Massless particles in five and higher dimensions

We describe a five-dimensional analogue of Wigner’s operator equation Wa = λPa, where Wa is the Pauli-Lubanski vector, Pa the energy-momentum operator, and λ the helicity of a massless particle. Higher dimensional generalisations are also given.


Introduction
The unitary representations of the Poincaré group in four dimensions were classified by Wigner in 1939 [1], see [2] for a recent review. Our modern understanding of elementary particles is based on this classification.
Unitary representations of the Poincaré group ISO 0 (d − 1, 1) in higher dimensions, d > 4, have been studied in the literature, see, e.g., [3]. However, there still remain some aspects that are not fully understood, see, e.g., [4] for a recent discussion. In this note we analyse the irreducible massless representations of ISO 0 (4, 1) with a finite (discrete) spin.
We recall that the Poincaré algebra iso(d − 1, 1) in d dimensions is characterised by the commutation relations 1 In any unitary representation of (the universal covering group of) the Poincaré group, the energy-momentum operator P a and the Lorentz generators J ab are Hermitian. For every dimension d, the operator P a P a is a Casimir operator. Other Casimir operators are dimension dependent.
In four dimensions, the second Casimir operator is W a W a , where is the Pauli-Lubanski vector. Using the commutation relations (1.1), it follows that the Pauli-Lubanski vector is translationally invariant, and possesses the following properties: The irreducible massive representations are characterised by the conditions where the quantum number s is called spin. Its possible values in different representations are s = 0, 1/2, 1, 3/2 . . . . The massless representations are characterised by the condition P a P a = 0. For the physically interesting massless representations, it holds that where the parameter λ determines the representation and is called the helicity. Its possible values are 0, ± 1 2 , ±1, and so on. The parameter |λ| is called the spin of a massless particle. In this paper we present a generalisation of Wigner's equation (1.5) to five and higher dimensions.
2 Unitary representations of ISO 0 (4, 1) The five-dimensional analogue of (1.2) is the Pauli-Lubanski tensor and possesses the following properties: Making use of W ab allows one to construct two Casimir operators, which are

Irreducible massive representations
The irreducible massive representations of the Poincaré group ISO 0 (4, 1) are characterised by two conditions in addition to (1.4a). Here s 1 and s 2 are two spin values corresponding to the two SU(2) subgroups of the universal covering group Spin(4) ∼ = SU(2) × SU(2) of the little group. 2

Irreducible massless representations
It turns out that all irreducible massless representations of ISO 0 (4, 1) with a finite spin are characterised by the condition Both Casimir operators (2.4) are equal to zero in these representations, W ab W ab = 0 and W ab J ab = 0.
Let |p, σ be an orthonormal basis in the Hilbert space of one-particle states, where p a denotes the momentum of a particle, P a |p, σ = p a |p, σ , and σ stands for the spin degrees of freedom. For a massless particle, we choose as our standard 5-momentum k a = (E, 0, 0, 0, E). On this eigenstate: Running through the elements of W ab , one finds: The equations (2.5) were independently derived during the academic year 1992-93 by Arkady Segal and David Zinger, who were undergraduates at Tomsk State University at the time.
If we rescale these generators and define: then these new operators satisfy: These are the commutation relations for the three-dimensional Euclidean algebra, iso (3).
The irreducible unitary representations of iso(3) are labelled by a continuous parameter µ 2 , corresponding to the value the Casimir operator R i R i takes. Since R i commute among themselves the operators can be simultaneously diagonalised, and the eigenvectors |r i taken as a basis. However the only restriction on these is that r i r i = µ 2 , which for non-zero µ 2 permits a continuous basis and is thus an infinite dimensional representation. Because we want only finite-dimensional representations, we must take: We are therefore restricted to those representations in which the translation component is trivial, and so only the generators J i remain, which generate the algebra so(3). The algebra of the little group on massless representations is thus so(3) which is isomorphic to su(2). As stated previously, the irreducible representations of su(2) are labelled by a non-negative (half) integer s and have a single Casimir operator J i J i which takes the value s(s + 1)½. This analysis leads to (2.6).
The spin value of a massless representation can still be found using a 'spin' operator. The following relation holds on massless representations: where J 2 = J i J i is the Casimir operator for the so(3) generators in (2.9). The parameter s is the spin of a massless particle. Its possible values in different representations are s = 0, 1/2, 1, and so on. Equation (2.12) naturally holds for massless spinor and vector fields [5].
In general, the operator S a is not translationally invariant, It is only for the massless representations with finite spin that the quantity on the right vanishes so that the spin operator commutes with the momentum operators. Equation (2.12) is the five-dimensional analogue of the operator equation (1.5). Its consistency condition is (2.6). 3

Generalisations
The results of section 2.2 can be generalised to d > 5 dimensions. The Pauli-Lubanski tensor (2.1) turns into The condition (2.6) is replaced with This equation is very similar to another that has appeared in the literature using the considerations of conformal invariance [6][7][8][9]. One readily checks that (3.2) is equivalent to The latter is solved on the momentum eigenstates by J ab p b ∝ p a , which is of the form considered in [6][7][8][9]. 4 Equation (3.2) characterises all irreducible massless representations of ISO 0 (d − 1, 1) with a finite (discrete) spin. Finally, the spin equation (2.12) turns into where J 2 = 1 2 J ij J ij is the quadratic Casimir operator of the algebra so(d − 2), with i, j = 1, . . . , d − 2. For every irreducible massless representation of ISO 0 (d − 1, 1) with a finite spin, it holds that J 2 ∝ ½.
It is possible to derive a five-dimensional analogue of the operator equation defining the N = 1 superhelicity κ in four dimensions [10]. The latter has the form 5 L a = κ + 1 4 P a , (3.5) where the operator L a is defined by L a = W a − 1 16 (σ a )α α Q α ,Qα . (3.6) The fundamental properties of the operator L a (the latter differs from the supersymmetric Pauli-Lubanksi vector [11]) are that it is translationally invariant and commutes with the supercharges Q α andQα in the massless representations of the N = 1 super-Poincaré group. 6 The superhelicity operator (3.6) was generalised to higher dimensions in [12,13]. Generalisations of (3.5) to five and higher dimension will be discussed elsewhere.