arXiv subjects

Chopin Soo

Publications and source records attributed to Chopin Soo.

At least 19 recordsLinked to original sources

arXiv2023

Cosmic time and the initial state of the universe

The exact solution of the Hamiltonian constraint in canonical gravity and the resultant reduction of Einstein's theory reveal the synergy between gravitation and the intrinsic cosmic clock of our expanding universe. Intrinsic Time Geometrodynamics advocates a paradigm shift from four covariances to just spatial diffeomorphism invariance. Consequently, causal time-ordering and quantum Schrodinger-Heisenberg evolution in cosmic time become meaningful. The natural addition of a Cotton-York term to the physical Hamiltonian changes the initial data problem radically. In the classical context, this is studied with the Lichnerowicz-York equation; quantum mechanically, it lends weight to the origin of the universe as an exact Chern-Simons Hartle-Hawking state, which features Euclidean-Lorentzian instanton tunneling. At the level of expectation values, this quantum state yields a low-entropy hot smooth Robertson-Walker beginning in accord with Penrose's Weyl Curvature Hypothesis. The Chern-Simons Hartle-Hawking state also manifests transverse traceless quantum metric fluctuations, with, at the lowest approximation, scale-invariant two-point correlations as one of its defining characteristics.

gr-qc↗

arXiv2021

Quantization of constrained systems as Dirac first class versus second class: a toy model and its implications

A toy model (suggested by Klauder) is analyzed from the perspective of First Class and Second Class Dirac constrained systems. The comparison is made by turning a First Class into a Second Class system with the introduction of suitable auxiliary conditions. The links between Dirac's system of constraints, the Faddeev-Popov canonical functional integral method and the Maskawa-Nakajima procedure to reduced phase space are explicitly illustrated. The model reveals stark contrasts and physically distinguishable results between First and Second class routes. Physically relevant systems such as the relativistic point particle and electrodynamics are briefly recapped. Besides its pedagogical value, the article also advocates the route of rendering First Class into Second Class systems prior to quantization. Second Class systems lead to well-defined reduced phase space and physical observables; absence of inconsistencies in the closure of quantum constraint algebra; and consistent promotion of fundamental Dirac brackets to quantum commutators. As First Class systems can be turned into well-defined Second Class ones, this has implications for the soundness of "Dirac quantization" of First Class constrained systems by simple promotion of Poisson, rather than Dirac brackets, to commutators without proceeding through Second Class procedures.

quant-ph↗

arXiv2021

Intrinsic time gravity, heat kernel regularization, and emergence of Einstein's theory

The Hamiltonian of Intrinsic Time Gravity is elucidated. The theory describes Schrodinger evolution of our universe with respect to the fractional change of the total spatial volume. Gravitational interactions are introduced by extending Klauder's momentric variable with similarity transformations, and explicit spatial diffeomorphism invariance is enforced via similarity transformation with exponentials of spatial integrals. In analogy with Yang-Mills theory, a Cotton-York term is obtained from the Chern-Simons functional of the affine connection. The essential difference is the fundamental variable for geometrodynamics is the metric rather than a gauge connection; in the case of Yang-Mills, there is also no analog of the integral of the spatial Ricci scalar curvature. Heat kernel regularization is employed to isolate the divergences of coincidence limits; apart from an additional Cotton-York term, a prescription in which Einstein's Ricci scalar potential emerges naturally from the positive-definite self-adjoint Hamiltonian of the theory is demonstrated.

gr-qc↗

arXiv2018

Gravitational waves in Intrinsic Time Geometrodynamics

Gravitational waves are investigated in Intrinsic Time Geometrodynamics. This theory has a non-vanishing physical Hamiltonian generating intrinsic time development in our expanding universe, and four-covariance is explicitly broken by higher spatial curvature terms. Linearization of Hamilton's equations about the de Sitter solution produces transverse traceless excitations, with the physics of gravitational waves in Einstein's General Relativity recovered in the low curvature low frequency limit. A noteworthy feature of the theory is that gravitational waves always carry positive energy density, even for compact spatial slicings without any energy contribution from boundary Hamiltonian. This study of gravitational waves in compact $k= +1$ cosmological de Sitter spacetime is in contradistinction to, and complements, previous $k= -1$ investigations of Hawking, Hertog and Turok and other more familiar $k=0$ works. In addition, possible non-four-covariant Horava gravity contributions are considered (hence the use of canonical Hamiltonian, rather than Lagrangian, methods). Recent explicit $S^3$ transverse-traceless mode spectrum of Lindblom, Taylor and Zhang are also employed to complete the discussion.

gr-qc↗

arXiv2018

Cosmic time and reduced phase space of General Relativity

In an ever-expanding spatially closed universe, the fractional change of the volume is the preeminent intrinsic time interval to describe evolution in General Relativity. The expansion of the universe serves as a subsidiary condition which transforms Einstein's theory from a first class to a second class constrained system when the physical degrees of freedom (d.o.f.) are identified with transverse traceless excitations. The super-Hamiltonian constraint is solved by eliminating the trace of the momentum in terms of the other variables, and spatial diffeomorphism symmetry is tackled explicitly by imposing transversality. The theorems of Maskawa-Nishijima appositely relate the reduced phase space to the physical variables in canonical functional integral and Dirac's criterion for second class constraints to nonvanishing Faddeev-Popov determinants in the phase space measures. A reduced physical Hamiltonian for intrinsic time evolution of the two physical d.o.f. emerges. Freed from the first class Dirac algebra, deformation of the Hamiltonian constraint is permitted, and natural extension of the Hamiltonian while maintaining spatial diffeomorphism invariance leads to a theory with Cotton-York term as the ultraviolet completion of Einstein's theory.

gr-qc↗

arXiv2016

Quantum geometrodynamics with intrinsic time development

Quantum geometrodynamics with intrinsic time development is presented. Paradigm shift from full space-time covariance to spatial diffeomorphism invariance yields a non-vanishing Hamiltonian, a resolution of the `problem of time', and gauge-invariant temporal ordering in an ever expanding universe. Einstein's general relativity is a particular realization of a wider class of theories; and the framework prompts natural extensions and improvements, with the consequent dominance of Cotton-York potential at early times when the universe was small.

gr-qc↗

arXiv2016

Intrinsic time geometrodynamics: explicit examples

Intrinsic time quantum geometrodynamics resolved `the problem of time' and bridged the deep divide between quantum mechanics and canonical quantum gravity with a Schrodinger equation which describes evolution in intrinsic time variable. In this formalism, Einstein's general relativity is a particular realization of a wider class of theories. Explicit classical black hole and cosmological solutions and the motion of test particles are derived and analyzed in this work in the context of constant three-curvature solutions in intrinsic time geometrodynamics; and we exemplify how this formalism yields results which agree with the predictions of Einstein's theory.

gr-qc↗

arXiv2016

New commutation relations for quantum gravity

A new set of fundamental commutation relations (CR) for quantum gravity is presented. The basic variables are the eight components of the unimodular part of the spatial dreibein and eight SL(3,R) generators which correspond to Klauder's momentric variables. The commutation relations are not canonical, but they have well-defined group-theoretic meanings. Explicit unitary irreducible infinite-dimensional representations are constructed both in the metric as well as dreibein representations. The dreibein components commute among themselves; but momentric, unlike momenta, do not. This differs starkly from the usual canonical CR that allow wave functionals to be realized in either of the conjugate representations; and it may explain why our universe seems fundamentally and intuitively `metric' in nature, and not `conjugately realized'.

gr-qc↗

arXiv2015

Intrinsic Time Quantum Geometrodynamics

Quantum Geometrodynamics with intrinsic time development and momentric variables is presented. An underlying SU(3) group structure at each spatial point regulates the theory. The intrinsic time behavior of the theory is analyzed, together with its ground state and primordial quantum fluctuations. Cotton-York potential dominates at early times when the universe was small; the ground state naturally resolves Penrose's Weyl Curvature Hypothesis, and thermodynamic and gravitational `arrows of time' point in the same direction. Ricci scalar potential corresponding to Einstein's General Relativity emerges as a zero-point energy contribution. A new set of fundamental commutation relations without Planck's constant emerges from the unification of Gravitation and Quantum Mechanics.

gr-qc↗

arXiv2014

Exact solutions of the Wheeler-DeWitt equation and the Yamabe construction

Exact solutions of the Wheeler-DeWitt equation of the full theory of four dimensional gravity of Lorentzian signature are obtained. They are characterized by Schrödinger wavefunctionals having support on 3-metrics of constant spatial scalar curvature, and thus contain two full physical field degrees of freedom in accordance with the Yamabe construction. These solutions are moreover Gaussians of minimum uncertainty and they are naturally associated with a rigged Hilbert space. In addition, in the limit the regulator is removed, exact 3-dimensional diffeomorphism and local gauge invariance of the solutions are recovered.

gr-qc↗

arXiv2014

Affine group formulation of the Standard Model coupled to gravity

This work demonstrates that a complete description of the interaction of matter and all forces, gravitational and non-gravitational, can in fact be realized within a quantum affine algebraic framework. Using the affine group formalism, we construct elements of a physical Hilbert space for full, Lorentzian quantum gravity coupled to the Standard Model in four spacetime dimensions. Affine algebraic quantization of gravitation and matter on equal footing implies a fundamental uncertainty relation which is predicated upon a non-vanishing cosmological constant.

gr-qc↗

arXiv2014

General Relativity without paradigm of space-time covariance, and resolution of the problem of time

The framework of a theory of gravity from the quantum to the classical regime is presented. The paradigm shift from full spacetime covariance to spatial diffeomorphism invariance, together with clean decomposition of the canonical structure, yield transparent physical dynamics and a resolution of the problem of time. The deep divide between quantum mechanics and conventional canonical formulations of quantum gravity is overcome with a Schrödinger equation for quantum geometrodynamics that describes evolution in intrinsic time. Unitary time development with gauge-invariant temporal ordering is also viable. All Kuchar observables become physical; and classical spacetime, with direct correlation between its proper times and intrinsic time intervals, emerges from constructive interference. The framework not only yields a physical Hamiltonian for Einstein's theory, but also prompts natural extensions and improvements towards a well behaved quantum theory of gravity. It is a consistent canonical scheme to discuss Horava-Lifshitz theories with intrinsic time evolution, and of the many possible alternatives that respect 3-covariance (rather than the more restrictive 4-covariance of Einstein's theory), Horava's ``detailed balance" form of the Hamiltonian constraint is essentially pinned down by this framework.

gr-qc↗

arXiv2013

Exact Abelian and Non-Abelian Geometric Phases

The existence of Hopf fibrations S^{2N+1}/S^1 = CP^N and S^{4K+3}/S^3 = HP^K allows us to treat the Hilbert space of generic finite-dimensional quantum systems as the total bundle space with respectively $U(1)$ and $SU(2)$ fibers and complex and quaternionic projective spaces as base manifolds. This alternative method of studying quantum states and their evolution reveals the intimate connection between generic quantum mechanical systems and geometrical objects. The exact Abelian and non-Abelian geometric phases, and more generally the geometrical factors for open paths, and their precise correspondence with geometric Kahler and hyper-Kahler connections will be discussed. Explicit physical examples are used to verify and exemplify the formalism.

quant-ph↗

arXiv2013

Affine group representation formalism for four dimensional, Lorentzian, quantum gravity

Within the context of the Ashtekar variables, the Hamiltonian constraint of four-dimensional pure General Relativity with cosmological constant, $Λ$, is reexpressed as an affine algebra with the commutator of the imaginary part of the Chern-Simons functional, $Q$, and the positive-definite volume element. This demonstrates that the affine algebra quantization program of Klauder can indeed be applicable to the full Lorentzian signature theory of quantum gravity with non-vanishing cosmological constant; and it facilitates the construction of solutions to all of the constraints. Unitary, irreducible representations of the affine group exhibit a natural Hilbert space structure, and coherent states and other physical states can be generated from a fiducial state. It is also intriguing that formulation of the Hamiltonian constraint or Wheeler-DeWitt equation as an affine algebra requires a non-vanishing cosmological constant; and a fundamental uncertainty relation of the form $\frac{Δ{V}}{<{V}>}Δ{Q}\geq 2πΛL^2_{Planck}$ (wherein $V$ is the total volume) may apply to all physical states of quantum gravity.

gr-qc↗

arXiv2012

Intrinsic time gravity and the Lichnerowicz-York equation

We investigate the effect on the Hamiltonian structure of general relativity of choosing an intrinsic time to fix the time slicing. 3-covariance with momentum constraint is maintained, but the Hamiltonian constraint is replaced by a dynamical equation for the trace of the momentum. This reveals a very simple structure with a local reduced Hamiltonian. The theory is easily generalised; in particular, the square of the Cotton-York tensor density can be added as an extra part of the potential while at the same time maintaining the classic 2 + 2 degrees of freedom. Initial data construction is simple in the extended theory; we get a generalised Lichnerowicz-York equation with nice existence and uniqueness properties. Adding standard matter fields is quite straightforward.

gr-qc↗

arXiv2011

Generalized Painleve-Gullstrand descriptions of Kerr-Newman black holes

Generalized Painleve-Gullstrand metrics are explicitly constructed for the Kerr-Newman family of charged rotating black holes. These descriptions are free of all coordinate singularities; moreover, unlike the Doran and other proposed metrics, an extra tunable function is introduced to ensure all variables in the metrics remain real for all values of the mass M, charge Q, angular momentum aM, and cosmological constant Λ> - 3/(a^2). To describe fermions in Kerr-Newman spacetimes, the stronger requirement of non-singular vierbein one-forms at the horizon(s) is imposed and coordinate singularities are eliminated by local Lorentz boosts. Other known vierbein fields of Kerr-Newman black holes are analysed and discussed; and it is revealed that some of these descriptions are actually not related by physical Lorentz transformations to the original Kerr-Newman expression in Boyer-Lindquist coordinates - which is the reason complex components appear (for certain ranges of the radial coordinate) in these metrics. As an application of our constructions the correct effective Hawking temperature for Kerr black holes is derived with the method of Parikh and Wilczek.

gr-qc↗

arXiv2011

New formulation of Horava-Lifshitz quantum gravity as a master constraint theory

Both projectable and non-projectable versions of Horava-Lifshitz gravity face serious challenges. In the non-projectable version, the constraint algebra is seemingly inconsistent. The projectable version lacks a local Hamiltonian constraint, thus allowing for an extra scalar mode which can be problematic. A new formulation of non-projectable Horava-Lifshitz gravity, naturally realized as a representation of the master constraint algebra studied by loop quantum gravity researchers, is presented. This yields a consistent canonical theory with first class constraints. It captures the essence of Horava-Lifshitz gravity in retaining only spatial diffeomorphisms (instead of full space-time covariance) as the physically relevant non-trivial gauge symmetry; at the same time the local Hamiltonian constraint needed to eliminate the extra mode is equivalently enforced by the master constraint.

gr-qc↗

arXiv2010

New length operator for loop quantum gravity

An alternative expression for the length operator in loop quantum gravity is presented. The operator is background independent, symmetric, positive semidefinite, and well defined on the kinematical Hilbert space. The expression for the regularized length operator can moreover be understood both from a simple geometrical perspective as the average of a formula relating the length to area, volume and flux operators, and also consistently as the result of direct substitution of the densitized triad operator with the functional derivative operator into the regularized expression of the length. Both these derivations are discussed, and the origin of an undetermined overall factor in each case is also elucidated.

gr-qc↗