On the precanonical structure of the Schrödinger wave functional

Kanatchikov, Igor V.

doi:10.48550/arxiv.1312.4518

Cited by 18 publications

(65 citation statements)

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“…And what does the Schrödinger equation look like, once we know the classical Hamilton-Jacobi equation (20). Although these questions have not been addressed in general, let us note that there have been studies of quantization in the De Donder-Weyl Hamiltonian theory, where the quantization of momenta is based on generalized Poisson brackets, and a field-theoretic generalization of the Schrödinger equation is proposed that features a Clifford-valued wave function, and reduces to the De Donder-Weyl Hamilton-Jacobi equation in the classical limit [6,14].…”

Section: Discussionmentioning

confidence: 99%

“…One possible method to approach the canonical equations ( 13) is the following. Suppose P (q) is given, that obeys the Hamiltonian constraint H(q, P (q)) = 0 (14) on some open subset of the configuration space C. By differentiation, we obtain, according to the chain rule (A41), ∂q H( q, P (q)) + ∂q Ṗ (q) • ∂ P H(q, P (q)) = 0, (15) and using the first canonical equation (13a), we find λ ∂q H( q, P (q)) = − ∂q Ṗ (q) • dΓ.…”

Section: Local Hamilton-jacobi Theorymentioning

confidence: 99%

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Classical field theories from Hamiltonian constraint: Canonical equations of motion and local Hamilton–Jacobi theory

Zatloukal

2016

Int. J. Geom. Methods Mod. Phys.

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Section: Discussionmentioning

confidence: 99%

Section: Local Hamilton-jacobi Theorymentioning

confidence: 99%

Classical field theories from Hamiltonian constraint: Canonical equations of motion and local Hamilton–Jacobi theory

Zatloukal

2016

Int. J. Geom. Methods Mod. Phys.

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“…Without a space-like foliation, the polysymplectic formalism provides a manifestly spacetime covariant formulation. The polymomentum phase-space is endowed with a canonical (n + 1)-form, called the polysymplectic form [16,17,18,19], which plays a similar role to the one of the standard symplectic 2-form. A bracket may be induced in the polymometum phasespace by this polysymplectic form, which results to be a graded generalization of the Poisson bracket, called the Poisson-Gerstenhaber bracket [19,20].…”

Section: Introductionmentioning

confidence: 99%

“…A bracket may be induced in the polymometum phasespace by this polysymplectic form, which results to be a graded generalization of the Poisson bracket, called the Poisson-Gerstenhaber bracket [19,20]. The Hamiltonian field equations encountered within this formalism, known as the De Donder-Weyl equations, can be written in terms of the Poisson-Gerstenhaber bracket [16,17,18,19,20] with all spacetime variables treated on an equal footing.…”

Section: Introductionmentioning

confidence: 99%

“…This paper is organized as follows: In Section 2 we review the main aspects of the polysymplectic approach in order to set up our notation and definitions, following as close as possible references [16,17,18,19,20,21,22,23] in which a well behaved and intuitive framework of the De Donder-Weyl Hamiltonian formalism is introduced. In section 3, we introduce the MacDowell-Mansouri action following the notation developed in [1] and [2].…”

Section: Introductionmentioning

confidence: 99%

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De Donder–Weyl Hamiltonian formalism of MacDowell–Mansouri gravity

Berra─Montiel

Molgado

Serrano-Blanco

2017

Class. Quantum Grav.

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We analyse the behaviour of the MacDowell-Mansouri action with internal symmetry group SO(4, 1) under the De Donder-Weyl Hamiltonian formulation. The field equations, known in this formalism as the De Donder-Weyl equations, are obtained by means of the graded Poisson-Gerstenhaber bracket structure present within the De Donder-Weyl formulation. The decomposition of the internal algebra so(4, 1) ≃ so(3, 1) ⊕ R 3,1 allows the symmetry breaking SO(4, 1) → SO(3, 1), which reduces the original action to the Palatini action without the topological term. We demonstrate that, in contrast to the Lagrangian approach, this symmetry breaking can be performed indistinctly in the polysymplectic formalism either before or after the variation of the De Donder-Weyl Hamiltonian has been done, recovering Einstein's equations via the Poisson-Gerstenhaber bracket.The equivalence between the action proposed by MacDowell and Mansouri and the Palatini action is made possible by the fact that the internal Lie algebra admits the orthogonal splitting so(4, 1) ≃ so(3, 1) ⊕R 3,1 , so that the symmetry breaking is achieved by projecting the associated SO(4, 1)-connection to its SO(3, 1) components, which turn out to be the standard Lorentz connection. As pointed out by Wise [2], a concise geometrical meaning can be given to the symmetry breaking process by identifying the SO(4, 1) gauge field as a connection of a Cartan geometry [11,12].Being a theory with a strong geometric background and also physically relevant due to its relation with General Relativity, the MacDowell-Mansouri model is a perfect candidate for analysis under the inherently geometric classical formulation known as the polysymplectic formalism. Based on the early work of De Donder, Weyl, Carathéodory, among others [13,14,15], the polysymplectic approach of field theory consists in a De Donder-Weyl Hamiltonian formalism endowed with a generalization of the symplectic structure which enables to define a Poisson-like bracket. This issue, in particular, sets an important distinction with most of the various versions of the multisymplectic formalism. One of the key points within the polysymplectic approach relies in changing the definition of the standard Hamiltonian momenta, here considered not as variations of the Lagrangian density with respect to time derivatives of the field variables but as variations with respect to the derivatives of the fields in every spacetime direction. These new fields, known within this context as polymomenta, define a De Donder-Weyl Legendre transformation which associates to the Lagrangian density a new function H DW called De Donder-Weyl Hamiltonian, and it dictates the physical behaviour of the system in the so-called polymomentum phase-space. Without a space-like foliation, the polysymplectic formalism provides a manifestly spacetime covariant formulation. The polymomentum phase-space is endowed with a canonical (n + 1)-form, called the polysymplectic form [16,17,18,19], which plays a similar role to the one of the standard symplectic 2-form. A ...

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Hamiltonian Constraint Formulation of Classical Field Theories

Zatloukal

2016

Adv. Appl. Clifford Algebras

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Classical field theory is considered as a theory of unparametrized surfaces embedded in a configuration space, which accommodates, in a symmetric way, spacetime positions and field values. Dynamics is defined via the (Hamiltonian) constraint between multivector-valued generalized momenta, and points in the configuration space. Starting from a variational principle, we derive the local equations of motion, that is, differential equations that determine classical surfaces and momenta. A local Hamilton-Jacobi equation applicable in the field theory then follows readily. In addition, we discuss the relation between symmetries and conservation laws, and derive a Hamiltonian version of the Noether theorem, where the Noether currents are identified as the classical momentum contracted with the symmetry-generating vector fields. The general formalism is illustrated by two examples: the scalar field theory, and the string theory.Throughout the article, we employ the mathematical formalism of geometric algebra and calculus, which allows us to perform completely coordinate-free manipulations. * Electronic address: zatlovac@fjfi.cvut.cz; URL: http://www.zatlovac.eu arXiv:1602.00468v1 [math-ph]

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On the precanonical structure of the Schrödinger wave functional

Cited by 18 publications

References 17 publications

Classical field theories from Hamiltonian constraint: Canonical equations of motion and local Hamilton–Jacobi theory

Classical field theories from Hamiltonian constraint: Canonical equations of motion and local Hamilton–Jacobi theory

De Donder–Weyl Hamiltonian formalism of MacDowell–Mansouri gravity

Hamiltonian Constraint Formulation of Classical Field Theories

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