Path integral representation for inverse third order wave operator within the Duffin-Kemmer-Petiau formalism. Part I
Abstract 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 conne...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2020-07-01
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| Series: | Journal of High Energy Physics |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1007/JHEP07(2020)094 |
| _version_ | 1819025766263816192 |
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| author | Yu. A. Markov M. A. Markova A. I. Bondarenko |
| author_facet | Yu. A. Markov M. A. Markova A. I. Bondarenko |
| author_sort | Yu. A. Markov |
| collection | DOAJ |
| description | Abstract 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 ± $$ {a}_k^{\pm } $$ and for an additional operator a 0 obeying para-Fermi statistics of order 2 based on the Lie algebra so $$ \mathfrak{so} $$ (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 derivative D ̂ μ $$ {\hat{D}}_{\mu } $$ to the time-dependent Hamilton operator ℋ ̂ τ $$ \hat{\mathrm{\mathscr{H}}}\left(\tau \right) $$ , is calculated in an explicit form. For this purpose the matrix elements of the operators a 0, a 0 2 $$ {a}_0^2 $$ , the commutators [a 0 , a n ± $$ {a}_n^{\pm } $$ ], [ a 0 2 , a n ± $$ {a}_0^2,{a}_n^{\pm } $$ ], and the product A ̂ $$ \hat{A} $$ [a 0 , a n ± $$ {a}_n^{\pm } $$ ] with A ̂ $$ \hat{A} $$ ≡ exp( − i 2 π 3 a 0 $$ -i\frac{2\pi }{3}{a}_0 $$ ), were preliminary defined. |
| first_indexed | 2024-12-21T05:15:54Z |
| format | Article |
| id | doaj.art-a01311ff8e8a4d5eae15ac5e42556f2b |
| institution | Directory Open Access Journal |
| issn | 1029-8479 |
| language | English |
| last_indexed | 2024-12-21T05:15:54Z |
| publishDate | 2020-07-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj.art-a01311ff8e8a4d5eae15ac5e42556f2b2022-12-21T19:14:55ZengSpringerOpenJournal of High Energy Physics1029-84792020-07-012020713410.1007/JHEP07(2020)094Path integral representation for inverse third order wave operator within the Duffin-Kemmer-Petiau formalism. Part IYu. A. Markov0M. A. Markova1A. I. Bondarenko2Matrosov Institute for System Dynamics and Control Theory SB RASMatrosov Institute for System Dynamics and Control Theory SB RASMatrosov Institute for System Dynamics and Control Theory SB RASAbstract 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 ± $$ {a}_k^{\pm } $$ and for an additional operator a 0 obeying para-Fermi statistics of order 2 based on the Lie algebra so $$ \mathfrak{so} $$ (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 derivative D ̂ μ $$ {\hat{D}}_{\mu } $$ to the time-dependent Hamilton operator ℋ ̂ τ $$ \hat{\mathrm{\mathscr{H}}}\left(\tau \right) $$ , is calculated in an explicit form. For this purpose the matrix elements of the operators a 0, a 0 2 $$ {a}_0^2 $$ , the commutators [a 0 , a n ± $$ {a}_n^{\pm } $$ ], [ a 0 2 , a n ± $$ {a}_0^2,{a}_n^{\pm } $$ ], and the product A ̂ $$ \hat{A} $$ [a 0 , a n ± $$ {a}_n^{\pm } $$ ] with A ̂ $$ \hat{A} $$ ≡ exp( − i 2 π 3 a 0 $$ -i\frac{2\pi }{3}{a}_0 $$ ), were preliminary defined.http://link.springer.com/article/10.1007/JHEP07(2020)094Effective Field TheoriesGauge Symmetry |
| spellingShingle | Yu. A. Markov M. A. Markova A. I. Bondarenko Path integral representation for inverse third order wave operator within the Duffin-Kemmer-Petiau formalism. Part I Journal of High Energy Physics Effective Field Theories Gauge Symmetry |
| title | Path integral representation for inverse third order wave operator within the Duffin-Kemmer-Petiau formalism. Part I |
| title_full | Path integral representation for inverse third order wave operator within the Duffin-Kemmer-Petiau formalism. Part I |
| title_fullStr | Path integral representation for inverse third order wave operator within the Duffin-Kemmer-Petiau formalism. Part I |
| title_full_unstemmed | Path integral representation for inverse third order wave operator within the Duffin-Kemmer-Petiau formalism. Part I |
| title_short | Path integral representation for inverse third order wave operator within the Duffin-Kemmer-Petiau formalism. Part I |
| title_sort | path integral representation for inverse third order wave operator within the duffin kemmer petiau formalism part i |
| topic | Effective Field Theories Gauge Symmetry |
| url | http://link.springer.com/article/10.1007/JHEP07(2020)094 |
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