scholarly journals The quantum holonomy-diffeomorphism algebra and quantum gravity

2016 ◽  
Vol 31 (10) ◽  
pp. 1650048 ◽  
Author(s):  
Johannes Aastrup ◽  
Jesper Møller Grimstrup
Keyword(s):  
Quantum Gravity ◽  
Semiclassical Limit ◽  
Three Dimensional ◽  
Type Operator ◽  
Dimensional Manifold ◽  
Unbounded Operators ◽  
Yang Mills ◽  
Semiclassical States ◽  
Gns Construction ◽  
Canonical Quantum Gravity

We introduce the quantum holonomy-diffeomorphism ∗-algebra, which is generated by holonomy-diffeomorphisms on a three-dimensional manifold and translations on a space of SU(2)-connections. We show that this algebra encodes the canonical commutation relations of canonical quantum gravity formulated in terms of Ashtekar variables. Furthermore, we show that semiclassical states exist on the holonomy-diffeomorphism part of the algebra but that these states cannot be extended to the full algebra. Via a Dirac-type operator we derive a certain class of unbounded operators that act in the GNS construction of the semiclassical states. These unbounded operators are the type of operators, which we have previously shown to entail the spatial three-dimensional Dirac operator and Dirac–Hamiltonian in a semiclassical limit. Finally, we show that the structure of the Hamilton constraint emerges from a Yang–Mills-type operator over the space of SU(2)-connections.

scholarly journals The Yang-Mills heat equation with finite action in three dimensions

2022 ◽  
Vol 275 (1349) ◽  
Author(s):  
Leonard Gross
Keyword(s):  
Heat Equation ◽  
Three Dimensional ◽  
Strong Solutions ◽  
Three Dimensions ◽  
Dimensional Manifold ◽  
Uniqueness Of Solutions ◽  
Reconstruction Procedure ◽  
Content Type ◽  
Yang Mills ◽  
Gauge Functions

The existence and uniqueness of solutions to the Yang-Mills heat equation is proven over R 3 \mathbb {R}^3 and over a bounded open convex set in R 3 \mathbb {R}^3 . The initial data is taken to lie in the Sobolev space of order one half, which is the critical Sobolev index for this equation over a three dimensional manifold. The existence is proven by solving first an augmented, strictly parabolic equation and then gauge transforming the solution to a solution of the Yang-Mills heat equation itself. The gauge functions needed to carry out this procedure lie in the critical gauge group of Sobolev regularity three halves, which is a complete topological group in a natural metric but is not a Hilbert Lie group. The nature of this group must be understood in order to carry out the reconstruction procedure. Solutions to the Yang-Mills heat equation are shown to be strong solutions modulo these gauge functions. Energy inequalities and Neumann domination inequalities are used to establish needed initial behavior properties of solutions to the augmented equation.

scholarly journals MEASUREMENTS AND INFORMATION IN SPIN FOAM MODELS

2012 ◽  
Vol 27 (28) ◽  
pp. 1250164
Author(s):  
J. MANUEL GARCÍA-ISLAS
Keyword(s):  
Information Theory ◽  
Quantum Gravity ◽  
Cosmological Constant ◽  
Shannon Entropy ◽  
Three Dimensional ◽  
Spin Network ◽  
Spin Foam Models ◽  
Foam Model ◽  
Spin Foam Model ◽  
Spin Foam

In the three-dimensional spin foam model of quantum gravity with a cosmological constant, there exists a set of observables associated with spin network graphs. A set of probabilities is calculated from these observables, and hence the associated Shannon entropy can be defined. We present the Shannon entropy associated with these observables and find some interesting bounded inequalities. The problem relates measurements, entropy and information theory in a simple way which we explain.

scholarly journals Three-dimensional super Yang-Mills with unquenched flavor

2015 ◽  
Vol 2015 (7) ◽  
Author(s):  
Antón F. Faedo ◽  
David Mateos ◽  
Javier Tarrío
Keyword(s):  
Three Dimensional ◽  
Yang Mills

scholarly journals CONSTRAINTS IN QUANTUM GEOMETRODYNAMICS

2004 ◽  
Vol 19 (10) ◽  
pp. 1609-1638 ◽  
Author(s):  
ADRIAN P. GENTLE ◽  
NATHAN D. GEORGE ◽  
ARKADY KHEYFETS ◽  
WARNER A. MILLER
Keyword(s):  
Quantum Gravity ◽  
General Setting ◽  
Square Root ◽  
Dynamical Equations ◽  
Dynamic Content ◽  
Quantization Formalism ◽  
Canonical Quantum Gravity ◽  
First Time ◽  
Quantum Geometrodynamics ◽  
Canonical Quantum

We compare different treatments of the constraints in canonical quantum gravity. The standard approach on the superspace of 3-geometries treats the constraints as the sole carriers of the dynamic content of the theory, thus rendering the traditional dynamical equations obsolete. Quantization of the constraints in both the Dirac and ADM square root Hamiltonian approaches leads to the well known problems of time evolution. These problems of time are of both an interpretational and technical nature. In contrast, the geometrodynamic quantization procedure on the superspace of the true dynamical variables separates the issues of quantization from the enforcement of the constraints. The resulting theory takes into account states that are off-shell with respect to the constraints, and thus avoids the problems of time. We develop, for the first time, the geometrodynamic quantization formalism in a general setting and show that it retains all essential features previously illustrated in the context of homogeneous cosmologies.

scholarly journals REVISED CANONICAL QUANTUM GRAVITY VIA THE FRAME FIXING

2004 ◽  
Vol 13 (01) ◽  
pp. 165-186 ◽  
Author(s):  
SIMONE MERCURI ◽  
GIOVANNI MONTANI
Keyword(s):  
Quantum Dynamics ◽  
Fundamental Problem ◽  
Semiclassical Limit ◽  
Momentum Density ◽  
Hamiltonian Constraint ◽  
Additional Energy ◽  
Canonical Quantum Gravity ◽  
Quantum Geometrodynamics ◽  
Canonical Quantum ◽  
Natural Way

We present a new reformulation of the canonical quantum geometrodynamics, which allows one to overcome the fundamental problem of the frozen formalism and, therefore, to construct an appropriate Hilbert space associate to the solution of the restated dynamics. More precisely, to remove the ambiguity contained in the Wheeler–DeWitt approach, with respect to the possibility of a (3+1)-splitting when space–time is in a quantum regime, we fix the reference frame (i.e. the lapse function and the shift vector) by introducing the so-called kinematical action. As a consequence the new super-Hamiltonian constraint becomes a parabolic one and we arrive to a Schrödinger-like approach for the quantum dynamics. In the semiclassical limit our theory provides General Relativity in the presence of an additional energy–momentum density contribution coming from non-zero eigenvalues of the Hamiltonian constraints. The interpretation of these new contributions comes out in natural way that soon as it is recognized that the kinematical action can be recasted in such a way that it describes a pressureless, but, in general, non-geodesic perfect fluid.

scholarly journals Fourier transform from momentum space to twistor space

2020 ◽  
pp. 2150032
Author(s):  
Jun-ichi Note
Keyword(s):  
Fourier Transform ◽  
Momentum Space ◽  
Twistor Space ◽  
Three Dimensional ◽  
Yang Mills ◽  
The Fourier Transform ◽  
Complex Integral ◽  
Cech Cohomology ◽  
Complex Projective ◽  
Operator Representations

Several methods use the Fourier transform from momentum space to twistor space to analyze scattering amplitudes in Yang–Mills theory. However, the transform has not been defined as a concrete complex integral when the twistor space is a three-dimensional complex projective space. To the best of our knowledge, this is the first study to define it as well as its inverse in terms of a concrete complex integral. In addition, our study is the first to show that the Fourier transform is an isomorphism from the zeroth Čech cohomology group to the first one. Moreover, the well-known twistor operator representations in twistor theory literature are shown to be valid for the Fourier transform and its inverse transform. Finally, we identify functions over which the application of the operators is closed.

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