De Donder-Weyl theory and a hypercomplex extension of quantum mechanics to field theory

Kanatchikov, Igor V.

doi:10.1016/s0034-4877(99)80024-x

Cited by 59 publications

(162 citation statements)

References 11 publications

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“…This generalizes to the present formulation of field theory the familiar statement in the analytical mechanics that Hamilton's canonical function generates the time evolution. Note that this observation largely underlies our hypothesis (2.6) regarding the form of a generalized Schrödinger equation within the precanonical quantization approach [26,47,48].…”

Section: Classical Theorysupporting

confidence: 64%

“…This philosophy leads to the following (covariant, "multi-temporal", hypercomplex) generalization of the Schrödinger equation to the precanonical framework [47][48][49] …”

Section: Precanonical Quantizationmentioning

confidence: 99%

“…The main argument in favour of a generalized Schrödinger equation (2.6) is that it satisfies at least two important aspects of the correspondence principle [48,49]:…”

Section: Precanonical Quantizationmentioning

confidence: 99%

“…The elements of field quantization based on the aforementioned PoissonGerstenhaber brackets on differential forms have been discussed in [26,[47][48][49] and will be briefly summarized in Section 2.2. 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.…”

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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Section: Classical Theorysupporting

confidence: 64%

“…This philosophy leads to the following (covariant, "multi-temporal", hypercomplex) generalization of the Schrödinger equation to the precanonical framework [47][48][49] …”

Section: Precanonical Quantizationmentioning

confidence: 99%

“…The main argument in favour of a generalized Schrödinger equation (2.6) is that it satisfies at least two important aspects of the correspondence principle [48,49]:…”

Section: Precanonical Quantizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Kanatchikov

2001

International Journal of Theoretical Physics

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“…The expectation value of an operator A is defined by 87) where the invariant measure (see (4.16) in the membrane space M with the metric (7.4) is…”

Section: The Expectation Valuesmentioning

confidence: 99%

The Landscape of Theoretical Physics: A Global View

Pavšič¹

2002

Fundamental Theories of Physics

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This is a book for those who would like to learn something about special and general relativity beyond the usual textbooks, about quantum field theory, the elegant Fock-Schwinger-Stueckelberg proper time formalism, the elegant description of geometry by means of Clifford algebra, about the fascinating possibilities the latter algebra offers in reformulating the existing physical theories, and quantizing them in a natural way. It is shown how Clifford algebra provides much more: it provides room for new physics, with the prospects of resolving certain long standing puzzles. The theory of branes and the idea of how a 3-brane might represent our world is discussed in detail. Much attention is paid to the elegant geometric theory of branes which employs the infinite dimensional space of functions describing branes. Clifford algebra is generalized to the infinite dimensional spaces. In short, this is a book for anybody who would like to explore how the "theory of everything" might possibly be formulated. The theory that would describe all the known phenomena, could not be formulated without taking into account "all" the theoretical tools which are available. Foundations of those tools and their functional interrelations are described in the book. Note: This book was published by Kluwer Academic Publishers in 2001.The body of the posted version is identical to the published one, except for corrections of misprints, and some minor revisions that I have found necessary.

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Dynamics without Time for Quantum Gravity: Covariant Hamiltonian Formalism and Hamilton-Jacobi Equation on the Space G

Rovelli

2003

Decoherence and Entropy in Complex Systems

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Hamiltonian mechanics of field theory can be formulated in a generally covariant and background independent manner over a finite dimensional extended configuration space. The physical symplectic structure of the theory can then be defined over a space G of three-dimensional surfaces without boundary, in the extended configuration space. These surfaces provide a preferred over-coordinatization of phase space. I consider the covariant form of the Hamilton-Jacobi equation on G, and a canonical function S on G which is a preferred solution of the Hamilton-Jacobi equation. The application of this formalism to general relativity is equivalent to the ADM formalism, but fully covariant. In the quantum domain, it yields directly the Ashtekar-Wheeler-DeWitt equation. Finally, I apply this formalism to discuss the partial observables of a covariant field theory and the role of the spin networks -basic objects in quantum gravity-in the classical theory.

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De Donder-Weyl theory and a hypercomplex extension of quantum mechanics to field theory

Cited by 59 publications

References 11 publications

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The Landscape of Theoretical Physics: A Global View

Dynamics without Time for Quantum Gravity: Covariant Hamiltonian Formalism and Hamilton-Jacobi Equation on the Space G

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