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

Abstract

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 ${\hat{T}}^{(b_0,{\varphi }_0)}$ was also considered as possible self-adjoint versions of $\hat{T}$. They represent the restrictions of $\hat{T}$ on their respective domains $D({\hat{T}}^{(b_0,{\varphi }_0)})$ 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 $m=0,\pm 1,\pm 2,...$.

25 pages, under review

References in corpus (28)

• Quantum Nature of the Big Bang: Improved dynamics
• Quantum Nature of the Big Bang
• Quantum Nature of the Big Bang: An Analytical and Numerical Investigation
• Loop quantum cosmology of k=1 FRW models
• Corrections to the Friedmann Equations from LQG for a Universe with a Free Scalar Field
• Are loop quantum cosmos never singular?
• Loop Quantum Cosmology: An Overview
• Quantum cosmology: a review
• Loop quantum cosmology and inhomogeneities
• Loop Quantization of Vacuum Bianchi I Cosmology
• Physical evolution in Loop Quantum Cosmology: The example of vacuum Bianchi I
• Loop quantization of spherically symmetric midi-superspaces
• Loop Quantum Cosmology in Bianchi Type I Models: Analytical Investigation
• Inhomogeneous Loop Quantum Cosmology: Hybrid Quantization of the Gowdy Model
• Are the spectra of geometrical operators in Loop Quantum Gravity really discrete?
• Loop quantization of spherically symmetric midi-superspaces : the interior problem
• Numerical evolution of squeezed and non-Gaussian states in loop quantum cosmology
• How unimodular gravity theories differ from general relativity at quantum level
• Open FRW model in Loop Quantum Cosmology
• Generalized effective description of loop quantum cosmology
• Comment on "Are the spectra of geometrical operators in Loop Quantum Gravity really discrete?" by B. Dittrich and T. Thiemann
• Brief review on black hole loop quantization
• Loop quantum cosmology with self-dual variables
• Loop quantum cosmology of Bianchi IX: Effective dynamics
• Anisotropic loop quantum cosmology with self-dual variables
• The Bender-Dunne basis operators as Hilbert space operators
• A one-particle time of arrival operator for a free relativistic spin-0 charged particle in (1+1) dimensions
• A Relativistic One-Particle Time of Arrival Operator for a Free Spin-1/2 Particle in (1 + 1) Dimensions

Cited by in corpus (1)

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