Jacobi-type identities in algebras and superalgebras

Лавров, П. М.; Радченко, О. В.; Tyutin, I. V.

doi:10.1007/s11232-014-0161-2

Cited by 3 publications

(8 citation statements)

References 15 publications

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“…Perhaps one reason of a purely algebraic nature is the fact that of the four basic identities (B.1) -(B.4), only two ones are independent, namely (B.2) and (B.3). This circumstance and its consequence were analyzed in detail in the paper by Lavrov et al [46]. In particular, the Jacobi identity (B.1) is a consequence of the generalized identity (B.2).…”

Section: Resultsmentioning

confidence: 99%

Unitary Quantization and Para-Fermi Statistics of Order 2

Markov

Markova

Гитман

2018

J. Exp. Theor. Phys.

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A connection between a unitary quantization scheme and para-Fermi statistics of order 2 is considered. An appropriate extension of Green's ansatz is suggested. This extension allows one to transform bilinear and trilinear commutation relations for the annihilation and creation operators of two different para-Fermi fields φ a and φ b into identity. The way of incorporating para-Grassmann numbers ξ k into a general scheme of uniquantization is also offered. For parastatistics of order 2 a new fact is revealed, namely, the trilinear relations containing both the para-Grassmann variables ξ k and the field operators a k , b m under a certain invertible mapping go over into the unitary equivalent relations, where commutators are replaced by anticommutators and vice versa. It is shown that the consequence of this circumstance is the existence of two alternative definitions of the coherent state for para-Fermi oscillators. The Klein transformation for Green's components of the operators a k , b m is constructed in an explicit form that enables us to reduce the initial commutation rules for the components to the normal commutation relations of ordinary Fermi fields. A nontrivial connection between trilinear commutation relations of the unitary quantization scheme and so-called Lie-supertriple system is analysed. A brief discussion of the possibility of embedding the Duffin-Kemmer-Petiau theory into the unitary quantization scheme is provided. *

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Section: Resultsmentioning

confidence: 99%

Unitary Quantization and Para-Fermi Statistics of Order 2

Markov

Markova

Гитман

2018

J. Exp. Theor. Phys.

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“…The following theorem shows that the torsion tensor (2.6) of product semi-symmetric connection satisfies Jacoby's identity. For Jacoby's identity you can see the article [3].…”

Section: Theorem 21 ([21])mentioning

confidence: 99%

Notes on product semisymmetric connection in a locally decomposable Riemannian space

Maksimović¹,

Stankovic²

2021

Turk J Math

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The purpose of this paper is to investigate the product semi-symmetric connection in a locally decomposable Riemannian space. The curvature tensors of this connection were considered. Some properties of almost product structure, some properties of torsion tensor of product semi-symmetric connection and some relations between curvature tensors and almost product structure are given. Also, the paper checks a special case of such connection when its generator is a gradient vector.

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“…The canonical commutation relation [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] is the fundamental relation between canonical conjugate quantities. For example, a set of equal-time commutation relations are introduced as xi , pj…”

Section: Canonical Commutation Relationmentioning

confidence: 99%

“…The operator corresponding to any mechanical quantity F is expressed as F in quantum mechanics, so there is F → F , namely it's shown as [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]…”

Section: Groenewold's Theoremmentioning

confidence: 99%

“…As we can see that the interaction picture is an intermediate representation between the Schrödinger picture and the Heisenberg picture. Whereas in the other two pictures either the state vector |ψ (t) or the operators F carry time dependence, in the interaction picture both carry part of the time dependence of observables [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15].…”

Section: Three Picturesmentioning

confidence: 99%

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Generalized geometric commutator theory and quantum geometric bracket and its uses

Wang¹

2020

Preprint

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Inspired by the geometric bracket for the generalized covariant Hamilton system, we abstractly define a generalized geometric commutator [a, b] = [a, b] cr + G (s, a, b) formally equipped with geomutator G (s, a, b) = −G (s, b, a) defined in terms of structural function s related to the structure of spacetime or manifolds itself for revising the classical representation [a, b] cr = ab − ba for any elements a and b of any algebra. We find that geomutator G (s, a, b) of any manifold which can be automatically chosen by G (s, a, b) = a[s, b] cr − b[s, a] cr .Then we use the generalized geometric commutator to define quantum covariant Poisson bracket that is closely related to the quantum geometric bracket defined by geomutator as a generalization of quantum Poisson bracket all associated with the structural function generated by the manifolds to study the quantum mechanics. We find that the covariant dynamics appears along with the generalized Heisenberg equation as a natural extension of Heisenberg equation and G-dynamics based on the quantum geometric bracket, meanwhile, the geometric canonical commutation relation is naturally induced. As an application of quantum covariant Poisson bracket, we reconsider the canonical commutation relation and the quantization of field to be more complete and covariant, as a consequence, we obtain some useful results.

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Jacobi-type identities in algebras and superalgebras

Cited by 3 publications

References 15 publications

Unitary Quantization and Para-Fermi Statistics of Order 2

Unitary Quantization and Para-Fermi Statistics of Order 2

Notes on product semisymmetric connection in a locally decomposable Riemannian space

Generalized geometric commutator theory and quantum geometric bracket and its uses

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