= Solution
Use four-dimensional <Minkowski spacetime> with $\{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}$ and $\gamma^{\mu\nu\rho}=\gamma^{[\mu}\gamma^\nu\gamma^{\rho]}$. The <Clifford algebra> duality identity makes the displayed epsilon-form equation equivalent, up to a nonzero convention-dependent phase, to
$$
E^\mu:=\gamma^{\mu\nu\rho}\partial_\nu\Psi_\rho=0.
$$
This is the <Euler-Lagrange equation> of the first-order <Rarita-Schwinger field> action $-\tfrac12\bar\Psi_\mu\gamma^{\mu\nu\rho}\partial_\nu\Psi_\rho$, with the overall phase adjusted to the chosen Majorana convention. Varying the <Majorana spinor> action accounts for both occurrences of the field and cancels the one-half normalization.
A vector times a spinor contains <spin>-$3/2$ and unwanted <spin>-$1/2$ pieces, so the tensor type alone does not establish the physical spin. The field equation has <gauge invariance>
$$
\Psi_\mu\longmapsto\Psi_\mu+\partial_\mu\epsilon,
$$
since the two derivatives contract an antisymmetric pair of gamma indices. Put $\chi=\gamma^\mu\Psi_\mu$ and $d=\partial^\mu\Psi_\mu$. <Gamma matrix> multiplication gives
$$
E^\mu=\not\!\partial\Psi^\mu-\partial^\mu\chi
+\gamma^\mu(\not\!\partial\chi-d),\qquad
\gamma_\mu E^\mu=2(\not\!\partial\chi-d).
$$
Hence $d=\not\!\partial\chi$. Use the <gauge transformation>, under which $\delta\chi=\not\!\partial\epsilon$, to impose $\chi=0$. The equations then imply
$$
\boxed{\gamma\cdot\Psi=0,\qquad\partial\cdot\Psi=0,\qquad
\not\!\partial\Psi_\mu=0,}
$$
with residual gauge freedom satisfying $\not\!\partial\epsilon=0$.
For a <plane wave> with nonzero momentum satisfying the <null condition> along the third axis, this residual freedom sets the temporal component to zero. Transversality then removes the longitudinal component, leaving the <tensor product> of two transverse vector polarizations with the two massless spinor <helicities>. The possible total <helicities> are $\pm3/2$ and $\pm1/2$. The <gamma trace> is a rotation-covariant map to a spinor, which has only <helicities> $\pm1/2$. It therefore vanishes on the aligned $\pm3/2$ combinations and removes the two mixed combinations. Equivalently, the transverse <gamma trace> condition has rank two on the four remaining complex positive-frequency coefficients. The <massless Rarita-Schwinger polarization count> is thus
$$
\boxed{h=\pm\tfrac32:\quad2\text{ physical states for a Majorana gravitino}.}
$$
The <Majorana spinor> reality condition relates negative-frequency coefficients to their conjugates; it does not introduce a separate antiparticle multiplet. A complex vector-spinor would also have independent antiparticle states.
A nonzero standard <Rarita-Schwinger field> mass term destroys the massless <gauge invariance> and adds the two longitudinal <helicities>. To see the constraints explicitly, choose its momentum-space sign so the equation is
$$
\gamma^{\mu\nu\rho}p_\nu u_\rho+m\gamma^{\mu\rho}u_\rho=0.
$$
Contracting with $p_\mu$ gives $\not\!p(\gamma\cdot u)-p\cdot u=0$. Its <gamma trace> then gives $3m\,\gamma\cdot u=0$. For $m\ne0$ these imply $\gamma\cdot u=p\cdot u=0$, and the remaining <Dirac equation> is $(\not\!p-m)u_\mu=0$. In the rest frame $u_0=0$. Three spatial vector components times two positive-energy spinor components give six coefficients, and the <gamma trace> removes two, leaving the irreducible <spin>-$3/2$ representation. Therefore
$$
\boxed{\text{a massive Majorana gravitino has }4\text{ physical states},
\quad h=\pm\tfrac32,\ \pm\tfrac12.}
$$
In <supergravity> a consistent spontaneous mass generation is the <super-Higgs mechanism>, which supplies those extra states from the <goldstino>.
The massless <graviton> also has two physical <helicities>, so the <graviton> and <gravitino> already match on shell. The reason for <auxiliary fields> is the <off-shell component count of minimal supergravity> mismatch, not a mismatch of these physical states. After removing gauge functions but before using field equations, a metric has $10-4=6$ independent bosonic components, while a Majorana vector-spinor has $16-4=12$ fermionic components. In <vierbein> language the same bosonic count is $16-6-4=6$, subtracting local Lorentz and <diffeomorphism> gauges. <Old-minimal supergravity> adds a complex scalar and a real vector, supplying six real nonpropagating bosonic components. This gives \b[$12+12$ off-shell components and closure without imposing equations of motion]. Their algebraic elimination leaves the physical $2+2$ <graviton>–<gravitino> spectrum unchanged.
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