Unitary Quantization and Para-Fermi Statistics of Order 2

Markov, Yu. G.; Markova, M. A.; Гитман, Д. М.

doi:10.1134/s1063776118090054

Cited by 3 publications

(4 citation statements)

References 46 publications

(80 reference statements)

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“…From here, it is also clear that the actions of paraoperators with relative paraboson relations f ± i , b ± i are related to the actions of those with relative parafermion relations f ± i , b ± i by means of a certain phase factor. This indicates that these two sets of paraoperators could be related to each other by a Klein transformation (see [40][41][42][43] for the first ideas in this respect), as it is for example the case for gl(m|n) and a Z 2 × Z 2 -graded version gl(m 1 , m 2 |n 1 , n 2 ) [44].…”

Section: The Fock Representations ṽ(P ∞) Of Pso(∞|∞)mentioning

confidence: 99%

The Z2×Z2 -graded Lie superalgebras pso(2n+1|2n) and pso(∞|∞) , and parastatistics Fock spaces

Stoilova¹,

Jeugt²

2022

J. Phys. A: Math. Theor.

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Section: The Fock Representations ṽ(P ∞) Of Pso(∞|∞)mentioning

confidence: 99%

The Z2×Z2 -graded Lie superalgebras pso(2n+1|2n) and pso(∞|∞) , and parastatistics Fock spaces

Stoilova¹,

Jeugt²

2022

J. Phys. A: Math. Theor.

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“…Here, we consider only action on the ground state. It follows from the expression (5.2) that 20) and the condition (2.14) for p = 2 yields…”

Section: Matrix Element Of the Operator A 2mentioning

confidence: 99%

“…However, a preliminary analysis [20] has shown that the use of parafermion algebra in the standard form is insufficient for solving some applied problems, for example in the construction of the path integral representation in parasuperspace for the Green's function of a spin-1 massive particle in external Maxwell's field, and here, a generalization of this algebra would be required. As is well known, trilinear commutation relations for the para-Fermi statistics generate algebra which is isomorphic to the Lie algebra so(2M +1) [21].…”

Section: Introductionmentioning

confidence: 99%

Generalization of Geyer’s commutation relations with respect to the orthogonal group in even dimensions

Markov

Markova

2020

Eur. Phys. J. C

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A connection between the deformed Duffin–Kemmer–Petiau (DKP) algebra and an extended system of the parafermion trilinear commutation relations for the creation and annihilation operators $$a^{\pm }_{k}$$ a k ± and for an additional operator $$a_{0}$$ a 0 obeying para-Fermi statistics of order 2 based on the Lie algebra $${\mathfrak {s}}{\mathfrak {o}}(2M+2)$$ s o ( 2 M + 2 ) is established. An appropriate system of the parafermion coherent states as functions of para-Grassmann numbers is introduced. The representation for the operator $$a_{0}$$ a 0 in terms of generators of the orthogonal group SO(2M) correctly reproducing action of this operator on the state vectors of Fock space is obtained. A connection of the Geyer operator $$a_{0}^{2}$$ a 0 2 with the operator of so-called G-parity and with the CPT- operator $${\hat{\eta }}_{5}$$ η ^ 5 of the DKP-theory is established. In a para-Grassmann algebra a noncommutative, associative star product $$*$$ ∗ (the Moyal product) as a direct generalization of the star product in the algebra of Grassmann numbers is introduced. Two independent approaches to the calculation of the Moyal product $$*$$ ∗ are considered. It is shown that in calculating the matrix elements in the basis of parafermion coherent states of various operator expressions it should be taken into account constantly that we work in the so-called Ohnuki and Kamefuchi’s generalized state-vector space $${\mathfrak {U}}_{\;G}$$ U G , whose state vectors include para-Grassmann numbers $$\xi _{k}$$ ξ k in their definition, instead of the standard state-vector space $${\mathfrak {U}}$$ U (the Fock space).

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“…However, a preliminary analysis [51] has shown that the use of parafermion algebra in the standard form is insufficient for solving the stated problem and here, a generalization of this algebra would be required. As is well known, trilinear commutation relations for the para-Fermi statistics generate algebra which is isomorphic to the Lie algebra so(2M + 1) [52].…”

Section: Introductionmentioning

confidence: 99%

Path integral representation for inverse third order wave operator within the Duffin-Kemmer-Petiau formalism. Part I

Markov

Markova

Bondarenko

2020

J. High Energ. Phys.

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Within the framework of the Duffin-Kemmer-Petiau (DKP) formalism with a deformation, an approach to the construction of the path integral representation in parasuperspace for the Green's function of a spin-1 massive particle in external Maxwell's field is developed. For this purpose a connection between the deformed DKP-algebra and an extended system of the parafermion trilinear commutation relations for the creation and annihilation operators a ± k and for an additional operator a 0 obeying para-Fermi statistics of order 2 based on the Lie algebra so(2M + 2) is established. The representation for the operator a 0 in terms of generators of the orthogonal group SO(2M) correctly reproducing action of this operator on the state vectors of Fock space is obtained. An appropriate system of the parafermion coherent states as functions of para-Grassmann numbers is introduced. The procedure of the construction of finite-multiplicity approximation for determination of the path integral in the relevant phase space is defined through insertion in the kernel of the evolution operator with respect to para-supertime of resolutions of the identity. In the basis of parafermion coherent states a matrix element of the contribution linear in covariant derivativeD µ to the time-dependent Hamilton operatorĤ(τ), is calculated in an explicit form. For this purpose the matrix elements of the operators a 0 , a 2 0 , the commutators [a 0 , a ± n ], [a 2 0 , a ± n ], and the productÂ[a 0 , a ± n ] withÂ ≡ exp −i 2π 3 a 0 , were preliminary defined.

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Unitary Quantization and Para-Fermi Statistics of Order 2

Cited by 3 publications

References 46 publications

The Z2×Z2 -graded Lie superalgebras pso(2n+1|2n) and pso(∞|∞) , and parastatistics Fock spaces

The Z2×Z2 -graded Lie superalgebras pso(2n+1|2n) and pso(∞|∞) , and parastatistics Fock spaces

Generalization of Geyer’s commutation relations with respect to the orthogonal group in even dimensions

Path integral representation for inverse third order wave operator within the Duffin-Kemmer-Petiau formalism. Part I

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