Rarita–Schwinger equation

In theoretical physics, the Rarita–Schwinger equation is the relativistic field equation of spin-3/2 fermions in a four-dimensional flat spacetime. It is similar to the Dirac equation for spin-1/2 fermions. This equation was first introduced by William Rarita and Julian Schwinger in 1941.

In modern notation it can be written as:^[1]

${\displaystyle \left(\epsilon ^{\mu \kappa \rho \nu }\gamma _{5}\gamma _{\kappa }\partial _{\rho }-im\sigma ^{\mu \nu }\right)\psi _{\nu }=0}$

where ${\displaystyle \epsilon ^{\mu \kappa \rho \nu }}$ is the Levi-Civita symbol, ${\displaystyle \gamma _{\kappa }}$ are Dirac matrices (with ${\displaystyle \kappa =0,1,2,3}$) and ${\displaystyle \gamma _{5}=i\gamma _{0}\gamma _{1}\gamma _{2}\gamma _{3}}$, ${\displaystyle m}$ is the mass, ${\displaystyle \sigma ^{\mu \nu }\equiv {\frac {i}{2}}[\gamma ^{\mu },\gamma ^{\nu }]}$, and ${\displaystyle \psi _{\nu }}$ is a vector-valued spinor with additional components compared to the four component spinor in the Dirac equation. It corresponds to the (⁠1/2⁠, ⁠1/2⁠) ⊗ ((⁠1/2⁠, 0) ⊕ (0, ⁠1/2⁠)) representation of the Lorentz group, or rather, its (1, ⁠1/2⁠) ⊕ (⁠1/2⁠, 1) part.^[2]

This field equation can be derived as the Euler–Lagrange equation corresponding to the Rarita–Schwinger Lagrangian:^[3]

${\displaystyle {\mathcal {L}}=-{\tfrac {1}{2}}\;{\bar {\psi }}_{\mu }\left(\epsilon ^{\mu \kappa \rho \nu }\gamma _{5}\gamma _{\kappa }\partial _{\rho }-im\sigma ^{\mu \nu }\right)\psi _{\nu }}$

where the bar above ${\displaystyle \psi _{\mu }}$ denotes the Dirac adjoint.

This equation controls the propagation of the wave function of composite objects such as the delta baryons (
Δ
) or for the conjectural gravitino. So far, no elementary particle with spin 3/2 has been found experimentally.

The massless Rarita–Schwinger equation has a fermionic gauge symmetry: is invariant under the gauge transformation ${\displaystyle \psi _{\mu }\rightarrow \psi _{\mu }+\partial _{\mu }\epsilon }$, where ${\displaystyle \epsilon \equiv \epsilon _{\alpha }}$ is an arbitrary spinor field. This is simply the local supersymmetry of supergravity, and the field must be a gravitino.

"Weyl" and "Majorana" versions of the Rarita–Schwinger equation also exist.

Equations of motion in the massless case

Consider a massless Rarita–Schwinger field described by the Lagrangian density

${\displaystyle {\mathcal {L}}_{RS}={\bar {\psi }}_{\mu }\gamma ^{\mu \nu \rho }\partial _{\nu }\psi _{\rho },}$

where the sum over spin indices is implicit, ${\displaystyle \psi _{\mu }}$ are Majorana spinors, and

${\displaystyle \gamma ^{\mu \nu \rho }\equiv {\frac {1}{3!}}\gamma ^{[\mu }\gamma ^{\nu }\gamma ^{\rho ]}.}$

To obtain the equations of motion we vary the Lagrangian with respect to the fields ${\displaystyle \psi _{\mu }}$, obtaining:

${\displaystyle \delta {\mathcal {L}}_{RS}=\delta {\bar {\psi }}_{\mu }\gamma ^{\mu \nu \rho }\partial _{\nu }\psi _{\rho }+{\bar {\psi }}_{\mu }\gamma ^{\mu \nu \rho }\partial _{\nu }\delta \psi _{\rho }=\delta {\bar {\psi }}_{\mu }\gamma ^{\mu \nu \rho }\partial _{\nu }\psi _{\rho }-\partial _{\nu }{\bar {\psi }}_{\mu }\gamma ^{\mu \nu \rho }\delta \psi _{\rho }+{\text{ boundary terms}}}$

using the Majorana flip properties^[4] we see that the second and first terms on the RHS are equal, concluding that

${\displaystyle \delta {\mathcal {L}}_{RS}=2\delta {\bar {\psi }}_{\mu }\gamma ^{\mu \nu \rho }\partial _{\nu }\psi _{\rho },}$

plus unimportant boundary terms. Imposing ${\displaystyle \delta {\mathcal {L}}_{RS}=0}$ we thus see that the equation of motion for a massless Majorana Rarita–Schwinger spinor reads:

${\displaystyle \gamma ^{\mu \nu \rho }\partial _{\nu }\psi _{\rho }=0.}$

The gauge symmetry of the massless Rarita-Schwinger equation allows the choice of the gauge ${\displaystyle \gamma ^{\mu }\psi _{\mu }=0}$, reducing the equations to:

${\displaystyle \gamma ^{\nu }{\partial }_{\nu }\psi _{\mu }=0,\quad \partial ^{\mu }\psi _{\mu }=0,\quad \gamma ^{\mu }\psi _{\mu }=0.}$

A solution with spins 1/2 and 3/2 is given by:^[5]

${\displaystyle \psi _{0}=\kappa ,\quad \psi _{i}=\psi _{i}^{TT}+{\frac {\gamma ^{j}\partial _{j}}{\nabla ^{2}}}\gamma _{0}\partial _{i}\kappa ,}$

where ${\displaystyle \nabla ^{2}}$ is the spatial Laplacian, ${\displaystyle \psi _{i}^{TT}}$ is doubly transverse,^[6] carrying spin 3/2, and ${\displaystyle \kappa }$ satisfies the massless Dirac equation, therefore carrying spin 1/2.

Drawbacks of the equation

The current description of massive, higher spin fields through either Rarita–Schwinger or Fierz–Pauli formalisms is afflicted with several maladies.

Superluminal propagation

As in the case of the Dirac equation, electromagnetic interaction can be added by promoting the partial derivative to gauge covariant derivative:

${\displaystyle \partial _{\mu }\rightarrow D_{\mu }=\partial _{\mu }-ieA_{\mu }}$.

In 1969, Velo and Zwanziger showed that the Rarita–Schwinger Lagrangian coupled to electromagnetism leads to equation with solutions representing wavefronts, some of which propagate faster than light. In other words, the field then suffers from acausal, superluminal propagation; consequently, the quantization in interaction with electromagnetism is essentially flawed^[why?]. In extended supergravity, though, Das and Freedman^[7] have shown that local supersymmetry solves this problem^[how?].

References

1. S. Weinberg, "The quantum theory of fields", Vol. 3, Cambridge p. 335
2. S. Weinberg, "The quantum theory of fields", Vol. 1, Cambridge p. 232
3. S. Weinberg, "The quantum theory of fields", Vol. 3, Cambridge p. 335
4. Pierre Ramond - Field theory, a Modern Primer - p.40
5. Valenzuela, M.; Zanelli, J. (2024). "Massless Rarita-Schwinger equations: Half and three halves spin solution". SciPost Phys. 16 (3): 065. arXiv:2305.00106. doi:10.21468/SciPostPhys.16.3.065.
6. Deser, S.; Kay, J. H.; Stelle, K. S. (1977). "Hamiltonian Formulation of Supergravity". Phys. Rev. D. 16 (8): 2448–2455. doi:10.1103/PhysRevD.16.2448.
7. Das, A.; Freedman, D. Z. (1976). "Gauge quantization for spin-3/2 fields". Nuclear Physics B. 114 (2): 271. Bibcode:1976NuPhB.114..271D. doi:10.1016/0550-3213(76)90589-7.; Freedman, D. Z.; Das, A. (1977). "Gauge internal symmetry in extended supergravity". Nuclear Physics B. 120 (2): 221. Bibcode:1977NuPhB.120..221F. doi:10.1016/0550-3213(77)90041-4.

Sources

• Rarita, William; Schwinger, Julian (1941-07-01). "On a Theory of Particles with Half-Integral Spin". Physical Review. 60 (1). American Physical Society (APS): 61. Bibcode:1941PhRv...60...61R. doi:10.1103/physrev.60.61. ISSN 0031-899X.
• Collins P.D.B., Martin A.D., Squires E.J., Particle physics and cosmology (1989) Wiley, Section 1.6.
• Velo, Giorgio; Zwanziger, Daniel (1969-10-25). "Propagation and Quantization of Rarita-Schwinger Waves in an External Electromagnetic Potential". Physical Review. 186 (5). American Physical Society (APS): 1337–1341. Bibcode:1969PhRv..186.1337V. doi:10.1103/physrev.186.1337. ISSN 0031-899X.
• Velo, Giorgio; Zwanzinger, Daniel (1969-12-25). "Noncausality and Other Defects of Interaction Lagrangians for Particles with Spin One and Higher". Physical Review. 188 (5). American Physical Society (APS): 2218–2222. Bibcode:1969PhRv..188.2218V. doi:10.1103/physrev.188.2218. ISSN 0031-899X.
• Kobayashi, M.; Shamaly, A. (1978-04-15). "Minimal electromagnetic coupling for massive spin-two fields". Physical Review D. 17 (8). American Physical Society (APS): 2179–2181. Bibcode:1978PhRvD..17.2179K. doi:10.1103/physrevd.17.2179. ISSN 0556-2821.