Towards Lagrangian construction for infinite half-integer spin field

We formulate the conditions for the generalized fields in the space with additional commuting Weyl spinor coordinates which define the infinite half-integer spin representation of the four-dimensional Poincar\'e group. Using this formulation we develop the BRST approach and derive the Lagrangian for the half-integer infinite spin fields.


Introduction
Various aspects of massless infinite spin irreducible representations of the Poincaré group [1][2][3] attract much attention last time (see, e.g., [4]- [38]). Such representations contain an infinite number of states with all possible integer or half-integer helicities, in contrast to the usual massless representations describing the fields of fixed helicity. New approaches to the Lagrangian description of such fields have been recently developed in [17], [18], [19], [23], [33] combining the appropriate number of free massless fields with definite helicities. Also, we note the BRST approach [27] to Lagrangian construction for bosonic massless infinite spin fields. However, many of interesting points related to the Lagrangian formulation for massless infinite spin fields still deserve further study. In this paper, we consider the BRST approach to Lagrangian formulations of the fermionic massless infinite spin fields.
As known, it is convenient to realize a field description of the above representations in terms of space-time fields depending on the additional coordinates, so that the expansion in these coordinates gives an infinite number of the helicity states. Beginning with the pioneer papers [1][2][3], the space-time vector-like quantities are usually used as such additional coordinates (see review [21]).
However, there is another possibility for the field description of infinite spin particles, which uses the commuting Dirac or Majorana spinors as the additional coordinates. For the first time, this type of fields was considered in [4]. In our recent papers [26,27,31,37] (see also [32]) we constructed the new infinite (continuous) spin fields depending on spinor additional coordinates. It was shown that such fields are obtained as a result of a quantization of the special twistor particle models.
Following [26,27,31,37], the infinite integer spin representation is described by the field Ψ(x; ξ,ξ) , (1.1) which depends on the space-time coordinates x m and additional commuting Weyl spinor ξ α ,ξα = (ξ α ) * . The conditions that such field describes the infinite spin representation are written in the form 1 where P m = −i∂/∂x m and µ is a real dimensional parameter. Infinite integer-spin field (1.1) does not have any external vector or spinor indices (see the detailed analysis of helicity content of infinite integer-and half-integer-spin fields in Appendix B). Further, we will call the relations (1.2)-(1.5) the basic conditions. In works [26,31], in contrast to the field (1.1), it was proposed to describe the massless infinite half-integer spin representation by the field with an external spinor Dirac index A = 1, 2, 3, 4: Ψ A (x; ξ,ξ) .
(1. 6) In addition to the basic conditions (1.2)-(1.5) (written for the field Ψ A (x; ξ,ξ)), the field (1. 6) should satisfy also the additional condition in form of Dirac equation The Klein-Gordon equation (1.2) is evident consequence of the Dirac equation (1.7). In the representation γ m = 0 σ m σ m 0 (see (A.9)), the field (1.6) is represented in terms of the Weyl Υα(x; ξ,ξ Taking into account the basic conditions (1.2)-(1.5) and the condition (1.7), it is natural to assume that there exists the Lagrangian formulation where all the above conditions are the consequences of the Lagrangian equations of motion. Diverse approaches to the Lagrangian description of the infinite spin fields with vector additional coordinate were considered in works [9-12, 14, 16-21, 23, 25, 28, 30, 33]. One of the powerful general methods for studying the equations of motions and Lagrangian formulations in higher spin theories is the BRST construction which was applied to the massless infinite spin field theory in refs. [13,17,18,24,25,29,30].
In the recent paper [27] the BRST construction was used to derive the Lagrangian for the infinite integer spin fields (1.1) with additional spinorial coordinate. This approach is some generalization of the BRST construction which was used for finding the Lagrangians of the free fields of different types in flat and AdS spaces (see e.g. [42][43][44][45][46] and the references therein, see also the review [47]).
In the present paper we develop the generalization of the BRST method used in [27] to derive the Lagrangian for the infinite half-integer spin fields.
The paper is organized as follows. In Sect. 2, we describe the component structure of the space-time generalized fields (1.6), (1.8), depending on additional commuting spinor variables ξ α , ξα, and present the equations of motion for these component fields. In Sect. 3, we introduce the extended Fock space in terms of additional bosonic creation and annihilation operators and ghost operators. Then we construct the Hermitian BRST charge and the corresponding equation of motion which reproduce the conditions for the component fields. Taking into account this BRST charge, we derive the space-time Lagrangian for fermionic infinite spin field. The resulting Lagrangian contains both physical fields and auxiliary and gauge fields. In Sect. 4, we discuss the results and open issues. In Appendix A, we fix the spinor notations used in this paper. In These equations have been encoded in twistor formulation developed in [26,31,37]. In the Appendix C we demonstrate this statement. Further we derive the Lagrangian without considering these constraints, assuming that they can be somehow taken into account in the final result.
Appendix B, we describe in details the component decomposition of the infinite spin fields which are considered in the paper. Appendix C is devoted to construction of the solution to the equations (1.9) on the base of the twistor formalism and description of the irreducible representation of the infinite half-integer spin filed.
In this case the conditions (1.2)-(1.5) for the fieldΨ(x; ξ,ξ; ζ,ζ) are rewritten in the form whereas the massless Dirac equations (1.10) look like The equations (2.3) imply the equation In this representation for the fieldΨ(x; ξ,ξ; ζ,ζ), the Hermitian angular momentum operator M mn is written as follows M mn = σ αβ mn M αβ −σαβ mnMαβ , . (2.6) One can prove that the operator (2.5) satisfies the standard commutation relations for the Lorentz group generators. In spinor notation, the Pauli-Lubanski pseudovector W m = 1 2 ε mnkl M nk P l takes the form and the second Casimir operator W 2 is Using the expression (2.6) for the angular momentum operator, we obtain one of the possible forms for the second Casimir operator where the operator D in the last line is and (ξPξ) := ξ α P αβξβ , (ξPζ) := ξ α P αβζβ , etc.
Due to the first two equations in (2.2) and the equation (2.3), the Casimir operators of the Poincare group P 2 and W 2 , defined in (2.9), act on the field (2.1) as following Hence, the field (2.1) and therefore the fields (1.8) describe the infinite half-integer spin representation. The homogeneity operator (U(1)-charge), given by the last equation in (2.2), commutes with all Poincaré generators and is the superselection operator. Next, we solve the first equation in (2.2) (or the equations (1.3)) as After that, the second equation in (2.2) (or the equations (1.4)) and the equations (2.3) (or the equations (1.10)) for the fields Φ α ,Xα take the form We consider the solution of last equation in (2.2) (or the equations (1.5)) for the fields Φ α ,Xα in form of power expansion in ξ andξ: The equations (2.12), (2.13) lead to the following equations for the component fields: where we have used It is convenient to combine the fields ϕ γ α(s)β(s) andχ˙γ α(s)β(s) in a single four component object ϕ C a(s)ḃ(s) of the form and define the Dirac adjoint (with respect indices C = ( γ ,˙γ) and C = ( γ ,˙γ)) field as Hereafter, we will often omit the index C in four component field ϕ C α(s)β(s) (2.20).
In the next section we will derive the Lagrangian formulation for the fields under consideration using the BRST approach in terms of the four component field (2.20).

Generalized Fock space
In the previous section we have used the spinor variables ξ α ,ξα and the corresponding momenta given by the derivatives with respect to ξ,ξ. For this reason, we introduce the operators Hermitian conjugation yield the operators with the commutation relation Below, similar to (2.17), we use the notations Following (3.2) and (3.4) we consider the operators a α andbα as annihilation operators and define the "vacuum" state |0 , Let us define the auxiliary Fock space with the vectors of the form Then the conjugate vector to (3.8) is written as follows Let us introduce the following (4 × 4) matrix operators In what follows we will often omit the four-component indices C, D in the operators (3.10)-(3.13) also. The nonzero (anti)commutators of the above matrix operators are where {L, T } ≡ L · T + T · L, and All other (anti)commutators among the operators (3.10)-(3.13) vanish. One can show that the vector |ϕ C (3.8) reproduces the fermionic infinite spin equations (2.18), on the vector |ϕ C are imposed. Further, by using BRST procedure, we will construct the Lagrangian, which reproduces the conditions (3.17) as the equations of motion.

BRST charge
Let us consider the operators F a = (T 0 , L 0 , L 1 , L + 1 ), defined in (3.10)-(3.13), as operators of the constraints of some yet unknown Lagrangian theory. Since these operators form closed (super)algebra [F a , F b } = f ab c F c (3.15) we can build BRST charge in a standard way as where c a and P a are the ghosts and their momenta and n a = 0 or 1 is the parity of the operator F a . The next step is to construct such a vector that contains the physical fields under the equations (3.17). The constraint on this vector, stipulated by the operator L + 1 , is not imposed. The BRST procedure for such systems was studied in papers [40] - [48] and we apply it to the infinite spin system under consideration.
Thus, using the operators F a = (T 0 , L 0 , L 1 , L + 1 ) and the corresponding ghosts c a = (q 0 , η 0 , η 1 , η + 1 ) we construct Hermitian BRST charge Q = Q † in the form which is nilpotent by definition The BRST-charge acts in the extended Fock space, where the action of fermionic η 0 , η 1 , η + 1 and bosonic q 0 ghost "coordinates", as well as the corresponding ghost "momenta" P 0 , P + 1 , P 1 and q 0 , are defined earlier. These ghost operators obey the (anti)commutation relations and act on the "vacuum" vector as follows They possess the standard ghost numbers, gh("coordinates") = −gh("momenta") = 1, providing the property gh(Q) = 1. The operator (3.19) acts in the extended Fock space of the vectors The equation of motion of this BRST-field is postulated in the form Due to the nilpotency of the BRST charge the field (3.23) is defined up to the gauge transformations where the gauge parameter |Λ D has (since gh(Q) = 1 and gh(P + 1 ) = −1) the form The fields |ϕ C , |ϕ 1C , |ϕ 2C , |ϕ 3C and the gauge parameter |λ C in (3.23) and (3.26) have the decompositions similar with |ϕ C in (3.8).
We emphasize that we take "the momentum representation" with respect to the canonical pair of ghost variables (η 1 , P + 1 ), in contrast to "the coordinate representation" for other canonical pairs of ghosts. This prescription leads to the possibility to consider the corresponding constraint L + 1 by using gauge symmetry, as it is given in Appendix B. Description of such a treatment to use the constraints in the BRST approach was given in [24,29].
The equation of motion Q|Φ = 0 (3.24) can be rewritten in term of the vectors |ϕ , |ϕ i , i = 1, 2, 3 in the form (we omit here the Dirac indices C, ... in all quantities)

Construction of the Lagrangian
It is easy to see that the equations (3.35)-(3.37) are Lagrangian equations for the following Lagrangian We use this Rarita-Schwinger-like fields in the expansions of all "physical" fields ϕ and gauge field λ. By construction all "physical" and gauge fields are totally symmetric traceless tensor Dirac spinors η m 1 m 2 ϕ A m(s) = 0 . (3.42) One can check that As a result the BRST Lagrangian (3.39) yields the following component Lagrangian  Here we taken into account that the equations of motion are constructed for the traceless fields. We can remove the fields ϕ 3 m(s) using their gauge transformations and after that we can make the gauge transformations using restricted gauge parameters subjected to the conditions ∂ λ m(s) = 0.
Note that the equations (3.51) on fields ϕ 2 m(s) take the same form as the equations on the gauge parameters ∂ ϕ 2 m(s) = 0. Therefore we have enough gauge freedom to remove fields ϕ 2 m(s) . Thus, after removing the fields ϕ 2 m(s) and ϕ 3 m(s) , the equations of motion (3.50), (3.52) take the form ∂ ϕ m(s) = 0 , −2i(s + 1)∂ n ϕ nm(s) − µϕ m(s) = 0 (3.53) and coincide with (2.18) and (2.19). As a result, we have shown that the Lagrangian (3.46) describes the fermionic infinite spin field. We emphasize that this Lagrangian (3.46) has consistently derived in the framework of the general BRST construction. In fact, it is a direct consequence of the basic conditions (1.2)-(1.5). The only assumption we made was a homogeneity condition (1.5) (see also (2.14)) for the fields Φ α (x, ξ,ξ) andXα(x, ξ,ξ) in (2.11). We see that the Lagrangian (3.46) depends on three sets of traceless Dirac fields ϕ m(s) , ϕ 2 m(s) , ϕ 3 m(s) (3.42) and each traceless field can be decomposed into two γ-traceless fields thus Lagrangian (3.46) depends on six sets of Dirac γ-traceless fields. We emphasize that just such a Lagrangian corresponds to the fields satisfying the basic conditions (1.2)-(1.5).
Recently the Lagrangian for fermionic infinite spin field has been proposed in [18], [23] by combining the free massless fermionic fields with definite helicities and assuming the special gauge symmetry. This Lagrangian depends on one set of Dirac triple γ-traceless fields and each field also can be decomposed into three Dirac γ-traceless fields. Thus, one can say that the set of the fields of our Lagrangian (3.46) (as well as gauge parameters and, respectively, degrees of freedom) is twice as large as that of Lagrangian proposed in [18], [23]. Nevertheless we emphasize once more that Lagrangian (3.46) was consistently derived only on the base of the basic conditions (1.2)-(1.5) including the homogeneity condition. At present it is not clear how our Lagrangian (3.46) relates to the Lagrangian obtained in the works [18], [23].

Summary and outlook
We have constructed the Lagrangian for the infinite half-integer spin fields. This construction is characterized by the following: • Irreducible infinite half-integer spin representation is described by the fields (2.11), which depend on additional even spinor variables. The fields (2.11) contain the fields (2.15), (2.16) that satisfy the conditions (2.12), (2.13) and have the power expansion.
• The second Casimir operator (2.9), which acts in the space of infinite spin fields with additional spinor variables is derived.
• Without the presence of the δ-function in (2.11), the fields ( (3.19). After the elimination of some auxiliary states, we stay with the physical and gauge states, that are described by the equations of motion Let us note some comments on the constructed Lagrangian.
i) It is interesting to generalize the BRST approach for obtaining the Lagrangian for the irreducible representation of infinite spin field.
ii) It is interesting to generalize the BRST approach for obtaining field Lagrangian to supersymmetric infinite spin field theory.
iii) It would also be interesting to obtain the Lagrangian of such type for the infinite spin fields in the AdS space.
In this Appendix we present the notations used in this paper. The space-time metric is η mn = diag(−1, +1, +1, +1). The totally antisymmetric tensor ε mnkl has the component ε 0123 = −1. The two-component Weyl spinor indices are raised and lowered by ǫ αβ , ǫ αβ , ǫαβ, ǫαβ with the non-vanishing components ǫ 12 = −ǫ 21 = ǫ 21 = −ǫ 12 = 1: where σ 1 , σ 2 , σ 3 are the Pauli matrices. The matrices satisfy the relations The link between the Minkowski four-vector A m and bi-spinor A αβ is given by The σ-matrices with two vector indices are defined by They satisfy the identities Using the σ-matrices (A.5), we represent the antisymmetric second rank vector tensor in the form where the inverse expressions for the symmetric second rank spinor tensors are In the Weyl representation, the Dirac matrices (γ m ) A B , A, B = 1, 2, 3, 4 have the form We use the following notations:

Four-component Dirac spinor Ψ A is represented by two Weyl spinors
Dirac conjugate spinorΨ = Ψ † γ 0 has the components In case of the Majorana spinor, the equality χ α = ψ α holds.

Appendix B: Free infinite spin fields with additional spinor coordinates in the space-time description
To analyze the field contents let us consider the light-cone reference system, where p + := p 0 + p 3 = 2E, p − := p 0 − p 3 = 0, p 1 = p 2 = 0 and the four-momentum has the form In this system the helicity operator takes the form Taking into account the relations (see (2.5), (2.6)) we obtain the following expression for helicity operator in the light-cone system .
Note, to obtain this expression for helicity operator we do not use the equations of motion.

B.1.1 Field without δ-function
Let us consider the generalized field in momentum representation which satisfy the equations of motion in the system (B.1) take the form As we see, the independent fields (for below fields we point out their helicities, calculated by formula (B.4)) are We indicate the structure of this set of states: on s-th step in the expansion, the physical states have the helicities from 0 to ±s, on (s + 1)-th step they have the helicities from 0 to ±(s + 1) etc.
That is, at each step there arise the states with the same helicities as in the previous step, and the additional states with helicities which are one more modulo larger. So, in the spectrum, there are all helicities, and there is an infinite number of states with an arbitrary fixed helicity. That is, this representation of the infinite spin is not irreducible, it is infinitely degenerate. We get the same result using a slightly different procedure.
In the light-cone system (B.1) second equation in (B.6) has the general polynomial solution where the multipliers are modified Bessel functions [49] 12) and the polynomial functions with respect to the variable ξ 2 : In the case of a general field (B.5) with zero degree of homogeneity, the fields F (k) (E; ξ 1 ,ξ˙1), −∞ < k < ∞ in the expansion (B.11) have the following degrees of homogeneity (U(1)-charges): As result, the solutions of the equations (B.14) have the form In these expansions, the infinite number of the functions f The presence of δ-function in (B.16) implies that the equality ξpξ = µ is fulfilled in this expression. We pass on as before to the light-cone system (B.1). Then this equality takes the form Consequently, the terms are absorbed by the field ϕ 0 after its redefinition, the terms are absorbed by the term ξ 1ξ2 ϕ 12 , the terms

B.1.3 Gauged field
So, let us consider the field  We emphasize that the condition (B.40) can be taken into account due to local symmetry, as in the case of integer helicities considered in the Subsection B.1.3.
Note that the equation (C.7) zeroes the field (B.35) in the expansion (B.33). It should also be emphasized that in this paper we consider the fields which are the power series in the additional spinor variable ξ, while in the paper [31] there was considered different class of space-time fields (see Appendix B in [31]).