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

摘要

This study considers the operator (T) over cap corresponding to the classical spacetime four-volume (T) over tilde ( 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 fourvolume is canonically conjugate to the cosmological 'constant', the operator (T) over cap is constructed by solving its canonical commutation relation with (Lambda) over tilde -the operator corresponding to the classical cosmological constant on-shell (Lambda) over tilde. This conjugacy, along with the action of (T) over cap on definite volume states reducing to (T) over cap, allows us to interpret that (T) over cap is indeed a quantum spacetime four-volume operator. The discrete spectrum of (T) over cap is calculated by considering the set of all tau's where the eigenvalue equation has a solution Phi(tau) in the domain of (T) over cap. It turns out that, upon assigning the maximal domain D((T) over cap) to (T) over cap, we have Phi(tau) is an element of D((T) over cap) for all tau is an element of C so that the spectrum of (T) over cap is purely discrete and is the entire complex plane. A family of operators (T) over cap ((b0,phi 0)) was also considered as possible self-adjoint versions of (T) over cap. They represent the restrictions of (T) over cap on their respective domains D((T) over cap ((b0,phi 0))) 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 tau(m) for m = 0, +/- 1, +/- 2, ....