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

This study considers the operator $\hat{T}$ corresponding to the classical spacetime four-volume $\tilde{T}$ (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 $\hat{T}$ is constructed by solving its canonical commutation relation with $\hat{\Lambda}$ - the operator corresponding to the classical cosmological constant on-shell $\tilde{\Lambda}$. This conjugacy, along with the action of $\hat{T}$ on definite volume states reducing to $\tilde{T}$, allows us to interpret that $\hat{T}$ is indeed a quantum spacetime four-volume operator. The discrete spectrum of $\hat{T}$ is calculated by considering the set of all $\tau$'s where the eigenvalue equation has a solution $\Phi_{\tau}$ in the domain of $\hat{T}$. It turns out that, upon assigning the maximal domain $D(\hat{T})$ to $\hat{T}$, we have $\Phi_{\tau}\in D(\hat{T})$ for all $\tau\in\mathbb{C}$ so that the spectrum of $\hat{T}$ is purely discrete and is the entire complex plane. A family of operators $\hat{T}^{(b_0,\phi_0)}$ was also considered as possible self-adjoint versions of $\hat{T}$. They represent the restrictions of $\hat{T}$ on their respective domains $D(\hat{T}^{(b_0,\phi_0)})$ 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 $m=0,\pm1,\pm2,...$.