Spacetime Quanta?: The Discrete Spectrum of a Quantum Spacetime Four-Volume Operator in Unimodular Loop Quantum Cosmology

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This study considers the operator corresponding to the classical spacetime four-volume (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 operator is constructed by solving its canonical commutation relation with —the operator corresponding to the classical cosmological constant on-shell . This conjugacy, along with the action of on definite volume states reducing to , allows us to interpret that is indeed a quantum spacetime four-volume operator. The discrete spectrum of is calculated by considering the set of all τ's where the eigenvalue equation has a solution in the domain of . It turns out that, upon assigning the maximal domain to , we have for all so that the spectrum of is purely discrete and is the entire complex plane. A family of operators was also considered as possible self-adjoint versions of . They represent the restrictions of on their respective domains 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 for .