Twistorial and space-time descriptions of massless infinite spin (super)particles and fields

We develop a new twistorial field formulation of a massless infinite spin particle. Unlike our previous approach arXiv:1805.09706, the quantization of such a world-line infinite spin particle model is carried without any gauge fixing. As a result, we construct a twistorial infinite spin field and derive its helicity decomposition. Using the field twistor transform, we construct the space-time infinite (continuous) spin field, which depends on the coordinate four-vector and additional commuting Weyl spinor. The equations of motion for infinite spin fields in the cases of integer and half-integer helicities are derived. We show that the infinite integer-spin field and infinite half-integer-spin field form the $\mathcal{N}{=}\,1$ infinite spin supermultiplet. The corresponding supersymmetry transformations are formulated and their on-shell algebra is derived. As a result, we find the field realization of the infinite spin $\mathcal{N}{=}\,1$ supersymmetry.


Introduction
Recently, there has been a surge of interest in the description of particles and fields related to infinite (continuous) spin representations of the Poincaré group [1,2,3]. Although the physical status of such unitary representations is still not very clear, interest in them is caused by an identical spectrum of states of the infinite spin theory [4] and the higher-spin theory [5,6,7] (see also the reviews [8,9,10]) and by its potential relation to the string theory (see [11] and recent paper [12] and references in it). Various problems related to the quantum-mechanical and field descriptions of such states were considered in a wide range of works devoted to particles and fields of an infinite spin (see, e.g., [13] - [34]).
In this paper, we continue to develop an approach to the description of infinite spin particles and fields, initiated in our previous paper [30], where we constructed a new model of an infinite (continuous) spin particle, which is a generalization of the twistor formulation of standard (with fixed helicity) massless particle [35,36,37] to massless infinite spin representations. Making the quantization of the twistor model with gauge fixing and using some type of twistor transform, we reproduced these space-time infinite spin fields, which depend on the position four-vector and obey the Wigner-Bargmann equations [1,2,3].
In paper [30], when quantizing the constructed twistor model, we imposed partial gauge fixings for gauge symmetries generated by some first class constraints. This yielded after quantization "limited" fields describing the massless representations of the infinite spin. Now we will carry out the quantization of the twistor model without any gauge fixing and use a different field twistor transform. As a result of such a quantization procedure, we obtain infinite spin fields that have a more transparent decomposition into helicities in the twistor formulation. Besides, we show that our twistorial model reproduces the space-time-spinorial formulation of infinite spin fields with integer helicities proposed in [31]. In addition, using the field twistor transform, we can now get the space-time-spinorial formulation of infinite spin fields with half-integer helicities. Moreover, such a formulation allows us to construct a supermultiplet of infinite spins [13] (see also the recent development in [25]) and describe some of its properties. It is worth pointing out that the aspects of infinite spin supersymmetry have almost been unconsidered in the literature earlier.
The plan of this paper is as follows. In Sect. 2, we describe the world-line twistor formulation of the infinite (continuous) spin particle, which we built in [30]. We present the twistorial constraints of the model and coordinate twistor transform to space-time formulation. In Sect. 3, by using canonical transformation, we introduce suitable phase variables into the twistor formulation, in which all twistor constraints take a very simple form appropriate for quantization. The twistor field of the infinite spin particle is obtained as a solution of equations obtained from the canonical quantization of the first class constraints. The resulting twistor field has the U(1) charge that corresponds to the method of field description of the infinite spin particle. Integer or half-integer values of this charge correspond to infinite spin particles with integer or half-integer values of helicities. Also, the decomposition of the twistor field in an infinite number of states with fixed helicities is found. In Sect. 4, we construct a field twistor transform, which determines the spacetime field of the infinite (continuous) spin particle, according to the twistor field obtained in the previous section. These space-time fields depend, in addition to the four-vector coordinates, also on the components of the additional commuting Weyl spinor. We have found the equations of motion for the fields of infinite spin particles, both in the case of integer and half-integer helicities. In Sect. 5, the N = 1 infinite spin supermultiplet is constructed. On the mass-shell this supermultiplet consists of two complex infinite spin fields with integer and half-integer helicities, respectively. In Sect. 6, we give some comments on the results obtained. In two Appendices, we give the proof of the equations of motion for the space-time infinite spin fields and present the explicit form of the momentum wave function in the space-time formulation.
are the components of the Pauli-Lubański pseudovector and p m and M kl := (x k p l − x l p k ) + (y k q l − y l q k ) are the Poincaré algebra generators. The models with Lagrangians (2.3) and (2.15) are equivalent at the classical level (see [30]) and describe massless particles with the infinite (continuous) spin.
Following [35], [36], [37] we choose the norms of twistors (2.20), (2.21) as and write the constraint (2.6) in concise form The normZ A Z A of the twistor Z commutes with constraints (2.4), (2.5), (2.6) and therefore is independent of τ . For a massless particle with fixed helicity the normZ A Z A defines the helicity operator So in the considered model of the infinite (continuous) spin particle, in view of the constraint (2.25), the particle helicity is not fixed since it is proportional to −Ȳ A Y A . Let us make some comments on the role of the constraints (2.5). First of all, we recall the statement from [30] that the matrices where πρ := π α ρ α can be considered as elements of SL(2, C). Moreover, the Lorentz transformation with A ∈ SL(2, C) given in (2.27) converts test massless momentum to momentum (2.11)p where we assume |πρ| = √ 2k. Inverse Lorentz transformations that translatep into the basis of test momentumk are given by the matrix In the twistor realization the Pauli-Lubański pseudovector with components (2.23) has the form where p αβ is defined in (2.11). Therefore, this pseudovector in the basis of test momentum is defined by the relation and has the components Using the standard analysis of the massless representations of the Poincaré group, we find from expressions (2.34) that the quantities are the generators of the small subgroup E(2). The generators E andĒ yield translations in C which correspond to the parameter α in (2.8), (2.9). Then, constraints (2.5) fix the Casimir operator of the small subgroup, and therefore fix the Casimir operator W 2 of the Poincaré group.

Alternative quantization of twistorial model and infinite spin twistor field
The twistorial model with Lagrangian (2.3) possesses the gauge symmetry under transformations (2.9) generated by the first class constraints (2.4), (2.5) and (2.6). In [30], we performed the quantization of the model (2.3) after partial fixing the gauge as s = 1 =s (the definition of s,s is given below). In this paper, we quantize the model on the base of a more general procedure without any gauge fixing.
The representation (3.42) for the twistorial field of the infinite spin particle is convenient to clarify its helicity content. Considering the relation which holds for the operator (3.33) and using the generalized Cartan-Penrose representations (2.11), (2.12) for p 0 , q 0 , we obtain that the action of the Pauli-Lubański pseudovector on the field (3.42) gives In contrast to quantity (3.37), the operator (3.47) satisfies the property α=αD αα = α=α π απα Λ = P 0 Λ .
As a result, the helicity operator (3.39) acts on the twistorial field in the following way: As we see from this expression, the action of the helicity operator Λ is defined by the action of the operator Λ on the functions ψ (c+k) (π,π) (where k ∈ Z) which are parameterized by two dimensonal complex variable π α . Due to the conditions (3.44), the fields ψ (c+k) (π,π) are the eigenvectors of the helicity operator (3.34): Λ ψ (c+k) (π,π) = − c + k ψ (c+k) (π,π) . Note that the helicity content of the twistor field Ψ (c) is the same for all integer values of the U(1) charge c. The distinction is only in the shift of the infinite tower of states (with all possible integer helicities) in k by integer difference of the values of c. The same situation occurs for half-integer values of c, when the tower of states with half-integer helicities is also the same for various c ∈ (Z + 1/2). The choice of number c, which takes a fixed value, determines only a specific way of describing the same infinite spin representations for all integer (or half-integer) spins. In fact, there are only two independent values for c, namely c = 0 and c = − 1 2 . Recall that the twistorial wave function (3.42) is complex and therefore all component fields in the expansion (3.43) are also complex. Then, in the CPT-invariant theory we must consider together with the field Ψ (c) its complex conjugated field (Ψ (c) ) * which (due to the condition (3.28)) has the opposite charge (Ψ (c) ) * :=Ψ (−c) . (3.50) To describe the bosonic infinite spin representation related to all integer helicities, we put and consider the twistorial field Ψ (0) (π,π; ρ,ρ) . (3.52) The complex conjugate fieldΨ (0) has also zero charge. Similarly, to describe the infinite spin representation related to half-integer helicities we take for c the value The corresponding wave function Ψ (−1/2) (π,π; ρ,ρ) (3.54) contains in its expansion (3.43) all half-integer helicities. The complex conjugate field Ψ (+1/2) (π,π; ρ,ρ) (3.55) possesses the charge c = +1/2. To conclude this section, we stress that the choice of the integer or half-integer values of the U(1)-charge 3 c can be considered as fixing of different boundary conditions, by analogy with the choice of the Ramond or Nevew-Schwarz sectors in the fermionic string theories (see, e.g., [43]).

Twistor transform for infinite spin fields
In [30], performing a quantization of the twistor model with special gauge fixing, we obtained space-time fields which depended on the coordinates x m of position four-vector and obeyed the Wigner-Bargmann equations [2,3] following from Lagrangian (2.15). Now we show that our twistorial model reproduces the formulation of the infinite spin field model developed in [31].
The field (4.8) satisfies four equations The proof of these equations is given in Appendix A. Equations (4.9), (4.10), (4.11), (4.12) underlie the definition of the infinite spin fields proposed in [31]. The explicit form of the momentum wave function corresponding to the space-time field (4.8) is given in Appendix B.
The algebraic structure of the infinite spin supersymmetry was discussed in [13]. Our consideration is the explicit field description of the supermultiplet introduced in the paper [13].

Summary and outlook
In this paper, we have presented the new twistorial field formulation of the massless infinite spin particle and field. As opposed to the paper [30], we obtained the field description by the canonical quantization of the world-line twistor model without any gauge fixing. We gave the helicity decomposition of twistorial infinite spin fields and constructed the field twistor transform to define the space-time infinite (continuous) spin fields. These space-time fields, bosonic Φ(x; ξ,ξ) and fermionic Φ α (x; ξ,ξ), depend on the coordinate four-vector x m and on the commuting Weyl spinors ξ α ,ξα. We found the equations of motion for Φ(x; ξ,ξ) and Φ α (x; ξ,ξ). Moreover, we showed that these fields form the N = 1 infinite spin supermultiplet.
In subsequent works we will consider the construction of the Lagrangian field theory of continuous spin, both in the bosonic and fermionic cases and also in the supersymmetric case. One of the commonly used methods for this purpose is the BRST quantization method, which was used in the case of continuous spin particles in [19,28,29,31,32,33]. In a recent paper [31] the Lagrangian formulation of the infinite integer-spin field was constructed by using the methods developed in [46,47]. We plan to construct the Lagrangian formulation for the infinite half-integer field, as well as in the supersymmetric case. Another interesting problem is to develop the Lagrange description of the infinite spin supermultiplet in a superfield approach. We note in this regard that in [25] the Lagrangian formulation of the infinite spin supermultiplets in the d= 3 space-time was constructed by using the frame-like formalism for corresponding bosonic and fermionic fields.