# What about the (1, 1/2), or (3/2, 1/2) representations of the Lorentz group?

All irreducible finite dimensional complex representations of the Lorentz group can be specified by two positive half-integers, i.e. $$(j_1, j_2)$$. The $$(0,0)$$ representation is the trivial scalar representation, $$(\tfrac{1}{2}, 0)$$ is the left handed Weyl spinor representation, $$(0, \tfrac{1}{2})$$ is the right handed Weyl spinor representation, and $$(\tfrac{1}{2}, \tfrac{1}{2})$$ is the (complex) vector representation. Most QFT textbooks talk about these representations. And then they stop. What about the all the other options, like $$(0,1)$$, $$(1, \tfrac{1}{2})$$, $$(\tfrac{3}{2}, 1)$$, etc? Do these have any relevance at all? Have people ever speculated that such fields exist?

This also gets at a bigger question. Certainly, many different $$(j_1, j_2)$$ representations will have the same $$SO(3)$$ spin. It seems to me like there should be many interesting ways to make a "spin $$\tfrac{3}{2}$$" particle, for example, each behaving differently under parity.

• Possible duplicate here. – knzhou Sep 26 '18 at 21:08
• Of course people have speculated, ad nauseam. Comfortable with the canonical table? – Cosmas Zachos Sep 26 '18 at 22:33

The $$(0,1)$$ rep is an antisymmetric two-tensor that is either self-dual or anti-self-dual. The field strength $$F_{\mu \nu}$$ in the Maxwell theory is a sum of both reps $$(1,0) + (0,1)$$. Likewise the (in)famous Rarita-Schwinger fermion transforms in the $$(1,1/2) + (1/2,1)$$ representation.
In general you'll find theorems that only a finite number of representations of the Lorentz group appear, because higher-spin fields behave pathologically. This isn't quite true, in the sense that starting with a boson $$\phi$$, a Dirac fermion $$\Psi$$ and a gauge field $$A_\mu$$ you can easily build composite operators that transform in any representation you want. Those composite operators don't have their own dynamics though: their behavior is completely governed by the Lagrangians of the fundamental fields they're built out of.