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:
With
- the Levi Civita symbol
- and Dirac matrices
- the mass of the fermion
- 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 .
The Rarita-Schwinger equation can be derived from the following Lagrange density :
The adjoint spinor denotes to .
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.
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 .