Path integral representation for inverse third order wave operator within the Duffin-Kemmer-Petiau formalism. 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 connection between the deformed DKP-algebra and an extended system of the parafermion trilinear commutation relations for the creation and annihilation operators $a_k^{\pm }$ and for an additional operator $a_0$ obeying para-Fermi statistics of order 2 based on the Lie algebra $\mathfrak{so}(2M+2)$ 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 ${\hat{D}}_{\mu }$ to the time-dependent Hamilton operator $\hat{\mathcal{H}}(\tau )$, is calculated in an explicit form. For this purpose the matrix elements of the operators $a_0^{}$, $a_0^2$, the commutators $[a_0^{},a_n^{\pm }]$, $[a_0^2,a_n^{\pm }]$, and the product $\hat{A}[a_0^{},a_n^{\pm }]$ with $\hat{A}\equiv \exp (-i\frac{2\pi }{3}{a}_0)$, were preliminary defined.

arXiv comment33 pages, the original text is considerably reduced, version published in JHEP

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Cited by in corpus (1)

• On the electromagnetic interaction and the anomalous term in the Duffin-Kemmer-Petiau theory