Maxwell group and HS field theory

We consider the master fields for HS multiplets defined on 10-dimensional tensorial extension \tilde{\cal M} of D=4 space-time described as a coset \tilde{\cal M}={\cal M}/Sl(2;C) of 16-parameter Maxwell group {\cal M}. The tensorial coordinates provide a geometrization of the coupling to constant uniform EM fields. We describe the spinorial model in extended space-time \tilde{\cal M} and by its first quantization we obtain new infinite HS-Maxwell multiplets with their massless components coupled to each other through constant EM background. We conclude our report by observing that three-dimensional spinorial model with a pair of spinors should provide after quantization D=3 massive HS-Maxwell multiplets.


Introduction
In order to introduce the field-theoretic description of infinite number of relativistic quantum fields with all spins it is convenient to consider the enlargement of space-time. There were proposed various extensions of Minkowski space-time with vectorial [1,2] and tensorial [3,4,5,6,7] auxiliary coordinates, with master fields describing the infinite-dimensional spin multiplets by Taylor expansion in additional variables. In such a way one connects the master HS fields with enlarged D = 4 Poincare algebra, with new generators describing the shifts in auxiliary variables.
In this talk we shall follow old derivation of HS field multiplets by quantization of spinorial superparticle model, invariant under SUSY with six tensorial central charges 1 [4,5,7]. Analogous tensorial charges occur e.g. in D = 11 in M-algebra [8] which as it is postulated describes the algebraic structure of eleven-dimensional M-theory. The master fields in D = 4 are described in such approach by fields on extended space-time (x µ , z µν ), where z µν = z [µν] are six auxiliary translations generated by tensorial central charges. Further in order to describe integer and half-integer spins it is convenient to express the tensorial central charges as bilinears in D = 4 Weyl spinorial variables λ α ,λα = (λ α ), α = 1, 2 2 Z µν = λ α (σ µν ) α β λ β +λα(σ µν )αβλβ (1.1) in analogy to the Penrose formula for massless fourmomenta Finally we will arrive at master fields depending on spinorially extended space-time (x µ , λ α ,λα), with their Taylor expansions describing HS fields with arbitrary spin. The description of HS with tensorial coordinates has been further generalized by M. Vasiliev [6,9,10] who followed Fronsdal observation [3] that the infinite HS free multiplets can be described by single irrep of D = 4 generalized conformal algebra Sp (8), containing as its subalgebra the generalized Poincare algebra with tensorial central charges. Subsequently the extended space-time method was applied to the description in AdS space-time and there were derived the multiplets of free HS fields on AdS space [11,12]. At present only two free HS field multiplets -in Minkowski and AdS space-time -are explicitly described, and both were derived by the method of quantization of spinorial model of Shirafuji type 3 .
. 3 The formula (1.2) was firstly inserted in massless spinorial Shirafuji model [13] which was the first one describing the link between the model describing standard relativistic massless superparticle and the one describing free twistorial dynamics in D = 4 supertwistor space.
The formula (1.1) has been firstly employed in the superparticle model in [4].  [15,16,17]. Using Maurer-Cartan (MC) forms for corresponding Maxwell group one can introduce [7] the Maxwell-invariant spinorial particle model and perform its first quantization. 4 By analysis of the constraints in extended phase space we shall obtain in Schrödinger realization the master fields for Maxwell-HS free fields which are coupled to each other by constant uniform EM field. The plan of our talk is the following. In Sect. 2 we will recall the notion of Maxwell group, Maxwell algebra and present the corresponding MC one-forms. Using this geometric framework we present in Sect. 3 the spinorial model [7] and its first quantization with complete discussion of occurring first and second class constraints. We will use the conversion method [18] which interprets canonical pair of second class constraints as describing the system with gauge-fixed local gauge transformations (one constraint from the canonical pair is considered as first class constraint and generating the gauge transformations, second as introducing the gauge-fixing condition). In such a way we get gauge-equivalent formulation of our model with nine first class constraints. In Sect. 4 we consider in detail the wave function of such a model satisfying on enlarged space-time nine wave equations. Finally we express the solution of quantum-mechanical model in terms of local D = 4 HS fields, which will be named free Maxwell-HS fields. In such a way we obtain three infinite-dimensional coupled sets of Maxwell-HS fields: one describing all HS bosonic fields (s = 0, 1, 2, . . .) and two infinite set of chiral and antichiral fermionic HS fields with half-integer spins (s = 1/2, 3/2, 5/2, . . .). In Sect. 5 we will discuss the considered model in dual representation as the one describing more explicitly Minkowski HS fields interacting with constant EM field. Sect. 6 is devoted to the outlook. In particular we conjecture that suitable reduction D = 4 → D = 3 of considered in [7] D = 4 spinorial model can provide the infinite-dimensional D = 3 Maxwell HS massive multiplets.

Maxwell algebra and covariant Maurer-Cartan oneforms
The Maxwell algebra [15,16] is the semi-direct sum of Lorentz algebra with the generators and the ten-dimension sector with generators of the Poincare translations P αβ = (P βα ) + and six tensorial generators Z αβ = Z βα ,Zαβ = (Z αβ ) + . The last ten generators transform under Lorentz algebra as it is indicated in the algebraic relations
Differently than in the Poincare algebra with commuting translation operators, the commutators of the quantities P αβ yield new tensorial generators (see also (1.3)) [P αα , P ββ ] = 2i e ǫαβZ αβ + ǫ αβZαβ , where e is a constant interpreted further as describing EM coupling. Six tensorial charges Z αβ , Zαβ commute with each other and with the Poincare translation generators, i.e. the following commutators (see also (1.4)) are valid. The D = 4 Maxwell algebra define the Maxwell group M, in standard way by exponential representation. We define the D = 4 proper Maxwell group M 0 , determining Maxwell tensorial space, as ten-dimensional coset with generators P αβ , Z αβ ,Zαβ. The coset coordinates have the following transformations under the space-time translations (parameters a αβ ) and tensorial Maxwell shifts (parameters b αβ ,bαβ) δx αβ = a αβ , (2.6) The Lorentz transformations with the parameters ℓ αβ ,lαβ look as follows δx αβ = ℓ αγ xβ γ +lαγx β γ , δz αβ = 2ℓ αγ z β γ , δzαβ = 2lαβzβ γ . (2.7) Using the parametrization (2.5) and the algebraic relations (2.3), (2.4) we can define the Maurer-Cartan (MC) one-forms Explicit formulae for MC one-forms defined by (2.8) are (2.9) Corresponding covariant derivatives have the form (2.10) Since the space-time and tensorial translations are the shifts of the group parameters in Maxwell tensorial space, they do not change the MC one-forms (2.9) and the form of operators (2.10). We add that the Lorentz symmetry acts on MC one-forms and covariant derivatives in standard way, by the linear transformations.

Particle action, constraints and the Casimirs
We will consider the model of massless HS particle which propagates in the Maxwell tensorial space X = (x αα , z αβ ,zαβ) enlarged by the pair of spinorial variables. Our model is described by the following Maxwell-invariant particle action First term in (3.1) is the generalization of the D = 4 spinorial particle model defined on flat tensorial space-time in [4,5,6] (it is a tensorial generalization of Shirafuji model [13]). The components of the commuting Weyl spinor λ α ,λα = (λ α ) can be further considered as parts of the D = 4 twistor, and the model can be rewritten also as free D = 4 twistor particle model [4,5].
In the action (3.2) the parameter a is complex. Because e in (3.2) is dimensionless, we should choose the tensorial coordinates (z αβ ,zαβ) having mass dimensionality equal to −2, 1 2 , and one can deduce that the complex parameter a is mass-like, [a] = 1. This mass-like parameter can be chosen real, a =ā = m, if we take into account U(1) phase transformations of the spinors λ α = e iϕ λ α ,λα = e −iϕλα .
Inserting (3.8), (3.9) in (3.7) we obtain the following representation of the constraints (3.7)-(3.9) where are the classical counterparts of the Maxwell-covariant derivatives (2.10). We see that the constraints (3.10) are the Maxwell-covariant generalization of the constraints leading to so-called unfolded equations for HS fields [5,6].
We stress that present model has important difference in comparison with the one describing the HS particle of [5,6], because the vectorial constraints do not commute (do have nonvanishing PB). There are the following nonvanishing Poisson brackets of the vectorial constraints (3.7) Other tensorial constraints (3.8), (3.9) are the same as in [5,6] and commute with all the constraints (3.7)-(3.9). Thus, the tensorial constraints (3.8), (3.9) are first class whereas the vectorial constraints (3.7) are the superposition of two first class and two second class constraints.
To perform quantization of our model it is important to project out the first and second class constraints present in (3.7). If we wish to preserve the Lorentz covariance this separation requires the use of additional spinorial variables. In order to have a basis in two-dimensional spinor space, we introduce second spinor, u α , as firstly proposed in [5]. This auxiliary spinor satisfies the normalization condition The nonvanishing PBs of u α {u α , y β } P = u α u β (3.14) preserve the normalization (3.13). Using this spinorial basis (λ α , u α ) we can introduce in Lorentz-covariant way the following projections Their unique nonvanishing PB is the following and one can conclude that the constraints are first class, whereas are second class. Introduction of the Dirac brackets for the second class constraints leads to complicated structure of the quantum-mechanical algebra. As a way out we use the conversion method [18] in which the canonical pair of second class constraints can be considered as system where one second class constraint is interpreted as gauge-fixing condition for the gauge transformations which are generated by the other constraint. In the following we will consider the constraint T uū ≈ 0 as gauge fixing condition, and the constraint T λū + T uλ ≈ 0 as generating new gauge degree of freedom. Finally we will consider the classical gauge-equivalent system which is described by the constraints replacing the vectorial constraints (3.7). The constraints (3.19) are in fact linear combinations of the projections T αβλβ , λ β T βα of the constraints (3.7) on the Weyl spinor components λ α ,λα. So, we can avoid using the auxiliary spinor u α in definition of the constraints and as the result, we describe our model by the following set of first class constraints We observe that the four constraints (3.20), (3.21) are not independent, because they satisfy the relation i.e. we get only three independent first class constraints. It appears that the condition (3.24) does not enter into the derivation of the spectrum of our model. In the transition to this new system of the constraints, we should be careful not to loose any of the constraints. In particular, performing the projections (3.20), (3.21) we are omitting the contribution to the vectorial constraint (3.7) which does not depend on spinor variables. Such contribution is described by the following new constraint This quadratic constraint is of first class. Indeed, the constraint (3.25) can be represented by the formula T = T λλ T uū − T λū T uλ ≈ 0 and therefore due to (3.19) it is first class. Thus, we should add the constraint (3.25) to the first class constraints (3.20)-(3.23). After quantization this constraint will provide the Maxwell extension of the Klein-Gordon (KG) equation. One can find a physical interpretation of the system of the first class constraints (3.20)-(3.23), (3.25). For that purpose we will look for the values of the Casimir operators for the symmetry algebra in our model, i.e. the Casimirs of the the Maxwell algebra [16,19,20] C M ax 1 = P αβ P αβ + 4e M αβ Z αβ +MαβZαβ , (3.26) Using the transformations (2.6), (2.7) and the following transformations of spinors we obtain from the action (3.4) the Noether charges in our model

First quantization of the particle model and interacting HS fields
Phase space coordinates after quantization become the operators, and for simplicity we will denote them further by the same letter (without hats). The Poisson brackets algebra (3.5), (3.6) generates the following quantum-mechanical algebra Further we consider Schrödinger-type representation in which the operators x αβ , z αβ ,zαβ, y α ,ȳα are realized as multiplications by c-numbers whereas the operators of quantized momenta are represented as partial derivatives The physical spectrum of the wave function Φ = Φ(x αβ , z αβ ,zαβ, y α ,ȳα) (4.5) is defined by the quantum counterpart of first class constraints (3.20)- (3.22): where D αβ is the Maxwell-covariant derivative (see (3.11), (2.10)). The solutions of eqs. (4.8) are described by compact formula Φ(x, z,z, y,ȳ) = e −im(z αβ ∂α∂ β +zαβ∂α∂β ) Φ 0 (x, y,ȳ) . From expression (4.9) follows that the tensorial coordinates z µν = (z αβ ,zαβ) are the auxiliary gauge degrees of freedom and the gauge-independent degrees of freedom are present in the HS master field Φ 0 (x, y,ȳ). Residual equations (4.6), (4.7) for unconstrained field Φ 0 (x, y,ȳ) take the form Spinor variables y α ,ȳα are besides space-time coordinates, the additional spinorial variables. We consider the following Taylor expansion of HS master field with respect to the additional spinor variables where Maxwell-HS component fields φ Let us find now the solution of the equations (4.10), (4.11).
In the beginning we present two simple equations, which are a direct consequence of the equations (4.10), (4.11).
2) By considering the difference of the equations (4.10) multiplied by∂α and the equations (4.11) multiplied by ∂ α we obtain From (4.15) we get the following equations for the component fields The equations for lowest component fields described by Lorentz spins (2, 0) + (0, 2) are as follows They represent half of the Maxwell equations (self-dual part) for the free electromagnetic field strength.
Full set of the equations for component fields of the HS master field (4.12), generated by the conditions (4.6), (4.7), are They represent the Maxwell-invariant generalizations of well-known Dirac-Pauli-Fierz equations [21,22]. For completing the analysis of constraints it is necessary to impose on the wave function the scalar constraint (3.25). This additional condition leads to additional equation for first scalar component of the wave function (4.12). The constraint (3.25) implies the following equation (see details in [7]) for the HS master field (4.12). The equation (4.20) provides the following infinite set of field equations for the HS component fields for the scalar field φ (0,0) (x).
which have the form of the Dirac equations in a constant electromagnetic field, with electromagnetic potential A µ = f µν x ν . As generalization of the standard approach for Dirac spin-half field, the wave functions in (5.13), (5.14) depend also on continuous electromagnetic field strength coordinates f αβ ,fαβ. We do not see yet the relation of our description of HS fields interacting with constant EM field to the approaches proposed in recent papers on EM coupling of HS fields [23,24], however to find such a link would be desirable. Generalized spin-zero fieldΨ (0,0) (x, f,f ) is described by generalized Klein-Gordon equation, which follows from the constraintT ααT ααΨ ≈ 0 .
(5.15) Taking into account that we obtain the following generalized Klein-Gordon equation for "generalized spin zero" field It should be emphasized that due to the equations (5.14) and the constraints (5.12) the last term in the operator (5.16) does not contribute to the equation (5.17) and finally we obtain standard Klein-Gordon equation with coupling to constant EM field.
From the equations (5.13), (5.14) and (5.17) follows that the link of different spins is due to EM coupling proportional to electric charge e. Further one can show that if the torsion in six tensorial dimensions of Maxwell space-time depends only on the D = 4 space-time coordinates it can be interpreted as a coupling to D = 4 Abelian gauge potential. Let us introduce the following "block-diagonal" 10-bein E AB CD = δ γ α δδβ, E αβ γδ , E αβγδ in the tensorial space x αβ , z αβ ,zαβ and corresponding covariant derivatives where eA αβ (x) = E αβ γδ (x)f γδ + E αβγδ (x)f˙γδ. In Maxwell tensorial space additional tensorial coordinates are twisted by a constant torsion proportional to e, the functions E αβ γδ (x) and E αβγδ (x) are linear in x, and we obtain in (5.20) the Abelian gauge field four-potential A αβ describing constant electromagnetic field strength (fαβ = (f αβ ) † ) A αβ = f γ α x γβ +fγ β x αγ . where have polynomial dependence on z,z and the component HS fields φ

Outlook
In this paper we considered the spinorial particle model in ten-dimensional tensorial spacetime with torsion described by the tensorial coset space X = (x αα , z αβ ,zαβ) of D = 4 Maxwell group and additional spinorial variables λ α ,λα. We performed the canonical quantization of the model with supplemented kinetic term for λ α ,λα. By using the phase space formulation we specified the set of first and second class constraints. It appears that in first quantized theory the first class constraints will describe the set of field equations for new higher spin multiplets in the tensorial space X defining new HS Maxwell dynamics. Such equations describe the generalization of the known "unfolded equations" [5,6,9] for massless HS free fields with flat space-time derivatives ∂ αβ replaced by the Maxwell-covariant derivatives D αβ (see (2.10)). Note that the Maxwell-covariant description of D = 4 Maxwell-HS fields requires the presence of particular space-time-dependent coupling terms between different spin fields which can be also interpreted as following from the electromagnetic covariantization of space-time derivatives in the presence of constant EM background field strength.
As an interesting question which we plan to study is the derivation of massive HS fields from twistorial model of Shirafuji type, with two spinors which are necessary in D = 4 in order to define in spinorial framework the time-like four-momentum. The idea of describing massive particles by multispinors is well-known from the consideration of Penrose and his collaborators [25,26,27,28,29] and further was considered as well as in the supersymmetric case in [30,31,32,33,34,35,36]. In order to illustrate the derivation of massive spinorial model we shall argue following [30] that D = 3 massive Shirafuji model can be obtained by dimensional reduction of D = 4 spinorial model considered in [7].
We plan to consider in our next publication the massive extension of the model considered in [7]. The construction of massive Shirafuji model one can obtain for D = 3 (real Majorana spinors), D = 4 (complex Weyl spinors) or D = 6 (quaternionic Weyl spinors), by using respective complex and quaternionic generalization of the formulae (6.1)-(6.4) (see also [37,38]).