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?
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1$\begingroup$ Hi Hare Krishna, I have taken the liberty to clarify your question as I suspect English may not be your first language, and the question was very unclear. Please let me know if I have substantially changed what you meant, or feel free to rollback the edit here. $\endgroup$– Emilio PisantyCommented Jun 5, 2015 at 17:03
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$\begingroup$ Related question: physics.stackexchange.com/q/133195 $\endgroup$– BosoneandoCommented Jun 5, 2015 at 17:11
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1 Answer
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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.
answered Jun 5, 2015 at 17:31
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$\begingroup$ actually i am asking that whether there exist another representation like spinor of lorentz group. what is the conserved quantity associated of them. $\endgroup$ Commented Jun 5, 2015 at 17:43
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$\begingroup$ @HareKrishna In the first paragraph, I've given you examples of other representations, and in the rest of my answer I've explained that the conserved quantity is always spin: the difference is what the value of spin is (I will expand on this) $\endgroup$ Commented Jun 5, 2015 at 17:46
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$\begingroup$ is that all representation having gamma matrices anti commutation relation ?.then they are typically same.meaning changing spin does not change the entire representation. $\endgroup$ Commented Jun 5, 2015 at 18:03
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$\begingroup$ @HareKrishna No, gamma matrices are elements of the Clifford algebra associated with the spinorial (i.e., spin 1/2) represenatation of the Lorentz field. There are other representations (the ones that I talked about in my answer) that are not constructed from gamma matrices, but that follow the same Lie algebra. $\endgroup$ Commented Jun 5, 2015 at 18:48
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$\begingroup$ @HareKrishna You should check this answer: physics.stackexchange.com/a/63178 where this is explained very clearly $\endgroup$ Commented Jun 5, 2015 at 18:50