= Solution
Use the <Clifford algebra> $\{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}$ and unit-weight antisymmetrization. The duality identity between $\gamma_5\gamma_\nu$ and three antisymmetrized <gamma matrices> turns the equation into
$$
E^\mu:=\gamma^{\mu\nu\rho}\partial_\nu\Psi_\rho=0,
\qquad \gamma^{\mu\nu\rho}=\gamma^{[\mu}\gamma^\nu\gamma^{\rho]}.
$$
The overall nonzero factor in this identity depends on the orientation and $\gamma_5$ convention and does not change its solutions. This <Rarita-Schwinger field> equation has the <gauge symmetry>
$$
\Psi_\mu\longmapsto\Psi_\mu+\partial_\mu\epsilon,
$$
since $\gamma^{\mu\nu\rho}\partial_\nu\partial_\rho\epsilon=0$. It is the free-field limit of local <supersymmetry>, so it is suitable for the <gravitino>.
To check which <spin> it propagates, put $A=\gamma^\mu\Psi_\mu$ and $B=\partial^\mu\Psi_\mu$. Expanding the three <gamma matrices> using the <Clifford algebra> gives
$$
E^\mu=\not\partial\Psi^\mu-\partial^\mu A
+\gamma^\mu(\not\partial A-B),
\qquad
\gamma_\mu E^\mu=2(\not\partial A-B)
$$
in four dimensions. Thus the <Euler-Lagrange field equation> requires $B=\not\partial A$. Under the <gauge transformation>, $A\mapsto A+\not\partial\epsilon$; locally one may choose gamma-trace <gauge fixing>, $A=0$, by solving $\not\partial\epsilon=-A$. The <Euler-Lagrange field equation> then yields
$$
\gamma^\mu\Psi_\mu=0,
\qquad \partial^\mu\Psi_\mu=0,
\qquad \not\partial\Psi_\mu=0.
$$
These are the <massless Dirac equation>, transversality and gamma-trace constraints that remove the lower-<spin> part of a vector-spinor.
For a more explicit <massless Rarita-Schwinger polarization count>, take a positive-frequency <plane wave> with null momentum along the third axis, in signature $(-,+,+,+)$. Transversality relates its temporal and longitudinal components. Residual <gauge transformations> with $\not p\epsilon=0$ remove both. Write the surviving components as $u_1,u_2$; their gamma-trace condition reads $\gamma^1u_1+\gamma^2u_2=0$, hence
$$
u_2=\gamma^1\gamma^2u_1,
\qquad \not p\,u_1=0.
$$
The <massless Dirac equation> leaves two complex positive-frequency amplitudes in $u_1$. If $i\gamma^1\gamma^2u_1=s u_1$, $s=\pm1$, then $(u_1,u_2)$ is proportional to $(1,-is)$ in the transverse vector index. With spatial rotation generators $J_3^{\mathrm{vec}}{}_{12}=-i$ and $J_3^{\mathrm{spin}}=-i\gamma^1\gamma^2/2$, the vector has <helicity> $-s$ and the <Majorana spinor> has <helicity> $-s/2$, giving total <helicity> $-3s/2$. The <Majorana spinor> condition fixes the negative-frequency coefficients from these amplitudes. Therefore \b[a massless gravitino has spin $3/2$, with the two physical helicities $\pm3/2$]. A complex vector-spinor would also have independent antiparticle modes.
The requested count is instead the <off-shell component count of minimal supergravity>, before imposing any <Euler-Lagrange field equations>. A four-dimensional <Majorana spinor> has four real components. Its vector index supplies four copies, and local <supersymmetry> supplies four real gauge functions:
$$
\boxed{n_{\mathrm{gravitino,off}}=4\times4-4=12.}
$$
One must not additionally subtract transversality and the <massless Dirac equation> in this count: those are <on shell> conditions. The gamma-trace <gauge fixing> already uses the same four gauge functions subtracted above. For the <graviton>, a symmetric <metric tensor> has ten real components, with four <diffeomorphism> gauge functions. Equivalently, a <vierbein> has sixteen real components, minus six <local Lorentz transformation> and four <diffeomorphism> functions:
$$
\boxed{n_{\mathrm{graviton,off}}=10-4=16-6-4=6.}
$$
The <gravitino> and <graviton> each have two propagating states <on shell>, but their off-shell counts disagree by six.
An <auxiliary field> has an algebraic <Euler-Lagrange field equation> and no propagating kinetic term. In <old-minimal supergravity>, a complex scalar $M$ contributes two real bosonic components and a real vector $b_\mu$ contributes four, so the bosonic count becomes $6+2+4=12$, matching the <gravitino>. Their <supersymmetry> transformations also absorb the terms that would otherwise require the propagating <Euler-Lagrange field equations> to close the local algebra. Thus \b[auxiliary fields allow off-shell supersymmetry closure and matching component counts without adding physical particles]. Eliminating them recovers an <on shell> formulation.
Back to article page
