- #1
gazebo_dude
- 9
- 0
Hi all and merry christmas,
I've been wondering about how to construct spin 3/2 reps of the Lorentz group. The way I've seen it done is with vector-spinors ala the Rarita-Schwinger equations: [tex]\psi_{a\mu}[/tex], [tex]\mu[/tex] being a 4-vector index and a being a (bi)spinor index. Then a reduction is made by projecting out 8 of the 16 degrees of freedom leaving 4 particle+4 antiparticle. This corresponds to the group theory ((1/2,0)+(0,1/2))x(1/2,1/2)=(1,1/2)+(0,1/2)+(1/2,0)+(1/2,1). The desired bits are (1,1/2)+(1/2,1).
My question is: why not just use the (3/2,0)+(0,3/2) rep? The fields can be represented as objects having three symmetric spin indices: [tex]\psi_{abc}[/tex] for the left handed and [tex]\bar{\xi}^{\dot{a}\dot{b}\dot{c}}[/tex] for the right handed. I haven't tried any computations yet, but a Lagrangian kinetic term might look something like [tex]\bar{\psi}^{\dot{a}\dot{b}\dot{c}} i\partial^{a}_{\dot{a}} \psi_{abc} \eta^{bc}_{\dot{b}\dot{c}}[/tex] where the derivative and metric are in terms of spinor indices and index gymnastics have been done with abandon. There are certainly no issues with projecting out undesired degrees of freedom.
I haven't seen anything like this for spin 3/2 particles before. Has it been tried? Are there any well known problems with this theory?
Cheers,
Michael
I've been wondering about how to construct spin 3/2 reps of the Lorentz group. The way I've seen it done is with vector-spinors ala the Rarita-Schwinger equations: [tex]\psi_{a\mu}[/tex], [tex]\mu[/tex] being a 4-vector index and a being a (bi)spinor index. Then a reduction is made by projecting out 8 of the 16 degrees of freedom leaving 4 particle+4 antiparticle. This corresponds to the group theory ((1/2,0)+(0,1/2))x(1/2,1/2)=(1,1/2)+(0,1/2)+(1/2,0)+(1/2,1). The desired bits are (1,1/2)+(1/2,1).
My question is: why not just use the (3/2,0)+(0,3/2) rep? The fields can be represented as objects having three symmetric spin indices: [tex]\psi_{abc}[/tex] for the left handed and [tex]\bar{\xi}^{\dot{a}\dot{b}\dot{c}}[/tex] for the right handed. I haven't tried any computations yet, but a Lagrangian kinetic term might look something like [tex]\bar{\psi}^{\dot{a}\dot{b}\dot{c}} i\partial^{a}_{\dot{a}} \psi_{abc} \eta^{bc}_{\dot{b}\dot{c}}[/tex] where the derivative and metric are in terms of spinor indices and index gymnastics have been done with abandon. There are certainly no issues with projecting out undesired degrees of freedom.
I haven't seen anything like this for spin 3/2 particles before. Has it been tried? Are there any well known problems with this theory?
Cheers,
Michael