scholarly journals Canonical Quantum Gravity, Constructive QFT, and Renormalisation

2020 ◽  
Vol 8 ◽  
Author(s):  
Thomas Thiemann
Keyword(s):  
Quantum Gravity ◽  
Quantum Dynamics ◽  
Statistical Physics ◽  
Gravitational Interaction ◽  
Quantum Field ◽  
Constructive Quantum Field Theory ◽  
Canonical Approach ◽  
Quantum Statistical ◽  
Canonical Quantum Gravity ◽  
Canonical Quantum

The canonical approach to quantum gravity has been put on a firm mathematical foundation in the recent decades. Even the quantum dynamics can be rigorously defined, however, due to the tremendously non-polynomial character of the gravitational interaction, the corresponding Wheeler–DeWitt operator-valued distribution suffers from quantisation ambiguities that need to be fixed. In a very recent series of works, we have employed methods from the constructive quantum field theory in order to address those ambiguities. Constructive QFT trades quantum fields for random variables and measures, thereby phrasing the theory in the language of quantum statistical physics. The connection to the canonical formulation is made via Osterwalder–Schrader reconstruction. It is well known in quantum statistics that the corresponding ambiguities in measures can be fixed using renormalisation. The associated renormalisation flow can thus be used to define a canonical renormalisation programme. The purpose of this article was to review and further develop these ideas and to put them into context with closely related earlier and parallel programmes.

scholarly journals MINISUPERSPACE MODEL FOR REVISED CANONICAL QUANTUM GRAVITY

2004 ◽  
Vol 13 (08) ◽  
pp. 1703-1718 ◽  
Author(s):  
GIOVANNI MONTANI
Keyword(s):  
Quantum Gravity ◽  
Reference Frame ◽  
Quantum Dynamics ◽  
Cosmological Solution ◽  
Canonical Quantization ◽  
Hamiltonian Operator ◽  
Big Bang ◽  
Dust Fluid ◽  
Canonical Quantum Gravity ◽  
Canonical Quantum

We present a reformulation of the canonical quantization of gravity, as referred to the minisuperspace; the new approach is based on fixing a Gaussian (or synchronous) reference frame and then quantizing the system via the reconstruction of a suitable constraint; then the quantum dynamics is re-stated in a generic coordinates system and it becomes dependent on the lapse function. The analysis follows a parallelism with the case of the non-relativistic particle and leads to the minisuperspace implementation of the so-called kinematical action as proposed in Ref. 1 (here almost coinciding also with the approach presented in Ref. 2). The new constraint leads to a Schrödinger equation for the system, i.e. to non-vanishing eigenvalues for the super-Hamiltonian operator; the physical interpretation of this feature relies on the appearance of a "dust fluid" (non-positive definite) energy density, i.e. a kind of "materialization" of the reference frame. As an example of minisuperspace model, we consider a Bianchi type IX Universe, for which some dynamical implications of the revised canonical quantum gravity are discussed. We also show how, on the classical limit, the presence of the dust fluid can have relevant cosmological issues. Finally we upgrade our analysis by its extension to the generic cosmological solution, which is performed in the so-called long-wavelength approximation. In fact, near the Big-Bang, we can neglect the spatial gradients of the dynamical variables and proceed to implement, in each space point, the same minisuperspace paradigm valid for the Bianchi IX model.

scholarly journals GENERALIZED QUANTIZATION PRINCIPLE IN CANONICAL QUANTUM GRAVITY AND APPLICATION TO QUANTUM COSMOLOGY

2012 ◽  
Vol 27 (20) ◽  
pp. 1250106 ◽  
Author(s):  
MARTIN KOBER
Keyword(s):  
Quantum Gravity ◽  
Quantum Cosmology ◽  
Dynamical Behavior ◽  
Generalized Uncertainty Principle ◽  
Heisenberg Algebra ◽  
Polynomial Method ◽  
Canonical Approach ◽  
Canonical Quantum Gravity ◽  
Canonical Quantum ◽  
Quantization Principle

In this paper, a generalized quantization principle for the gravitational field in canonical quantum gravity, especially with respect to quantum geometrodynamics is considered. This assumption can be interpreted as a transfer from the generalized uncertainty principle in quantum mechanics, which is postulated as generalization of the Heisenberg algebra to introduce a minimal length, to a corresponding quantization principle concerning the quantities of quantum gravity. According to this presupposition there have to be given generalized representations of the operators referring to the observables in the canonical approach of a quantum description of general relativity. This also leads to generalized constraints for the states and thus to a generalized Wheeler–DeWitt equation determining a new dynamical behavior. As a special manifestation of this modified canonical theory of quantum gravity, quantum cosmology is explored. The generalized cosmological Wheeler–DeWitt equation corresponding to the application of the generalized quantization principle to the cosmological degree of freedom is solved by using Sommerfelds polynomial method.

Forming the Canon

2020 ◽  
pp. 160-192
Author(s):  
Dean Rickles
Keyword(s):  
General Relativity ◽  
Gravitational Field ◽  
Quantum Gravity ◽  
Early Development ◽  
Subsequent Development ◽  
Hamiltonian Form ◽  
Coordinate Transformations ◽  
Canonical Approach ◽  
Canonical Quantum Gravity ◽  
Canonical Quantum

This chapter charts the early development of the canonical quantum gravity (that is, the quantization of the gravitational field in Hamiltonian form). What we find in this period include: the establishment of a procedure for quantizing in curved spaces; the first expressions for the Hamiltonian of general relativity; recognition of the existence and importance of constraints (i.e. the generators of infinitesimal coordinate transformations); a focus on the problem of observables (and the realisation of conceptual implications in defining these for generally relativistic theories), and a (template of a) method for quantizing the theory. Although it commenced relatively early, the canonical approach was slow in its subsequent development. This had two sources: (1) it required the introduction of tools and concepts from outside of quantum gravity proper (namely, the constraint machinery and the parameter formalism); (2) by its very nature, it is highly rigorous in a conceptual sense, demanding lots of groundwork to be established, in terms of the structure of physical observables, before the actual issue of quantization can even be considered. Work was further complicated by the fact that these two sources of difficulty happened to be entangled. Particular emphasis is placed on the parameter formalism of Paul Weiss.

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.

Symmetry groups of state vectors in canonical quantum gravity

1986 ◽  
Vol 27 (2) ◽  
pp. 573-592 ◽  
Author(s):  
Donald M. Witt
Keyword(s):  
Quantum Gravity ◽  
Symmetry Groups ◽  
Canonical Quantum Gravity ◽  
Canonical Quantum

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.

ARITHMETIC QUANTUM GRAVITY

2010 ◽  
Vol 19 (14) ◽  
pp. 2305-2310 ◽  
Author(s):  
AXEL KLEINSCHMIDT ◽  
HERMANN NICOLAI
Keyword(s):  
Quantum Gravity ◽  
Algebraic Structure ◽  
Complete Resolution ◽  
Quantum Level ◽  
Cosmological Singularity ◽  
Quantum Billiard ◽  
Symmetry Structure ◽  
Canonical Quantum Gravity ◽  
The Universe ◽  
Canonical Quantum

The arithmetic chaos of classical (super)gravity near a spacelike singularity is elevated to the quantum level via the construction of a cosmological quantum billiard system. Its precise formulation, together with its underlying algebraic structure, allows for a general analysis of the wavefunction of the universe near the singularity. We argue that the extension of these results beyond the billiard approximation may provide a concrete mechanism for emergent space as well as new perspectives on several long-standing issues in canonical quantum gravity. The exponentially growing complexity of the underlying symmetry structure could introduce an element of non-computability that effectively "screens" the cosmological singularity from a complete resolution.

Matter Time in Canonical Quantum Gravity

1993 ◽  
pp. 201-221
Author(s):  
K. V. Kuchař
Keyword(s):  
Quantum Gravity ◽  
Canonical Quantum Gravity ◽  
Canonical Quantum
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