Spin and Tensor Representations

The label “spin” compresses several choices: Euclidean rotation or Lorentz representation, orthogonal group or Spin cover, tensor or spinor realization, chirality and reality conditions, parity, and statistics. A conformal tensor structure is computable only after those choices and the dimension are fixed. Symmetric traceless tensors admit an efficient null-polarization encoding; mixed-symmetry tensors need one polarization family per Young row or column; spinors require gamma-matrix and charge-conjugation conventions appropriate to the real form.

Required background. Primaries, Descendants, and Conformal Multiplets supplies the $(\Delta,R)$ module labels. Representations, Intertwiners, and Invariants supplies irreducible tensors and invariant maps. Helpful background. Spinors, Conjugations, Bilinears, and Fierz Identities supplies dimension- and signature-dependent spinor conventions.

The group represented by an operator

In Euclidean radial quantization, a local primary at the origin transforms under $SO(d)$ if it is bosonic and tensorial. A fermionic primary transforms under the double cover $\operatorname{Spin}(d)$. After Lorentzian continuation, local fields carry finite-dimensional representations of $\operatorname{Spin}(d-1,1)$, which are generally nonunitary because the Lorentz group is noncompact; unitarity belongs to the Hilbert-space representation of the full conformal group, not to the finite set of field components.

The following data are not determined by writing only a Young diagram:

Datum	Why it matters
Dimension and signature	Change the rotation or Lorentz real form, Hodge dualities, chirality, and reality conditions
Global cover	Determines whether spinorial and projective representations are allowed
Highest-weight convention	Fixes how Dynkin labels, Young diagrams, and physical spin are translated
Trace and Young projection	Removes reducible tensor components
Chirality and reality	Decide whether Weyl, Majorana, symplectic-Majorana, or only complex Dirac spinors exist
Parity convention	Determines which epsilon-tensor and gamma-matrix structures are even or odd
Statistics and internal representation	Fix exchange signs and which OPE structures survive for identical operators

For example, a four-dimensional Lorentzian primary may be labeled by $(j,\bar j)$ under $SL(2,\mathbb C)$, whereas its Euclidean continuation is labeled by $(j_L,j_R)$ under $SU(2)_L\times SU(2)_R$. The labels have parallel complexified algebras but different conjugation conditions. Treating them as the same real representation loses Hermiticity information.

Symmetric traceless tensors

A rank-$\ell$ symmetric traceless tensor $\mathcal O_{\mu_1\cdots\mu_\ell}(x)$ can be encoded by an auxiliary commuting vector $z^\mu$:

$$\mathcal O(x,z) =z^{\mu_1}\cdots z^{\mu_\ell} \mathcal O_{\mu_1\cdots\mu_\ell}(x), \qquad z^2=0.$$

Symmetry is automatic because the $z$‘s commute. The condition $z^2=0$ sets every trace term to zero: two contracted tensor indices produce a factor $z^2$. The polynomial is homogeneous,

$$\mathcal O(x,\lambda z)=\lambda^\ell\mathcal O(x,z),$$

so its degree records the spin. Tensor components can be recovered with the Todorov operator

$$D_z^\mu =\left(\frac d2-1+z\mathbin{\cdot}\partial_z\right)\partial_z^\mu -\frac12z^\mu\partial_z^2,$$

up to the standard normalization obtained by applying one $D_z$ per index. This differentiates intrinsically on the null cone and preserves tracelessness.

In the embedding formalism, a physical point is a null ray $P^A$, and a spin-$\ell$ operator uses a polarization $Z^A$ satisfying

$$P^2=0, \qquad Z^2=0, \qquad P\mathbin{\cdot}Z=0, \qquad Z\sim Z+\alpha P.$$

The last equivalence is transversality: components parallel to the projective ray are unphysical. Homogeneity in $P$ supplies $\Delta$, and homogeneity in $Z$ supplies $\ell$. Costa, Penedones, Poland, and Rychkov develop these conditions and the resulting tensor structures in Costa et al. 2011, §§ 2–4.

Mixed symmetry and spinors

A general bosonic irrep uses a Young diagram. Begin with one commuting polarization for each symmetric row, impose the Young condition that symmetrizing an index of a lower row into a higher row vanishes, and quotient all metric traces. An equivalent column construction uses Grassmann polarizations for antisymmetric columns. The chosen encoding must be stated because its homogeneity and gauge relations determine the allowed invariants. Dimension-dependent Hodge dualities can identify diagrams whose columns approach height $d/2$, and any column longer than $d$ vanishes. A dimension-independent diagram count therefore overcounts in low $d$. A systematic embedding-space construction for mixed symmetry appears in Costa and Hansen 2015, §§ 2–3.

For spinors, choose gamma matrices obeying

$$\{\gamma_\mu,\gamma_\nu\}=2g_{\mu\nu}.$$

The existence of a chirality matrix, a real structure, and an invariant bilinear depends on $d$ modulo $8$ and on signature. A Dirac spinor can always be complexified, but imposing Weyl and Majorana conditions simultaneously is dimension dependent. Moreover, the Euclidean adjoint used in reflection positivity is not generally the same algebraic operation as Lorentzian Dirac conjugation. A correlator must specify which spinor index is in $S$, $S^*$, or a charge-conjugate representation.

Auxiliary commuting spinors $s^\alpha$ efficiently contract totally symmetric spinor indices, while anticommuting polarizations can encode exterior products. Fierz identities then reduce apparently different bilinears. Because the available charge-conjugation matrices and gamma-matrix dualities depend on dimension, that reduction is part of the calculation rather than universal notation. A general-dimensional embedding treatment with polarization spinors is given in Isono 2017, §§ 2–3.

Constructing a vector–vector–scalar correlator

For three points, define physical-space building blocks

$$\begin{aligned} H_{12}&=x_{12}^2(z_1\mathbin{\cdot}z_2) -2(z_1\mathbin{\cdot}x_{12})(z_2\mathbin{\cdot}x_{12}),\\ V_{1,23}&= \frac{x_{12}^2(z_1\mathbin{\cdot}x_{13}) -x_{13}^2(z_1\mathbin{\cdot}x_{12})}{x_{23}^2},\\ V_{2,13}&= \frac{x_{21}^2(z_2\mathbin{\cdot}x_{23}) -x_{23}^2(z_2\mathbin{\cdot}x_{21})}{x_{13}^2}. \end{aligned}$$

Each is linear in the polarization belonging to the indicated vector, and $H_{12}$ and $V_{1,23}V_{2,13}$ have the same length dimension. Let $\tau_i=\Delta_i+\ell_i$, with $(\ell_1,\ell_2,\ell_3)=(1,1,0)$, and define

$$\begin{aligned} \mathcal D_{123}={}& (x_{12}^2)^{(\tau_1+\tau_2-\tau_3)/2} (x_{23}^2)^{(\tau_2+\tau_3-\tau_1)/2}\\ &\times (x_{31}^2)^{(\tau_3+\tau_1-\tau_2)/2}. \end{aligned}$$

The most general parity-even three-point function before conservation or exchange constraints is

$$\langle V_1(x_1,z_1)V_2(x_2,z_2)\mathcal O_3(x_3)\rangle =\frac{\lambda_H H_{12}+\lambda_V V_{1,23}V_{2,13}} {\mathcal D_{123}}.$$

The length degree of the numerator is two, so the full expression scales as $\lambda^{-\Delta_1-\Delta_2-\Delta_3}$ under a common rescaling. In a conformal frame $(x_1,x_2,x_3)=(0,e,\infty)$, the two structures reduce to the two $SO(d-1)$ invariants $z_1\mathbin{\cdot}z_2$ and $(z_1\mathbin{\cdot}e)(z_2\mathbin{\cdot}e)$. This gives an independent count of two.

If $V_1=V_2$ are identical bosonic operators, exchange symmetry relates the coefficients according to the scalar parity and internal channel. If either vector is conserved, applying the divergence gives linear relations among $\lambda_H$ and $\lambda_V$ and may eliminate one structure. In dimensions admitting an epsilon invariant, parity-odd structures may also appear. None of these reductions should be imposed before dimension, parity, conservation, and statistics are declared.

Constructing a spinor–spinor–scalar correlator

The conformal-frame method avoids pretending that one gamma-matrix formula covers every real form. Put the scalar at infinity and the two spinors at $0$ and a unit vector $e$. The subgroup preserving the three points is $\operatorname{Spin}(d-1)$. For complex Dirac spinors, candidate scalar intertwiners are generated by

$$\bar s_1s_2, \qquad \bar s_1\gamma(e)s_2, \qquad \gamma(e)=e^\mu\gamma_\mu.$$

They are respectively even and odd under $e\mapsto-e$ once the parity action on the spinors has been fixed. Restrict $S_d$ to $\operatorname{Spin}(d-1)$, retain the singlets in $S_1^*\otimes S_2$, and then impose:

1. chirality—some bilinears map equal chiralities, others opposite chiralities;
2. reality or charge conjugation—Majorana-type conditions can relate a structure to its complex conjugate;
3. exchange statistics—identical fermions acquire a minus sign;
4. scalar parity and internal representation; and
5. dimension-specific Fierz and gamma-duality identities.

Each surviving conformal-frame invariant lifts uniquely to a position-space conformal structure by the spinor parallel-transport matrices associated with inversion. This is a constructive algorithm: for example, if parity and chirality admit both displayed singlets, there are two independent structures; if a Weyl selection rule kills one, only the other lifts. Quoting “two” without these qualifications is false in special dimensions or for Majorana/Weyl operators.

Conversion and validation checklist

Input representation	Computational encoding	Conditions to impose	Check before using a correlator
Scalar	No polarization	Parity and internal labels	Correct homogeneity in every $x_i$
Symmetric traceless rank $\ell$	Null commuting $z$, degree $\ell$	$z^2=0$; embedding transversality $Z\sim Z+\alpha P$	Trace terms vanish and inversion matrices act on each index
Mixed Young tensor	Several row or column polarizations	Young symmetrizers, traces, low-$d$ dualities	Dimension of the encoded irrep matches the group-theory dimension
Dirac/Weyl/Majorana spinor	Polarization spinor and gamma intertwiners	Spin cover, chirality, conjugation, Fierz relations	Bilinear maps the actual representations and has the stated parity
Identical operators	Same as above	Bose or Fermi exchange and internal channel	Correlator acquires the required sign under point exchange

A useful counterexample is a rank-two symmetric polynomial with an unconstrained $z$: it represents a symmetric tensor, not a symmetric traceless tensor, because its $z^2$ coefficient retains the trace. Likewise, a formula written with $SO(d)$ matrices cannot act on a spinor for which a $2\pi$ rotation is $-1$; the Spin cover is not optional notation.

Unitarity Bounds and Null States applies positivity to these precise rotation representations. Spinning Correlators and Tensor Structures uses the same polarizations and intertwiners to classify two- and three-point structures; dimension-specific Fierz reduction remains part of that construction rather than a universal afterthought.

Exercises

Show that imposing $z^2=0$ removes the trace of a symmetric rank-two tensor.

Solution

Write $T_{\mu\nu}=T_{\mu\nu}^{\mathrm{STT}}+(1/d)\delta_{\mu\nu}T^\rho{}_{\rho}$. Then

$$z^\mu z^\nu T_{\mu\nu} =z^\mu z^\nu T_{\mu\nu}^{\mathrm{STT}} +\frac{z^2}{d}T^\rho{}_{\rho}.$$

The trace term vanishes on the auxiliary null cone.

Use the conformal frame to count parity-even structures for two vectors and one scalar before conservation.

Solution

The residual rotations are $SO(d-1)$ around the distinguished unit vector $e$. A bilinear linear in each polarization can contract them directly, $z_1\mathbin{\cdot}z_2$, or project each along $e$, $(z_1\mathbin{\cdot}e)(z_2\mathbin{\cdot}e)$. These are independent for generic $d\geq3$, giving two parity-even structures. Their covariant lifts are $H_{12}$ and $V_{1,23}V_{2,13}$.

References

• Costa, Miguel S., and Tobias Hansen. “Conformal Correlators of Mixed-Symmetry Tensors.” Journal of High Energy Physics 2015, no. 2 (2015): 151. DOI; Open PDF
• Costa, Miguel S., João Penedones, David Poland, and Slava Rychkov. “Spinning Conformal Correlators.” Journal of High Energy Physics 2011, no. 11 (2011): 071. DOI; Open PDF
• Isono, Hiroshi. “On Conformal Correlators and Blocks with Spinors in General Dimensions.” Physical Review D 96 (2017): 065011. DOI; Open PDF