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 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 ] with ≡ exp −i 2π 3 a 0 , were preliminary defined.

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

“…The lack of a back-transformation, which would allow us to obtain solutions of the first order DKP equation from solutions of the second order equation as it is in the Dirac theory does not give a possibility within the framework of the DKP theory to construct the path integral representation for the Green's function of a spin-1 particle in a background gauge field in a spirit of the approaches developed for a spin-(1/2) particle. This problem is discussed in detail in [18,19] and ways to construct the path integral representation are proposed. Considering the DKP equation interacting minimally with an electromagnetic field [16], an additional term appears, called the anomalous term, because it has no equivalent in the spin-(1/2) Dirac theory.…”