Ehrenfest Theorem in Precanonical Quantization of Fields and Gravity

Kanatchikov, Igor V.

doi:10.48550/arxiv.1602.01083

Cited by 7 publications

(29 citation statements)

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“…A validity of the Ehrenfest theorem in this formulation of quantum gravity is to be discussed elsewhere. 7 A relation of the above description to the standard QFT in the functional Schrödinger representation has been established in the limit of infinite κ or, more precisely, when 1 κ γ 0 → ̟ 0 . 8 In this limiting case one can construct the Schrödinger wave functional as the multidimensional Volterra's product integral 9 of precanonical wave functions and derive the canonical functional derivative Schrödinger equation from the precanonical Schrödinger equation ( 2).…”

Section: Precanonical Quantization Of Fieldsmentioning

confidence: 95%

On the “spin connection foam” picture of quantum gravity from precanonical quantization

Kanatchikov¹

2017

The Fourteenth Marcel Grossmann Meeting

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Precanonical quantization is based on a generalization of the Hamiltonian formalism to field theory, the so-called De Donder-Weyl (DW) theory, which does not require a spacetime splitting and treats the space-time variables on an equal footing. Quantum dynamics is described by a precanonical wave function on the finite dimensional space of field coordinates and space-time coordinates, which satisfies a partial derivative precanonical Schrödinger equation. The standard QFT in the functional Schrödinger representation can be derived from the precanonical quantization in a limiting case. An analysis of the constraints within the DW Hamiltonian formulation of the Einstein-Palatini vielbein formulation of GR and quantization of the generalized Dirac brackets defined on differential forms lead to the covariant precanonical Schrödinger equation for quantum gravity. The resulting dynamics of quantum gravity is described by the wave function or transition amplitudes on the total space of the bundle of spin connections over space-time. Thus, precanonical quantization leads to the "spin connection foam" picture of quantum geometry represented by a generally non-Gaussian random field of spin connection coefficients, whose probability distribution is given by the precanonical wave function. The normalizability of precanonical wave functions is argued to lead to the quantumgravitational avoidance of curvature singularities. Possible connections with LQG are briefly discussed.Keywords: Quantum gravity; precanonical quantization; De Donder-Weyl theory; vielbein gravity; Gerstenhaber algebra; Dirac brackets; Clifford algebra; Lévy processes. Precanonical quantization of fieldsContemporary quantum field theory originates from the canonical quantization which is based on the canonical Hamiltonian formalism. The latter dictates a picture of fields as infinite dimensional Hamiltonian systems. It also restricts the consideration to the globally hyperbolic space-times, as it implies a different role of the time dimension, along which the evolution proceeds, and the space dimensions, which label the continuum of degrees of freedom of fields. Many problems we encounter in quantum gravity theories can be traced back to this very origin of QFT.However, the canonical Hamiltonian formalism is not the only possibility to extend the Hamiltonian formalism from mechanics to field theory. The alternative "Hamiltonizations" (i.e. writing the field equations in the first order form using some generalization of the Legendre transform) known in the calculus of variations 1 are, in fact, inherently more geometrical than the canonical formalism, and they treat the space and time variables (i.e. the independent variables of the multiple integral variational problem) on the equal footing, i.e. essentially as multidimensional generalizations of the one-dimensional time parameter in mechanics.

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Section: Precanonical Quantization Of Fieldsmentioning

confidence: 95%

On the “spin connection foam” picture of quantum gravity from precanonical quantization

Kanatchikov¹

2017

The Fourteenth Marcel Grossmann Meeting

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“…The appearance of constraints is problematic for the straightforward application of DW Hamiltonian formalism. However, Kanatchikov [67] has generalised the Dirac's constraints algorithm to the covariant DW Hamiltonian theory which was applied to the Palatini formulation of vielbein gravity [47][48][49][50][51] and to other modified theories of gravity [68][69][70].…”

Section: The Constraintsmentioning

confidence: 99%

“…A generalization of Dirac bracket in the context of constrained DW Hamiltonian systems with 2-nd class constraint has been proposed by Kanatchikov in [67]. It has been used in his precanonical quantization of Einstein gravity in vielbein variables [47][48][49][50][51]. For (n − 1)-and 0-forms A and B…”

Section: Generalized Dirac Bracketsmentioning

confidence: 99%

“…De Donder [1] and H. Weyl [3] in the same way as the standard Hamilton-Jacobi theory in mechanics is related to the Hamiltonian formalism [19]. Later on precanonical quantization has been applied to gauge fields [33,[40][41][42] and general relativity in the standard metric formulation [43][44][45][46] and the Palatini vielbein formulation [47][48][49][50][51]. The relation of this approach with the standard canonical quantization in the Schrödinger representation has been established in [22,40,41,[52][53][54][55][56] both for quantum scalar fields in flat and curved space-times and Yang-Mills gauge theory.…”

Section: Introductionmentioning

confidence: 99%

“…The classical limit of precanonical quantization has been analysed by means of a generalization of the Ehrenfest theorem [30,31,33,50]: the De Donder-Weyl covariant Hamiltonian field equations are shown to be the equations satisfied by the expectation values of certain operators. However, the reduction of precanonical Schrödinger equation to the De Donder-Weyl covariant Hamilton-Jacobi equation so far has been demonstrated only for scalar fields in [30,57].…”

Section: Introductionmentioning

confidence: 99%

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On the Covariant Hamilton-Jacobi Equation for the Teleparallel Equivalent of General Relativity

Pietrzyk¹,

Barbachoux²

2022

Preprint

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The quantum waves of Minkowski space-time and the minimal acceleration from precanonical quantum gravity

Kanatchikov

2023

J. Phys.: Conf. Ser.

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We construct the simplest solutions of the previously obtained precanonical Schrödinger equation for quantum gravity, which correspond to the plane waves on the spin connection bundle and reproduce the Minkowski space-time on average. Quantum fluctuations lead to the emergence of the minimal acceleration a0 related to the range of the Yukawa modes in the fibers of the spin connection bundle. This minimal acceleration is proportional to the square root of the cosmological constant Λ generated by the operator re-ordering in the precanonical Schrödinger equation. Thus the mysterious connection between the minimal acceleration in the dynamics of galaxies as described by Milgrom’s MOND and the cosmological constant emerges as an elementary effect of precanonical quantum gravity. We also argue that the observable values of a0 and Λ can be obtained when the scale of the parameter ϰ introduced by precanonical quantization is subnuclear, in agreement with the previously established connection between the scale of ϰ and the mass gap in quantum SU(2) Yang-Mills theory.

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Ehrenfest Theorem in Precanonical Quantization of Fields and Gravity

Cited by 7 publications

References 0 publications

On the “spin connection foam” picture of quantum gravity from precanonical quantization

On the “spin connection foam” picture of quantum gravity from precanonical quantization

On the Covariant Hamilton-Jacobi Equation for the Teleparallel Equivalent of General Relativity

The quantum waves of Minkowski space-time and the minimal acceleration from precanonical quantum gravity

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