Spinor formalism in QFT

Question

We can describe fields by two formalisms: vector and spinor. This is the result of possibility of representation of the Lorentz's group irreducible rep as straight cross product of two $SU(2)$ or two $SO(3)$ irreducible representation.

Vector formalism is more porular, because working with it is more convenient. But there are some theories (models of interaction), where an introduction of spinor formalism has some advantages; sometimes we can't avoid introduction of spinors (in the case of half-integer spin of the field) and sometimes we can create an interaction by introducing $SU(2)$ fields (like in the case of Yang-Mills theory).

So, my question is follow: how often spinor formalism is suitable for using in quantum field theory (in addition to the above results)?

Answer 1

Spinors and vectors aren't two "competing formalisms". They are two inequivalent representations of the rotational or Lorentzian group. For fermions such as electrons, one absolutely needs spinors and it would be extremely awkward, nearly impossible, to produce the same physics just with vectors and tensors constructed out of vectors.

On the other hand, spinors are "more elementary" so vectors may be constructed as tensor products of two spinors (a spinor is morally a "[tensor] square root" of a vector). So all the vector indices in vectors and tensors may be converted to spinor indices – this is probably what you meant by the spinor formalism. Penrose and Rindler loved to write all equations like that. Their precise formalism is rarely used but it is absolutely obvious that spinors can't be lived without.