] the cosmological model under consideration is spatially flat, homogeneous and isotropic. We have before quantization of T 0 Ṫ0 (t) = {T 0 , H}.…”