Spinor-helicity representations of
(A)dS₄ particles of any mass

In the $N=1$ case, considered in [12, 13, 14], the dual algebra $\mathfrak{dual}^{(1)}_{\Lambda}$ is simply isomorphic to $\mathfrak{u}(1)$ generated by

$$K=\lambda_{a}\,\frac{\partial}{\partial\lambda_{a}}-\tilde{\lambda}_{\dot{a}}\,\frac{\partial}{\partial\tilde{\lambda}_{\dot{a}}}\,,$$

(3.1)

which is nothing but the standard helicity operator. The $K=s$ state describes massless helicity $s$ representations in Mink₄, AdS₄ and dS₄. This universal description is due to the conformal symmetry they enjoy: the SH representations of $\mathfrak{sym}_{\Lambda}$ can be lifted to a single irreducible representation, typically referred to as ‘singleton’, of the four-dimensional conformal group $\operatorname{\mathfrak{so}}(2,4)$ [24, 25, 26, 27] (see also [28, 29, 30] for the oscillator realization, where sometimes the representation is referred to as ‘doubleton’ for a historical reason). This special property of singleton can be easily understood in terms of the dual pair correspondence, as it was shown in [19]. We shall come back to this point in Section 4.3.

The $N=2$ case will turn out to be sufficient to describe all massive spin representations in four dimensions. The generators $M=M^{12}$ and $\tilde{M}=\tilde{M}_{12}$ commute with the subalgebra $\operatorname{\mathfrak{so}}(3)\simeq\operatorname{\mathfrak{su}}(2)\subset\operatorname{\mathfrak{u}}(2)$ generated by ${\cal K}^{I}{}_{J}=K^{I}{}_{J}-\frac{1}{2}\,\delta^{I}_{J}\,K^{K}{}_{K}$ while the $\operatorname{\mathfrak{u}}(1)$ part $K=K^{I}{}_{I}$ satisfies

$$[M,\tilde{M}]=-\Lambda\,K,\qquad[K,M]=2\,M\,,\qquad[K,\tilde{M}]=-2\,\tilde{M}\,.$$

(3.2)

Taking into account that $M^{\dagger}=\tilde{M}$ and $K^{\dagger}=K$,⁵⁵5Note that the Hermitian conjugation $\dagger$ is defined with respect to the $L^{2}(\mathbb{C}^{2N})$ norm, and hence $\lambda^{I}{}_{a}{}^{\dagger}=(\lambda^{I}{}_{a})^{*}=\tilde{\lambda}_{I\,\dot{a}}$ and $(\partial/\partial\lambda^{I}{}_{a})^{\dagger}=-\partial/\partial\tilde{\lambda}_{I\,\dot{a}}$. it is easy to show that the Hermitian generators $\frac{1}{2}\,K$, $\frac{1}{2}\,(M+\tilde{M})$ and $\frac{i}{2}\,(M-\tilde{M})$ form $\operatorname{\mathfrak{so}}(2,1)$ for $\Lambda>0$, $\operatorname{\mathfrak{so}}(3)$ for $\Lambda<0$ and $\operatorname{\mathfrak{iso}}(2)$ for $\Lambda=0$. The last case corresponds to the massive Mink₄ SH formulation [6, 7]. To summarize, we find that for $N=2$, the dual algebra is the direct sum,

$$\mathfrak{dual}^{(2)}_{\Lambda}\simeq\mathfrak{s}\oplus\mathfrak{m}_{\Lambda}\,,$$

(3.3)

where the two ideals $\mathfrak{s}$ and $\mathfrak{m}_{\Lambda}$ are

$$\mathfrak{s}=\mathfrak{so}(3)\,,\quad\mathfrak{m}_{\Lambda}=\left\{\begin{aligned} &\mathfrak{so}(2,1)\quad&[\Lambda>0]\\ &\mathfrak{so}(3)\quad&[\Lambda<0]\\ &\mathfrak{iso}(2)\quad&[\Lambda=0]\end{aligned}\right..$$

(3.4)

Below, we will show that the common ideal $\mathfrak{s}$ for any $\Lambda$ is responsible for the spin label of the $\mathfrak{sym}_{\Lambda}$ irreps, whereas the other subalgebra $\mathfrak{m}_{\Lambda}$ determines the mass. In order to see this identification, let us first exploit the relations between Casimir operators of $\mathfrak{sym}_{\Lambda}$ and $\mathfrak{dual}^{(2)}_{\Lambda}$.

Since $\mathfrak{sym}_{\Lambda}$ is a rank two Lie algebra $\mathfrak{so}(1,4)$ or $\mathfrak{so}(2,3)$ for $\Lambda\neq 0$, there are two independent Casimir operators: the quadratic and quartic ones, whose expressions in vector notation read

	$\displaystyle C_{2}(\mathfrak{sym}_{\Lambda})=-\frac{1}{2}\,J^{A_{1}}{}_{A_{2}}\,J^{A_{2}}{}_{A_{1}}\,,$		(3.5a)
	$\displaystyle C_{4}(\mathfrak{sym}_{\Lambda})=\frac{1}{2}\,W_{A}\,W^{A}\,,\qquad W_{A}=\frac{1}{2}\,\epsilon_{ABCDE}\,J^{BC}\,J^{DE}\,,$		(3.5b)

where the capital indices take the values $A,B,\dots=0,1,\dots,4$ and $J_{AB}=-J_{BA}$ are the generators of $\mathfrak{sym}_{\Lambda}$. Splitting $J_{AB}$ into Lorentz and translation generators as $J_{4\mu}=P_{\mu}/\sqrt{|\Lambda|}$ and $J_{\mu\nu}=L_{\mu\nu}$, the two Casimirs are

	$\displaystyle C_{2}(\mathfrak{sym}_{\Lambda})$	$\displaystyle=$	$\displaystyle-\frac{1}{2\,\Lambda}\,P^{2}+\frac{1}{4}\,(L^{2}+\tilde{L}^{2})\,,$		(3.6a)
	$\displaystyle C_{4}(\mathfrak{sym}_{\Lambda})$	$\displaystyle=$	$\displaystyle\frac{1}{4\,\Lambda}\,P^{a\dot{a}}\,P^{b\dot{b}}\,L_{ab}\,\tilde{L}_{\dot{a}\dot{b}}+\frac{1}{16\,\Lambda}\,P^{2}\,(L^{2}+\tilde{L}^{2})$		(3.6b)
			$\displaystyle-\,\frac{1}{4}\,(L^{2}+\tilde{L}^{2})-\frac{1}{64}\,(L^{2}-\tilde{L}^{2})^{2}\,.$

Here, $P^{2}=P_{a\dot{b}}\,P^{a\dot{b}}=-2\,P_{\mu}P^{\mu}$, $L^{2}=L_{ab}L^{ab}$ and $L_{\mu\nu}L^{\mu\nu}=\frac{1}{2}(L^{2}+\tilde{L}^{2})$, where we use the mostly-plus signature for $\eta_{\mu\nu}$. Note that $\Lambda\,C_{2}(\mathfrak{sym}_{\Lambda})$ and $\Lambda\,C_{4}(\mathfrak{sym}_{\Lambda})$ reproduce the familiar quadratic Casimir and the Pauli–Lubański vector squared in the $\Lambda\to 0$ limit.

On the other hand, the dual algebra is composed of two rank-one ideals, so we have one Casimir operator for each:

	$\displaystyle C_{2}(\mathfrak{s})$	$\displaystyle=$	$\displaystyle\frac{1}{2}{\cal K}^{I}_{J}\,{\cal K}^{J}_{I}\,,$		(3.7)
	$\displaystyle C_{2}(\mathfrak{m}_{\Lambda})$	$\displaystyle=$	$\displaystyle-\frac{1}{2\Lambda}\,\{M,\tilde{M}\}+\frac{1}{4}\,K^{2}\,.$		(3.8)

The SH representation of $\mathfrak{sym}_{\Lambda}$ (2.1) and (2.2) and that of $\mathfrak{dual}^{(2)}_{\Lambda}$ (2.9a) and (2.9b) relate these Casimir operators as

	$\displaystyle C_{2}(\mathfrak{sym}_{\Lambda})$	$\displaystyle=$	$\displaystyle C_{2}(\mathfrak{m}_{\Lambda})+C_{2}(\mathfrak{s})-2\,,$		(3.9a)
	$\displaystyle C_{4}(\mathfrak{sym}_{\Lambda})$	$\displaystyle=$	$\displaystyle-\,C_{2}(\mathfrak{m}_{\Lambda})\,C_{2}(\mathfrak{s})\,.$		(3.9b)

From the above relations, we can read off the Casimir eigenvalues of the unitary irreps of $\mathfrak{sym}_{\Lambda}$ by fixing an irrep of $\mathfrak{dual}^{{(2)}}_{\Lambda}\simeq\mathfrak{s}\oplus\mathfrak{m}_{\Lambda}$. For the ideal $\mathfrak{s}\simeq\mathfrak{so}(3)$, the $(2s+1)$-dimensional irreps with

$$C_{2}(\mathfrak{s})=s(s+1)\,,$$

(3.10)

account for all unitary irreps. About the ideal $\mathfrak{m}_{\Lambda}$, the quadratic Casimir operator can be parameterized as

$$C_{2}(\mathfrak{m}_{\Lambda})=\mu(\mu+1)\,,$$

(3.11)

which is invariant under

$$\mu\to-1-\mu,$$

(3.12)

and we have the following options:

• •

  For $\Lambda>0$, apart from the trivial irrep with $\mu(\mu+1)=0$, we have three series of unitary irreps for $\mathfrak{so}(2,1)$ :

  • –

    The principal series irreps ${\cal C}_{\mu}^{\pm}$ with complex $\mu$ satisfying

    $$\mu(\mu+1)<-\frac{1}{4}\,,$$

    (3.13)

    which is spanned by eigenstates of $K$ with even/odd integer eigenvalues, related to the label $+/-$ respectively. We can parametrize irreps in this series via $\mu=-\frac{1}{2}+i\,\rho$ with $\rho\in\mathbb{R}$. In this case, the map (3.12), $\rho\to-\rho$, is an isomorphism, and hence we may restrict to the case $\rho>0$.

  • –

    The complementary series irrep ${\cal C}_{\mu}$ with $-1<\mu<0$ satisfying

    $$-\frac{1}{4}\leq\mu(\mu+1)<0\,,$$

    (3.14)

    spanned by all even $K$-eigenstates. The map (3.12) is again an isomorphism.

  • –

    The positive/negative discrete series irrep ${\cal D}^{\pm}_{2\mu+2}$ with

    $$\mu=-\tfrac{1}{2},0,\tfrac{1}{2},1,\tfrac{3}{2},\ldots\,,$$

    (3.15)

    spanned by the $K$-eigenstates with eigenvalues $\pm 2(\mu+1),\pm 2(\mu+2)$, etc. These are lowest/highest weight irreps.

• •

  For $\Lambda\to 0$ , the “bosonic/fermionic” irrep of $\operatorname{\mathfrak{iso}}(2)$ with $|\mu|\to\infty$ while keeping finite

  $$m={\sqrt{-\Lambda\,\mu^{2}}}\,,$$

  (3.16)

  which is spanned by $K$-eigenstates with even/odd eigenvalues. These irreps can be thought of as the counterpart of the massive scalar and spinor representations of the Poincaré group (depending on the parity of the $K$-eigenstates).

  The trivial representation, with $m=0$, and which can be thought of as the counterpart of the zero-momentum irrep of the Poincaré group.

• •

  For $\Lambda<0$, the $(2\mu+1)$-dimensional irrep of $\operatorname{\mathfrak{so}}(3)$ with

  $$\mu=0,\tfrac{1}{2},1,\tfrac{3}{2},\ldots\,,$$

  (3.17)

  with basis composed of $K$-eigenstates with eigenvalues $-2\mu,-2\mu+2,\dots,+2\mu$.

These irreps of $\mathfrak{dual}^{{(2)}}_{\Lambda}$ are in one-to-one correspondence with the irreps of $\mathfrak{sym}_{\Lambda}$ with

	$\displaystyle C_{2}(\mathfrak{sym}_{\Lambda})$	$\displaystyle=\mu(\mu+1)+s(s+1)-2\,,$		(3.18a)
	$\displaystyle C_{4}(\mathfrak{sym}_{\Lambda})$	$\displaystyle=-\mu(\mu+1)\,s(s+1)\,,$		(3.18b)

and we can compare these values with those of known irreps of $\mathfrak{sym}_{\Lambda}$.

In the AdS₄ case with $\Lambda<0$, the irrep label $\mu$ of the dual algebra $\mathfrak{m}_{\Lambda}$ parameterizes the mass squared again as (3.21). The allowed $\mu$ for the unitarity of the lowest-energy irreps of $\mathfrak{sym}_{\Lambda}$ are $\mu=s-1,s,s+1,\ldots$ for spin $s=0,\frac{1}{2}\,1,\ldots$. The $\mu=s-1$ case corresponds to the massless spin $s$ field, and higher $\mu$ cases correspond to massive fields. The reason that we have a discrete mass spectrum is due to the fact that $\mu$ is an eigenvalue of the generator of the compact subgroup $SO(2)$ associated with rotations in the plane of temporal directions, and hence is quantized. These representations can be interpreted as the irreps of 3d conformal group: $\Delta=\mu+2$ and $s$ correspond to the conformal weight and spin of the conformal primaries, respectively. In the scalar case, the $\mu=-1$ and $\mu=0$ cases mapped by (3.12) are distinct irreps and correspond to different modes of the conformal scalar in AdS₄. Note that, moving to a covering group of $SO(1,4)$, the point $\mu=-\frac{3}{2}$ can be included for $s=0$, and it corresponds to the conformal scalar in 3d.

The unitarity of the lowest energy irreps of $\mathfrak{sym}_{\Lambda}\cong\mathfrak{so}(2,3)$ excludes the lower $\mu$ values with $\mu<s-1$ from (3.17), corresponding to partially-massless fields, together with all integer/half-integer values of $\mu$ for half-integral/integral spin.

Let us note that there are a few other types of $\mathfrak{sym}_{\Lambda}$ irreps with unbounded energy. These irreps would cover different ranges of $C_{2}(\mathfrak{sym}_{\Lambda})$ and $C_{4}(\mathfrak{sym}_{\Lambda})$ .

Let us consider first the case with $\Lambda>0$. Our aim is to obtain a dictionary between the irreps of $Sp(1,1)$ and $O^{*}(2N)$ appearing in the decomposition of the SH representation.

For $N=1$, the dual pair correspondence between $Sp(1,1)$ and $O^{*}(2)$ has been explicitly established in [19]. Here, we just quote the result. Since $O^{*}(2)$ is isomorphic to $U(1)$, it has only one-dimensional irreps, each labelled by an integer. This integer corresponds to twice the helicity of a $Sp(1,1)$ massless representation. The analysis is based on the decomposition of the $Sp(1,1)$ irrep into its maximal subgroup $Sp(1)\times Sp(1)$, and the SH representation restricted by the $O^{*}(2)$ irrep condition is shown to have the structure of the massless spin $s$ irrep of $Sp(1,1)$ demonstrated e.g. in [31].

For the $N=2$ case, we need to begin with identifying irreps of the dual group $O^{*}(4)$. Thanks to the isomorphism $O^{*}(4)\cong[SU(2)\times SL(2,\mathbb{R})]/\mathbb{Z}_{2}$ (here, $SU(2)$ and $SL(2,\mathbb{R})$ are simply the Lie groups associated with $\mathfrak{s}=\mathfrak{so}(3)$ and $\mathfrak{m}_{\Lambda>0}=\mathfrak{so}(2,1)$), we know everything about the unitary irreps of $O^{*}(4)$: the irreps of $SU(2)$ are all given by $(2s+1)$-dimensional representation, which will be denoted by $[2s]$ henceforth, while $SL(2,\mathbb{R})$ has three classes of unitary irreps, namely ${\cal C}^{\pm}_{\mu=-\frac{1}{2}+i\,\rho}$ (3.13), ${\cal C}_{\mu}$ (3.14) and ${\cal D}^{\pm}_{2\mu+2}$ (3.15). We will denote these $O^{*}(4)$ irreps as $\tilde{\pi}_{s,\mu}$.

In the previous section, we have seen that not all $O^{*}(4)$ irreps correspond to irreps of $Sp(1,1)$ based on the match of Casimir operators. We shall see below how they are restricted. For that, we first consider the dual pair $\big{(}Sp(1),O^{*}(4)\big{)}\subset Sp(8,\mathbb{R})$, whose representations are explicitly identified in [19, Sec. 5.4]: Since $Sp(1)\cong SU(2)$ the $Sp(1)$ irreps are again given by $[m]$ with non-negative integer $m$, and they correspond to the $O^{*}(4)$ irreps $[m]\otimes{\cal D}^{\pm}_{m+2}$ . Note that only discrete series representations appear in the $SL(2,\mathbb{R})$ side, with the highest/lowest weight $m+2$ tied with the dimension $m+1$ of the $SU(2)$ irrep (which is a consequence of the fact that the Howe dual is a compact group, namely $Sp(1)$). Whether the irrep ${\cal D}^{\pm}_{m+2}$ is a highest/lowest weight one is conventional at this stage, and only one sign is chosen depending on the convention of $SL(2,\mathbb{R})$.

Now we move on to the dS₄ group $Sp(1,1)$ and consider its maximal compact subgroup, which is $Sp(1)\times Sp(1)$. This subgroup forms its own dual pair in the same SH space (that is, in $Sp(16,\mathbb{R})$) with $O^{*}(4)\times O^{*}(4)$. The latter contains the original dual group $O^{*}(4)$ as the diagonal subgroup. The situation is conveniently depicted by the “seesaw” diagram,

(4.1)

where the arrows indicate the respective dual pairs. Any irrep of $Sp(1,1)$, say $\pi_{\sigma}$ with some label $\sigma$, can be decomposed into irreps of $Sp(1)\times Sp(1)$ as

$$\pi_{\sigma}=\bigoplus_{m,n}N_{\sigma}^{m,n}\,[m]\otimes[n]\,,$$

(4.2)

where $N^{m,n}_{\sigma}$ are the multiplicities of $[m]\otimes[n]$, and each of $[m]\otimes[n]$ correspond to the $O^{*}(4)\times O^{*}(4)$ irrep,

$$\Big{(}[m]\otimes{\cal D}^{-}_{m+2}\Big{)}\otimes\Big{(}[n]\otimes{\cal D}^{+}_{n+2}\Big{)}\,.$$

(4.3)

Here, we used the correspondence between the irreps of $Sp(1)$ and $O^{*}(4)$ that we introduced earlier. Note that the first $SL(2,\mathbb{R})$ irrep is a lowest-weight irrep, while the second is a heighest-weight irrep. This is because the $Sp(1)\times Sp(1)$ is embedded in the opposite signature parts of $Sp(1,1)$. The irrep (4.3) of $O^{*}(4)\times O^{*}(4)$ can be decomposed as well into the diagonal subgroup $O^{*}(4)$ :

$$\Big{(}[m]\otimes{\cal D}^{-}_{m+2}\Big{)}\otimes\Big{(}[n]\otimes{\cal D}^{+}_{n+2}\Big{)}=\bigoplus_{s,\mu}\tilde{N}^{s,\mu}_{m,n}\,\tilde{\pi}_{s,\mu}\,,$$

(4.4)

where $\tilde{N}^{s,\mu}_{m,n}$ are the multiplicities of the $O^{*}(4)$ irrep $\tilde{\pi}_{s,\mu}$ that we have introduced before. The crucial point assured by the seesaw duality (see [20, 21, 22] and also [19, Sec. 2.3]) is the equality between two multiplicities: for any $[m]\otimes[n]$,

$$N^{m,n}_{\sigma(s,\mu)}=\tilde{N}^{s,\mu}_{m,n}\,.$$

(4.5)

Here, $\sigma(s,\mu)$ is the label of the $Sp(1,1)$ irrep dual to the $O^{*}(4)$ irrep $\tilde{\pi}_{s,\mu}$.

Now let us identify the multiplicities $\tilde{N}^{s,\mu}_{m,n}$. The decomposition (4.4) comes in two parts: the decomposition of the $SU(2)$ irreps,

$$[m]\otimes[n]=[|m-n|]\oplus[|m-n|+2]\oplus\cdots\oplus[m+n]\,,$$

(4.6)

and the decomposition of the $SL(2,\mathbb{R})$ irreps [43] (see also [44]),

$${\cal D}^{-}_{m+2}\otimes{\cal D}^{+}_{n+2}=\int_{0}^{\infty}{\rm d}\rho\ {\cal C}_{-\frac{1}{2}+i\,\rho}^{(-1)^{m+n}}\oplus\bigoplus_{0\leq k<\frac{|m-n|}{2}}\,{\cal D}^{{\rm sgn}(m-n)}_{|m-n|-2k}\,.$$

(4.7)

We see that the multiplicities are either $1$ or $0$. Hence, for a fixed $\tilde{\pi}_{s,\mu}$ the above decomposition simply restricts the possible $[m]\otimes[n]$ which appear in the decomposition (4.2) of $\pi_{\sigma(s,\mu)}$. Moreover, we find that certain $\tilde{\pi}_{s,\mu}$’s do not admit any $[m]\otimes[n]$ implying that such irreps cannot correspond to any (even trivial) $Sp(1,1)$ irrep. In other words, they are simply not contained in the SH representation. Let us see the details now. By choosing the $SU(2)$ irrep as $[2s]$, $m$ and $n$ are restricted as

$$|m-n|\leq 2s\leq m+n\,,\qquad m+n-2s\in 2\,\mathbb{Z}\,.$$

(4.8)

For the $SL(2,\mathbb{R})$ irreps with label $\mu$, we have three choices, the principal series ${\cal C}^{\pm}_{\mu=-\frac{1}{2}+i\,\rho}$, the complementary series ${\cal C}_{\mu}$ and the discrete series ${\cal D}^{\pm}_{2\mu+2}$. We notice already that the complementary series is not available since it does not appear in the content of the tensor product decomposition, that is, in the RHS of (4.7).