Spacetime symmetry: fields and states

The representation attached to a field’s indices is not the representation carried by its physical states. This path connects those two questions without dropping spacetime signature, the covering group, gauge charges, or exchange statistics. Its running examples are a four-dimensional Weyl bilinear and a massive vector, followed by optional CFT and AdS modules.

Assume four-dimensional Lorentzian spin spacetime for the field and particle steps, with $\eta=\operatorname{diag}(-,+,+,+)$ and connected Lorentz transformations. Field components are complex and the coefficient field has characteristic zero. Parity, time reversal, curved-spacetime spin-structure choices and continuous-spin particles are outside the examples. Read Irreducible representations first if the category distinction is unfamiliar. The linked spinor lesson uses $(+---)$ for its explicit gamma matrices; its signature convention must be retained when reusing those matrices.

1. Fix the spin cover and the field indices

Start with Which representation describes the object? and Spinors, covers, and reality conditions. The Lorentz spin cover is $\operatorname{Spin}^+(1,3)\cong SL(2,\mathbb C)$ as a real Lie group. Its finite-dimensional complex field representations have labels $(j_L,j_R)$: a left Weyl field is $(1/2,0)$ and a vector is $(1/2,1/2)$. They have two and four components, respectively. Boosts do not act unitarily in a positive-definite inner product on these component spaces.

Carry forward: the Lorentzian spin group and Euclidean Spin(4) are different real groups. A compact $SU(2)\times SU(2)$ calculation can nevertheless recover the same finite polynomial invariant space after complexifying the algebras. The real-form lesson explains the boundary of that algebraic substitution.

2. Follow one Weyl field to its scalar bilinear

Read From representations to local fields. Take a gauge-singlet left Weyl field $N^\alpha$, with independent Grassmann components, and $\epsilon_{12}=+1$. The contraction

$$B=\epsilon_{\alpha\beta}N^\alpha N^\beta =N^1N^2-N^2N^1=2N^1N^2$$

is nonzero. Each component squares to zero, but their product need not. For $f$ flavors, the same exchange calculation gives $B_{ab}=B_{ba}$, so there are $f(f+1)/2$ complex bilinears. This is the anticommuting-spinor identity in Dreiner–Haber–Martin, §2, equation (2.60).

Carry forward: for three flavors the answer is six, with canonical operator dimension three. Reproduce “Sterile Weyl mass · three flavors” in the local-operator workbench. Six invariant polynomials does not mean six physical masses: Hermiticity and flavor-basis changes are additional operations. Then compare the lesson’s neutral weak doublet, whose extra antisymmetric gauge pairing reverses the flavor symmetry and removes the one-flavor mass.

3. Attach gauge charges and a physical setup

Continue to Matter representations and From fields to interactions. The full field space includes Lorentz, gauge and flavor factors, and statistics act on their product. A left-handed conjugate of a right-handed field has reversed abelian charges and conjugate nonabelian labels; it is not interchangeable with its Hermitian-conjugate letter in a polynomial.

Carry forward: every field must descend to the selected global gauge group before an interaction is counted. Test the Standard Model setup, then inspect the Weinberg operator as an example where weak, Lorentz and flavor exchange symmetries work together. The implemented operator count is derivative-free and does not supply a complete EFT basis.

4. Change categories when asking about particles

Return to the particle section of the spacetime guide. For a future massive momentum $p^2=-m^2$, $m>0$, the spin cover of the little group is $SU(2)$. A spin-$s$ particle has $2s+1$ polarizations at fixed momentum; its full one-particle Hilbert space also contains the continuous momentum orbit and is infinite-dimensional.

A free massive vector makes the distinction concrete. Its covariant polarization vector has four components, but the field equation imposes $p_\mu\varepsilon^\mu=0$. In the rest frame $p^\mu=(m,0,0,0)$ this sets $\varepsilon^0=0$, leaving the three components of spin one. This is the rank-one instance of Bekaert–Boulanger, §§5.1–5.2, equations (48)–(53).

Carry forward: neither the four-dimensional Lorentz vector nor its three-dimensional fixed-momentum polarization space is the entire Poincaré representation. For a massless momentum, the little-group algebra is instead $\mathfrak{iso}(2)$. Ordinary finite-helicity sectors assume its translation part acts trivially. A real Maxwell field combines the two helicities $+1$ and $-1$; those assumptions exclude the distinct continuous-spin class. Bekaert–Boulanger’s §3 distinguishes these cases.

5. Add a scaling dimension at a conformal fixed point

Read Conformal symmetry and representation categories. The vector Lorentz label $(1/2,1/2)$ does not determine a conformal multiplet. In a unitary four-dimensional CFT, a conserved global current has primary labels $(\Delta,j_L,j_R)=(3,1/2,1/2)$ and an infinite tower of descendants, subject to its null divergence.

Carry forward: a primary supplies both spin and scaling dimension. Work through the scalar norm calculation: at $\Delta=3/4$ it gives a negative-norm descendant, while at $\Delta=1$ the trace descendant is null. Do not substitute a canonical engineering dimension for an interacting scaling dimension or confuse a gauge potential with a gauge-invariant primary.

6. Match AdS symmetry with the boundary category

Finish with AdS isometries and holographic symmetry. The page separates the embedded AdS hyperboloid, its unwrapped time cover, and the spin covers of $SO_0(2,d)$. It then derives $m^2L^2=\Delta(\Delta-d)$ for a free bulk scalar and specifies when two boundary quantizations are possible.

Carry forward: for $AdS_4$ with $m^2L^2=-2$, both $\Delta=1$ and $\Delta=2$ are available in the strict alternate-quantization window. One bulk mass and one Lie algebra therefore do not select the boundary operator dimension without a boundary condition. Matching these symmetry labels does not establish a holographic duality.

A final category check

For each calculation, record the object and the relevant labels:

Object	Labels or data retained
Weyl field letter	Lorentz chirality, gauge labels and charges, flavor, statistics
Fixed local monomial	Field powers, conjugate letters, global group, imposed equivalences
Massive particle	Mass, little-group spin, positive-energy momentum orbit
Conformal primary	Scaling dimension, rotation spin, internal charges, null relations
AdS scalar state	Bulk dimension, radius and mass, time/spin cover, boundary quantization

Use the spacetime atlas for groups and maps, and the reproducibility guide to retain the inputs of supported calculations. None of these categories is replaced by a list of representation dimensions.