theoretical physics, the Rarita-Schwinger equation is the
field equationof spin-3/2 fermions. It is similar to the Dirac equationfor spin-1/2 fermions. This equation was first introduced by William Raritaand Julian Schwingerin 1941. In modern notation it can be written as::where is the Levi-Civita symbol, and are Dirac matrices, is the mass and is a vector-valued spinorwith additional components compared to the four component spinor in the Dirac equation. It corresponds to the representation of the Lorentz group, or rather, its part.This field equation can be derived from the following Lagrangian::where the bar above denotes the Dirac adjoint.As in the case of the Dirac equation, electromagnetic interaction can be added simply by promoting the partial derivative to gauge covariant derivative::
The massless Rarita-Schwinger equation has a gauge symmetry, under the gauge transformation of , where is an arbitrary spinor field.
"Weyl" and "Majorana" versions of the Rarita-Schwinger equation also exist.
This equation is useful for the
wave functionof composite objects like Delta (Δ) baryons or for proposed elementary fields like the gravitino. So far, no fundamental particle with spin 3/2 has been found experimentally.
Drawbacks of the formalism
The current description of massive, higher spin fields through either
Rarita-Schwingeror Fierz–Pauliformalisms is afflicted with several maladies. Upon gauging, high spin fields suffer from acausal, superluminal propagation; besides, the quantizationof these systems in interaction with electromagnetism is essentially flawed. Also, algebraic inconsistencies appear upon gauging which can only be avoided by requiring that all equations involving derivatives be obtainable from a Lagrangian, a procedure that becomes involved because of the need of introducing auxiliary fields in order to obtain all constraints from the Lagrangian.
In 1969, Velo and Zwanziger showed that the Rarita–Schwinger lagrangian coupled to
electromagnetismleads to equation with solutions representing wavefronts, some of which propagate faster than light. There are some Lorentz framesthat allow the consistent formulation of quantum mechanicsand quantum field theory; there are some others that don’t.
* W. Rarita and J. Schwinger, " [http://prola.aps.org/abstract/PR/v60/i1/p61_1 On a Theory of Particles with Half-Integral Spin] Phys. Rev. 60, 61 (1941).
*Collins P.D.B., Martin A.D., Squires E.J., "Particle physics and cosmology" (1989) Wiley, "Section 1.6".
* G. Velo, D. Zwanziger, Phys. Rev. 186, 1337 (1969).
* G. Velo, D. Zwanziger, Phys. Rev. 188, 2218 (1969).
* M. Kobayashi, A. Shamaly, Phys. Rev. D 17,8, 2179 (1978).
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