In theoretical physics, the Rarita–Schwinger equation is the
relativistic field equation for spin-3/2
fermions. It is the spin-3/2 analogue of the Dirac equation for
spin-1/2 fermions. The equation was introduced by William Rarita and
Julian Schwinger in 1941.
The Rarita–Schwinger field is a vector-valued spinor, usually denoted
<math>\psi_\mu</math>. In four dimensions it transforms as the tensor product
of a Lorentz vector and a Dirac spinor,
,
and the spin-3/2 part is contained in the
representation.
The field therefore contains lower-spin components, which must be removed by
constraints.
In modern four-component notation the massive free equation may be written as
:<math>
\left(\epsilon^{\mu\kappa\rho\nu}\gamma_5\gamma_\kappa\partial_\rho
- i m \sigma^{\mu\nu}\right)\psi_\nu = 0,
</math>
where <math>\epsilon^{\mu\kappa\rho\nu}</math> is the
Levi-Civita symbol, <math>\gamma_\kappa</math> are the
Dirac matrices, <math>\gamma_5=i\gamma_0\gamma_1\gamma_2\gamma_3</math>,
<math>m</math> is the mass, and
:<math>
\sigma^{\mu\nu}\equiv {i\over 2}[\gamma^\mu,\gamma^\nu].
</math>
The equation follows from the Rarita–Schwinger Lagrangian
:<math>
\mathcal L
= -{1\over 2}\bar\psi_\mu
\left(\epsilon^{\mu\kappa\rho\nu}\gamma_5\gamma_\kappa\partial_\rho
- i m \sigma^{\mu\nu}\right)\psi_\nu ,
</math>
where the bar denotes the Dirac adjoint.
The Rarita–Schwinger equation is the standard relativistic field equation
for the gravitino, the spin-3/2 gauge field of supergravity. In
four-dimensional <math>\mathcal N=1</math> supergravity the gravitino may be
written as a Weyl vector-spinor, or equivalently as a Majorana
Rarita–Schwinger field. When supersymmetry is spontaneously broken, the
super-Higgs mechanism gives the gravitino a Majorana mass. In extended
supergravity there can be several gravitini; for example, in broken
<math>\mathcal N=2</math> supergravity two Majorana gravitini may be organized
as a Dirac gravitino if an appropriate residual symmetry is present.
Massless equation and gauge invariance
A massless Rarita–Schwinger field may be described by the Lagrangian density
:<math>
\mathcal L_{RS}=\bar\psi_\mu \gamma^{\mu\nu\rho}\partial_\nu\psi_\rho,
</math>
where
:<math>
\gamma^{\mu\nu\rho}\equiv \gamma^{[\mu}\gamma^\nu\gamma^{\rho]}
</math>
is the antisymmetrized product of gamma matrices. The corresponding equation
of motion is
:<math>
\gamma^{\mu\nu\rho}\partial_\nu\psi_\rho=0.
</math>
The massless equation has the fermionic gauge invariance
:<math>
\psi_\mu \rightarrow \psi_\mu+\partial_\mu\epsilon,
</math>
where <math>\epsilon</math> is an arbitrary spinor field. This gauge invariance
removes the unphysical spin-1/2 components of the vector-spinor. In supergravity this gauge invariance is the spin-3/2 part of local
supersymmetry, and the Rarita–Schwinger field is the gravitino.
Using this gauge invariance, one may impose the gamma-traceless gauge
:<math>
\gamma^\mu\psi_\mu=0.
</math>
Together with the equation of motion, this gives the transverse and Dirac-type
conditions
:<math>
\gamma^\nu\partial_\nu\psi_\mu=0,\qquad
\partial^\mu\psi_\mu=0,\qquad
\gamma^\mu\psi_\mu=0.
</math>
These equations describe the two helicity states of a massless spin-3/2 field.
Massive field and constraints
A massive spin-3/2 particle in four dimensions has
:<math>
2s+1=4
</math>
physical degrees of freedom. The vector-spinor <math>\psi_\mu</math>, however,
contains more components. A consistent massive Rarita–Schwinger system must
therefore imply subsidiary conditions that remove the lower-spin sector.
For the free massive theory, the equation of motion implies the constraints
:<math>
\gamma^\mu\psi_\mu=0,\qquad
\partial^\mu\psi_\mu=0,
</math>
together with the Dirac equation acting on each vector component,
:<math>
(\gamma^\nu\partial_\nu+m)\psi_\mu=0
</math>
up to convention-dependent factors of <math>i</math>. These constraints are
the spin-3/2 analogue of the Fierz–Pauli subsidiary conditions for higher-spin
fields.
Interactions and the Velo–Zwanziger problem
The question of coupling higher-spin particles to external fields goes back
at least to Dirac's 1936 work on relativistic wave equations for particles of
arbitrary spin. Dirac emphasized that such equations might be useful either
for future elementary particles with spin greater than one half, or as
approximate equations for composite particles.
For spin <math>3/2</math>, this problem is subtle because the vector-spinor
field contains unphysical spin-1/2 components. A consistent interacting
equation must propagate the constraints that remove these components, rather
than turning them into additional degrees of freedom.
In four-component notation the vector-spinor is denoted
<math>\psi_\mu</math>. Equivalently, in two-component notation one may write
it as a pair of Weyl vector-spinors,
:<math>
\psi_\mu =
\begin{pmatrix}
\chi_{\mu\alpha} \\
\bar\lambda_\mu{}^{\dot\alpha}
\end{pmatrix}.
</math>
The spin-3/2 constraints are then the two-component form of the
gamma-trace condition:
:<math>
\bar\sigma^\mu\chi_\mu=0,\qquad
\sigma^\mu\bar\lambda_\mu=0.
</math>
Together with the corresponding divergence constraints, these remove the
lower-spin sector and leave the four physical polarizations of a massive
spin-3/2 particle.
For a charged field, the simplest attempt is the minimal substitution
:<math>
\partial_\mu\rightarrow D_\mu=\partial_\mu-i e A_\mu .
</math>
Johnson and Sudarshan found an inconsistency in the canonical quantization of
the minimally coupled spin-3/2 field,
and Velo and Zwanziger showed that the corresponding classical equations in
an external electromagnetic background can have pathological characteristic
surfaces.
The problem may appear either as superluminal propagation or as the loss of a
constraint, leading to the propagation of an unphysical number of modes.
A useful way to state the Velo–Zwanziger problem is therefore the following:
one must find interacting equations of motion whose constraint chain closes
in the electromagnetic background. In other words, the primary spin-3/2
conditions, such as <math>\gamma^\mu\psi_\mu=0</math>, must imply consistent
secondary divergence constraints, and these constraints must be preserved by
the equations of motion.
Non-minimal electromagnetic couplings can restore this closure in special
backgrounds. In a constant electromagnetic field, a point-particle
Fierz–Pauli system for a massive charged spin-3/2 field may be written as
:<math>
i\bar\sigma^n D_n\chi_m+M\bar\lambda_m=0,
</math>
:<math>
i\sigma^n D_n\bar\lambda_m+M\chi_m
=-\,i{2e\over M}F_{mn}\chi^n,
</math>
together with
:<math>
\bar\sigma^m\chi_m=0,\qquad
\sigma^m\bar\lambda_m=0.
</math>
Taking traces and divergences of these equations gives a closed
Fierz–Pauli constraint chain in the constant-field background. The
non-minimal Pauli term is essential, and the corresponding point-particle
gyromagnetic ratio is
:<math>
g=2.
</math>
The value <math>g=2</math> is not universal for composite spin-3/2 particles:
the baryon, for example, has a measured magnetic
moment corresponding to a gyromagnetic ratio different from 2. Such cases can
be described by effective equations of motion, valid in weak electromagnetic
backgrounds, in which the constraint chain is maintained perturbatively.
Supergravity gives another special realization of interacting spin-3/2
fields, namely the gravitino. It should not be regarded as a general solution
of the massive charged Rarita–Schwinger problem, because in supergravity the
gravitino mass, charge and gravitational coupling are not independent
parameters.