On the Particle Content of Moyal-Higher-Spin Theory

Abstract

The Moyal-Higher-Spin (MHS) formalism, involving fields dependent on spacetime and auxiliary coordinates, is an approach to studying higher-spin (HS)-like models. To determine the particle content of the MHS model of the Yang–Mills type, we calculate the quartic Casimir operator for on-shell MHS fields, finding it to be generally non-vanishing, indicative of infinite/continuous spin degrees of freedom. We propose an on-shell basis for these infinite/continuous spin states. Additionally, we analyse the content of a massive MHS model.

1. Introduction

There are both purely theoretical and phenomenological reasons for constructing consistent higher-spin (spin $s>2$) quantum field theories (QFTs) in the Minkowski background, which still remains an elusive goal. For the status of research on higher-spin theory and continuous spin theory, one can consult reviews [1,2,3,4,5], in which the limitations of all approaches are discussed. Higher-spin and continuous spin theory could also provide a suitable framework for the description of dark matter [6,7]. Gauge theories in noncommutative spacetime naturally emerge as effective field theories in the context of string theory [8] on a non-trivial B-field background, and have been extensively studied (see, e.g., [9] and references within) for their capability to capture in a consistent framework properties such as renormalizability and nonlocality, which are desirable in a consistent quantum gravity theory. On the other hand, in recent years, the so-called ambient space formalism has been developed as a valuable technique to compactly describe the higher-spin tensor fields emerging in the tensionless limit of string theory (see, e.g., [10,11]), and in particular to encode their minimal coupling to standard matter theory in terms of a Moyal-like product [12]. Taking inspiration from this approach, an effective action approach to massless higher-spin interactions has been discussed [13,14]. A related development also based on emerging noncommutative geometry is the one in which higher-spins emerge in the low-energy limit of the IKKT matrix model [15,16,17,18,19,20,21].

Theories with higher-spins are expected to contain an infinite tower of higher-spin fields/particles, a property which could ensure a better (softer) UV behaviour and, in this way, avoid some of the problems present in standard QFTs (e.g., Landau pole). If the HS fields are described by totally symmetric Lorentz tensor spacetime fields ${\varphi }_{{\mu }_1\cdots {\mu }_s}\left(x\right)$, the infinite tower can be packed into a field on a $2d$-dimensional domain by using an auxiliary space with the coordinates $u=u_{\mu },\mu =0,\dots ,d-1$, transforming as a Lorentz vector, in the following way

$$\varphi (x,u)=\sum _{s=0}^{\infty }{\varphi }^{{\mu }_1\cdots {\mu }_s}\left(x\right)u_{{\mu }_1}\cdots u_{{\mu }_s}.$$

(1)

We refer to fields defined in the $2d$-dimensional space spanned by x and u as master fields. Usually, master fields are used just as a formal construct to write equations for free HS spacetime fields in a compact way, in particular without any requirements of convergence for the infinite series in (1). We observe here that requiring convergence in (1) automatically implies constraints on spacetime fields, which means that they cannot be independent. However, recently, in the literature, there have appeared constructions in which master fields are used as fundamental objects in the off-shell descriptions [22,23,24,25]. Here, we shall be interested in the approaches in which master fields are required to be square-integrable in the auxiliary space

$$\int d^{d}u\varphi {(x,u)}^{†}\varphi (x,u)<\infty ,$$

(2)

an example being given by the Moyal-Higher-Spin (MHS) gauge theories [13,14,23,24,26,27,28,29]. This constraint makes the models well-defined, off-shell, and ensures the finiteness of the observables, and, as we will see below, provides a unitary representation of the Lorentz group. In this case, a Taylor expansion (1) is not much of use, as the spacetime fields defined in this way cannot be treated as independent degrees of freedom. Consequently, one cannot read off the particle content of the theory by plugging (1) into the linearised equations of motion, so one has to use different methods to understand how Lorentz and Poincaré groups are represented. These representations can be studied in general for any dependence on u, as well as on a particular basis in the auxiliary space u. The Poincaré transformations, in general, act on master fields in the following way

$${\varphi }_r^{\prime }(x,u)=\sum _s\mathcal{D}{{\left(\mathsf{\Lambda }\right)}_r}^s{\varphi }_s({\mathsf{\Lambda }}^{-1}x-\zeta ,{\mathsf{\Lambda }}^{-1}u),$$

(3)

where r and s represent a finite number of Lorentz indices and $\mathcal{D}\left(\mathsf{\Lambda }\right)$ is some standard finite-dimensional IRREP of the Lorentz group (which, for a scalar master field, is trivial). It was noted in [30] that one can use an infinite-dimensional representation of the Lorentz group induced by the transformation of the auxiliary coordinate u in (3). It was further noted in [23] that the restriction of master field dependence on u to the linear vector space $L_2\left({\mathbb{R}}^d\right)$ (square-integrable functions over the auxiliary space) is both infinite-dimensional and unitary. The explicit expression of the matrix representation, written on the basis of d-dimensional Hermite functions, was presented in [31]. What remained to be analysed was the perturbative spectrum (i.e., particle content of a free theory) of a master field that is square integrable in the auxiliary space, in terms of the IRREP of the Poincaré group by the Wigner construction, and this is what we do in the present paper. To avoid unnecessary complications, we shall demonstrate the construction on the example of the master field gauge potential in the YM theory, both for the massless and massive cases.

We begin by reviewing the MHS formalism, and then display several approaches to analysing the theory’s spacetime content. The first one, a Taylor expansion in the auxiliary space, is conventional and enables a direct comparison to the standard higher-spin models. However, as mentioned, it is not consistent with the requirement of the square integrability of the master fields. The second approach is an expansion in terms of the orthonormal basis of functions (for discrete bases, orthonormality is defined using Kroneckers, and, for continuous bases, using Dirac delta functions) in the auxiliary space [22]. We employ modes given by the products of Hermite functions (momentum-independent). Such modes furnish an infinite-dimensional representation of the Lorentz group ([23,31]). We also employ modes that are momentum-dependent and are solutions to the differential equations posed by the little-group generators.

By analysing the polarisation structure of the solutions of the linear equations of motion of the MHSYM model (linearised around a Minkowski vacuum) expanded in terms of Hermite functions, we learn about supported helicities and obtain an indication that, in general, we can have a non-vanishing value for the quartic Casimir. In addition, by analysing the polarisation structure, now in terms of functions which are solutions to the differential equations posed by the little-group generators, we find a complete on-shell (momentum-dependent) basis which enables us to read off the particle content of the solutions. In the Appendix A, we analyse the particle content of a massive MHS master field, working directly in the space of one-particle states.

2. Master Field Yang–Mills Theories

As a working example, let us use the generalisation of Yang–Mills theories defined on the flat Minkowski background and write here just the basic aspects and equations important for the context of the present paper. An explicit example of such a class of theories is the MHS gauge theory, developed in [22]. While the theory can be defined for an arbitrary number of spacetime dimensions d, we shall focus mostly on the case $d=4$. The basic object is the gauge potential master field $h_a(x,u)$, where $x=\{x^{\mu },\mu =0,\dots d-1\}$ are spacetime coordinates, $u=\{u_{\mu },\mu =0,\dots d-1\}$ are (dimensionless) coordinates in the auxiliary space transforming as a Lorentz (covariant) vector, and $a=\{0,\dots ,d-1\}$ is a frame index. Infinitesimal gauge transformations are given by

$$\delta h_a(x,u)={\partial }_a\epsilon (x,u)+O\left(h\right),$$

(4)

where the gauge parameter $\epsilon (x,u)$ is a generic (infinitesimal) function satisfying some appropriate boundary conditions. Note that we assume here the simplest case of the Maxwell-like theories based on the $U\left(1\right)$ internal gauge symmetry. The formalism can be naturally extended to noncommutative internal gauge groups. However, the structure of the terms linear in the gauge potential depends on the theory in question, and is not important for this paper. In the example of the MHS theory, the YM algebraic structure is defined by the Moyal product, which introduces a noncommutative structure between the spacetime and the auxiliary space. This structure guarantees not only good classical behaviour, but also enables one to formally use standard quantisation methods. Generalising the Yang–Mills procedure, one defines the master field strength as follows:

$$F_{ab}(x,u)={\partial }_a^x{h}_b(x,u)-{\partial }_b^x{h}_a(x,u)+O\left(h^2\right),$$

(5)

and the YM action

$$S[h,\psi ]=S_{\mathrm{ym}}\left[h\right]+S_{\mathrm{matt}}[h,\psi ],$$

(6)

where $\psi$ denotes (minimally coupled) matter in the form of master fields and/or ordinary spacetime fields, and

$$S_{\mathrm{ym}}=-{\displaystyle \frac{1}{4}}\int d^{d}x{d}^{d}u{F}^{ab}(x,u)F_{ab}(x,u).$$

(7)

The equations of motion for the gauge master field are

$$\begin{array}{c}\hfill {\square }_x{h}_a-{\partial }_a^x{\partial }_b^x{h}^b+m_h^2{h}_a+O\left(h^2\right)={\displaystyle \frac{\delta S_{\mathrm{matt}}^{\prime }}{\delta h^a}},\end{array}$$

(8)

where $S_{\mathrm{matt}}^{\prime }$ is part of $S_{\mathrm{matt}}$, which does not include the mass term, possibly generated by the Higgs mechanism. The energy in the linear approximation is given by

$$E\approx {\displaystyle \frac{1}{2}}\int d\mathbf{x}\int d^{d}u\left(\sum _j{F}_{0j}{(x,u)}^2+\sum _{j<k}{F}_{jk}{(x,u)}^2+m_h^2\left(h_0{(x,u)}^2+\sum _i{h}_i{(x,u)}^2\right)\right)+U_{\mathrm{matt}},$$

(9)

where $j,k\in 1,\dots ,d-1$. From (8) and (9), we can conclude that the theory is classically consistent, at least in the perturbative domain: the vacuum $h_a=0$ is stable, and the energy is positive-definite and finite.

Furthermore, there are no problems with ghosts in perturbative expansions. To understand this, let us first rewrite the MHS gauge field transformation under Poincaré transformations in the following way

$$h_a^{\prime }(x,u)={{\mathsf{\Lambda }}_a}^b\left(D\left(\mathsf{\Lambda }\right)h_b\right)({\mathsf{\Lambda }}^{-1}x-\zeta ,u),$$

(10)

$$\left(D\left(\mathsf{\Lambda }\right)h_b\right)(x,u)=h_b(x,{\mathsf{\Lambda }}^{-1}u).$$

(11)

$D\left(\mathsf{\Lambda }\right)$ defines an infinite-dimensional linear representation of the Lorentz group on $L_2\left({\mathbb{R}}^d\right)$, the vector space of square-integrable functions over the auxiliary space. Moreover, this representation is unitary with respect to the standard inner product on $L_2\left({\mathbb{R}}^d\right)$

$$\mathsf{\langle }\Psi |\mathsf{\Phi }\rangle =\int d^{d}u\Psi {\left(u\right)}^{†}\mathsf{\Phi }\left(u\right).$$

(12)

The proof is

$$\int d^{d}u{\left(D\left(\mathsf{\Lambda }\right)\Psi \left(u\right)\right)}^{†}D\left(\mathsf{\Lambda }\right)\mathsf{\Phi }\left(u\right)=\int d^{d}u{\Psi }^{†}\left(u\mathsf{\Lambda }\right)\mathsf{\Phi }\left(u\mathsf{\Lambda }\right)=\int d^{d}u{\Psi }^{†}\left(u\right)\mathsf{\Phi }\left(u\right),$$

(13)

where we used $d^d\left(u\mathsf{\Lambda }\right)=d^{d}u$. As $\mathsf{\Phi }$ and $\Psi$ are arbitrary elements of $L_2\left({\mathbb{R}}^d\right)$, it follows from (13) that

$$D{\left(\mathsf{\Lambda }\right)}^{†}=D{\left(\mathsf{\Lambda }\right)}^{-1}.$$

(14)

The unitarity of $D\left(\mathsf{\Lambda }\right)$ guarantees that the master gauge symmetry is large enough to deal with all physically unacceptable modes, i.e., (4) is sufficient to remove them. In fact, it was shown [26,29] that the theory can be formally quantised by using the standard methods of YM theory. Let us add here that C, P and T symmetries have natural extensions on the master field description (the action of P is already described above by putting $\mathsf{\Lambda }=\mathcal{P}\equiv \mathrm{diag}(1,-1,1,\dots ,1)$ and taking into account the phase factor), which means that $CPT$ is the symmetry of the MHS theory.

The next task is to investigate the spectrum (particle content) of such theories. In what follows, we shall assume that the master fields, as functions of auxiliary coordinates u, are essentially restricted only by the condition of square integrability.

3. Spacetime Fields

3.1. Taylor Expansion

In general, when dealing with higher-spin fields, it is useful to pack a complete tower of higher-spin fields into a single structure by using an auxiliary Lorentz vector as a book-keeping device (see, e.g., [4,5,32]).

By reversing this logic, it may appear that the master potential should be understood, from the purely spacetime viewpoint, as an infinite collection of HS fields obtained from

$$h_a(x,u)=\sum _{n=0}^{\infty }{h}_a^{\left(n\right){\mu }_1\cdots {\mu }_n}\left(x\right)u_{{\mu }_1}\dots u_{{\mu }_n},$$

(15)

where we use a Latin index for the master field and Greek indices for variables of expansion. The coefficients in the expansion are spacetime fields that are Lorentz tensors of rank $n+1$, symmetric in their n (Greek) indices, and which, by (8), in the massless case, satisfy Maxwell-type equations of motion

$$\square h_a^{\left(n\right){\mu }_1\cdots {\mu }_n}-{\partial }_a{\partial }^b{h}_b^{\left(n\right){\mu }_1\cdots {\mu }_n}+O\left(h^2\right)=0,$$

(16)

while from (4), we can deduce that the gauge transformations, obtained by expanding the gauge parameter as in (15), are of the form

$${\delta }_{\epsilon }{h}_a^{\left(n\right){\mu }_1\cdots {\mu }_n}\left(x\right)={\partial }_a{\epsilon }^{{\mu }_1\cdots {\mu }_n}\left(x\right)+O\left(h\right).$$

(17)

While there is a priori nothing wrong with the manifestly Lorentz covariant power expansion (15), it cannot by itself be used for the purpose of uncovering the spectrum of physical excitations (particle spectrum) of the theory. This is because when substituting (15) into the action (7) or the energy (9), and organising terms by the order of $u_{\mu }$, we encounter divergent integrations over the auxiliary space u of the form $\int d^{d}u{u}_{{\mu }_1}\dots u_{{\mu }_n}$ at each order n. Therefore, the requirement of square integrability in the auxiliary space forces us to abandon (15) as useful means.

3.2. Orthogonal Functions Expansion

An alternative to the Taylor expansion above comes from relaxing the notion of how Lorentz covariance is to be achieved and giving priority to the fact that integrals over the auxiliary space should be finite. We can then use a complete orthonormal set of functions in the auxiliary space $\left\{f_r\left(u\right)\right\}$, indexed by some formal parameter r, to expand the master potential as

$$h_a(x,u)=\sum _r{h}_a^{\left(r\right)}\left(x\right)f_r\left(u\right),$$

(18)

where

$$\int d^{d}u{f}_r\left(u\right)f_s\left(u\right)={\delta }_{rs}.$$

(19)

Using such an expansion, one arrives at the purely spacetime off-shell description, with the free (quadratic) part of the Lagrangian given by

$$S_0\left[h\right]=-{\displaystyle \frac{1}{4g_{\mathrm{ym}}^2}}\sum _{r,s}\int d^{d}x\left({\partial }_a{h}_b^{\left(r\right)}-{\partial }_b{h}_a^{\left(r\right)}\right){\eta }^{ac}{\eta }^{bd}{\delta }_{rs}\left({\partial }_c{h}_d^{\left(s\right)}-{\partial }_d{h}_c^{\left(s\right)}\right).$$

(20)

On a linear level, the gauge symmetry acts on spacetime fields $h_a^{\left(r\right)}\left(x\right)$ as

$${\delta }_{\epsilon }{h}_a^{\left(r\right)}\left(x\right)={\partial }_a{\epsilon }^{\left(r\right)}\left(x\right)+O\left(h\right),$$

(21)

where ${\epsilon }^{\left(r\right)}\left(x\right)$ are obtained from the master gauge parameter $\epsilon (x,u)$ by expanding, as in (18). We see that the free theory is equivalent to an infinite collection of Maxwell fields. This form shows explicitly the absence of physically non-acceptable modes (ghosts, runaway modes or negative energy modes) in the spectrum of the free theory. To this we may add that the MHS theory is formally eligible to BRST quantisation [26,29], so one does not expect problems with unitarity, at least not in the perturbative domain.

The appearance of the Kronecker in (19), however, leads to another effect. The positive definite product of two basis functions in (19) might seem in disagreement with the condition of Lorentz covariance, since intuition usually leads us to expect the Minkowski metric on the right-hand side of equations such as (19) if Lorentz covariance is to be achieved. However, as we have shown in the previous section, the Lorentz covariance is still present in the form of a unitary infinite-dimensional representation of the Lorentz group acting on the index r. One particularly convenient choice for the orthonormal basis of functions is using the multi-dimensional Hermite functions that we used in [31]. We repeat the definition here with modifications due to the fact that the auxiliary space variables are $u_a$, not $u^a$. $H_n\left(u\right)$ are the Hermite polynomials

$$H_n\left(u\right)={(-1)}^n{e}^{u^2}{\displaystyle \frac{d^n}{d{u}^n}}{e}^{-u^2},$$

(22)

where the index n can attain arbitrary non-negative integer values. The Hermite functions are defined as

$$f_n\left(u\right)={\displaystyle \frac{1}{\sqrt{2^{n}n!\sqrt{\pi }}}}{e}^{-{\displaystyle \frac{u^2}{2}}}{H}_n\left(u\right).$$

(23)

The multi-dimensional Hermite function that we will use for the expansion in the auxiliary space is defined as

$$f_{n_0\cdots n_{d-1}}\left(u\right)=f_{n_0}\left(u_0\right)\cdots f_{n_{d-1}}\left(u_{d-1}\right).$$

(24)

They satisfy the orthonormality condition

$$\int d{u}_0\cdots d{u}_{d-1}{f}_{n_0\cdots n_{d-1}}\left(u\right)f_{m_0\cdots m_{d-1}}\left(u\right)={\delta }_{n_0}^{m_0}\cdots {\delta }_{n_{d-1}}^{m_{d-1}}.$$

(25)

The MHS potential $h_a(x,u)\equiv h_a(x^b,u_c)$ is now expanded on the basis of d-dimensional Hermite functions as

$$h_a(x,u)=\sum _{\left\{n\right\}=0}^{\infty }{h}_a^{n_0\cdots n_{d-1}}\left(x\right)f_{n_0\cdots n_{d-1}}\left(u\right).$$

(26)

The spin (i.e., helicity, as fields are massless) operator is derived in Section 3.3 and given by Equation (46). However, in Section 6, we find that the quartic Casimir of the Poincaré group is nonzero, i.e., these fields correspond to continuous spin particles, so the helicity of these fields is not a Lorentz invariant number, as is also noted in [30]. Following the transformation property

$$h_a^{\prime }(x^{\prime },u^{\prime })={{\mathsf{\Lambda }}_a}^b{h}_b(x,u),$$

(27)

we can deduce the rules for the Lorentz transformations of the component spacetime fields $h_a^{n_0\cdots n_{d-1}}\left(x\right)$

$$h_a^{\prime }(x,u)={{\mathsf{\Lambda }}_a}^b{h}_b({\mathsf{\Lambda }}^{-1}x,u\mathsf{\Lambda })$$

(28)

and we can expand both sides of the equation on the Hermite basis

$$\sum _{\left\{n\right\}=0}^{\infty }{h}_a^{\prime n_0\cdots n_{d-1}}\left(x\right)f_{n_0\cdots n_{d-1}}\left(u\right)={{\mathsf{\Lambda }}_a}^b\sum _{\left\{m\right\}=0}^{\infty }{h}_b^{m_0\cdots m_{d-1}}\left({\mathsf{\Lambda }}^{-1}x\right)f_{m_0\cdots m_{d-1}}\left(u\mathsf{\Lambda }\right).$$

(29)

Due to (25), we can multiply both sides with $f_{r_0\cdots r_d}\left(u\right)$, integrate them over the auxiliary space, and conclude

$$\begin{array}{c}\hfill h_a^{\prime r_0\cdots r_{d-1}}\left(x\right)={{\mathsf{\Lambda }}_a}^b\sum _{\left\{m\right\}=0}^{\infty }{L}_{m_0\cdots m_{d-1}}^{r_0\cdots r_{d-1}}\left(\mathsf{\Lambda }\right)h_b^{m_0\cdots m_{d-1}}\left({\mathsf{\Lambda }}^{-1}x\right),\end{array}$$

(30)

where

$$L_{m_0\cdots m_{d-1}}^{r_0\cdots r_{d-1}}\left(\mathsf{\Lambda }\right)=\int d{u}_0\cdots d{u}_{d-1}{f}_{r_0\cdots r_{d-1}}\left(u\right)f_{m_0\cdots m_{d-1}}\left(u\mathsf{\Lambda }\right)$$

(31)

is a representation matrix of the Lorentz group in the space of the Hermite functions.

We explicitly constructed the representation matrices in [31]. Since, here, the variables of integration are auxiliary space coordinates with lower Lorentz indices, while in [31], we worked with variables with upper Lorentz indices, (31) differs slightly from the convention in [31]. We now adapt the representation matrices to the current situation. For simplicity, we restrict this to two dimensions. From [31], we have

$$D_{n_0{n}_1}^{m_0{m}_1}\left(\mathsf{\Lambda }\right)=\int d{u}^{0}d{u}^1{f}_{m_0}\left(u^0\right)f_{m_1}\left(u^1\right)f_{n_0}\left({\left({\mathsf{\Lambda }}^{-1}u\right)}^0\right)f_{n_1}\left({\left({\mathsf{\Lambda }}^{-1}u\right)}^1\right),$$

(32)

while explicitly in (31) we have

$$L_{n_0{n}_1}^{m_0{m}_1}\left(\mathsf{\Lambda }\right)=\int d{u}_{0}d{u}_1{f}_{m_0}\left(u_0\right)f_{m_1}\left(u_1\right)f_{n_0}\left({\left(u\mathsf{\Lambda }\right)}_0\right)f_{n_1}\left({\left(u\mathsf{\Lambda }\right)}_1\right).$$

(33)

In the mostly plus signature (${\eta }^{00}=-1$, ${\eta }^{ii}=1$) that we are using, we can re-express

$$u_0=-u^0,u_1=u^1,{\left(u\mathsf{\Lambda }\right)}_0=-{\left(u\mathsf{\Lambda }\right)}^0,{\left(u\mathsf{\Lambda }\right)}_1={\left(u\mathsf{\Lambda }\right)}^1$$

(34)

and further we realise

$${\left(u\mathsf{\Lambda }\right)}_{\mu }=u_{\nu }{{\mathsf{\Lambda }}^{\nu }}_{\mu },{\left(u\mathsf{\Lambda }\right)}^{\mu }=u_{\nu }{{\mathsf{\Lambda }}^{\nu }}^{\mu }=u^{\nu }{{\mathsf{\Lambda }}_{\nu }}^{\mu }={{\left({\mathsf{\Lambda }}^{-1}\right)}^{\mu }}_{\nu }{u}^{\nu }={\left({\mathsf{\Lambda }}^{-1}u\right)}^{\mu }.$$

(35)

With the property of the Hermite functions $f_n(-u)={(-1)}^n{f}_n\left(u\right)$, we can finally relate

$$\begin{array}{c}\hfill L_{n_0{n}_1}^{m_0{m}_1}\left(\mathsf{\Lambda }\right)={(-1)}^{n_0+m_0}{D}_{n_0{n}_1}^{m_0{m}_1}\left(\mathsf{\Lambda }\right).\end{array}$$

(36)

With these conventions, the prefactor ${(-1)}^{n_0+m_0}$ does not depend on the number of spacetime dimensions. The representation matrices $D_{n_0{n}_1}^{m_0{m}_1}\left(\mathsf{\Lambda }\right)$ can be found in [31].

For further convenience, we explicitly write down the generators of the Lorentz group in $d=4$ in the infinite-dimensional representation over the Hermite functions, adapted to the purposes of this paper (here, we also make the operators Hermitean so the rotation generators $J_i$ differ from those in [31] by a global factor of $-i$, while the boost generators $K_i$ differ by a global factor of i).

$$\begin{array}{c}\hfill {K_1}_{n_0{n}_1{n}_2{n}_3}^{m_0{m}_1{m}_2{m}_3}=i{\delta }_{-n_0+n_1+n_2+n_3}^{-m_0+m_1+m_2+m_3}{\delta }_{n_2}^{m_2}{\delta }_{n_3}^{m_3}\left({\delta }_{n_1+1}^{m_1}\sqrt{(n_0+1)(n_1+1)}-{\delta }_{n_1-1}^{m_1}\sqrt{n_1{n}_0}\right)\end{array}$$

(37)

$$\begin{array}{c}\hfill {K_2}_{n_0{n}_1{n}_2{n}_3}^{m_0{m}_1{m}_2{m}_3}=i{\delta }_{-n_0+n_1+n_2+n_3}^{-m_0+m_1+m_2+m_3}{\delta }_{n_1}^{m_1}{\delta }_{n_3}^{m_3}\left({\delta }_{n_2+1}^{m_2}\sqrt{(n_0+1)(n_2+1)}-{\delta }_{n_2-1}^{m_2}\sqrt{n_2{n}_0}\right)\end{array}$$

(38)

$$\begin{array}{c}\hfill {K_3}_{n_0{n}_1{n}_2{n}_3}^{m_0{m}_1{m}_2{m}_3}=i{\delta }_{-n_0+n_1+n_2+n_3}^{-m_0+m_1+m_2+m_3}{\delta }_{n_1}^{m_1}{\delta }_{n_2}^{m_2}\left({\delta }_{n_3+1}^{m_3}\sqrt{(n_0+1)(n_3+1)}-{\delta }_{n_3-1}^{m_3}\sqrt{n_3{n}_0}\right)\end{array}$$

(39)

$$\begin{array}{c}\hfill {J_1}_{n_0{n}_1{n}_2{n}_3}^{m_0{m}_1{m}_2{m}_3}=i{\delta }_{-n_0+n_1+n_2+n_3}^{-m_0+m_1+m_2+m_3}{\delta }_{n_1}^{m_1}{\delta }_{n_0}^{m_0}\left({\delta }_{n_2+1}^{m_2}\sqrt{(n_2+1)n_3}-{\delta }_{n_2-1}^{m_2}\sqrt{n_2(n_3+1)}\right)\end{array}$$

(40)

$$\begin{array}{c}\hfill {J_2}_{n_0{n}_1{n}_2{n}_3}^{m_0{m}_1{m}_2{m}_3}=i{\delta }_{-n_0+n_1+n_2+n_3}^{-m_0+m_1+m_2+m_3}{\delta }_{n_2}^{m_2}{\delta }_{n_0}^{m_0}\left({\delta }_{n_3+1}^{m_3}\sqrt{(n_3+1)n_1}-{\delta }_{n_3-1}^{m_3}\sqrt{n_3(n_1+1)}\right)\end{array}$$

(41)

$$\begin{array}{c}\hfill {J_3}_{n_0{n}_1{n}_2{n}_3}^{m_0{m}_1{m}_2{m}_3}=i{\delta }_{-n_0+n_1+n_2+n_3}^{-m_0+m_1+m_2+m_3}{\delta }_{n_3}^{m_3}{\delta }_{n_0}^{m_0}\left({\delta }_{n_1+1}^{m_1}\sqrt{(n_1+1)n_2}-{\delta }_{n_1-1}^{m_1}\sqrt{n_1(n_2+1)}\right)\end{array}$$

(42)

3.3. Linear Solutions and Helicity

In this subsection, we focus on the particular number of dimensions, i.e., $d=4$, and find the helicities of the plane wave solutions for the massless MHSYM model. We can use the expansion (26) and insert it into the linearised EoM obtained from (8). The component fields in the expansion satisfy

$$\square h_a^{n_0{n}_1{n}_2{n}_3}\left(x\right)-{\partial }_a{\partial }^b{h}_b^{n_0{n}_1{n}_2{n}_3}\left(x\right)=0$$

(43)

and they enjoy a linearised gauge symmetry of the form

$${\delta }_{\epsilon }{h}_a^{n_0{n}_1{n}_2{n}_3}\left(x\right)={\partial }_a{\epsilon }^{n_0{n}_1{n}_2{n}_3}\left(x\right).$$

(44)

To find out about the helicity of the field, we can write down a plane wave solution to the EoM (43), use the freedom available through (44) to fix the gauge and choose a direction of propagation (conventionally, we choose the z-direction). Such a solution is given by

$$h_{\pm }^{a{n}_0{n}_1{n}_2{n}_3}\left(x\right)={ϵ}_{(\pm )}^a{p}^{n_0{n}_1{n}_2{n}_3}{e}^{ikx},$$

(45)

where $k^2=0$, meaning that the field is massless, and where ${ϵ}_{(\pm )}^a={\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{cccc}0& 1& \pm i& 0\end{array}\right)$, and $p^{n_0{n}_1{n}_2{n}_3}$ is an a priori unconstrained polarisation factor in the infinite-dimensional unitary representation of the Lorentz group.

The helicity of a plane wave can be calculated as the eigenvalue of the rotation generator around the propagation axis. As we have followed the convention to choose the z-axis as the axis of propagation, we want to find the eigenvalue of the rotation operator $J_3$. When acting on (45), which is in a mixed representation, each generator will have two parts; one belonging to the finite-dimensional representation (indices $a,b$), and one belonging to the infinite-dimensional representation (indices $m_0,\cdots ,m_3$ and $n_0,\cdots ,n_3$ in case of the Hermite expansion in $d=4$), i.e.,

$${{{\left(J_3\right)}_{n_0{n}_1{n}_2{n}_3}^{m_0{m}_1{m}_2{m}_3}}^a}_b={\left(J_3\right)}_{n_0{n}_1{n}_2{n}_3}^{m_0{m}_1{m}_2{m}_3}{\delta }_b^a+{\delta }_{n_0{n}_1{n}_2{n}_3}^{m_0{m}_1{m}_2{m}_3}{{\left(J_3\right)}^a}_b,$$

(46)

where ${{\left(J_3\right)}^a}_b$ is in the fundamental (vector) representation of the Lorentz group, given explicitly by

$$J_3=\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& -i& 0\\ 0& i& 0& 0\\ 0& 0& 0& 0\end{array}\right).$$

(47)

We have found the eigenvectors of ${\left(J_3\right)}_{n_0{n}_1{n}_2{n}_3}^{m_0{m}_1{m}_2{m}_3}$ in [31]. They are given by

$$p^{n_0{n}_1{n}_2{n}_3}=d^{n_0{n}_3}{C}_{(r,\lambda )}^{n_1{n}_2},$$

(48)

with the coefficients $d^{n_0{n}_3}$ being arbitrary and $C_{(r,\lambda )}^{n_1{n}_2}$ given by

$$\begin{array}{c}\hfill C_{(r,\lambda )}^{n_1,n_2}={\delta }_r^{n_1+n_2}{i}^{k-n_1-1}{d^{r/2}}_{k-r/2,n_1-r/2}\left({\displaystyle \frac{\pi }{2}}\right),\end{array}$$

(49)

where ${d^l}_{m,n}\left(\beta \right)$ are the Wigner d-functions, $\lambda =(2k-r)$ with $r=n_1+n_2$, and with k having possible values $k=0,1,\cdots ,r.$ These coefficients, when contracted with the basis vectors, produce the expected azimuthal dependence in the auxiliary space, $C_{(r,\lambda )}^{n_1,n_2}{f}_{n_1}\left(u_x\right)f_{n_2}\left(u_y\right)\sim e^{i\lambda u_{\varphi }}$, where the summation convention for $n_1$ and $n_2$ is assumed. It is then straightforward to see that

$${{{\left(J_3\right)}_{n_0{n}_1{n}_2{n}_3}^{m_0{m}_1{m}_2{m}_3}}^a}_b{ϵ}_{(\pm )}^b{p}^{n_0{n}_1{n}_2{n}_3}=(\pm 1+\lambda ){ϵ}_{(\pm )}^a{p}^{m_0{m}_1{m}_2{m}_3},$$

(50)

where the polarisation coefficients of definite $(r,\lambda )$ are

$$p^{n_0{n}_1{n}_2{n}_3}=d^{n_0{n}_3}{C}_{(r,\lambda )}^{n_1{n}_2}.$$

(51)

All together, this means that the plane waves (45) with $p^{n_0{n}_1{n}_2{n}_3}$, given by (51) for some given $r=0,1,\dots$ and $\lambda =-r,-r+2,\dots ,r-2,r$, can carry helicity (50) in the range $-r-1,-r+1,\dots ,r-1,r+1$. A single value of helicity can appear in infinitely many polarisation factors (51), e.g., helicity 3 can appear for each even value of $r\ge 2$. We can also see that helicity is not a Lorentz invariant quantity for solutions such as (45). For example, if we boost the solution in the x direction, the $n_0$ and $n_1$ indices will get mixed, and the final expression will no longer have a well-defined helicity, i.e., it will be a superposition of terms with various values of $\lambda$. This is a characteristic of continuous spin particles, as emphasised in [30].

Another approach to learning about supported helicities already noted in [23] involves using spherical harmonics in the auxiliary space which are diagonal in the rotation generator around the axis of propagation. The axis of propagation $\widehat{\mathbf{k}}$ in spacetime defines the preferred axis ${\widehat{\mathbf{u}}}_{\widehat{\mathbf{k}}}\equiv \widehat{\mathbf{k}}$ in the auxiliary space, as well as corresponding spherical coordinates $u_r$, $u_{\widehat{\mathbf{k}},\theta }$ and $u_{\widehat{\mathbf{k}},\varphi }$ (where ${\widehat{\mathbf{u}}}_{\widehat{\mathbf{k}}}$ is tangent to the direction, where $u_{\widehat{\mathbf{k}},\theta }=0$).

$$g_{r_{0}nlm}(u,\widehat{\mathbf{k}})=f_{n_0}\left(u_0\right)F_n\left(u_r\right)Y_l^m(u_{\widehat{\mathbf{k}},\theta },u_{\widehat{\mathbf{k}},\varphi }),$$

(52)

where $F_n$ are Laguerre functions, $Y_l^m$ are spherical harmonics, and $n_0=0,1,2,\dots$, $n=0,1,2,\dots$, $l=0,1,2,\dots$, $m=-l,-l+1,\dots ,l$. The plane wave solutions for the MHS potential can then be expanded as

$${\mathbf{ϵ}}_{\sigma r_{0}nlm}\left(\mathbf{k}\right)e^{ik·x}{g}_{r_{0}nlm}(u,\widehat{\mathbf{k}}),\mathbf{k}·{\mathbf{ϵ}}_{\sigma r_{0}nlm}\left(\mathbf{k}\right)=0,k^2=0,$$

(53)

with $\sigma =\pm 1$. Helicity is $\sigma +m$, showing an infinite number of degrees of freedom for each helicity value.

4. Wigner’s Classification

The field we worked with above is massless, and to classify the possible excitations further, we need to find the value of the quartic Casimir operator of the Poincaré group. Here, we would like to review the basics of Wigner’s method for classifying elementary particles and highlight the relationship to the plane wave solutions of the linear equations of motion. This exposition closely follows the first volume of Weinberg’s Quantum Theory of Fields [33] while additional details can be found in [34,35,36]. Wigner’s classification of elementary particles [37] is a classification of a spacetime isometry group (in our case, the Poincaré group) represented in the space of one-particle states (see

Table 1. Representations of the Poincaré group.

${\mathit{C}}_2=-{\mathit{p}}^2$	Standard Momentum ${\mathit{k}}^{\mathit{\mu }}$	Little Group	Example
$p^2=-m^2$	$(\pm m,0,0,0)$	$SO\left(3\right)$	$W^{\pm }$ and Z bosons
$p^2=0$	$(\pm \omega ,0,0,\omega )$	$ISO\left(2\right)$	photons
$p^2=n^2$	$(0,0,0,n)$	$SO{(2,1)}^{\uparrow }$	tachyons
$p^2=0$	$(0,0,0,0)$	$SO{(3,1)}^{\uparrow }$	vacuum

).

In $d=4$, the Poincaré group has a quadratic and a quartic Casimir operator [38]

$$\begin{array}{c}\hfill C_2=-P^{\mu }{P}_{\mu }=-P^2,C_4=W^{\mu }{W}_{\mu }=W^2,\end{array}$$

(54)

where $P^{\mu }$ are the translation generators, the Pauli–Lubanski vector $W^{\mu }$ is defined as

$$W_{\rho }={\displaystyle \frac{1}{2}}{ϵ}_{\mu \nu \rho \kappa }{M}^{\mu \nu }{P}^{\kappa }$$

(55)

and the Lorentz generators are denoted by $M^{\mu \nu }$. It is convenient to label the single-particle states with the eigenvalues of the Casimir operator. Further, since momentum operators form an abelian subgroup, we work with their eigenvectors and label them as $|p^2,w^2,p^{\mu },\sigma \rangle$ such that

$$P^2|p^2,w^2,p^{\mu },\sigma \rangle =p^2|p^2,w^2,p^{\mu },\sigma \rangle ,W^2|p^2,w^2,p^{\mu },\sigma \rangle =w^2|p^2,w^2,p^{\mu },\sigma \rangle$$

(56)

and

$$P^{\mu }|p^2,w^2,p^{\mu },\sigma \rangle =p^{\mu }|p^2,w^2,p^{\mu },\sigma \rangle$$

(57)

with $\sigma$ labelling other degrees of freedom that are to be determined.

While translations act on the basis vectors as

$$U(\Delta x)|p^2,w^2,p^{\mu },\sigma \rangle =e^{-ip·\Delta x}|p^2,w^2,p^{\mu },\sigma \rangle ,$$

(58)

it can be shown that homogeneous Lorentz transformations act as

$$U\left(\mathsf{\Lambda }\right)|p^2,w^2,p^{\mu },\sigma \rangle =\mathcal{N}\sum _{{\sigma }^{\prime }}{\mathcal{D}}_{{\sigma }^{\prime }\sigma }\left(W(\mathsf{\Lambda },p)\right)|p^2,w^2,{{\mathsf{\Lambda }}^{\mu }}_{\nu }{p}^{\nu },{\sigma }^{\prime }\rangle ,$$

(59)

where $\mathcal{N}=\mathcal{N}\left(p^0\right)$ is a normalisation factor and $W(\mathsf{\Lambda },p)=S^{-1}\left(\mathsf{\Lambda }p\right)\mathsf{\Lambda }S\left(p\right)$ is an element of the little group for a particular standard momentum with $S\left(p\right)$ defined to satisfy $p^{\mu }={S^{\mu }}_{\nu }{k}^{\nu }$. Matrices ${\mathcal{D}}_{{\sigma }^{\prime }\sigma }\left(W\right)$ furnish an irreducible representation of the little group, and their construction is sufficient to properly characterise the one-particle state. Within a choice of standard momentum, the quartic Casimir of the Poincaré group is equal to the Casimir operator of the little group. We will be especially interested in the massless case, so we will focus in more detail on the group $ISO\left(2\right)$.

If we choose the momentum to be standard $p^{\mu }=k^{\mu }=(\omega ,0,0,\omega )$, we can input this momentum into (55) and explicitly find the components of the Pauli–Lubanski vector, which are the generators of the little group for the case of massless particles

$$W^{\mu }=\omega (J_3,J_1-K_2,J_2+K_1,J_3).$$

(60)

where $J_i$ are generators of rotations and $K_i$ are generators of boosts. It is convenient to name the generators

$$A=\omega (J_1-K_2),B=\omega (J_2+K_1).$$

(61)

It is easy to check that $A,B$ and $J_3$ span the Lie algebra $\mathfrak{iso}\left(2\right)$

$$\begin{array}{cc}\hfill [A,B]& =0,[J_3,A]=iB,[J_3,B]=-iA.\hfill \end{array}$$

(62)

The quartic Casimir in this choice of standard momentum is then given by

$$\begin{array}{cc}\hfill W_{\mu }{W}^{\mu }\equiv W^2=& {\omega }^2{(J_1-K_2)}^2+{\omega }^2{(J_2+K_1)}^2\hfill \end{array}$$

(63)

$$\begin{array}{cc}\hfill =& A^2+B^2=w^2.\hfill \end{array}$$

(64)

The faithful irreducible unitary representations of the little group $ISO\left(2\right)$, which have the non-vanishing value of the Casimir operator $W^2$, are necessarily infinite-dimensional [38]. If written in a basis diagonal in the rotation operator around the standard momentum, it can be seen that each irreducible representation contains an infinite tower of helicity states mixing under Lorentz transformations. For that reason, such representations are usually named “infinite-spin”. A different basis choice is possible, which motivates a different name—“continuous spins”. This class of representations was originally considered by Wigner to be unsuitable for physical use, since the infinite tower of helicities would have to correspond to an infinite heat capacity. However, in recent years, there has been a renewed interest in this class of particles and in analysing their kinematic and dynamical aspects ([5,25,30,39,40,41,42,43,44,45]).

There is also a possibility of a non-faithful representation of the little group $ISO\left(2\right)$ where the operators $A,B$ act trivially. In this case, $W^2$ gives a vanishing value, and the little group becomes isomorphic to $SO\left(2\right)$. The representations are one-dimensional, with the only non-trivial operator being the rotation around the standard momentum. The eigenvectors of this rotation are the ordinary helicity states (due to the eigenvalues of A and B being zero, helicity is now Lorentz-invariant) describing particles corresponding to massless fields of a fixed spin, such as the Maxwell field, linear Einstein gravity, higher-spin fields of the Fronsdal type, etc.

There is an important relationship between the matrices ${\mathcal{D}}_{{\sigma }^{\prime }\sigma }\left(W\right)$, which, as we have seen, act on the one-particle states, and the Lorentz transformation matrices we use to express quantum field components in different inertial frames. From the creation and annihilation operators, we can build a quantum field

$$h^r\left(x\right)=\sum _{\sigma }\int {\displaystyle \frac{d^3\mathbf{p}}{{\left(2\pi \right)}^{3/2}\sqrt{2{\omega }_{\mathbf{p}}}}}\left(u^r(\mathbf{p},\sigma )a(\mathbf{p},\sigma )e^{-ipx}+v^r(\mathbf{p},\sigma )a^{†}(\mathbf{p},\sigma )e^{ipx}\right),$$

(65)

where r stands for any set of Lorentz indices (e.g., for the MHS model, this includes both a and $\mu$ indices in the case of the Taylor expansion, and a and n indices in the case of the Hermite expansion). Under a Lorentz transformation, it transforms as

$$U\left(\mathsf{\Lambda }\right)h^r\left(x\right)U^{-1}\left(\mathsf{\Lambda }\right)=\sum _{s}D{\left({\mathsf{\Lambda }}^{-1}\right)}^{rs}{h}^s\left(\mathsf{\Lambda }x\right),$$

(66)

where $D\left(\mathsf{\Lambda }\right)$ is a representation of the Lorentz group (finite or infinite-dimensional) on the space of the fields. This equation can be seen as a compatibility condition [33,35,38] between the infinite-dimensional unitary Fock-space representation on the one-particle states and the representation of the homogeneous Lorentz group on the space of the fields. In a straightforward manner, it can be brought down to compatibility equations for the polarisation functions in the standard momentum $\mathbf{k}$ on the level of the Lie algebra of the little group

$$\begin{array}{c}\hfill \sum _{{\sigma }^{\prime }}{u}^r(\mathbf{k},{\sigma }^{\prime }){\mathcal{J}}_{{\sigma }^{\prime }\sigma }=\sum _s{J}^{rs}{u}^s(\mathbf{k},\sigma )\end{array}$$

(67)

$$\begin{array}{c}\hfill \sum _{{\sigma }^{\prime }}{v}^r(\mathbf{k},{\sigma }^{\prime }){\mathcal{J}}_{{\sigma }^{\prime }\sigma }^{*}=-\sum _s{J}^{rs}{v}^s(\mathbf{k},\sigma ),\end{array}$$

(68)

which follows from the expansion ${\mathcal{D}}_{{\sigma }^{\prime }\sigma }\approx {\delta }_{\sigma {\sigma }^{\prime }}+i\theta {\mathcal{J}}_{\sigma {\sigma }^{\prime }}$, and $L^{rs}\approx {\delta }^{rs}+i\theta J^{rs}$.

The polarisation functions in the standard momentum thus carry the representation of the little group, as do the one-particle states (56), and also contain information about the quartic Casimir operator. By classifying polarisation functions of a certain field, through solving the eigensystem

$$\begin{array}{c}\hfill \sum _{s}D{\left(W^2\right)}^{rs}{u}^s(\mathbf{k},\sigma )=w^2{u}^r(\mathbf{k},\sigma ),\end{array}$$

(69)

where $w^2$ on the right-hand side is constant due to (56), we can learn about supported particle types, i.e., the values of the Casimir operator $W^2$, associated with that particular field.

5. The Quartic Casimir in the Hermite Expansion

Our first goal is to build an explicit expression for (64) for the case of the plane wave solution (45) and then attempt to find possible eigenvalues. For the sake of the clarity of our argument, we will demonstrate the procedure on the Maxwell field before following these steps for the MHSYM component field.

In the case of electrodynamics, a plane-wave solution of the equations

$$\square A^{\mu }-{\partial }^{\mu }\partial ·A=0,$$

(70)

with momentum oriented in the z direction $k^{\mu }=(\omega ,0,0,\omega )$, is given by $A^{\mu }\left(x\right)={ϵ}_{\pm }^{\mu }{e}^{ikx}$, where ${ϵ}_{\pm }^{\mu }=(0,1,\pm i,0)$. Since $A^{\mu }\left(x\right)$ is a Lorentz vector, we can use the vector representation of the Lorentz generators ${{\left(M^{\mu \nu }\right)}^{\rho }}_{\kappa }=i\left({\eta }^{\mu \rho }{\delta }_{\kappa }^{\nu }-{\eta }^{\nu \rho }{\delta }_{\kappa }^{\mu }\right)$. The quartic Casimir element (64) is then explicitly given by

$$W^2={\omega }^2\left(\begin{array}{cccc}-2& 0& 0& 2\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ -2& 0& 0& 2\end{array}\right),$$

(71)

so it is readily visible by applying (69) that there is a single eigenvalue of the quartic Casimir, and it is vanishing ${{\left(W^2\right)}^{\mu }}_{\nu }{A}^{\nu }=0.$

5.1. Quartic Casimir of the On-Shell MHS Field

As stated, the MHS field is in a mixed representation of the Lorentz group—a direct product of the finite-dimensional vector representation and the infinite-dimensional unitary representation. As in (46), the generators will be a direct sum of two parts: one belonging to the finite-dimensional representation and one belonging to the infinite-dimensional representation, e.g.,

$${{{\left(J_1\right)}_{n_0{n}_1{n}_2{n}_3}^{m_0{m}_1{m}_2{m}_3}}^a}_b={\left(J_1\right)}_{n_0{n}_1{n}_2{n}_3}^{m_0{m}_1{m}_2{m}_3}{\delta }_b^a+{\delta }_{n_0{n}_1{n}_2{n}_3}^{m_0{m}_1{m}_2{m}_3}{{\left(J_1\right)}^a}_b.$$

(72)

Using capital $\mathbf{N}$ instead of the tuple $\left\{n_0{n}_1{n}_2{n}_3\right\}$

$${{{\left(J_1\right)}_{\mathbf{N}}^{\mathbf{M}}}^a}_b={\left(J_1\right)}_{\mathbf{N}}^{\mathbf{M}}{\delta }_b^a+{\delta }_{\mathbf{N}}^{\mathbf{M}}{{\left(J_1\right)}^a}_b.$$

(73)

The Casimir element (64) is then given by (repeated indices summed over)

$$\begin{array}{cc}\hfill {{{\left(W^2\right)}_{\mathbf{N}}^{\mathbf{M}}}^a}_c=& {\omega }^2\left({J_1^a}_b{J_1^b}_c+{J_2^a}_b{J_2^b}_c+{K_1^a}_b{K_1^b}_c+{K_2^a}_b{K_2^b}_c-2{J_1^a}_b{K_2^b}_c+2{K_1^a}_b{J_2^b}_c\right){\delta }_{\mathbf{N}}^{\mathbf{M}}\hfill \\ \hfill +& {\omega }^2\left({J_1}_{\mathbf{R}}^{\mathbf{M}}{J_1}_{\mathbf{N}}^{\mathbf{R}}+{J_2}_{\mathbf{R}}^{\mathbf{M}}{J_2}_{\mathbf{N}}^{\mathbf{R}}+{K_1}_{\mathbf{R}}^{\mathbf{M}}{K_1}_{\mathbf{N}}^{\mathbf{R}}+{K_2}_{\mathbf{R}}^{\mathbf{M}}{K_2}_{\mathbf{N}}^{\mathbf{R}}-2{J_1}_{\mathbf{R}}^{\mathbf{M}}{K_2}_{\mathbf{N}}^{\mathbf{R}}+2{K_1}_{\mathbf{R}}^{\mathbf{M}}{J_2}_{\mathbf{N}}^{\mathbf{R}}\right){\delta }_c^a\hfill \\ \hfill +& {\omega }^2(2{J_1^a}_c{J_1}_{\mathbf{N}}^{\mathbf{M}}+2{J_2^a}_c{J_2}_{\mathbf{N}}^{\mathbf{M}}+2{K_1^a}_c{K_1}_{\mathbf{N}}^{\mathbf{M}}+2{K_2^a}_c{K_2}_{\mathbf{N}}^{\mathbf{M}}\hfill \\ \hfill -& 2{K_2^a}_c{J_1}_{\mathbf{N}}^{\mathbf{M}}-2{K_2}_{\mathbf{N}}^{\mathbf{M}}{J_1^a}_c+2{K_1^a}_c{J_2}_{\mathbf{N}}^{\mathbf{M}}+2{K_1}_{\mathbf{N}}^{\mathbf{M}}{J_2^a}_c).\hfill \end{array}$$

(74)

The first bracket contains the finite-dimensional vector representation of the $W^2$, which is multiplied by ${\delta }_{\mathbf{N}}^{\mathbf{M}}$. As in the case of a finite-dimensional massless vector field, this will give zero when acting on the polarisation vector ${ϵ}^a$ found in (45).

The mixed contributions to the Casimir can be rewritten as

$$\begin{array}{cc}\hfill {{{\left(W_{mixed}^2\right)}_{\mathbf{N}}^{\mathbf{M}}}^a}_c=& 2{A^a}_c{A}_{\mathbf{N}}^{\mathbf{M}}+2{B^a}_c{B}_{\mathbf{N}}^{\mathbf{M}},\hfill \end{array}$$

where A and B were defined in (61). When ${A^a}_c$ or ${B^a}_c$ act on the polarisation vector ${ϵ}^a$, the result will be proportional to the momentum, e.g., for ${A^a}_c$

$$i\left(\begin{array}{cccc}0& 0& -1& 0\\ 0& 0& 0& 0\\ -1& 0& 0& 1\\ 0& 0& -1& 0\end{array}\right){\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{c}0\\ 1\\ \pm i\\ 0\end{array}\right)={\displaystyle \frac{\pm 1}{\sqrt{2}}}\left(\begin{array}{c}1\\ 0\\ 0\\ 1\end{array}\right)\propto p^a,$$

(75)

i.e., a pure gauge contribution in the finite-dimensional sector. A non-trivial eigenvalue of the quartic Casimir for the MHS field can thus come only from the second line of (74), which contains the infinite-dimensional part

$$\begin{array}{c}\hfill {\left(W_{inf}^2\right)}_{\mathbf{N}}^{\mathbf{M}}{{\delta }^a}_c={\omega }^2{\delta }_c^a({J_1}_{\mathbf{R}}^{\mathbf{M}}{J_1}_{\mathbf{N}}^{\mathbf{R}}+{J_2}_{\mathbf{R}}^{\mathbf{M}}{J_2}_{\mathbf{N}}^{\mathbf{R}}+{K_1}_{\mathbf{R}}^{\mathbf{M}}{K_1}_{\mathbf{N}}^{\mathbf{R}}+{K_2}_{\mathbf{R}}^{\mathbf{M}}{K_2}_{\mathbf{N}}^{\mathbf{R}}-2{K_2}_{\mathbf{R}}^{\mathbf{M}}{J_1}_{\mathbf{N}}^{\mathbf{R}}+2{J_2}_{\mathbf{R}}^{\mathbf{M}}{K_1}_{\mathbf{N}}^{\mathbf{R}}).\end{array}$$

(76)

We now use the explicit expressions for the infinite-dimensional generators (37–42) and arrive at the result written out without the use of the compact notation.

$$\begin{array}{cc}\hfill {\left(W_{inf}^2\right)}_{n_0{n}_1{n}_2{n}_3}^{m_0{m}_1{m}_2{m}_3}=& {\omega }^2{\delta }_{-n_0+n_1+n_2+n_3}^{-m_0+m_1+m_2+m_3}\times \hfill \\ \hfill (& 2{\delta }_{n_0}^{m_0}{\delta }_{n_1}^{m_1}{\delta }_{n_2}^{m_2}{\delta }_{n_3}^{m_3}(1+n_0+n_3)(1+n_1+n_2)\hfill \\ \hfill -& {\delta }_{n_0}^{m_0}{\delta }_{n_1}^{m_1}{\delta }_{n_2+2}^{m_2}{\delta }_{n_3-2}^{m_3}\sqrt{(n_2+1)(n_2+2)(n_3-1)n_3}\hfill \\ \hfill -& {\delta }_{n_0}^{m_0}{\delta }_{n_1}^{m_1}{\delta }_{n_2-2}^{m_2}{\delta }_{n_3+2}^{m_3}\sqrt{(n_2-1)n_2(n_3+2)(n_3+1)}\hfill \\ \hfill -& {\delta }_{n_0}^{m_0}{\delta }_{n_1-2}^{m_1}{\delta }_{n_2}^{m_2}{\delta }_{n_3+2}^{m_3}\sqrt{(n_1-1)n_1(n_3+1)(n_3+2)}\hfill \\ \hfill -& {\delta }_{n_0}^{m_0}{\delta }_{n_1+2}^{m_1}{\delta }_{n_2}^{m_2}{\delta }_{n_3-2}^{m_3}\sqrt{(n_1+1)(n_1+2)(n_3-1)n_3}\hfill \\ \hfill -& {\delta }_{n_0+2}^{m_0}{\delta }_{n_1+2}^{m_1}{\delta }_{n_2}^{m_2}{\delta }_{n_3}^{m_3}\sqrt{(n_0+1)(n_0+2)(n_1+1)(n_1+2)}\hfill \\ \hfill -& {\delta }_{n_0-2}^{m_0}{\delta }_{n_1-2}^{m_1}{\delta }_{n_2}^{m_2}{\delta }_{n_3}^{m_3}\sqrt{(n_0-1)n_0(n_1-1)n_1}\hfill \\ \hfill -& {\delta }_{n_0+2}^{m_0}{\delta }_{n_1}^{m_1}{\delta }_{n_2+2}^{m_2}{\delta }_{n_3}^{m_3}\sqrt{(n_0+1)(n_0+2)(n_2+2)(n_2+1)}\hfill \\ \hfill -& {\delta }_{n_0-2}^{m_0}{\delta }_{n_1}^{m_1}{\delta }_{n_2-2}^{m_2}{\delta }_{n_3}^{m_3}\sqrt{(n_0-1)n_0(n_2-1)n_2}\hfill \\ \hfill +& 2{\delta }_{n_0+1}^{m_0}{\delta }_{n_1}^{m_1}{\delta }_{n_2+2}^{m_2}{\delta }_{n_3-1}^{m_3}\sqrt{(n_0+1)(n_2+1)(n_2+2)n_3}\hfill \\ \hfill -& 2{\delta }_{n_0+1}^{m_0}{\delta }_{n_1}^{m_1}{\delta }_{n_2}^{m_2}{\delta }_{n_3+1}^{m_3}\left(\sqrt{(n_0+1)n_2{n}_2(n_3+1)}+\right.\hfill \\ \hfill & \left.+\sqrt{(n_0+1)(n_1+1)(n_1+1)(n_3+1)}\right)\hfill \\ \hfill -& 2{\delta }_{n_0-1}^{m_0}{\delta }_{n_1}^{m_1}{\delta }_{n_2}^{m_2}{\delta }_{n_3-1}^{m_3}\left(\sqrt{n_0(n_2+1)(n_2+1)n_3}+\sqrt{n_0{n}_1{n}_1{n}_3}\right)\hfill \\ \hfill +& 2{\delta }_{n_0-1}^{m_0}{\delta }_{n_1}^{m_1}{\delta }_{n_2-2}^{m_2}{\delta }_{n_3+1}^{m_3}\sqrt{n_0(n_2-1)n_2(n_3+1)}\hfill \\ \hfill +& 2{\delta }_{n_0+1}^{m_0}{\delta }_{n_1+2}^{m_1}{\delta }_{n_2}^{m_2}{\delta }_{n_3-1}^{m_3}\sqrt{(n_0+1)(n_1+1)(n_1+2)n_3}\hfill \\ \hfill +& 2{\delta }_{n_0-1}^{m_0}{\delta }_{n_1-2}^{m_1}{\delta }_{n_2}^{m_2}{\delta }_{n_3+1}^{m_3}\sqrt{n_0(n_1-1)n_1(n_3+1)}).\hfill \end{array}$$

(77)

When acting on the field polarisation factors $p^{n_0{n}_1{n}_2{n}_3}$ in (45), we obtain

$$\begin{array}{cc}\hfill (W_{inf}^2& {)}_{n_0{n}_1{n}_2{n}_3}^{m_0{m}_1{m}_2{m}_3}{p}^{n_0{n}_1{n}_2{n}_3}=\hfill \\ \hfill 2& p^{m_0{m}_1{m}_2{m}_3}\left[(1+m_0+m_3)(1+m_1+m_2)\right]\hfill \\ \hfill -& p^{m_0{m}_1(m_2-2)(m_3+2)}\sqrt{(m_2-1)m_2(m_3+1)(m_3+2)}\hfill \\ \hfill -& p^{m_0{m}_1(m_2+2)(m_3-2)}\sqrt{(m_2+1)(m_2+2)m_3(m_3-1)}\hfill \\ \hfill -& p^{m_0(m_1+2)m_2(m_3-2)}\sqrt{(m_1+1)(m_1+2)(m_3-1)m_3}\hfill \\ \hfill -& p^{m_0(m_1-2)m_2(m_3+2)}\sqrt{(m_1-1)m_1(m_3+1)(m_3+2)}\hfill \\ \hfill -& p^{(m_0-2)(m_1-2)m_2{m}_3}\sqrt{(m_0-1)m_0(m_1-1)m_1}\hfill \\ \hfill -& p^{(m_0+2)(m_1+2)m_2{m}_3}\sqrt{(m_0+1)(m_0+2)(m_1+1)(m_1+2)}\hfill \\ \hfill -& p^{(m_0-2)m_1(m_2-2)m_3}\sqrt{(m_0-1)m_0{m}_2(m_2-1)}\hfill \\ \hfill -& p^{(m_0+2)m_1(m_2+2)m_3}\sqrt{(m_0+1)(m_0+2)(m_2+1)(m_2+2)}\hfill \\ \hfill +2& p^{(m_0-1)m_1(m_2-2)(m_3+1)}\sqrt{m_0(m_2-1)m_2(m_3+1)}\hfill \\ \hfill -2& p^{(m_0-1)m_1{m}_2(m_3-1)}\sqrt{m_0{m}_2{m}_2{m}_3)}\hfill \\ \hfill -2& p^{(m_0+1)m_1{m}_2(m_3+1)}\sqrt{(m_0+1)(m_2+1)(m_2+1)(m_3+1)}\hfill \\ \hfill +2& p^{(m_0+1)m_1(m_2+2)(m_3-1)}\sqrt{(m_0+1)(m_2+1)(m_2+2)m_3}\hfill \\ \hfill -2& p^{(m_0-1)m_1{m}_2(m_3-1)}\sqrt{m_0(m_1+1)(m_1+1)m_3}\hfill \\ \hfill +2& p^{(m_0-1)(m_1-2)m_2(m_3+1)}\sqrt{m_0(m_1-1)m_1(m_3+1)}\hfill \\ \hfill +2& p^{(m_0+1)(m_1+2)m_2(m_3-1)}\sqrt{(m_0+1)(m_1+1)(m_1+2)m_3}\hfill \\ \hfill -2& p^{(m_0+1)m_1{m}_2(m_3+1)}\sqrt{(m_0+1)m_1{m}_1(m_3+1)}.\hfill \end{array}$$

(78)

5.2. Casimir Eigenvalue Problem

To learn about the particle spectrum of our theory, following (69), we should solve the eigensystem

$${\left(W_{inf}^2\right)}_{n_0{n}_1{n}_2{n}_3}^{m_0{m}_1{m}_2{m}_3}{p}^{n_0{n}_1{n}_2{n}_3}=w^2{p}^{m_0{m}_1{m}_2{m}_3}.$$

(79)

Even before explicitly trying to find eigenvectors and eigenvalues in (79), we can conclude from (78) that there will exist non-trivial states, i.e., the expression (78) shows that a polarisation factor $p^{n_0{n}_1{n}_2{n}_3}$ used in (45) will in general not give a vanishing eigenvalue, through which we can confirm that the MHS formalism supports a description of the infinite spin particles.

One way of tackling the eigenvalue problem is by computer-assisted iterative solving, which could give us a hint for an appropriate ansatz. It can be seen by the structure of the Casimir that any eigenvector should necessarily have an infinite number of components (terms such as ${\delta }_{n_0-1}^{m_0}{\delta }_{n_1}^{m_1}{\delta }_{n_2}^{m_2}{\delta }_{n_3-1}^{m_3}$ will always simultaneously raise the values of the 0th and 3rd index; thus a closed solution cannot have a finite number of terms), so we could only hope for hints coming from a truncated calculation. The Casimir operator can be rewritten in a basis of eigenvectors of $J_3$, which we found in [31], but the complexity of the problem remains. So far, we are left with making educated guesses, and one of them comes from the “massless limit” of the eigenvectors of ${\overrightarrow{J}}^2$.

5.2.1. Massless Limit of Massive States

In the case of massive particles, the little group is $SO\left(3\right)$, and the Casimir operator is simply ${\overrightarrow{J}}^2$. An appropriately performed Inönü–Wigner contraction of a representation of $SO\left(3\right)$ can give us a representation of $ISO\left(2\right)$, which is the little group in the case of massless particles. The limiting procedure entails the limits $m\to 0,v\to 1$ while keeping fixed $m\gamma ={\displaystyle \frac{m}{\sqrt{1-v^2}}}=\omega$. In the representation by Hermite functions, we have

$$\begin{array}{cc}\hfill {{\left(\overrightarrow{J}\right)}^2}_{n_0{n}_1{n}_2{n}_3}^{m_0{m}_1{m}_2{m}_3}=& {\delta }_{-n_0+n_1+n_2+n_3}^{-m_0+m_1+m_2+m_3}{\delta }_{n_0}^{m_0}\times \hfill \\ \hfill (2& {\delta }_{n_1}^{m_1}{\delta }_{n_2}^{m_2}{\delta }_{n_3}^{m_3}(n_1+n_2+n_3+n_1{n}_2+n_2{n}_3+n_3{n}_1)\hfill \\ \hfill -& {\delta }_{n_1}^{m_1}{\delta }_{n_2-2}^{m_2}{\delta }_{n_3+2}^{m_3}\sqrt{(n_2-1)n_2(n_3+1)(n_3+2)}\hfill \\ \hfill -& {\delta }_{n_1}^{m_1}{\delta }_{n_2+2}^{m_2}{\delta }_{n_3-2}^{m_3}\sqrt{(n_2+1)(n_2+2)(n_3-1)n_3}\hfill \\ \hfill -& {\delta }_{n_1+2}^{m_1}{\delta }_{n_2}^{m_2}{\delta }_{n_3-2}^{m_3}\sqrt{(n_3-1)n_3(n_1+1)(n_1+2)}\hfill \\ \hfill -& {\delta }_{n_1-2}^{m_1}{\delta }_{n_2}^{m_2}{\delta }_{n_3+2}^{m_3}\sqrt{(n_3+1)(n_3+2)(n_1-1)n_1}\hfill \\ \hfill -& {\delta }_{n_1-2}^{m_1}{\delta }_{n_2+2}^{m_2}{\delta }_{n_3}^{m_3}\sqrt{(n_1-1)n_1(n_2+1)(n_2+2)}\hfill \\ \hfill -& {\delta }_{n_1+2}^{m_1}{\delta }_{n_2-2}^{m_2}{\delta }_{n_3}^{m_3}\sqrt{(n_1+1)(n_1+2)(n_2-1)n_2}).\hfill \end{array}$$

(80)

The simplest simultaneous eigenvector of ${\overrightarrow{J}}^2$ (80) and $J_3$ (42) in the representation over Hermite functions is

$${\mathsf{\Phi }}_{n=0,s=0,\lambda =0}\left(u\right)={\delta }_0^{n_0}{\delta }_0^{n_1}{\delta }_0^{n_2}{\delta }_0^{n_3}{f}_{n_0{n}_1{n}_2{n}_3}\left(u\right),$$

(81)

where $n=n_1+n_2+n_3$ and s corresponds to the eigenvalue of ${\overrightarrow{J}}^2=s(s+1)$, while $\lambda$ is an eigenvalue of $J_3$. We can boost (81) with velocity v in the z direction to prepare it for the massless limit. The transformation matrices for a finite boost are (see [31] and (36))

$$\begin{array}{cc}\hfill L_{n_0{n}_1{n}_2{n}_3}^{m_0{m}_1{m}_2{m}_3}& \left(v\right)={(-1)}^{m_0+m_3}\sqrt{{\displaystyle \frac{m_3!n_0!}{n_3!m_0!}}}{\delta }_{-n_0+n_1+n_2+n_3}^{-m_0+m_1+m_2+m_3}{\delta }_{n_1}^{m_1}{\delta }_{n_2}^{m_2}\times \hfill \\ \hfill & \sum _{j=0}^{m_0}\left({\displaystyle \genfrac{}{}{0pt}{}{m_0}{j}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{n_3}{m_3-j}}\right){(-1)}^j{\sqrt{1-v^2}}^{m_3+m_0+1-2j}{v}^{2j-m_3+n_3}.\hfill \end{array}$$

(82)

Boosting the chosen eigenvector, we obtain

$$\begin{array}{c}\hfill \sum _{n_0,n_1,n_2,n_3=0}^{\infty }{L}_{n_0{n}_1{n}_2{n}_3}^{m_0{m}_1{m}_2{m}_3}\left(v\right)·{\delta }_0^{n_0}{\delta }_0^{n_1}{\delta }_0^{n_2}{\delta }_0^{n_3}={(-1)}^{m_0}{\delta }_0^{m_1}{\delta }_0^{m_2}{\delta }^{m_0,m_3}\sqrt{1-v^2}{(-v)}^{m_3}.\end{array}$$

(83)

Since $m\gamma \to \omega$ while $v\to 1$, we can divide this result by m to obtain a useful result in the limiting procedure

$$p^{n_0{n}_1{n}_2{n}_3}\to {\delta }_0^{m_1}{\delta }_0^{m_2}{\delta }^{m_0,m_3}.$$

(84)

A more general case could be an eigenvector of $J_3$ and $J^2$ of the form

$${\mathsf{\Phi }}_{n=n,s=n,\lambda =n}\left(u\right)=z^{\left(n_0\right)}{C}_{(n,n)}^{n_1{n}_2}{\delta }_0^{n_3}{f}_{n_0{n}_1{n}_2{n}_3}\left(u\right).$$

(85)

In the sector $n_1+n_2+n_3=n$, where $s=n$ and $\lambda =n$, there is only one such vector, and the factor $C_{(n,n)}^{n_1{n}_2}$ is given in (49). The index $n_3$ has to be equal to 0, while $n_0$ is arbitrary (or fixed by a choice of $N=-n_0+n$), meaning that the factor $z^{\left(n_0\right)}$ must have the form

$$z^{\left(n_0\right)}=const.·{\delta }_{n-N}^{n_0}.$$

(86)

We boost (85) in the z direction to prepare it for the massless limit

$$\begin{array}{cc}\hfill L_{n_0{n}_1{n}_2{n}_3}^{m_0{m}_1{m}_2{m}_3}\left(v\right)z^{\left(n_0\right)}{C}_{n,n}^{n_1{n}_2}{\delta }_0^{n_3}=& {(-1)}^{m_0}\sqrt{{\displaystyle \frac{m_0!}{m_3!(m_0-m_3)!}}}{z}^{m_0-m_3}{C}_{(m_1+m_2,m_1+m_2)}^{m_1{m}_2}{\sqrt{1-v^2}}^{m_0-m_3+1}{(-v)}^{m_3}.\hfill \end{array}$$

(87)

5.2.2. Eigenvector Candidates

The result in (87) motivates an ansatz of the form

$$\begin{array}{c}\hfill p^{n_0{n}_1{n}_2{n}_3}={\delta }^{n_0,n_3}{C}_{r,r}^{n_1{n}_2}{c}^{\left(n_3\right)}.\end{array}$$

(88)

Upon inserting this into (79), we obtain two independent equations for $c^{\left(m_3\right)}$

$$\begin{array}{c}\hfill c^{\left(m_3\right)}(1+2m_3)-c^{(m_3-1)}{m}_3-c^{(m_3+1)}(m_3+1)=-{\displaystyle \frac{w^2}{2(1+r)}}{c}^{\left(m_3\right)},\end{array}$$

(89)

$$c^{(m_3+2)}+c^{\left(m_3\right)}-2c^{(m_3+1)}=0.$$

(90)

The solution is given by $w^2=0$ and

$$c^{\left(m_3\right)}=c^{\left(0\right)}.$$

(91)

This gives the polarisation factor

$$\begin{array}{c}\hfill p^{n_0{n}_1{n}_2{n}_3}={\delta }^{n_0,n_3}{C}_{(r,r)}^{n_1{n}_2},\end{array}$$

(92)

which is simultaneously an eigenvector of $J_3$ with helicity $\lambda =r$. The norm of this solution is not finite, and we can explicitly see that in the sum over Hermite functions in the auxiliary space, the eigenvector will contain a delta function. As an example with $r=0$, from the completeness identity of Hermite functions, we find

$$\sum _{n_0,n_1,n_2,n_3=0}^{\infty }{\delta }^{n_0,n_3}{\delta }_0^{n_1}{\delta }_0^{n_2}{f}_{n_0}\left(u_0\right)f_{n_1}\left(u_1\right)f_{n_2}\left(u_2\right)f_{n_3}\left(u_3\right)={\delta }^{\left(2\right)}(u_0-u_3)e^{-{\displaystyle \frac{u_1^2+u_2^2}{2}}}.$$

(93)

Through this approach, we were able to obtain a solution to the Equation (79) with a vanishing eigenvalue of the quartic Casimir $W^2$, and with an arbitrary integer helicity. This would, by definition, correspond to an ordinary higher-spin massless field. Observe that this solution is not square-integrable, which means that if it is present in the physical spectrum, it must belong to a continuous spectrum. To see this, a different approach will be used for the complete characterisation of the particle spectrum, which we show in the next section.

6. The Quartic Casimir for an On-Shell Master Field

Consider the linearised equations of motion one obtains if the integration over the auxiliary space is not performed prior to extremising the action

$$\square h_a(x,u)-{\partial }_a{\partial }^b{h}_b(x,u)=0.$$

(94)

As in the previous section, we can fix the gauge to ${\partial }^a{h}_a(x,u)=0$ and consider solutions representing plane waves directed along the z-axis

$$h_a(x,u)={ϵ}_a\mathsf{\Phi }\left(u\right)e^{ikx},$$

(95)

where $k^a=(\omega ,0,0,\omega )$ and ${ϵ}_a={\displaystyle \frac{1}{\sqrt{2}}}(0,1,\pm i,0)$. Now, let us consider an active Lorentz transformation following the transformation properties (28)

$$h_a^{\prime }(x^c,u_d)={{\mathsf{\Lambda }}_a}^b{h}_b({\left({\mathsf{\Lambda }}^{-1}x\right)}^c,{(u·\mathsf{\Lambda })}_d).$$

(96)

If we expand the Lorentz transformation matrix up to the first-order

$${{\mathsf{\Lambda }}^a}_b\approx {\delta }_b^a+i\psi {G^a}_b,$$

with $\psi$ being the expansion parameter (the parameter can be an angle if $\mathsf{\Lambda }$ is a rotation, or rapidity in case of boosts), then ${{\left({\mathsf{\Lambda }}^{-1}\right)}^a}_b\approx {\delta }_b^a-i\psi {G^a}_b$, and since ${{\left({\mathsf{\Lambda }}^{-1}\right)}^a}_b={{\mathsf{\Lambda }}_b}^a$, it is true that ${G^a}_b=-{G_b}^a$. Through a simple expansion, we obtain

$$\begin{array}{c}\hfill h_a^{\prime }(x^c,u_d)\approx h_a(x,u)+i\psi ({G_a}^b{h}_b(x,u)+{G_c}^b{x}^c{\partial }_b^x{h}_a(x,u)+{G^b}_c{u}_b{\partial }_u^c{h}_a(x,u)).\end{array}$$

(97)

In the case of our solution (95), the action of a generator of the Lorentz group, where $D\left(G\right)$ is a representation of the generator G, becomes

$$D\left(G\right)·h_a(x,u)=\left({G_a}^b{ϵ}_b\mathsf{\Phi }\left(u\right)+{G_c}^b{x}^{c}i{k}_b\mathsf{\Phi }\left(u\right)+{G^b}_c{u}_a{\partial }_u^c\mathsf{\Phi }\left(u\right){ϵ}_b\right)e^{ikx}.$$

(98)

We would now like to examine the behaviour of (95) under the action of the generators $A,B$ of the little group $ISO\left(2\right)$ with the reference momentum $k^a=(\omega ,0,0,\omega )$. If we are able to find the eigenfunctions of the mentioned generators, they will be the on-shell basis for the representation of the little group. Since $A,B$ commute, their eigenfunctions will correspond to the plane-wave basis of $ISO\left(2\right)$ seen in [38].

It is straightforward to find the explicit vector representations for the operators $A=J_1-K_2$ and $B=J_2+K_1$.

$$A=\omega \left(\begin{array}{cccc}0& 0& i& 0\\ 0& 0& 0& 0\\ i& 0& 0& i\\ 0& 0& -i& 0\end{array}\right),B=\omega \left(\begin{array}{cccc}0& -i& 0& 0\\ -i& 0& 0& -i\\ 0& 0& 0& 0\\ 0& 0& i& 0\end{array}\right).$$

(99)

We see from (98) the three possible terms, of which only one will be non-trivial. Since ${A^a}_b{k}^b={B^a}_b{k}^b=0$, and ${A^a}_b{ϵ}^b\propto k^b,{B^a}_b{ϵ}^b\propto k^b$, which is a pure gauge contribution, the only important term in the equation (98) in the case of the generators A and B is the last one, of the form ${G^b}_c{u}_b{\partial }_u^c\mathsf{\Phi }\left(u\right)$. We now explicitly state the differential equations for A and B.

$$\begin{array}{cc}\hfill A·\mathsf{\Phi }\left(u\right)=& u_a{A^a}_b{\partial }_u^b\mathsf{\Phi }\left(u\right)\hfill \end{array}$$

(100)

$$\begin{array}{cc}\hfill =& i\omega (u_t-u_z){\displaystyle \frac{\partial }{\partial u_y}}+i\omega u_y\left({\displaystyle \frac{\partial }{\partial u_t}}+{\displaystyle \frac{\partial }{\partial u_z}}\right)\mathsf{\Phi }\left(u\right).\hfill \end{array}$$

(101)

In null-coordinates $u_{+}=u_t+u_z,u_{-}=u_t-u_z$, it becomes somewhat simpler

$$\begin{array}{c}\hfill A·\mathsf{\Phi }\left(u\right)=i\omega \left[u_{-}{\displaystyle \frac{\partial }{\partial u_y}}+2u_y{\displaystyle \frac{\partial }{\partial u_{+}}}\right]\mathsf{\Phi }(u_{+},u_{-},u_x,u_y).\end{array}$$

(102)

The equation for B is similarly

$$\begin{array}{c}\hfill B·\mathsf{\Phi }\left(u\right)=-i\omega \left[u_{-}{\displaystyle \frac{\partial }{\partial u_x}}+2u_x{\displaystyle \frac{\partial }{\partial u_{+}}}\right]\mathsf{\Phi }(u_{+},u_{-},u_x,u_y).\end{array}$$

(103)

Similarly to (69), we want to find functions $\mathsf{\Phi }\left(u\right)$ that satisfy the eigensystem

$$\begin{array}{cc}\hfill A·\mathsf{\Phi }\left(u\right)& =\alpha \mathsf{\Phi }\left(u\right)\hfill \end{array}$$

(104)

$$\begin{array}{cc}\hfill B·\mathsf{\Phi }\left(u\right)& =\beta \mathsf{\Phi }\left(u\right).\hfill \end{array}$$

(105)

The solutions to these equations for A and B, separately, are

$$\begin{array}{c}\hfill {\mathsf{\Phi }}_A\left(u\right)=exp\left({\displaystyle \frac{-i\alpha u_y}{\omega u_{-}}}\right)G_1\left(u_{-},u_x,-{\left(u_t\right)}^2+{\left(u_y\right)}^2+{\left(u_z\right)}^2\right)\end{array}$$

(106)

$$\begin{array}{c}\hfill {\mathsf{\Phi }}_B\left(u\right)=exp\left({\displaystyle \frac{i\beta u_x}{\omega u_{-}}}\right)G_2\left(u_{-},u_y,-{\left(u_t\right)}^2+{\left(u_x\right)}^2+{\left(u_z\right)}^2\right),\end{array}$$

(107)

where $G_1$ and $G_2$ are arbitrary functions of their respective variables. We can write down a simultaneous solution with $G_r$, an arbitrary function, as

$${\mathsf{\Phi }}_{\alpha \beta r}\left(u\right)=exp\left(i{\displaystyle \frac{\beta u_x-\alpha u_y}{\omega u_{-}}}\right)G_r\left(u_{-},u_{\mu }{u}^{\mu }\right),$$

(108)

where $\alpha$ and $\beta$ stand for the eigenvalues of A and B, and r stands for any additional indices that may be used to discriminate between different solutions. The explicit representation for $W^2$ is

$$\begin{array}{c}{W}^2=A^2+B^2\hfill \end{array}$$

(109)

$$\begin{array}{cc}\hfill =& -{\omega }^2\left(u_{-}^2\left({\displaystyle \frac{{\partial }^2}{\partial u_x^2}}+{\displaystyle \frac{{\partial }^2}{\partial u_y^2}}\right)+4u_{-}\left(u_x{\displaystyle \frac{\partial }{\partial u_x}}+u_y{\displaystyle \frac{\partial }{\partial u_y}}+1\right){\displaystyle \frac{\partial }{\partial u_{+}}}+4(u_x^2+u_y^2){\displaystyle \frac{{\partial }^2}{\partial u_{+}^2}}\right)\hfill \end{array}$$

(110)

and we can immediately see that solutions (108) are eigenfunctions of the Casimir operator

$$W^2·\mathsf{\Phi }\left(u\right)=({\alpha }^2+{\beta }^2)\mathsf{\Phi }\left(u\right)=w^2\mathsf{\Phi }\left(u\right).$$

(111)

As expected from the properties of the little group, the eigenvalues of the Casimir $W^2$ are non-negative. Analogous solutions were obtained in [30] by examining a scalar master field as a wave function of the continuous spin particle. The analysis here tells us that, due to gauge invariance (44), having the (frame) vector index on a master field does not change the result for the form obtained in [30] for the scalar master field.

A complete orthonormal basis in the auxiliary space can be built from functions of the form (108) for a specific choice of the standard momentum. One possibility is to define

$$f_{\alpha \beta nl}\left(u\right)={\displaystyle \frac{1}{\sqrt{2{\pi }^2}}}exp\left(i{\displaystyle \frac{\beta u_x-\alpha u_y}{\omega u_{-}}}\right)h_n\left(\omega u_{-}\right)h_l\left(\omega u_{-}{u}^2\right),$$

(112)

where $h_n\left(x\right)$ are any orthonormal and complete functions defined on $\mathbb{R}$, such as Hermite functions.

For $\alpha =\beta =0$, one has particle states with the vanishing eigenvalue of the second Casimir, which means that they belong to standard massless IRREP in the Wigner classification. We now see explicitly how such states appear inside the continuous spectrum that is parametrised by the eigenvalue of the second Casimir $w^2={\alpha }^2+{\beta }^2$.

We prove that the functions $f_{\alpha \beta nl}\left(u\right)$ are orthonormal:

$$\begin{array}{cc}\hfill \int d^{4}u{f}_{{\alpha }^{\prime }{\beta }^{\prime }{n}^{\prime }{l}^{\prime }}{\left(u\right)}^{*}{f}_{\alpha \beta nl}\left(u\right)={\displaystyle \frac{1}{{\left(2\pi \right)}^2}}& {\int }_{-\infty }^{\infty }d{u}_{-}{h}_{n^{\prime }}{\left(\omega u_{-}\right)}^{*}{h}_n\left(\omega u_{-}\right)\hfill \\ \hfill \times & {\int }_{-\infty }^{\infty }d{u}_1{\int }_{-\infty }^{\infty }d{u}_2{e}^{-i{\displaystyle \frac{(\alpha -{\alpha }^{\prime })u_2-(\beta -{\beta }^{\prime })u_1}{\omega u_{-}}}}\hfill \end{array}$$

(113)

$$\begin{array}{cc}\hfill \times & {\int }_{-\infty }^{\infty }d{u}_{+}{h}_{l^{\prime }}{\left(\omega u_{-}{u}^2\right)}^{*}{h}_l\left(\omega u_{-}{u}^2\right).\hfill \end{array}$$

(114)

We can use a substitution

$$w\equiv \omega u_{+}{u}^2=\omega u_{+}(-u_{+}{u}_{-}+u_1^2+u_2^2)$$

(115)

to write the third integral as

$${\int }_{-\infty }^{\infty }d{u}_{+}{h}_{l^{\prime }}{\left(\omega u_{+}{u}^2\right)}^{*}{h}_l\left(\omega u_{+}{u}^2\right)={\displaystyle \frac{1}{\omega {\left(u_{-}\right)}^2}}{\int }_{-\infty }^{\infty }dw{h}_{l^{\prime }}{\left(w\right)}^{*}{h}_l\left(w\right)={\displaystyle \frac{{\delta }_{l^{\prime }l}}{\omega {\left(u_{-}\right)}^2}}.$$

(116)

The second integral gives

$${\int }_{-\infty }^{\infty }d{u}_1{\int }_{-\infty }^{\infty }d{u}_2{e}^{-i{\displaystyle \frac{(\alpha -{\alpha }^{\prime })u_2-(\beta -{\beta }^{\prime })u_1}{\omega u_{-}}}}={\left(2\pi \omega u_{-}\right)}^2\delta ({\alpha }^{\prime }-\alpha )\delta ({\beta }^{\prime }-\beta )$$

(117)

and in the first integral, we have

$${\int }_{-\infty }^{\infty }d{u}_{-}{h}_{n^{\prime }}{\left(\omega u_{-}\right)}^{*}{h}_n\left(\omega u_{-}\right)={\displaystyle \frac{1}{{\omega }^2}}{\delta }_{n{n}^{\prime }}.$$

(118)

Finally, we confirm that the basis functions are orthonormal

$$\int d^{4}u{f}_{{\alpha }^{\prime }{\beta }^{\prime }{n}^{\prime }{l}^{\prime }}{\left(u\right)}^{*}{f}_{\alpha \beta nl}\left(u\right)=\delta ({\alpha }^{\prime }-\alpha )\delta ({\beta }^{\prime }-\beta ){\delta }_{l^{\prime }l}{\delta }_{n^{\prime }n}.$$

(119)

We can also prove that the choice (112) is complete

$$\begin{array}{c}\hfill \sum _{n=0}^{\infty }\sum _{l=0}^{\infty }{\int }_{-\infty }^{\infty }d\alpha {\int }_{-\infty }^{\infty }d\beta f_{\alpha \beta nl}{\left(u^{\prime }\right)}^{*}{f}_{\alpha \beta nl}\left(u\right)={\displaystyle \frac{1}{2{\pi }^2}}\sum _l{h}_l{\left(\omega u_{-}^{\prime }{u}^{\prime 2}\right)}^{*}{h}_l\left(\omega u_{-}{u}^2\right)\\ \hfill \times {\int }_{-\infty }^{\infty }d\alpha {\int }_{-\infty }^{\infty }d\beta e^{i{\displaystyle \frac{\alpha (u_2^{\prime }-u_2)-\beta (u_1^{\prime }-u_1)}{\omega u_{-}}}}\sum _n{h}_n{\left(\omega u_{-}^{\prime }\right)}^{*}{h}_n\left(\omega u_{-}\right).\end{array}$$

(120)

Elementary functions such as Hermite functions satisfy completeness relations

$$\begin{array}{cc}\hfill & \sum _{n=0}^{\infty }{h}_n{\left(\omega u_{-}^{\prime }\right)}^{*}{h}_n\left(\omega u_{-}\right)=\delta \left(\omega (u_{-}^{\prime }-u_{-})\right)={\displaystyle \frac{1}{\left|\omega \right|}}\delta (u_{-}-u_{-}^{\prime })\hfill \end{array}$$

(121)

$$\begin{array}{cc}\hfill & \sum _{l=0}^{\infty }{h}_l{\left(\omega u_{-}^{\prime }{u}^{\prime 2}\right)}^{*}{h}_l\left(\omega u_{-}{u}^2\right)=\delta (\omega u_{-}{u}^2-\omega u_{-}^{\prime }{u}^{\prime 2}).\hfill \end{array}$$

(122)

With the exponential functions, we have

$$\begin{array}{cc}\hfill & {\int }_{-\infty }^{\infty }d\beta e^{i{\displaystyle \frac{\beta (u_1-u_1^{\prime })}{\omega u_{-}}}}=2\pi \left|\omega u_{-}\right|\delta (u_1-u_1^{\prime })\hfill \end{array}$$

(123)

$$\begin{array}{cc}\hfill & {\int }_{-\infty }^{\infty }d\alpha e^{i{\displaystyle \frac{\alpha (u_2^{\prime }-u_2)}{\omega u_{-}}}}=2\pi \left|\omega u_{-}\right|\delta (u_2-u_2^{\prime }).\hfill \end{array}$$

(124)

We can insert the results into (120) and obtain

$$\begin{array}{cc}\hfill \sum _n\sum _l& \int d\alpha \int d\beta f_{\alpha \beta nl}{\left(u^{\prime }\right)}^{*}{f}_{\alpha \beta nl}\left(u\right)=\hfill \\ \hfill =& 2\omega {\left(u_{-}\right)}^2\delta (u_{-}-u_{-}^{\prime })\delta (u_1-u_1^{\prime })\delta (u_2-u_2^{\prime })\delta (\omega u_{-}{u}^2-\omega u_{-}^{\prime }{u}^{\prime 2})\hfill \\ \hfill =& 2\delta (u_{-}-u_{-}^{\prime })\delta (u_1-u_1^{\prime })\delta (u_2-u_2^{\prime })\omega {\left(u_{-}\right)}^2\delta \left(\omega {\left(u_{-}\right)}^2(u_{+}^{\prime }-u_{+})\right)\hfill \\ \hfill =& 2\delta (u_{-}-u_{-}^{\prime })\delta (u_1-u_1^{\prime })\delta (u_2-u_2^{\prime })\delta (u_{+}^{\prime }-u_{+})\hfill \end{array}$$

(125)

$$={\delta }^4(u-u^{\prime }),$$

(126)

which is the completeness relation.

The indices $n,l$ are little-group-invariant, as is the combination ${\alpha }^2+{\beta }^2$, so we conclude that the content of the massless theory is two polarisations × infinite × infinite number of a continuous number of infinite-spin particles. On-shell, a classical solution corresponding to a non-vanishing value of the quartic Casimir $W^2={\alpha }^2+{\beta }^2$, can be chosen, such as, e.g.,

$$h_{a\left(\alpha \beta nlk\right)}^{\pm }(x,u)={ϵ}_a^{\pm }{f}_{\alpha \beta nl}(k,u)e^{ikx},$$

(127)

where we have emphasised that the polarisation functions have an implicit dependence on the momentum $k^{\mu }$. To construct a field variable which is square integrable in the auxiliary space, we can form a superposition such as

$$\begin{array}{c}\hfill h_{a\left(nlk\right)}^{\pm }(x,u)=\int d\alpha d\beta c(\alpha ,\beta )h_{a\left(\alpha \beta nlk\right)}^{\pm }(x,u),\end{array}$$

(128)

where ${\int d\alpha d\beta \left|c(\alpha ,\beta )\right|}^2<\infty$. Such a superposition goes over various values of $w^2={\alpha }^2+{\beta }^2$, reminiscent of unparticle physics [46] where fields can be thought of as having a continuous distribution of mass [47,48].

7. Discussion

We explore the particle content of massive and massless free theories of a master field, assuming the requirement that it is square integrable in the auxiliary space. The motivation for such a requirement is the following: (1) it ensures a well-defined integration over the auxiliary space without using a nonstandard measure of integration that could violate MHS gauge invariance, (2) it ensures that auxiliary space integration gives a finite value which is required to have finiteness of observables such as energy, and (3) it ensures the unitarity of the representation of the part of the Lorentz group that acts on the auxiliary space. The consequence is that it leads to nonstandard methods of unpacking the spacetime content of the master field. We examine various choices of bases in the auxiliary space where the coefficients of expansion serve as the usual spacetime fields. The technical difficulty is that the representations of the Lorentz group are infinite-dimensional. The result is that in the massless case, the theory contains continuous spin particles in different Poincaré representations specified by two discrete and one continuous label (the continuous label being the eigenvalue of the quartic Casimir of the Poincaré group). In the massive case, the theory contains an infinite number of towers (specified by two discrete labels) of particles containing all spins.

For the analysis in this paper, we only needed the free part of the action. Let us end this paper with some remarks on the special case of the full (interacting) MHS gauge theory, which is an unconventional case of Yang–Mills theory and can, therefore, be formally quantised using standard methods such as BRST [26,29]. With the already mentioned properties of a perturbatively stable vacuum and the absence of ghosts, this leads to the expectation of (perturbative) unitarity. One may ask whether this contradicts the well-known no-go theorems [49]. The answer is no because the MHS theory has several properties, such as an infinite number of states below any mass scale in the particle spectrum consisting of infinite spin particles, an unusual (infinite-dimensional) representation of the Lorentz group acting on spacetime fields and (weak) nonlocality stemming from the higher derivatives in the Moyal product, which violate some assumptions of the no-go theorems (see also discussion in [5,42]). Moreover, there is the property of the UV finiteness of individual Feynman diagrams with loops consisting of MHS particles defined by an expansion in a discrete basis, a property that is partially evident from the structure of the propagators and vertices calculated in [23]. The problem is that summations over the complete orthonormal basis in $L_2\left({\mathbb{R}}^4\right)$ lead to Dirac delta function divergences resulting from the use of the completeness relation, and this should be dealt with in some way (the singular factors of the type ${\sum }_{n}1$ appear also in other higher-spin theories, e.g., chiral higher-spin gravity [50]). We are currently investigating two approaches to deal with the problem in a way that preserves the full MHS gauge symmetry. One idea is to impose additional constraints on master fields (apart from the square integrability over the auxiliary space) by changing the integration measure, in the spirit of the Schuster–Toro approach [25,30,39,40,42,43,44]. In this way, one arrives at a complicated (weakly) nonlocal free field action that is difficult to analyse. Another idea is to regularise the completeness relation by taking into account the fact that the spacetime fields are not completely independent, since they are connected by the square-integrability condition. This is beyond the scope of this paper and is part of our current and future research. We hope that we have provided evidence that the MHS theory has sufficient merit and potential to be the subject of future investigations.