scholarly journals Hilbert space structure in quantum gravity: an algebraic perspective

2015 ◽  
Vol 2015 (12) ◽  
pp. 1-21 ◽  
Author(s):  
Steven B. Giddings
Keyword(s):  
Hilbert Space ◽  
Quantum Gravity ◽  
Space Structure

scholarly journals Clock-dependent spacetime

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Sung-Sik Lee
Keyword(s):  
Hilbert Space ◽  
General Relativity ◽  
Quantum Gravity ◽  
Hilbert Spaces ◽  
Coordinate Systems ◽  
Physical Laws

Abstract Einstein’s theory of general relativity is based on the premise that the physical laws take the same form in all coordinate systems. However, it still presumes a preferred decomposition of the total kinematic Hilbert space into local kinematic Hilbert spaces. In this paper, we consider a theory of quantum gravity that does not come with a preferred partitioning of the kinematic Hilbert space. It is pointed out that, in such a theory, dimension, signature, topology and geometry of spacetime depend on how a collection of local clocks is chosen within the kinematic Hilbert space.

scholarly journals Real algebras with a Hilbert space structure

1966 ◽  
Vol 6 (4-5) ◽  
pp. 459-465
Author(s):  
Lars Ingelstam
Keyword(s):  
Hilbert Space ◽  
Space Structure ◽  
Real Algebras

scholarly journals POLYMER GEOMETRY AT PLANCK SCALE AND QUANTUM EINSTEIN EQUATIONS

1996 ◽  
Vol 05 (06) ◽  
pp. 629-648 ◽  
Author(s):  
ABHAY ASHTEKAR
Keyword(s):  
Quantum Gravity ◽  
Planck Scale ◽  
Coarse Graining ◽  
Space Structure ◽  
Einstein Equations ◽  
Hamiltonian Constraint ◽  
Recent Approach ◽  
Canonical Approach ◽  
Background Fields ◽  
Discrete Spectra

Over the last two years, the canonical approach to quantum gravity based on connections and triads has been put on a firm mathematical footing through the development and application of a new functional calculus on the space of gauge equivalent connections. This calculus does not use any background fields (such as a metric) and thus well-suited to a fully non-perturbative treatment of quantum gravity. Using this framework, quantum geometry is examined. Fundamental excitations turn out to be one-dimensional, rather like polymers. Geometrical observables such as areas of surfaces and volumes of regions are purely discrete spectra. Continuum picture arises only upon coarse graining of suitable semi-classical states. Next, regulated quantum diffeomorphism constraints can be imposed in an anomaly-free fashion and the space of solutions can be given a natural Hilbert space structure. Progress has also been made on the quantum Hamiltonian constraint in a number of directions. In particular, there is a recent approach based on a generalized .Wick transformation which maps solutions to the Euclidean quantum constraints to those of the Lorentzian theory. These developments are summarized. Emphasis is on conveying the underlying ideas and overall pictures rather than technical details.

scholarly journals Discrete Hilbert space, the Born Rule, and quantum gravity

2021 ◽  
Vol 36 (03) ◽  
pp. 2150013
Author(s):  
Stephen D. H. Hsu
Keyword(s):  
Hilbert Space ◽  
Quantum Gravity ◽  
Discrete Models ◽  
Born Rule ◽  
Gravitational Effects ◽  
Spacetime Geometry ◽  
Decoherent Histories ◽  
Lattice Quantum ◽  
Spacetime Interval ◽  
Matter Fields

Quantum gravitational effects suggest a minimal length, or spacetime interval, of order of the Planck length. This in turn suggests that Hilbert space itself may be discrete rather than continuous. One implication is that quantum states with norm below some very small threshold do not exist. The exclusion of what Everett referred to as maverick branches is necessary for the emergence of the Born Rule in no collapse quantum mechanics. We discuss this in the context of quantum gravity, showing that discrete models (such as simplicial or lattice quantum gravity) indeed suggest a discrete Hilbert space with minimum norm. These considerations are related to the ultimate level of fine-graining found in decoherent histories (of spacetime geometry plus matter fields) produced by quantum gravity.

Polymer quantization effects on gauge-flation

2018 ◽  
Vol 15 (10) ◽  
pp. 1850169
Author(s):  
M. Mardaani ◽  
K. Nozari
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Gravity ◽  
Gauge Field ◽  
Quantum Cosmology ◽  
Loop Quantum Gravity ◽  
Loop Quantum Cosmology ◽  
Abelian Gauge ◽  
Friedmann Equations ◽  
Cosmological Inflation

Polymer quantum mechanics, as a non-standard representation of quantum mechanics, is based on a symmetric sector of loop quantum gravity known as loop quantum cosmology. In this work, by analyzing the Hamiltonian and Friedmann equations in the standard Hilbert space and polymer Hilbert space, we show that polymer quantization is a successful formalism for a non-Abelian gauge field driving the cosmological inflation, the so-called gauge-flation, in order to remove initial singularity and also keeping the inflationary trajectories in this model as attractors of dynamics after the bounce.

scholarly journals A simple characterization of commutativeH*-algebras

1999 ◽  
Vol 22 (4) ◽  
pp. 885-888
Author(s):  
Parfeny P. Saworotnow
Keyword(s):  
Hilbert Space ◽  
Space Structure ◽  
Simple Characterization

CommutativeH*-algebras are characterized without postulating the existence of Hilbert space structure.

scholarly journals Hilbert space structure and positive operators

2005 ◽  
Vol 305 (2) ◽  
pp. 560-565 ◽  
Author(s):  
Dimosthenis Drivaliaris ◽  
Nikos Yannakakis
Keyword(s):  
Hilbert Space ◽  
Space Structure ◽  
Positive Operators

scholarly journals THE SOq(N, R)-SYMMETRIC HARMONIC OSCILLATOR ON THE QUANTUM EUCLIDEAN SPACE ${\bf R}_q^N$ AND ITS HILBERT SPACE STRUCTURE

1993 ◽  
Vol 08 (26) ◽  
pp. 4679-4729 ◽  
Author(s):  
GAETANO FIORE
Keyword(s):  
Hilbert Space ◽  
Euclidean Space ◽  
Harmonic Oscillator ◽  
Mechanical Model ◽  
Scalar Product ◽  
Space Structure ◽  
Quantum Mechanical ◽  
Quantum Space ◽  
Isotropic Harmonic Oscillator ◽  
Definition Of

We show that the isotropic harmonic oscillator in the ordinary Euclidean space RN (N≥3) admits a natural q-deformation into a new quantum-mechanical model having a q-deformed symmetry (in the sense of quantum groups), SO q(N, R). The q-deformation is the consequence of replacing RN by [Formula: see text] (the corresponding quantum space). This provides an example of quantum mechanics on a noncommutative geometrical space. To reach the goal, we also have to deal with a sensible definition of integration over [Formula: see text], which we use for the definition of the scalar product of states.

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