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## Main Question or Discussion Point

Hi,

from Srednickis QFT textbook, we know the following coupling of Lorentz group representations: [tex](2,1)\otimes (2,1) = (1,1)_A \oplus (3,1)_S,[/tex] which yields [tex]\epsilon_{a b}[/tex] as an invariant symbol. Generalising, we can look at [tex](2,1)\otimes (2,1) \otimes (2,1) \otimes (2,1) = (1,1) \oplus (1,1)\oplus ...[/tex]

which implies the existence of two invariant symbols with four spinor indices. One is just the trivial coupling [tex]\epsilon_{a b}\epsilon_{c d},[/tex] but what is the other one? It should be symmetric in the first two indices, and in the last two, and (I think) be further symmetric under the exchange of the two pairs. Could someone please confirm that it is just [tex](S_L^{\mu\nu})_{a b} (S_L^{\rho\sigma})_{c d} \epsilon_{\mu\nu\rho\sigma}? [/tex]

(or in a different notation

[tex](\sigma^{\mu\nu})_{a b} (\sigma_{\mu\nu})_{c d}.) [/tex]

Thanks,

torus

from Srednickis QFT textbook, we know the following coupling of Lorentz group representations: [tex](2,1)\otimes (2,1) = (1,1)_A \oplus (3,1)_S,[/tex] which yields [tex]\epsilon_{a b}[/tex] as an invariant symbol. Generalising, we can look at [tex](2,1)\otimes (2,1) \otimes (2,1) \otimes (2,1) = (1,1) \oplus (1,1)\oplus ...[/tex]

which implies the existence of two invariant symbols with four spinor indices. One is just the trivial coupling [tex]\epsilon_{a b}\epsilon_{c d},[/tex] but what is the other one? It should be symmetric in the first two indices, and in the last two, and (I think) be further symmetric under the exchange of the two pairs. Could someone please confirm that it is just [tex](S_L^{\mu\nu})_{a b} (S_L^{\rho\sigma})_{c d} \epsilon_{\mu\nu\rho\sigma}? [/tex]

(or in a different notation

[tex](\sigma^{\mu\nu})_{a b} (\sigma_{\mu\nu})_{c d}.) [/tex]

Thanks,

torus