Canonical noncommutativity algebra for the tetrad field in general relativity

Kober, Martin

doi:10.1088/0264-9381/28/22/225021

Cited by 6 publications

(7 citation statements)

References 101 publications

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“…This model with the gauge symmetries intact was generalized to NC spacetime (via the * -product and the SeibergWitten map) but it turned out that due to symmetry requirements the O(θ ) corrections miraculously cancel and hence to first order in θ the Einstein action and its NC extension are identical [70][71][72][73][74][75]. Thus we come to the conclusion that symmetries of canonical noncommutative spacetime naturally lead to the noncommutative version of unimodular gravity for O(θ ) results.…”

Section: Cosmological Implicationsmentioning

confidence: 90%

Noncommutative geometry and fluid dynamics

Das

Ghosh

2016

Eur. Phys. J. C

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In the present paper we have developed a NonCommutative (NC) generalization of perfect fluid model from first principles, in a Hamiltonian framework. The noncommutativity is introduced at the Lagrangian (particle) coordinate space brackets and the induced NC fluid bracket algebra for the Eulerian (fluid) field variables is derived. Together with a Hamiltonian this NC algebra generates the generalized fluid dynamics that satisfies exact local conservation laws for mass and energy, thereby maintaining mass and energy conservation. However, nontrivial NC correction terms appear in the charge and energy fluxes. Other non-relativistic spacetime symmetries of the NC fluid are also discussed in detail. This constitutes the study of kinematics and dynamics of NC fluid. In the second part we construct an extension of the FriedmannRobertson-Walker (FRW) cosmological model based on the NC fluid dynamics presented here. We outline the way in which NC effects generate cosmological perturbations bringing about anisotropy and inhomogeneity in the model. We also derive a NC extended Friedmann equation.

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Section: Cosmological Implicationsmentioning

confidence: 90%

Noncommutative geometry and fluid dynamics

Das

Ghosh

2016

Eur. Phys. J. C

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“…where ψ * (x) denotes the conjugated quaternionic wave function, which is defined by (22) and the definition of quaternionic conjugation (3). Also hermitian conjugation of operators is defined in analogy to usual quantum mechanics by replacing complex conjugation by quaternionic conjugation.…”

Section: Quaternionic Quantization In Quantum Mechanicsmentioning

confidence: 99%

“…The wave function in the generalized Dirac equation (27) represents a quaternionic spinor wave function. This means that the spinor structure is the same as in usual quantum field theory, but the wave function is quaternionic and thus it is of a shape as defined in (22). With respect to usual complex wave functions one can perform phase transformation, ψ C (x) → e iα ψ C (x).…”

Section: B Quaternionic Gauge Principlementioning

confidence: 99%

“…If this is postulated, then these generalized quantization principles have also to be transferred to the quantization of general relativity an thus the gravitational field what differs from the formulation of clasical general relativity on noncommutative space-time [10] or even usual quantum general relativity on noncommutative space-time [11], [12], [13], [14]. In [15], [16], [17], [18], [19] ideas to transfer the concept of a generalized uncertainty principle to gravity can be found, but in [20] and [21] the generalized uncertainty principle principle has really been transferred to the variables of canonical quantum gravity and quantum cosmology, whereas in [22] the concept of noncommutative geometry has been transferred to the components of the tetrad field. An extension of the field theoretic quantization principle to a nonlocal quantization principle has been considered in [23].…”

Section: Introductionmentioning

confidence: 99%

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Quaternionic quantization principle in general relativity and supergravity

Kober

2016

Int. J. Mod. Phys. A

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A generalized quantization principle is considered, which incorporates nontrivial commutation relations of the components of the variables of the quantized theory with the components of the corresponding canonical conjugated momenta referring to other space-time directions. The corresponding commutation relations are formulated by using quaternions. At the beginning, this extended quantization concept is applied to the variables of quantum mechanics. The resulting Dirac equation and the corresponding generalized expression for plane waves are formulated and some consequences for quantum field theory are considered. Later, the quaternionic quantization principle is transferred to canonical quantum gravity. Within quantum geometrodynamics as well as the Ashtekar formalism the generalized algebraic properties of the operators describing the gravitational observables and the corresponding quantum constraints implied by the generalized representations of these operators are determined. The generalized algebra also induces commutation relations of the several components of the quantized variables with each other. Finally, the quaternionic quantization procedure is also transferred to N = 1 supergravity. Accordingly, the quantization principle has to be generalized to be compatible with Dirac brackets, which appear in canonical quantum supergravity.

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“…Such a generalization of the quantization principle in canonical quantum gravity as it is implied by the noncommutativity of space-time presupposed in this paper, has in other manifestations been considered in [36], [37], [38], [39], [40]. A canonical noncommutativity algebra between the components of the gravitational field and the corresponding generalization of general relativity has been considered in [41]. After formulating the generalized canonical quantum theory of gravity by referring to quantum geometrodynamics, a transition to Ashtekars formalism is performed in this paper.…”

Section: Introductionmentioning

confidence: 99%

Canonical quantum gravity on noncommutative space–time

Kober¹

2015

Int. J. Mod. Phys. A

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In this paper canonical quantum gravity on noncommutative space-time is considered. The corresponding generalized classical theory is formulated by using the moyal star product, which enables the representation of the field quantities depending on noncommuting coordinates by generalized quantities depending on usual coordinates. But not only the classical theory has to be generalized in analogy to other field theories. Besides, the necessity arises to replace the commutator between the gravitational field operator and its canonical conjugated quantity by a corresponding generalized expression on noncommutative space-time. Accordingly the transition to the quantum theory has also to be performed in a generalized way and leads to extended representations of the quantum theoretical operators. If the generalized representations of the operators are inserted to the generalized constraints, one obtains the corresponding generalized quantum constraints including the Hamiltonian constraint as dynamical constraint. After considering quantum geometrodynamics under incorporation of a coupling to matter fields, the theory is transferred to the Ashtekar formalism. The holonomy representation of the gravitational field as it is used in loop quantum gravity opens the possibility to calculate the corresponding generalized area operator.

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Canonical noncommutativity algebra for the tetrad field in general relativity

Cited by 6 publications

References 101 publications

Noncommutative geometry and fluid dynamics

Noncommutative geometry and fluid dynamics

Quaternionic quantization principle in general relativity and supergravity

Canonical quantum gravity on noncommutative space–time

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