Rarita-Schwinger equation

In theoretical physics , the Rarita-Schwinger equation (after William Rarita and Julian Schwinger , who formulated it in 1941 ) is a relativistic field equation for spin -3/2 fermions . It is commonly used to describe and study composite particles like the delta baryon , and sometimes it is used for hypothetical particle fields like the gravitino . So far, however, no stable elementary particle with spin 3/2 has been experimentally proven.

The Rarita-Schwinger equation has a similar structure to the Dirac equation for spin 1/2 fermions and can be derived from this. In modern notation , it is written as follows:

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

With

• ${\displaystyle \epsilon ^{\mu \kappa \rho \nu }}$the Levi Civita symbol
• ${\displaystyle \gamma _{5}}$and Dirac matrices ${\displaystyle \gamma _{\nu }}$
• ${\displaystyle m}$the mass of the fermion
• ${\displaystyle \sigma ^{\mu \nu }\equiv i/2\left[\gamma ^{\mu },\gamma ^{\nu }\right]}$
• ${\displaystyle \psi _{\nu }}$a wave function with the Lorentz index . The wave function transforms with respect to this index like an ordinary four-vector . Each of the four individual components of the wave function also transforms like a Dirac spinor . The representation corresponds to that or representation of the Lorentz group . ${\displaystyle \nu }$ ${\displaystyle \left({\tfrac {1}{2}},{\tfrac {1}{2}}\right)\otimes \left(\left({\tfrac {1}{2}},0\right)\oplus \left(0,{\tfrac {1}{2}}\right)\right)}$ ${\displaystyle \left(1,{\tfrac {1}{2}}\right)\oplus \left({\tfrac {1}{2}},1\right)}$

The Rarita-Schwinger equation can be derived from the following Lagrange density :

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

The adjoint spinor denotes to . ${\displaystyle {\bar {\psi }}_{\mu }=\psi _{\mu }^{\dagger }\gamma ^{0}}$ ${\displaystyle \psi _{\mu }}$

The Rarita-Schwinger equation has a gauge symmetry for particles with mass 0 with respect to the gauge transformation . There is a freely selectable, fermionic Majorana field that belongs to a calibrated supersymmetry transformation. ${\displaystyle \psi _{\mu }\rightarrow \psi _{\mu }+\partial _{\mu }\epsilon }$ ${\displaystyle {\mathcal {\epsilon }}}$

Weyl and Majorana representations of the Rarita-Schwinger equation also exist, which do not differ from the original equation in terms of the physical results.

literature

• W. Rarita and J. Schwinger, On a Theory of Particles with Half-Integral Spin. Phys. Rev. 60, 61 (1941).
• Collins PDB, Martin AD, Squires EJ, Particle physics and cosmology (1989) Wiley, Section 1.6 .
• G. Velo, D. Zwanziger, Propagation and Quantization of Rarita-Schwinger Waves in an External Electromagnetic Potential , Phys. Rev. 186, 1337 (1969).
• G. Velo, D. Zwanziger, Noncausality and Other Defects of Interaction Lagrangians for Particles with Spin One and Higher , Phys. Rev. 188, 2218 (1969).
• M. Kobayashi, A. Shamaly, Minimal Electromagnetic coupling for massive spin-two fields , Phys. Rev. D 17, 8, 2179 (1978).

Books

• Walter Greiner : Theoretical Physics. Volume 6: Relativistic Quantum Mechanics. Wave equations. 2nd revised and expanded edition. Deutsch, Thun et al. 1987, ISBN 3-8171-1022-7 .

Individual evidence

1. ^ S. Weinberg, "The quantum theory of fields", Volume 3, Cambridge p. 335
2. ^ S. Weinberg, "The quantum theory of fields", Volume 1, Cambridge p. 232
3. ^ S. Weinberg, "The quantum theory of fields", Volume 3, Cambridge p. 335