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I am reading Peskin and Schroeder's An introduction to field theory. They first describe the spinor representation of the Lorentz group, and then they mention the fact that different particles have different spins, and go on to describe the angular momentum associated with those particles. This is for the spinor representation of the Lorentz group, but: if there are other representations of the Lorentz group, like the spinor representation, will those other representations have another angular-momentum-like quantity associated with them?

Qmechanic
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Hare Krishna
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1 Answers1

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The spin of a quantum field is related to the representation of the Lorentz group they transform under: scalar fields transform under the trivial representation, spinors transform under the spinorial representation, gauge bosons under the vectorial representation, gravitons (if they exist) under the second-rank tensorial representation...

If you restrict to spatial rotations, any infinitesimal transformation can be written in terms of the infinitesimal generation of the transformation, namely the [total] angular moment: $$\phi(x) \to \phi'(x') = \left(1 - \frac{i}{2} \omega_{\mu\nu}J^{\mu\nu}\right)\phi(x)$$

What is the source of the change between the original and the rotated fields? There are two:

  • The transformation of the points $x$ to $x'$: this contribution is present in every field, and the related generator is what we call orbital angular moment
  • If the field has more than one component, these components transform in each other in a non-trivial way. The generator associated with this fact is called spin, and naturally, depends on the representation of the Lorentz group for the field.

EDIT: To clarify, in every representation the conserved quantity is spin. The only difference is the value of the spin of particles: Scalar fields have spin 0, spinorial fields have spin 1/2, gauge bosons have spin 1 and second-rank tensor have spin 2.

Bosoneando
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