Precanonical quantization and the Schrödinger wave functional

Kanatchikov, Igor V.

doi:10.1016/s0375-9601(01)00225-0

Cited by 31 publications

(96 citation statements)

References 24 publications

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“…This is evident, on the classical level, from the fact that the DW Hamiltonian equations are equivalent to the field equations and, on the quantum level, from the observation that our generalized Schrödinger equation, eq. (2.6), reproduces the field equations in the classical limit and also can be related to the standard functional differential Schrödinger equation [50].…”

mentioning

confidence: 79%

“…(2.6) allows us to derive the standard functional differential Schrödinger equation once a suitable physically motivated ansatz relating the Schrödinger wave functional and the wave function in (2.6) is constructed [50].…”

Section: Precanonical Quantizationmentioning

confidence: 99%

“…Some details on the application of the present precanonical quantization scheme to the case of scalar fields can be found in [48][49][50]. It should be noted that a capability of eq.…”

Section: Precanonical Quantizationmentioning

confidence: 99%

“…Similarly to the classical level, the customary "infinite-dimensionality" goes over into a "multi-parametricity". Further details on the interplay between precanonical and canonical field quantization have been discussed recently in [50].…”

Section: Precanonical Quantizationmentioning

confidence: 99%

“…Unfortunately, many fundamental aspects of the corresponding precanonical approach to field quantization, as we suggest to call it, so far remain poorly understood (see [47,49] and Section 4 for a discussion) and require a further analysis. This particularly concerns an interplay with the standard formalism and notions of quantum field theory (see [50] for a recent progress). Nevertheless, the already elaborated part of the theory points to intriguing features and as yet unexplored potential which, hopefully, are capable to make the precanonical approach a useful complement to the presently available concepts and techniques of quantum field theory.…”

Section: Introductionmentioning

confidence: 99%

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Kanatchikov

2001

International Journal of Theoretical Physics

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A nonpertubative approach to quantum gravity using precanonical field quantization originating from the covariant De Donder-Weyl Hamiltonian formulation which treats space and time variables on equal footing is presented. A generally covariant "multi-temporal" generalized Schrödinger equation on the finite dimensional space of metric and space-time variables is obtained. An important ingredient of the formulation is the "bootstrap condition" which introduces a classical space-time geometry as an approximate concept emerging as the quantum average in a self-consistent with the underlying quantum dynamics manner. An independence of the theory from an arbitrarily fixed background is ensured in this way. The prospects and unsolved problems of precanonical quantization of gravity are outlined. *

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confidence: 79%

Section: Precanonical Quantizationmentioning

confidence: 99%

“…Some details on the application of the present precanonical quantization scheme to the case of scalar fields can be found in [48][49][50]. It should be noted that a capability of eq.…”

Section: Precanonical Quantizationmentioning

confidence: 99%

Section: Precanonical Quantizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Kanatchikov

2001

International Journal of Theoretical Physics

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Lectures on Loop Quantum Gravity

2003

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Quantum General Relativity (QGR), sometimes called Loop Quantum Gravity, has matured over the past fifteen years to a mathematically rigorous candidate quantum field theory of the gravitational field. The features that distinguish it from other quantum gravity theories are 1) background independence and 2) minimality of structures.Background independence means that this is a non-perturbative approach in which one does not perturb around a given, distinguished, classical background metric, rather arbitrary fluctuations are allowed, thus precisely encoding the quantum version of Einstein's radical perception that gravity is geometry.Minimality here means that one explores the logical consequences of bringing together the two fundamental principles of modern physics, namely general covariance and quantum theory, without adding any experimentally unverified additional structures such as extra dimensions, extra symmetries or extra particle content beyond the standard model. While this is a very conservative approach and thus maybe not very attractive to many researchers, it has the advantage that pushing the theory to its logical frontiers will undoubtedly either result in a successful theory or derive exactly which extra structures are required, if necessary. Or put even more radically, it may show which basic principles of physics have to be given up and must be replaced by more fundamental ones.QGR therefore is, by definition, not a unified theory of all interactions in the standard sense since such a theory would require a new symmetry principle. However, it unifies all presently known interactions in a new sense by quantum mechanically implementing their common symmetry group, the four-dimensional diffeomorphism group, which is almost completely broken in perturbative approaches.

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Precanonical quantization of Yang-Mills fields and the functional Schrödinger representation

Kanatchikov¹

2004

Reports on Mathematical Physics

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Precanonical quantization of pure Yang-Mills fields, which is based on the covariant De Donder-Weyl (DW) Hamiltonian formulation, and its connection with the functional Schrödinger representation in the temporal gauge are discussed. The mass gap problem is related to a finite dimensional spectral problem for a generalized Clifford-valued magnetic Schrödinger operator which represents the DW Hamiltonian operator.

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Precanonical quantization and the Schrödinger wave functional

Cited by 31 publications

References 24 publications

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Lectures on Loop Quantum Gravity

Precanonical quantization of Yang-Mills fields and the functional Schrödinger representation

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