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

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 $a^{\pm}_{k}$ and an additional operator $a_{0}$ obeying para-Fermi statistics of order 2 based on the Lie algebra $\mathfrak{so}(2M+2)$ is established. On the strength of this connection 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. 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. A connection of the Geyer operator $a_{0}^{2}$ with the operator of so-called $G$-parity and with the $CPT$-operator $\hat{\boldsymbol{\eta}}_{5}$ of the DKP-theory is established. 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 $\hat{\cal H}(\tau)$, is calculated in an explicit form. For this purpose the matrix elements of the operators $a_0$, $a_0^2$, the commutators $[a_{0}, a^{\pm}_{n}]$, $[a^{2}_{0}, a^{\pm}_{n}]$, and the product $\hat{A}[a_{0}, a^{\pm}_{n}]$, where $\hat{A}\equiv\exp\bigl(-i\frac{2\pi}{3}\,a_{0}\bigr)$ were preliminary defined.


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 an additional operator a 0 obeying para-Fermi statistics of order 2 based on the Lie algebra so(2M + 2) is established. On the strength of this connection 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. 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. A connection of the Geyer operator a 2 0 with the operator of so-called G-parity and with the CP T -operatorη 5 of the DKP-theory is established. In the basis of parafermion coherent states a matrix element of the contribution linear in covariant derivativeD µ to the time-dependent Hamilton operator H(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 ], whereÂ ≡ exp −i 2π 3 a 0 were preliminary defined.

Introduction
The propagators (the Green's functions) for free quantized fields involved in the interaction processes and their generalization to the case of external classical fields in a system are important structural elements in the calculation of the Feynman diagrams in quantum field theory. However, in a number of problems it is convenient to have an alternative to the standard technique in quantum field theory. One of such alternatives is a possibility to present the Green's functions in the form of quantum-mechanical path integrals and thereby to reformulate quantum field theory in the language of world-lines of particles.
The representations in the form of path integrals were constructed for the scalar propagator [1], the electron propagator in an external Maxwell field [2][3][4][5][6][7][8][9] and for the quark propagator in an external Yang-Mills gauge field [10][11][12][13]. In constructing the desired representations the variety of approaches and methods were used. The case of propagators for particles with halfinteger spin (electrons and quarks) in external gauge fields and also their generalization to the case of supersymmetric theories [14][15][16][17] were studied in greater detail. We note that the representations of the Green's functions (and the one-loop effective actions closely connected with them) in the form of path integrals enable one to obtain by a more simple way some wellknown results of quantum field theory and in particular, quantum electrodynamics, for example, the Euler-Heisenberg Lagrangian for the case of strong constant or slowly varying field [18]. Moreover, this approach was successfully used in calculating the two-loop effective action that enables one to calculate a correction to the effective Euler-Heisenberg Lagrangian [19][20][21]. Finally, the exact calculation of functional integrals for special configurations of external fields gives an alternative possibility to study a problem of vacuum stability perturbed by external Maxwell's or Yang-Mills' fields.
Whereas in principle, one can construct the representation in the form of path integral for propagators of free fields with an arbitrary spin, such an attempt for fields with a spin, which is greater than 1/2 interacting with an external (Abelian or non-Abelian) gauge field encounters a problem of consistency [22][23][24][25]. In future, we focus on the propagator of a field with the spin 1, more exactly, on the propagator of a charged massive vector particle in the external Maxwell's field.
In this paper we would like to propose an approach to the construction of the representation for the Green's function of a vector particle in an external field in the form of path integral based on a well-known Duffin-Kemmer-Petiau (DKP) formalism [26][27][28] developed for describing relativistic scalar and vector particles. One of the most important advantages of this formalism is a possibility of using a well-developed technique for the case of the electron and quark propagators. In constructing such a representation for the vector particle we will follow mainly approaches suggested by Halpern, Jevicki and Senjanović [10], Borisov and Kulish [11], Fradkin and Shvartsman [29], Fradkin and Gitman [8] and van Holten [30]. We study in more detail a connection between para-Fermi quantization based on the Lie algebra of the orthogonal group SO(2M + 2) and the Duffin-Kemmer-Petiau theory with a deformation early suggested in [31], where as the deformation parameter a primitive cubic root of unity is used and the wave function of the particle with spin 1 obeys the third order wave equation. Note that an analysis of this connection is of particular mathematical interest without an application to a specific physical problem, since the connection represents nontrivial synthesis of various subjects such as algebra, the theory of classical Lie groups and theoretical aspects of (para)quantization of fields. In the present paper and in its second part [32] on the basis of this connection we will develop convenient mathematical technique which enables us within the framework of DKP-theory with the deformation to construct the representation of the Green's function for a massive charged vector particle in external electromagnetic field in the form of path integral in a certain parasuperspace.
There is a large number of papers devoted to various aspects of the DKP-formalism. Below we will mention just a few of them, which are concerned to the object of the given research.
As was mentioned above, the DKP-formalism deals with a field of 0 and 1 spins. The equation of motion represents the first order matrix-differential equation looking very similar to the Dirac equation. Analogue of Dirac's γ-matrices is so-called β-matrices obeying a more complicated algebra in contrast to the Dirac-Clifford algebra, namely the Duffin-Kemmer-Petiau algebra β µ β ν β λ + β λ β ν β µ = δ µν β λ + δ λν β µ . (1.1) Mathematical aspects of the DKP-algebra were studied in greater detail in the fundamental papers by Kemmer [33], Harish-Chandra [34], Fujiwara [35], Tokuoka and Tanaka [36,37], Chernikov [38], Fischbach et.al [39,40], Filippov, Isaev and Kurdikov [41], Isaev [42] etc. In particular, it was shown that classification of the representations of the DKP-algebra can be reduced to the classification of irreducible representations of the Lie algebra so(2M + 1) of the orthogonal group SO(2M + 1). This DKP-algebra for physically more important case M = 2 has 126 independent elements and admits the irreducible matrix representations of dimensions of 1 (trivial case), 5 and 10. Umezawa [43] has constructed the expressions for the projection operators on the sectors with spins 0 and 1. Finally, it was shown that the DKP-algebra admits supersymmetric generalization [44]. As was discussed above, the ten-dimensional representation of the DKP-algebra describes fields with spin 1. Relativistic particle theories with spin 1 was studied since that time, when Dirac has written out his famous equation for a particle with spin 1/2 [27,[45][46][47][48][49][50][51][52]. In particular, it was shown that the well-known Proca equation for a massive vector field can be rewritten in the matrix form of DKP-relativistic wave equation. The description of spin degree of freedom of a massive non-Abelian vector field based on DKP-approach can be found in the papers by Bogush and Zhirkov [53], Okubo and Tosa [54], and Gribov [55].
For the first time, the interaction with an external gauge (electromagnetic) field within the framework of the DKP-formalism was considered in the pioneering paper by Kemmer [27]. The interaction with the external field was introduced within the framework of the minimal coupling scheme that thereby actually provides gauge invariance of the DKP Lagrangian. Further, in a number of papers [56][57][58] a question of the interaction of a charged vector particle with electromagnetic field was analysed in more depth. In particular, it was explained that the main difference of the DKP-equation from the Dirac equation is that it involves redundant components. Some interaction terms in the Hamilton form of the DKP equation do not have a physical meaning and will not affect the calculation of physical observables. Furthermore, Nowakowski [56] pointed out that the DKP-equation of the second order obtained by Kemmer [27] by analogy with the second order Dirac equation has a rather limited physical applicability, since (1) it is only one of a class of second order equations which can be derived from the original DKP-equation in external electromagnetic field and (2) it has not 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. These results are true for an arbitrary representation of β-matrices (even not necessarily irreducible). All these principal issues arising in the problem of interacting DKP-field with an external Abelian one (and also with non-Abelian one) would have to take into account in solving the problem stated in the present paper.
Further, the Duffin-Kemmer-Petiau algebra closely related to an entirely different branch of theoretical physics, namely, the theory of parastatistics, more exactly, to the para-Fermi statistics of order p = 2. This nontrivial fact was noted for the first time in the papers by Volkov [59], Chernikov [38] and independently by Ryan and Sudarshan [60]. This connection provided an opportunity to present the DKP-algebra within the framework of an operator formalism (see section 3) in the form of parafermion algebra of order p = 2 and to realize a spin space of vector particle as a Fock space for a system of para-Fermi operators [61].
However, a preliminary analysis [62] 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) [63]. Geyer in the paper [64] has suggested to extend this isomorphism to the Lie algebra so(2M + 2). The extension is of great value for us, since in the corresponding algebra of para-Fermi operators an additional operator a 0 arises. This operator in the case of parastatistics of order 2 can be related to within a sign to the Schrödinger "pseudomatrix" ω [50] playing a key role in constructing the divisor for the first order DKP operator of a vector particle in an external gauge field [31]. This divisor enables us in particular, to write an operator expression for the inverse propagator of the vector particle in the form of the Fock-Schwinger proper parasupertime representation.
There are a few papers, where a question of the construction of path integral for a system of identical particles obeying parastatistics was considered (see, e.g. Polychronakos [65], Chaichian and Demichev [66], Greenberg and Mishra [67]). In this direction of researches the papers by Omote and Kamefuchi [68] and Ohnuki and Kamefuchi [69] are of particular interest for us. For a generalization of the notion of path integral to the case of parafermion variables in these papers the first step was to suggest an generalization of the well-known Grassmann algebra to the so-called para-Grassmann algebra [70]. This generalization is a direct analogue of generalization of the Fermi operators to the case of the para-Fermi operators in parastatistics. The authors have introduced the definition of the para-Grassmann algebra of arbitrary order p, the notions of integration and differentiation in this algebra, change of variables in integrals, Fourier transformation and so on. They also have defined the notions of coherent states for the para-Fermi operators and written out the formula of resolution of the identity (the completeness relation). These parafermion coherent states and resolution of the identity are of fundamental importance in a procedure of the construction of path integrals. The authors have constructed the path integral for the para-Fermi fields using para-Grassmann variables following the definition of the path integral as the limit of a product of time evolution operators for small time intervals. In formulating the theory the authors actively used the so-called Green ansatz [71]. Note that the papers [68,69] are a direct generalization of the paper by Ohnuku and Kashiwa [72], in which the construction of path integrals over Grassmann variables was presented, and are decisive in solving the problem stated in the given work. Essentially all the mathematical apparatus constructed by these authors will be actively used in the suggested research.
It should be also noted that there exists another direction of the description of massive and massless spinning particles within the framework of the so-called pseudoclassical mechanics using odd ("spinning") Grassmann or para-Grassmann variables in addition to usual even variables (coordinate and momentum). The results of these researches are also important for us, since the Lagrangians analysed there (and correspondingly, the classical actions) of free particles or particles in an external field, massive or massless ones possessing symmetries of various kinds, need to appear in one form or another in the exponential in the path integral representation of propagators of these particles in quantum field theory, thus forming a connection between relativistic mechanics of classical spinning particles and the Green's functions in quantum field theory.
In the paper by Gershun and Tkach [73] in particular it was shown that for the description of classical and quantum dynamics of a particle with spin 1 it is necessary to introduce two real Grassmann-valued vector variables ψ k µ , k = 1, 2 (instead of one variable as in the case of spin 1/2). Superspace formulation of the given approach with the so-called doubly supersymmetry can be found in [74][75][76]. Further, in the paper by Barducci and Lussanna [77] the pseudoclassical description of a massless particle with helicity ±1 in terms of complex conjugate pair of Grassmann 4-vectors ψ µ and ψ * µ was presented. With the use of canonical quantization, onephoton wave function in the Lorentz gauge was obtained and based on quantization within the framework of path integration non-covariant transverse propagator for a free field was derived. The authors have also considered the case of describing massive photon within the framework of pseudoclassical mechanics [78]. They have suggested a set of a first-class constraints, which after quantization reproduce the Proca equation for a massive vector field.
In two subsequent papers Gershun and Tkach [79,80] have analysed more closely a case of vector particles. It was cleared up that for a massless particle the descriptions by using a set of two Grassmann variables ψ k µ and with the help of one para-Grassmann variable ψ µ of order p = 2 (i.e. (ψ µ ) 3 = 0) are fully equivalent, whereas the description of a massive particle with the spin 1 is possible only with the para-Grassmann variables ψ µ and ψ 5 . The Lagrangian, which describes the motion of the free massive particle with spin 1 in terms of the para-Grassmann variables, has the following form: where Here, µ = 1, 2, 3, 4, the dot denotes differentiation with respect to τ , the fields e(τ ), λ(τ ) and V (τ ) are (one-dimensional) vierbein, gravitino and vector fields, respectively, and play the role of the Lagrange multipliers. The Lagrangian is invariant up to a total derivative under the coordinate transformation of the parameter τ , the infinitesimal supersymmetry transformations with an arbitrary Grassmann-valued function α = α(τ ) and local O(2) internal transformations. A set of the classical para-Grassmann variables (ψ µ , ψ 5 ) obeys trilinear relation ψ µ ψ ν ψ λ + ψ λ ψ ν ψ µ = 0, which after quantization passes into the operator relation of the algebra of para-Fermi fields 1 where now µ, ν, λ = 1, 2, 3, 4, 5. The pseudoclassical Lagrangian (1.2) has a direct relationship to our problem, and therefore is of greater interest for us. In the papers by Korchemsky [81,82], the Lagrangian (1.2) in the case, when B = 0 was used for the first quantization of a relativistic spinning particle. The author has shown that in the massless case, i.e. for L m = 0, after quantization the physical subspace of the parasupersymmetric particle whose spinning coordinates belong to the irreducible representations of the Duffin-Kemmer-Petiau algebra labelled by integer number is described by the strength tensors of antisymmetrical gauge fields and topological gauge fields.
Marnelius and Mårtensson [83], Lin and Ni [84], Rivelles and Sandoval [85] and Marnelius [86] have considered the BRST-quantization (within the framework of the Batalin-Fradkin-Vilkovisky procedure) of a model of relativistic spinning particle with N = 2 extended local supersymmetry on the worldline, which after quantization describes a particle with spin 1. Further, Gitman, Gonçalves and Tyutin [25] suggested a consistent procedure for canonical quantization of the pseudoclassical model of a spin 1 relativistic particle. They have shown that the quantum mechanics obtained after quantization for the massive case is equivalent to the Proca theory, and for the massless case, to the Maxwell theory. In this paper the case of the interaction with an electromagnetic field was also considered and it was shown that for an arbitrary external field the corresponding Lagrange equations become inconsistent. Only in the case of a constant external field (the authors in particular have considered an external constant magnetic field) one can obtain the consistent equations of motion.
A possibility of introducing the interaction with external electromagnetic field in the model with N = 2 extended supersymmetry on the worldline was also considered in the paper by P. Howe et al. [23]. The authors have shown also impossibility of the self-consistent description of interaction of the charged vector particle with the electromagnetic field. In addition, it could, however, be said that pseudoclassical models for a particle with spin 1 admit the interaction with an external gravitation field [24,87,88].
By this means within the framework of standard approaches such as the pseudoclassical mechanics, the usual Duffin-Kemmer-Petiau theory, an approach based on the Bargmann-Wigner equations and so on it is impossible in a consistent manner to introduce the interaction of the charged vector particle with external gauge fields. Our approach will allow one to get around this problem by the increasing complexity of the first order differential operator acting on a wave function of the vector particle.
The paper is organized as follows. In section 2, a brief review of our work [31] devoted to deriving the third order wave equation within the framework of Duffin-Kemmer-Petiau theory with a deformation, is presented. In section 3, for constructing the path integral representation we give all necessary formulae of operator formalism: the trilinear relations to which the operators of creation and annihilation of parafermions, the basis of parafermion coherent states in the spin space L obey, the normalization and completeness relations for the coherent states and so on. The generalized Hamilton operatorĤ =Ĥ(τ ) explicitly depending on the evolution 1 We have redefined the para-Grassmann numbers and operators from [80] as follows: parameter τ and containing linear, quadratic and cubic terms in the covariant derivativeD µ is taken into consideration. On the basis of the Hamiltonian the proper-time evolution oper-atorÛ (T, 0) used in constricting the scheme of finite multiplicity approximations is defined. In section 4, the form of the initial for further analysis matrix element of contribution to the generalized Hamilton operator linear in covariant derivative is written out. Section 5 is devoted to calculation of the matrix element for the Geyer operator a 2 0 , an analysis of its structure and derivation of its a more compact and visual representation. In this section we have defined the resolvent operator R of the a 2 0 on the basis of which an integral representation of the operator a 0 from the Lie algebra so(2M + 2) is written out. In section 6 we have shown that this integral representation of the operator a 0 incorrectly reproduces action of this operator on the state vectors of the Fock space. In the same section another representation for the operator a 0 in terms of the generators of the group SO(2M) correctly reproducing action on the state vectors is suggested. A connection of this operator with the pseudoclassical DKP-operatorω is obtained.
Section 7 is concerned with the calculation of the matrix element for the operator a 0 in the basis of parafermion coherent states. At the end of this section a proof of the operator relation a 3 0 = a 0 in terms of the matrix elements is given. In section 8, a connection between the Harish-Chandra operatorω 2 and the Geyer operator a 2 0 is analyzed. As a secondary result the connection between the pseudoscalar DKP-operatorω and the so-called CP T -operatorη 5 in the DKP theory is obtained. Section 9 is devoted to the calculation of the matrix element of the commutators [a 0 , a ± n ], [a 2 0 , a ± n ], which arise within the framework of finite-multiplicity approximation in constructing the required path integral representation of the Green function for a vector particle. Two different forms of representation for the matrix elements of the commutators [a 2 0 , a ± n ] are considered. In section 10 a similar calculation of the matrix elements of the productÂ[a 0 , a ± n ], whereÂ ≡ exp −i 2π 3 a 0 , is performed. More compact representations for these matrix elements are defined. On the basis of the obtained expressions for the matrix elements in this and previous sections a complete expression for matrix element (k) ′ p |[χ,L(z,D)]|(k − 1) x from section 4, is given. In section 11 a connection between operator a 2 0 and operator of so-called G-parity (the operator of parafermion parity (−1) n , where n is the parafermion number operator) is established. In the same section, a brief analysis of a connection between two approaches in constructing Lie algebra of the group SO(2M + 2), namely, an approach of Geyer [64] and an approach of Fukutome [89], is performed. In section 12 we prove the validity of the operator relation (−1) n a 0 = a 0 based on an analysis of its matrix element or in other words we show that the matrix element of the operator a 0 in a basis of parafermion coherent states is even function with respect to change of the sign of para-Grassmann variables ξ 1 and ξ 2 (orξ ′ 1 andξ ′ 2 ) entering into the definition of the coherent states. In the concluding section 13 the key points of our work are specified and inconsistency of two different representations of the operator a 0 is briefly discussed.
In Appendix A all of the necessary formulae of algebra of the matrices ω and β µ are listed. In Appendix B a brief review of the Geyere article [64] on the Lie algebra of the orthogonal group O(2M + 2) is given. Appendix C is devoted to the formulation of the definition of a para-Grassmann algebra in a spirit of Omote and Kamefuchi [68]. The trilinear relations between the para-Grassmann numbers ξ k of order 2 and the creation and annihilation para-Fermi operators a ± n of parastatistics of order p = 2 is also written out. All necessary formulae of differentiation with respect to para-Grassmann variables are given. In Appendix D a list of the commutation relations between the generators L kl , M kl and N kl of the group SO(2M) and between these generators and the operators a ± n are written out. Finally, in Appendix E we give a proof of turning into identity the commutation relations from Appendix B containing the operator a 0 , when the latter is written in terms of the generators L kl , M kl and N kl .
2 Third-order wave operator As already mentioned in Introduction in the paper by Nowakowski [56] devoted to the problem of electromagnetic coupling in the Duffin-Kemmer-Petiau theory, unusual circumstance relating to a second order DKP equation has been pointed out. It is connected with the fact that the second order Kemmer equation [27] lacks a back-transformation which would allow one to obtain solutions of the first order DKP equation from solutions of the second order equation, as is the case in Dirac's theory. The reason of the latter is that the Klein-Gordon-Fock divisor [90,91] in the spin-1 case 2 ceases to be commuted with the original DKP operator when we introduce the interaction with an external electromagnetic field within the framework of the minimal coupling scheme: Here, I is the unity matrix; ≡ ∂ µ ∂ µ , ∂ µ ≡ ∂/∂x µ , and the matrices β µ obey the trilinear relation (1.1). One of the negative consequences of this fact is impossibility to construct the Green function representation of (massive) vector particle in an external gauge field in the form of path integral in a certain (para)superspace remaining only within the framework of the original DKP-theory.
Nowakowski has suggested a way how this problem may be circumvented. To achieve the commutativity of the divisor d(D) and the DKP operator L(D) in the presence of an external electromagnetic field we have to give up the requirement that the product of these two operators is an operator of the Klein-Gordon-Fock type, i.e.
where G [A µ ] is a functional of the potential A µ , which vanishes in the absence of interaction. In other words it is necessary to introduce into consideration not the second order, but a higher order wave equation which would have the same virtue as the second order Dirac equation, i.e. a back-transformation to the solutions of the first order equation. In the paper [56] from heuristic considerations such a higher (third) order wave equation possessing a necessary property of the reversibility was proposed. However, by virtue of that the higher order equation does not reduce to the Klein-Gordon-Fock equation in the interaction free case, this leads to the delicate question of physical interpretation of the terms in such a higher order equation.
In our paper [31] this approach was analysed in more detail. We have suggested a scheme of systematic deriving the wave equation of third order and obtained the most general form of this equation in comparison with a similar equation in the paper by Nowakowski [56]. This scheme enables one in principle to obtain the wave equations of higher order in derivatives for a description of particles with a spin greater than 1 (the case of s = 3/2 was discussed in [92]).
We have established that the construction of the required divisor d(D), which would commute with the L(D)-operator, is closely related with a problem of constructing a cubic root of the third order (massive) wave operator in the interaction free case. By a direct calculation we have shown that by using only the algebra of Duffin-Kemmer-Petiau matrices, it is impossible to calculate the required cubic root and thereby eventually -the required divisor d(D). For solving this problem we had to introduce into consideration an additional algebraic object, the so-called q-commutator (q is a deformation parameter, representing a primitive cubic root of unity) and a new set of matrices η µ (z) instead of the original β µ -matrices of the DKP-algebra. These matrices depend in a general case on an arbitrary complex parameter z and Schrödinger's "pseudomatrix" ω and are not connected by any unitary transformation with the β µ -matrices. We have shown that based on new algebraic objects a procedure of constructing cubic root of the third order wave operator can be reduced to a few simple algebraic transformation and operation of the passage to the limit z → q. In other words, the third order wave operator (without interaction) is obtained as a formal limit of the cube of some first order differential operatorL(z, D) singular at z = q. The definitions of this operator, of the matrices η µ (z), and of the pseudomatrix ω will be given just below.
We have made corresponding generalization of the result obtained to the case of the presence of an external electromagnetic field in the system and performed a detail comparison with the result of Nowakowski. This gives us the possibility to have a new way of looking at the problem of constructing the propagator of a massive vector particle in an external gauge field in the form of path integral in parasuperspace within the framework of Duffin-Kemmer-Petiau theory with the deformation. As discussed above, the lack of commutativity of the Klein-Gordon-Fock divisor in the case of spin-1 particle with the original DKP-operator L(D) in the presence of a gauge field in the system leads to that we can not define the Fock-Schwinger proper-time representation for the inverse DKP-operator L −1 (D), i.e. already at the very first step of constructing the desired integral representation we are faced with the problem of a fundamental character, and we can overcome it only by redefining the original DKP-operator L(D) and corresponding divisor d(D).
This a rather drastic step has allowed us [31] to write almost immediately the Fock-Schwinger proper-time representation for the inverse operatorL −1 (z): is the Hamilton operator, D µ = ∂ µ + ieA µ (x) is the covariant derivative. The Greek letters µ, ν, . . . run from 1 to 2M unless otherwise stated, and χ is a para-Grassmann variable of order p = 2 (i.e. χ 3 = 0) with the rules of an integration [68]: In (2.2) we have introduced the function where q and q 2 are primitive cubic roots of unity As a proper para-supertime it is necessary to take a triple (T, χ, χ 2 ). Note that the representation (2.1) implicitly supposes the validity of the following relations: It is far less trivial to prove (2.7) and really it is a good test to check the self-consistency of the approach under consideration as a whole 3 . The operatorL(z, D) represents the cubic root of some third order wave operator in an external electromagnetic field. Matrix element of the inverse operatorL −1 (z, D) in the corresponding basis of states can be considered as a propagator of a massive vector particle in the background gauge field. Further, the matrices η µ (z) are defined by the matrices β µ obeying the Duffin-Kemmer-Petiau algebra (1.1) and by the complex deformation parameter z as follows: In fact an analysis of the relations of the type (2.7) even in the case of spin 1/2 in the presence of an external electromagnetic field is not quite simple and this delicate point for some reason is not discussed at all in literature (see, for example, [8]). Here, instead of (2.2) and (2.3) we havê (2.8) The reason of complication in the analysis of the first relation in (2.7) is that, for example, in the operator realization of the Dirac-Clifford algebra in terms of Grassmann variables and their derivatives the operatorŝ γ µ are Grassmann-odd (fermionic) operators while the realization ofγ 5 ≡ −(1/4!) ǫ µνλσγµγνγλγσ results in a Grassmann-even (bosonic) operator. Van Holten in the paper [30] was the first to point out this fact of mixing the terms with different Grassmann parity by a non-zero mass term in (2.8). It is precisely this circumstance that leads the first relation in (2.7) to require the Maxwell background field to satisfy equation of motion. In the case of a spin-1 particles the situation becomes more entangled. We will consider all these points in our subsequent papers, when mathematical technique required for this purpose will be developed. where (2.10) In Appendix A all of the necessary formulae of algebra of the matrices ω and β µ are listed. The matrix A in the expression (2.2) was determined by us [31] in the form of the expansion in powers of ω: where the coefficients are and I is the unit matrix. In the expansion (2.11) the property (A.1) was taken into account.
Here, in addition we would like to give once more representation of the matrix (2.11), which sometimes is more convenient in concrete calculations. It is easy to show by using the property (A.1) that the following formula where t is an arbitrary real number, holds. In particular, for t = 2π we have We are mainly interested in two important special cases: 1. in the case when t = 2π/3, we have 2. in the case when t = 4π/3, we have Thus the matrix A/α is a cubic root of the unit matrix (2.12).
At the end of all calculations, it should be necessary to proceed to the limit z → q and in particular, in this limit the operatorĤ(z), Eq. (2.3), defines the third-order wave operator in an external electromagnetic field An explicit form of this limit is given in the paper [31]. We note that the argument of the exponential in the Fock-Schwinger proper-time representation (2.1) is in a good agreement with the structure of the action for a relativistic classical spin-1 particle (1.2), defined in terms of para-Grassmann variables. However, a kinetic part of the action (1.2) was chosen in a complete analogy with the kinetic parts of classical and quantum models of Dirac's particle, whereas we expect based on a general formula of the representation (2.1) that the situation here can be more complicated since the operatorĤ(z) contains the third order derivatives with respect to x µ .
We adopt the Fock-Schwinger representation (2.1) for the inverse operator L −1 (z, D) with the deformation as an initial expression for constructing representation in the form of path integral with the use of corresponding system of coherent states in a close analogy with the paper by Borisov and Kulish [11] for the case of spin 1/2. One of the main goal of this study is the development of a convenient mathematical technique that would enable us to construct the desired path integral representation in a certain parasuperspace using the Duffin-Kemmer-Petiau approach. Here, we can effectively use a connection between the DKP-algebra of β µ -matrices and para-Fermi algebra of order p = 2 mentioned above. In particular, the connection gives us a possibility to employ a well-developed technique for the construction of a system of parafermion coherent states, resolution of the identity in a parasuperspace and so on, as it was defined in the paper by Omote and Kamefuchi [68]. However, in this case instead of the original β µ -matrices of DKP-algebra we have matrices η µ (z) explicitly depending on the deformation parameter z and pseudomatrix ω. Trilinear relation for the η µ (z) matrices coincides with the trilinear relation for β µ -matrices (1.1) only in the limit z → q and therefore here, we need to develop somewhat more subtle approach. In the paper [62] we attempt to construct such an approach within the framework of Govorkov's unitary quantization formalism [93], i.e. the quantization of fields based on the Lie algebra relations for the unitary group SU(2M + 1). Unfortunately, in spite of certain similarity between the DKP theory and the unitary quantization, there were a number of contradictions between two formalisms. All of these were discussed in detail in [62].
In this paper we would like to realize a further possibility of the construction of the required technique based on the parafermion quantization in accordance with the Lie algebra of the orthogonal group SO(2M + 2). Such a quantization in due time was considered by Geyer, [64]. Since this paper is a fundamental one in our consideration, in Appendix B we give all information from this work, which is necessary for the further consideration.

The operator formalism
The starting point of our study is the Fock-Schwinger proper-time representation (2.1). The problem of finding the Green's function D αβ (x ′ , x; z) of a massive vector particle in an external electromagnetic field reduces to the construction of an operator that is the inverse of is the highest rank of the irreducible representations of the DKP algebra with an even number 2M of the elements β µ . Hereinafter, we use the notation of quantities with hat above for those operators, which need to be distinguished from their matrix analogue. We restrict our consideration to the most important case M = 2 that corresponds to the four-dimension Euclidean space-time.
The operatorL −1 (z,D) acts on the space H of the representation of the algebras The space H is determined in the form of the tensor product of two spaces H and L, which realize representations of each algebra (3.1) and (3.2). The Green's function To construct the path integral, we require a basis of coherent states in the spin-1 space L. In L, the representation space of the Duffin-Kemmer-Petiau operator algebra (3.3) in accordance with (B.2) we introduce the parafermion creation and annihilation operators These operators by virtue of (3.3) obey the following algebra: and the space L can be realized as a finite Fock space for the para-Fermi operators (a ± 1 , a ± 2 ). As coherent states of the para-Fermi operators we take the coherent states as they were defined by Omote and Kamefuchi [68]. For papastatistics p = 2 they have the form (in the case when M = 2): are para-Grassmann numbers obeying algebra (C.2). For brevity sometimes we will write and moreover since we are interested in only the case parastatistics of order 2, then we will omit the symbol 2 in the notation of the parafermion coherent states, i.e.
The overlap function and completeness relation for the coherent states (3.8) are given by The transition from the matrix elements in the coherent basis to the representation in which the DKP matrices β µ have a specific form is realized as follows: The calculation of the explicit form of the transition functions α| ξ and ξ |β will be considered in Part III [94].
To present the propagator D αβ (x ′ , x; z) in the form of a path integral in parasuperspace of an exponential whose argument is the classical action for the massive vector particle, we use the operator formalism and the Fock-Schwinger proper time representation for the inverse operator L −1 (z), Eq. (2.1). We rewrite the matrix element of the inverse operatorL −1 (z) in the form Further, in accordance with Tobocman [95], we have to divide the interval [0, T ] into N parts, T = ∆τ N and to represent the exponential in matrix element (3.12) in the form of a product of N exponential multiplies Such a factorization of the exponential is well defined for the part linear in T . However, here we have also the term quadratic in T that is already qualitatively different from the standard consideration. Let us analyse this important point in more detail. We introduce a generalized Hamiltonian operator explicitly depending from "time" τ : In the paper by Mizrahi [96] the problem of path integral representation for a system in which a Hamiltonian explicitly depends on time, was considered. Here, we will follow the approach presented in this work. For the construction of the required representations it is necessary to ensure that the following condition holds: (3.14) By virtue of the definition (3.13) this requirement reduces to the first relation in (2.7). For further formalization of the task it is convenient to define an evolution operator aŝ The condition (3.14) assures a correctness of the following decomposition: or with regard to (3.13), we havê (3. 16) In the limit N → ∞, ∆τ → 0 it should be considered that i.e. an effective Lagrangian in the classical action for the massive vector particle will depend on the additional continuous parameter τ . Thus instead of the standard decomposition [95] e −iTĤ = e −i∆τĤ e −i∆τĤ . . . e −i∆τĤ N times , ∆τ N = T in our case we will use the decomposition (3.15) with (3.16) and insert resolutions of the identity in H ⊗ L between the evolution operators U(τ j , τ j−1 ). Following Borisov and Kulish [11] in the k-th position, we insert Since the evolution operatorŪ(τ j , τ j−1 ) contains the noncommuting operatorsp µ ,x µ , a ± n , for obtaining the explicit form of the matrix elements it is necessary to use an additional resolution of the identity: Thus the matrix element of evolution operatorÛ (T, 0) takes the form: pÛ (τ 1 ,τ 0 )|x, ξ and the following analysis in view of (3.17) reduces to the calculation of the matrix element (3. 19) We recall that in (3.18) the Dirac brackets designate In the given paper and in Part II [32] we restrict our consideration to an analysis of the matrix element of term linear with respect to the covariant derivative, i.e. of the term [χ,L(z, D)] in (3.18). The calculation of the matrix elements for more complicated contributionsĤ(z) and [χ,L(z, D)] 2 will be presented in Part III [94] after the development of all required mathematical technique.
At the end of this section we will write out in an expanded form the expression for the term η µ (z)D µ , which is included into the definition of the operatorL(z,D), Eq. (3.1). Taking into account (2.9), we getη In (3.20) we have turned to the creation and annihilation operators in accordance with (3.4).

Matrix element
In this section we give a detail form for the matrix element of term linear in the operator L(z,D) in the general expression (3.18). Since the variable χ is a para-Grassmann number, then by virtue of relation (C.5) and definition of the parafermion coherent states (3.8), it can be factored out from the Dirac brackets (k) ′ p | and | (k − 1) x : By using the representation (3.20), (3.21) for the first term in the last line, we have Matrix element in the mass term on the right-hand side of (4.1) by virtue of the expansion (2.11) has the following form: Thus in analysis of the expression (4.1) we face with the necessity of calculating matrix elements for the operatorsω,ω 2 ,Âa ± n andÂ[ω, a ± n ] in the basis of parafermion coherent states. We carry this out in several stages, the first of which is to define a connection between the operatorω, which within the framework of the DKP theory is given by expression (2.10) and the operator a 0 arising in the scheme of quantization based on the Lie algebra of the orthogonal group SO(2M + 2), Eq. (B.3).

Operator a 2 0
We begin our analysis of a connection between operatorω and a 0 with the construction of matrix element for the operator a 2 0 . In the paper by Geyer [64] the explicit form of this operator is given in Appendix B, Eq. (B.17). If one introduce the para-Fermi number operator (for parastatistics p = 2) then the expression (B.17) can be presented in the following form: Hereinafter, for convenience of further construction we redefine the operator a 0 : Let us determine action of the operator a 2 0 on the coherent state (3.8). For this purpose, we find a rule of action of the para-Fermi number operator n k on | ξ : Here, the last term vanishes by virtue of the definition of vacuum state. The commutator in the first term is easily calculated by using the operator identity and commutation relations (C.3) and (C.4). This commutator equals 1 2 [a + k , ξ k ] and thus we have Recall that there is no summation over repeated Latin indices. Similar calculation for n 2 k gives In view of the definition (5.2), it follows from (5.4) and (5.5) that and thus the required matrix element has the form (5.6) We will analyze the structure of this expression in more detail. For the sake of convenience of further reasoning we introduce the notations: These variables by the algebra of para-Grassmann numbers (C.1) and (C.2) satisfy the following relations: In terms of x and y, the matrix element (5.6) is written out in the form of a polynomial in x and y: Here, we state the problem of representation of the expression in the square brackets in the form of the exponential of some function U, which we present as U = U(x, y) = α(x + y) + β(x 2 + y 2 ) + γ xy + δ(x 2 y + y 2 x) + ρx 2 y 2 , where α, β, γ . . . are unknown coefficients. By virtue of algebra (5.8) we have 11) i.e. the power series exactly terminates with the fourth-order term. Let us substitute (5.10) into the right-hand side of (5.11), raise to the corresponding power with allowance for (5.8) and collect similar terms. Equating such obtained expression to the expression in the square brackets in (5.9), we get an algebraic system for the unknown coefficients: An unique solution of this system has the form and thus If we remember the expression for the overlap function then matrix element (5.9) goes into In spite of a more compact form in comparison with the initial expression (5.6), this formula is not convenient in concrete calculations by virtue of nonlinear character in x and y of the argument in the exponential. The form of the matrix element ξ ′ | a 2 0 | ξ can be further simplified if we note that We return to the operator a 2 0 and analyze some of its properties. By analogy with (5.7) we introduce the notationsx Instead of algebra (5.8) now we have the operator algebrâ In terms of (5.14) the operator a 2 0 takes the form Note that this operator is self-adjoint. Taking into account (5.15), it is not difficult to see that i.e. the operator has the property of a projector. Another projector orthogonal to P 1 has the obvious form It is worth pointing out that there exist one more structure orthogonal to P 1 , namely, which, however, doesn't possess the property of a projector. Now we consider the problem of defining an explicit form of the resolvent of the operator a 2 0 , i.e. of the operator (a 2 0 − λ) −1 . For this purpose, we analyze the following equation: where µ is unknown constant, and operatorÛ is defined by expression (5.10) with the replacements x →x, y →ŷ. Equation (5.17) with algebra (5.15) results in a simple system of algebraic equations for the unknown coefficients in (5.10) Hence, the resolvent of the operator a 2 0 has the form The resolvent is defined for all values of the parameter λ with the exception of two points: 0 and 1, i.e. the spectrum is σ(a 2 0 ) = {0, 1}. In particular, it immediately follows that the operator a 2 0 is irreversible. Further, we can define an arbitrary analytic function of a 2 0 using for this purpose the representation [97] ϕ(a 2 where the contour Γ a 2 0 surrounds the spectrum σ(a 2 0 ). We are interested in the special case of choosing the function ϕ ϕ(λ) = √ λ, i.e. formally we have the expression for the operator a 0 , which enters into the commutation relation (B.7) -(B.11) and in the condition (B.14) on the vacuum state vector |0 . By the spectral mapping theorem [97] the spectrum of this operator is σ(a 0 ) = [σ(a 2 0 )] 1/2 = {0, ±1}. Let us rewrite expression (5.18) in a somewhat different form Now we consider the question of action of the operator a 0 on state vectors of the system under consideration. At the end of the section we consider only action on the ground state. It follows from the expression (5.2) that 20) and the condition (B.14) for p = 2 yields (with the replacement a 0 → 2a 0 ) On the other hand from the representation (5.19) by virtue of (5.20) it follows that Hence, uncertainty in the sign in the condition (5.21) is connected with twovaluedness of the function λ 1/2 in the domain of 0 < | λ| < ∞. Indeed, let us consider the contour Γ a 2 0 consisting of the circle |λ| = 2 and the segments [−2, 0] and [0, −2], which lie on the upper and lower banks, respectively. The function √ λ splits in the domain into two regular branches, g 1 (λ) and g 2 (λ). This means that the integrand splits into two regular branches, f 1 (λ) = g 1 (λ)/(λ − 1) and f 2 (λ) = g 2 (λ)/(λ − 1). Let g 1 (λ) be the branch of the root on which g 1 (1) = 1, then g 2 (1) = −1. Each function f 1,2 (λ) is regular in the domain being considered except at point λ = 1, which is a simple pole. By the residue theorem we have 6 Operator a 0 For convenience of further reference we write out all independent state vectors spanned by the operators a ± k . In our case, when M = 2 according to Chernikov [38] the number of these state vectors equals C 2 5 = 10, and the number of para-Fermi particles in each state is n 0 = C 0 2 C 0 2 = 1, n 1 = C 1 2 C 0 2 = 2, n 2 = C 1 2 C 1 2 = 4, n 3 = C 2 2 C 1 2 = 2, n 4 = C 2 2 C 2 2 = 1.
From the other hand, if we act by the operator a 0 in representation (5.19) on the state vectors (6.1), then in view of (6.2) for the states with an odd number of paraparticles we will have: and for states with an even number of paraparticles we have If we fix the positive branch in the integral (5.22) for consistency with our choice of the sign in (6.3), and besides simply 5 set 1 2πi Γ a 2 0 dλ λ 1/2 = 0, (6.9) then we reproduce relations (6.3), (6.4), (6.7) and (6.8). However, differences in the signs for (6.5), and most importantly, complete disagreement with (6.6), take place. The possible reason for this lies in the fact that the expression for the operator a 2 0 , Eq. (5.2), suggested by Geyer [64] is not likely to be the square of the operator a 0 , i.e. in other words, and thus the representation (5.18) is not correct. This delicate matter will be discussed in detail in Part II.
In the remainder of this section we consider another approach to defining an explicit form of 5 The integral (6.9) is badly defined since one of points of the spectrum σ(a 2 0 ) is the branch point for the function λ 1/2 . It can be trivially estimated as follows. As the contour Γ a 2 0 we take the circle C R : |λ| = R. Further, we set λ = Re iϕ and therefore dλ = iRe iϕ dϕ, λ 1/2 = R 1/2 e iϕ/2+iπn , n = 0, 1. Purely formal calculation results in the expression which vanishes only in the limit R → 0.
the operator a 0 based on making use of the generators L kl , M kl and N kl of the group SO(2M) as they were defined in Appendix D, Eq. (D.1). In the special case M = 2 we have the following components of these generators different from zero: By using the definition (6.10) and the algebra (3.5) -(3.7) it is easy to check the validity of the following relations: and, correspondingly, If one accepts that the operator a 0 have the following structure: then a 0 |12 = −|21 , a 0 |21 = −|12 . (6.14) By doing so, we reproduce equality (6.6). However, the action of the operator (6.8) on vacuum state vector gives us that is in contradiction with (6.3). To determine the action of the operator (6.13) on the other state vectors, we need the rules of commutation of the group generators (6.10) with the operators a ± k . These rules follow from general relations (D.3) for M = 2: Based on these relations and rules of action on the vacuum state (6.15), it is not difficult to obtain for (6.13) a 0 |1 = a 0 |2 = 0, a 0 |112 = 0, Here, we also observe appearance of undesirable factor 1 2 on the right-hand side as it also takes place in (6.15).
Let us take, instead of (6.13), an operator We choose an operator function U(N 1 , N 2 ) ≡ U(n 1 − 1, n 2 − 1) so that the operator a 0 would reproduce correctly relations (6.15) and (6.17), while retaining (6.14). As a general expression for U one takes (5.10) with the replacements x → n 1 − 1 and y → n 2 − 1. We obtain the following system of linear algebraic equations for the unknown coefficients in (5.10): The solution of this system is and therefore, the desired operator function U has the form Thus the operator a 0 as a function of the generators (6.10), correctly reproducing the relations (6.3) -(6.8) is of the following final structure: In closing this section let us define a connection between the operator a 0 and the matrix ω, Eq. (2.10). In particular, for M = 2 within the framework of the operator formalism, we havê ω = 1 4 ǫ µνλσβµβνβλβσ or in an equivalent form:ω We rewrite the expression in the last line in terms of the creation and annihilation operators by using the connection (3.4) and the definition (6.10). It is easy to show that the following relations hold: The minus sign on the right-hand side is caused by the choice of the sign in (6.3). Now we can supplement algebra (3.5) -(3.7) for the para-Fermi operators a ± k of order p = 2, having included in it also the operator a 0 . This addition follows from the relations (A.1) -(A.5) in view of the relationship (6.21), Nevertheless, we note that with respect to the operator relation (6.22c) in Part II (section 10) as well as with respect to the matrix relation (A.3) we will have an important refinement.
7 Matrix element of the operator a 0 Given an explicit form of the operator a 0 , Eq. (6.18), we can define its matrix element in the basis of the para-Fermi coherent states. We need this matrix element, in particular for determining the matrix element of the operatorÂ, Eq. (4.3), where in accordance with (6.21) we should perform the replacementω → −a 0 . In section 4 we have identified action of the operator n k = N k + 1 on the parafermion coherent state, Eq. (5.4), therefore Let us consider action of the product N 2 N 1 on the coherent state By using the commutation rules (B.4) and (C.3) it is not difficult to show that the double commutator on the right-hand side vanishes, and consequently we have and matrix element for the anticommutator is Further we consider the operator expression {L 12 , M 12 }, which with the help of the second expression in (6.11), can be presented as Recalling the definition of the operators L 12 and M 12 , Eq. (6.10), taking into account the relation (7.1), it is easy to obtain the desired matrix element It remains for us only to define matrix element for the anticommutator {N 12 , N 21 }. Action of the generator N 21 on the coherent state is defined as From here we get Here, for computing the double commutator we use the Jacobi identity and commutation rules (B.4), (C.3): , and therefore By analogy, we have Making use of the expressions obtained we define the matrix element for {N 12 , N 21 }: With allowance made for the expressions (7.2), (7.3) and (7.5), the desired matrix element of the operator a 0 takes the following form: This matrix element along with the matrix element for the operator a 2 0 , Eq. (5.6), enables us to fully define the expression for matrix element of operatorÂ as it was defined by equation (4.3) (with the replacementω → −a 0 ). In particular, from here we have immediately a consequence. By virtue of the fact that the expressions (7.6) and (5.6) are defined only by the commutators of para-Grassmann numbers, the first relation in (C.1) leads to that the last term in (4.1) vanishes, i.e.
In closing this section let us consider circumstantial proof of the operator relation (6.22a) for our presentation (6.18). Matrix element of this relation can be presented as follows: Here, we have used the completeness relation (3.10). For the matrix element of the operator a 2 0 it is convenient to use the representation (5.13), then Substituting the last expression into (7.7) and taking into account that [68] where the δ-function for parastatistics p = 2 is we obtain, instead of (7.7), Thereby, in order that the preceding expression turns into identity, the following equality must be true By virtue of (7.6), the matrix element on the left-hand side has the following form: We see that the sign of terms linear in commutators and in the overlap function, has changed. It is not at all obvious that the equality (7.10) will take place. We shall defer the proof of (7.10) until section 11.
8 Connection between the operatorsω 2 and a 2 0 In section 6 we have defined a connection between the operatorsω and a 0 . Recall that the first of these operators arises naturally within the framework of the Duffin-Kemmer-Petiau formalism, whereas the second one enters into a generating set of the orthogonal group SO(2M + 2). In this section, we would like to analyse independently a connection between the operatorsω 2 and a 2 0 . According to conclusions of section 5 the operator a 2 0 introduced by Geyer [64] generally speaking, is not the square of the operator a 0 at least in the form given by expression (6.18).
In view of the general formula (67) from Harish-Chandra's paper [34] for M = 2 we will have the representation for the squared matrix ω 2 : where the indices µ 1 , µ 2 , µ 3 and µ 4 are all different and (P) denotes a sum over all permutations (1,2,3,4). Let us rewrite this expression in terms of Kemmer's matrices 6 [27,33] η µ = 2β 2 µ − 1 (8.1) possessing the properties then Based on (8.2), the sum on the right-hand side can be presented in the following form: where with the property On the other hand, for the square of the sum η µ i in (8.4), by virtue of the definition (8.1), we obtain Here we have used the definition of the matrix B in Appendix A, the first formula in (A.8), and property (A.1). Substituting (8.6) into (8.4) and further into (8.3), we obtain finally It must be especially noted that we have not seen anywhere in literature such simple relation between the matrices ω and η 5 . The most intriguing thing here is that two quantities entering into the relation have a rather different physical meaning. This difference has so clearly underlined in the paper by Krajcik and Nieto [103]. The matrix ω = 1 4 ǫ µνλσ β µ β ν β λ β σ plays a role of the "pseudoscalar operator" used in pseudoscalar coupling (in the Dirac theory analog of this matrix is (1/4!)ǫ µνλσ γ µ γ ν γ λ γ σ ) while the matrix η 5 , Eq. (8.5), plays a role of CP T operator in the DKP theory (in the Dirac theory its analogue is the matrix γ 5 = γ 1 γ 2 γ 3 γ 4 ). In the Dirac case the pseudoscalar and CP T operators are the same operator γ 5 by virtue of the purely algebraic peculiarities of the γ-matrices. However, in the DKP theory ω = η 5 , and the relation (8.7) shows us how these two different operators correlate among themselves. Note, moreover, that relation (8.7) correctly reproduces formula (A.3) by virtue of the property {η 5 , β µ } = 0. Now we turn to the consideration of the operator a 2 0 as it was defined by Geyer. Here, we give once again its explicit form where, we recall that N k = 1 2 [a + k , a − k ]. Let us rewrite the operator a 2 0 in terms of the operators η µ as they follows from the matrix definition (8.1). By virtue of the representation (3.4), we have ] and therefore, due to the DKP operation algebra (3.3) and the properties (8.2), we derive and similar we get Substituting the obtained expressions (8.9) and (8.10) into (8.8), we define a connection between the operators a 2 0 and η 5 : Comparing this expression with (8.7), we can conclude that However, as it will be shown further, the operator relation (8.12) is true only in a some limited sense, and within the framework of our problem it is not correct and requires a principle improvement that will be done in section 6 of Part II .
9 Matrix elements of the commutators [a 0 , a ± n ] and [a 2 0 , a ± n ] Let us return to matrix element (4.2). The first term in braces on the right-hand side has the form A similar term with the creation operator a + n has somewhat a more complicated structure, sincê and therefore ξ ′(k) |Âa + n | ξ (k−1) = (9.2) . Here, we have taken into account the representation of the operatorÂ in the form (2.11) with the replacement ω → −a 0 . Therefore, we are confronted by the task of deriving matrix elements of the commutators [a 0 , a + n ] and [a 2 0 , a + n ]. Let us consider the first of them. By virtue of the representation of the operator a 0 , Eq. (6.18), we have and commutation rules (6.16), it is not difficult to obtain a more simple form of the commutators on the right-hand side (9.3). We write them in two representations: the first of them is and the second one is In Appendix E we use the first representation in the proof of turning into identity the commutation relations including the operator a 0 , Eqs. (B.7) -(B.11). The second one is more convenient for deriving the required matrix element ξ ′ | [a 0 , a + n ]| ξ (and also ξ ′ |Â[a 0 , a + n ]| ξ , see the next section).
We need the matrix elements of the generators L 12 , M 12 , . . . , which can be easily obtained from their definitions (6.10): (9.7) Hereinafter, for the sake of simplification of the notations, we omit the iteration numbers (k) and (k − 1) ofξ ′ and ξ. Substituting (9.6) into (9.3) and taking into account (9.7), we obtain the desired matrix element The expression can be presented in a more compact and visual form. For this purpose we write matrix element of operator a 0 in the following form: (9.9) where in accordance with (7.6) we have (9.10) Let us take the derivative of the function Ω with respect to ξ n by making use of the rules of differentiation (C.9), (C.10) Comparing the last expression with (9.8), we obtain that Similar reasoning for the commutator [a 0 , a − n ] leads us to the representation of corresponding matrix element Now we turn to an analysis of the matrix element ξ ′ | [a 2 0 , a + n ]| ξ . By virtue of Geyer's representation (8.8) we have the starting expression By using the last two formulae in the commutation rules (6.16), we obtain [N 2 k , a + n ] = δ kn a + n + 2δ kn a + n N k .
Matrix element of this commutator equals Further, the commutator with the product N 2 1 N 2 2 in (9.15) has the form The last term here vanishes for M = 2. We need a matrix element of operator N 2 k . It can easily be obtained from the formulae (5.4) and (5.5) with regard to the definition (5.1) As a consequence of commutativity of the operators N 1 and N 2 we have and a similar expression for the product N 1 N 2 2 with the replacement 1 ⇄ 2. Substituting the obtained expressions into (9.15), we derive the explicit form of the desired matrix element This expression can be given in a more visual form if one takes into account the fact that The relation holds by virtue of algebra (5.8). Then by using the notations x and y introduced in section 5, Eq. (5.7), instead of (9.17) we will have We note also that the matrix element ξ ′ | a 2 0 | ξ defined by expression (5.6), can be given in a similar form ξ ′ | a 2 0 | ξ = (9.18) By straightforward calculation one can easily check the correctness of the following relations: Taking into account these relations and the form of the overlap function (5.12), we get, instead of (9.18), ξ ′ | a 2 0 | ξ = 1 2 1 + e −2(x+y) e x+y = cosh(x + y).
Thus we reproduce the simple formula (5.13) obtained in section 5 on the basis of completely different considerations. Let us return to the expression (9.17). We will present it in the form similar to the form (9.12) for the matrix element of commutator the [a 0 , a + n ]. For this purpose, we write out the matrix element of the operator a 2 0 in the form ξ ′ | a 2 0 | ξ = Ω ξ ′ | ξ , (9.19) where in accordance with (5.6), we have By a direct calculation, using the formulae of differentiation (C.9) and (C.10), it is easy to verify that the following relation is true. The same reasoning leads to Let us return to the matrix element (9.2). We write the matrix element of the operatorÂ in a form similar to the form of expressions (9.9) and (9.19): (9.23) The function A = A(ξ ′ , ξ) can be written out based on the expression (4.3) (with the replacementω → −a 0 ) and with allowance made for (9.9), (9.10) and (9.19), (9.20). This will be done in Part II, where we will consider in detail a question of a connection between the operator a 2 0 defined by the expression (8.8) and the square of the operator a 0 (i.e. (a 0 ) 2 ≡ a 0 · a 0 ) defined by the expression (6.18). For the remaining two terms in (9.2) we use the representations (9.12) and (9.21), correspondingly. As a result, instead of (9.2), we have Follow the same procedure, we can write out the matrix element for the product a − The last two expressions will be used in the following section. In the accepted notations the matrix element (9.1) is rewritten in the form Asymmetry of this expression with (9.25) is connected with the fact that operatorsÂ and a − n are not commutative.
10 Matrix elements of the productÂ[a 0 , a ± n ] Now we proceed to analysis of the remaining matrix elements in the initial expression (4.2), namely, to analysis of ξ ′(k) |Â[a 0 , a ± n ]| ξ (k−1) . As in the previous section, for brevity the indices (k) and (k − 1), namely the iteration numbers are omitted.
We need the following representations for the commutators [a 0 , a − n ] and [a 0 , a + n ]: , A proof of the second representation was given in section 9, Eqs. (9.3) and (9.6), the first one is proved in a similar way. Further we consider action of the commutator (10.1) on the parafermion coherent state Action of the generators N 1 , N 2 , N 12 and N 21 on the coherent state was defined by us in section 7, Eqs. (7.1) and (7.4). Let us write out the expressions obtained there for convenience of further references: We note that the following relation is true: since the operator a 0 consists of only the commutators of the operators a + k and a − k , and by virtue of (C.3) and (C.4), the following relationships hold: A trivial consequence of (10.5) is the relation We note once more that here in the last term we write exactly (a 0 ) 2 ≡ a 0 · a 0 to distinguish it from the symbol a 2 0 , which we keep for the notation of Geyer's operator (8.8). Taking into account the expressions (10.3), (10.4) and relation (10.6), we can present the matrix element of the productÂ[a 0 , a − n ] as follows: Thus we have been able to reduce the calculation of the initial matrix element ξ ′ |Â[a 0 , a − n ]| ξ to that of the matrix elements ξ ′ |Â| ξ and ξ ′ |Âa + n | ξ , which in turn are given by (9.23) and (9.24), correspondingly. Collecting similar terms and recalling the definition of the derivative ∂ Ω/∂ξ ′ n , Eq. (9.14), we can write the expression (10.8) in a more compact form In the second term on the right-hand side instead of variablesξ ′ n in the derivative (9.14) it is necessary to substitute ∂A/∂ξ n .
Finally, we consider the remaining term in (4.2) containing the productÂ[a 0 , a + n ]. We present the matrix element of this product similar to (9.2) in the following form: We perform analysis of the first term in the same way as it was just done for the matrix element ξ ′ |Â[a 0 , a − n ]| ξ . By using the representation (10.2), we obtain Further, instead of (10.4), we will need the expressions With these relations and (10.6) the first term on the right-hand side of (10.10) takes the form Here, we have introduced the notations The first term on the right-hand side of (10.14) with the function A = A(ξ ′(k) , ξ (k−1) ) has a quite reasonable form. On the structure it corresponds to the initial operator expressionÂη µ (z)D µ . The remaining terms are connected with the presence of additional commutators on the righthand sides of (9.2) and (10.10), which inevitably violate symmetry of the expressions with respect to the creation a + n and annihilation a − n operators. The consequence of this is appearing the terms in (10.14) of the type (∂ Ω/∂ξ ′ n )ξ ′ n = ∂A/∂ ξn , which cannot be easily interpreted. In Part II we will consider somewhat different formalism which enables us at least on the formal level to write the expression (10.14) in a more symmetric and visual form.
11 Another representation for the operator a 2 0 In this section we establish a connection between the operator a 2 0 and the parafermion number counter (−1) n , where n = n 1 + n 2 . (11.1) We begin our consideration with a reminder of how a similar connection arises for parastatistics p = 1, i.e. in the Dirac theory and then we extend it to the case p = 2. In usual Fermi statistics the operator N k = 1 2 [a + k , a − n ] satisfies the condition by virtue of {a + k , a − n } = 1 and (a ± k ) 2 = 0. Further we introduce the operator By straightforward calculation using the condition (11.2), it is not difficult to verify a validity of the relations and therefore we can write Here, at the last step we have taken into account the relation (B.16) from Appendix B. On the other hand, for p = 1 we have a 0 ≡γ 5 = i 2γ 1γ2γ3γ4 . If we introduce the fermion number operators for particles of the kind k then in terms of (11.1) we derive finally (−1) n = a 0 = (2n 1 − 1)(2n 2 − 1). (11.5) By doing so we reproduce the expression given in the paper by Dilkes, McKeon and Schubert [104] (see also D'Hoker and Gagné [105]). Now we turn to the case of parastatistics p = 2. Here, instead of the condition (11.2) we have As in the case of (11.4), we perform an analysis separately for even and odd powers of the sum N 1 + N 2 . By using (11.6) it is easy to obtain the explicit expressions for the first few even powers, which we write as follows: It is not difficult to define a general form of the coefficient of the last term on the right-hand side if one notes that for the power 4 we have for the power 6 we have 2 k=0 C 2k+1 etc. Here C k n are the binomial coefficients. By this means we have where n = 1, 2, 3, . . . . By using formula for the sum of the binomial coefficients from Prudnikov et al. [106], finally we define Now we consider odd powers. The first three nontrivial terms can be reduced to the following form: (11.9) The coefficient before the last term on the right-hand side for an arbitrary odd power 2n + 1 equals a sum of all binomial coefficients (with the exception of the first and the last those) divided by 2, i.e.
It is easy to check that this formula correctly reproduces the coefficients in (11.9) and thus we get We turn to the general expansion (11.3), which can be rewritten as follows: Let us substitute the above expressions (11.7), (11.8) and (11.10) into (11.11). Then for the sum of even powers we obtain and we get a similar expression for the sum of odd powers: Certainly, vanishing the contribution with odd powers is a consequence of evenness of the initial expression (−1) N 1 +N 2 with respect to the sum of operators N 1 + N 2 . Verifying this fact by a direct calculation serves to show of consistency of the calculation scheme. Thus from (11.11) it follows The final step is to pass on the left-hand side of (11.12) to the parafermion number operators n k = N k + 1 such that and for the right-hand side we recall the definition of the operator a 2 0 , Eq. (8.8). As a result, instead of (11.12), we obtain This expression is an immediate generalization of formula (11.5) to the case of parastatistics p = 2. It is interesting to note that the operator a 2 0 in the representation (11.13) coincides in its structure with the Gliozzi, Scherk and Olive operator [107] (the GSO projection), which projects onto states of even (para)fermion number.
A few consequences of the relation (11.13) can be obtained. Let us consider the matrix element of the operator a 2 0 , then due to (11.13) we have (11.14) Here, we have taken into account the equality (−1) n | ξ = |−ξ (11.15) and therefore the overlap function is By doing so, we have shown by the third way a correctness of the expression (5.13) (the second way was considered in section 9). Further, we consider the matrix element ξ ′ | [a 0 , a + k ]| ξ . Taking into account that we get Here, we have used the differentiation rules (C.9), (C.10) and the definition of the function Ω: 17) In this way we reproduce the result (9.21). We would like to draw some parallel between Geyer's [64] and Fukutome's [89] approaches. In the latter the problem of the construction of the algebra so(2M + 2) from the algebra so(2M + 1) was considered. For this purpose Fukutome has introduced the projectors with the properties P 2 ± = P ± , P + P − = 0. By virtue of the relations (11.13) and (5.16), in our notations these projectors have the form: The Lie algebra so(2M + 1) in [89] consists of the elements a + n , a − k , E k l , E kl , E kl , which correspond to the generators 7 For an extension of the algebra so(2M + 1) to the algebra so(2M + 2) Fukutome add new (11.18) By this means, he constructs the algebra so(2M + 2) simply adding " by hand" new generating elements to the algebra so(2M + 1), whereas Geyer [64] immediately considers the algebra so(2M + 2), in which there is already a new element β 2M +1 ≡ a 0 , Eq. (B.3), and this element in principle is not reduced to one of the elements (11.18). Therefore, in spite of some similarity of two approaches in the determination of the algebra so(2M + 2), one can state that they do not coincide literally. The requirement of consistency of the representation (11.13) and the property a 3 0 = a 0 lead us to the relations The commutativity of the operators (−1) n and a 0 is a simple consequence of the representation (6.18), property (11.16) and the operator identity 7 Note that in the construction of the Lie algebra so(2M + 1) Fukutome has used usual fermion creation and annihilation operators. In the case of para-Fermi statistics, we use the definition of the generator of algebra so(2M + 1) following the paper Bracken and Green [108].
A proof of the relation (11.19) is much more nontrivial. We consider it only in terms of the matrix elements.
In section 6 we written out the rules of action of the operator a 0 on the state vectors. The operator (−1) n changes a sign for the states with odd numbers of para-Fermi particles. However, as we see from the formulae (6.4) and (6.7), it is these states that turn to zero under the action of the operator a 0 . In this sense the operator a 0 and the product (−1) n a 0 are equivalent within the framework of the usual Fock space of the system under consideration. The situation changes qualitatively, when we use the para-Fermi coherent state | ξ in the form of (3.8). The fact is that this definition of the coherent state in principle does not admit an expansion in the number basis (a + i ) n (a + j ) m | 0 , i, j = 1, 2, n, m ≤ 2. (11.20) where ∆Ω(ξ ′ , ξ) = − 1 2 The notations x and y was introduced by us in section 5, Eq. (5.7). Then the expression (12.3) can be written as ∆Ω(ξ ′ , ξ) − (x + y − 1) e x+y = ∆Ω(ξ ′ , ξ) − (x + y − 1) e −(x+y) or collecting similar terms it takes the form ∆Ω(ξ ′ , ξ) + 1) tanh(x + y) = x + y. (12.5) In view of the algebra (5.8), further we obtain tanh(x + y) = (x + y) − 1 3 (x + y) 3 = (x + y) − (x 2 y + xy 2 ).
Taking into account the expansion and the explicit form of the function ∆Ω, Eq. (12.4), instead of (12.5) we obtain The terms linear in x and y were cancelled. In further analysis of the expression (12.6) for the para-Grassmann numbers we have to use, instead of the general relations (C.1), the particular relation (C.2) valid only for para-Grassmann numbers of order p = 2.
At first, we deal with the expression Let us consider the first contribution in the product on the left-hand side of (12.6) At the last step we have used relation (12.7). The remaining three contributions in a product in (12.6)  Substituting the obtained expressions into (12.6), we see that it turns into identity.

Conclusion
In this paper we have taken initial steps to develop a mathematical formalism needed to construct the path integral representation for the Green's function of a massive vector particle within the framework of the Duffin-Kemmer-Petiau theory with deformation. One of the key point in our approach is the use of the connection between the deformed DKP-algebra and an extended system of parafermion trilinear commutation relations for the creation and annihilation operators a ± k obeying para-Fermi statistics of order 2 and an additional operator a 0 . We have considered two representations of the operators a 0 . The first of them is an "indirect" representation based on employing the resolvent R of the Geyer operator a 2 0 . The second is an "explicit" representation constructed from the generators of the group SO(2M). It was shown that the former in contrast to the latter leads to incorrect formulae determining the rules of action of the operator a 0 on the state vectors of the corresponding finite Fock space. We have suggested that the reason of such an inconsistency is that Geyer's expression for the operator a 2 0 in terms of the parafermion number operators is most probably not the square of the initial operator a 0 . We recall that the latter appears as some additional abstract element of the algebra so(2M + 2). In our subsequent paper [32] we will take a second look at this rather nontrivial point.
As a secondary result we have obtained a simple elegant representation for the operator a 2 0 in terms of the parafermion parity operator (−1) n , where n is the parafermion number operator. This representation in particular enabled us to obtain the expression for some matrix elements in the basis of parafermion coherent states by a simple way in contrast to an approach based on the Geyer representation, Eq. (B.17). Besides we found an intriguing connection between the a 2 0 operator and the so-called CP T operatorη 5 , Eq. (8.11). We have calculated all necessary matrix elements, which will be used in analysis of the contributions of the second and third orders with respect to the covariant derivativeD µ in generalized Hamiltonian (3.13). Although these matrix elements are presented in the most compact and visual form, the final expression for the whole matrix element of the contribution linear in the covariant derivative, Eq. (10.14), ultimately proved to be cumbersome. One of the purposes of our next paper [32] is to give to the obtained expression a more symmetric and simple form.
In the paper [64] a general relation, which connects the operator a 0 with the operators N 1 , . . . , N M is also given (without a proof), where

Appendix C Para-Grassmann numbers
In this Appendix we will list the most important formulae of commutation and differentiation with para-Grassmann numbers. We follow the definition of a para-Grassmann algebra suggested by Omote and Kamefuchi [68]: a set of independent numbers ξ 1 , ξ 2 , . . . , ξ M are said to form a para-Grassmann algebra of order p when these numbers satisfy the following relations: a ± i a ∓ j ξ k + ξ k a ∓ j a ± i = 2δ ij ξ k , a ± i ξ k a ∓ j + a ∓ j ξ k a ± i = 0. (C.6) By direct calculations, one can verify a validity of these equalities using Green's decomposition [71] a ± i = For the commutation rules (C.4) and (C.5) we can write out similar relations for the particular case p = 2 a ± i a ± j ξ k + ξ k a ± j a ± i = 0, a ± i ξ k a ± j + a ± j ξ k a ± i = 0 (C.7) and ξ i ξ j a ± k + a ± k ξ j ξ i = 0, ξ i a ± k ξ j + ξ j a ± k ξ i = 0. (C.8) Let us present the formulae of differentiation with respect to a para-Grassmann number ξ. Throughout this text we mean left differentiation. The required formulae [41,69]  In particular, from the last formula for m = 1 and p = 2 it follows that ∂ ∂ ξ ξ = 2. (C.12) Substituting these expressions into (E.2) and collecting similar terms with respect to Kronecker