If the orbital angular momentum of an electron in an atomic orbital is associated with (generated by) an asymmetry in the orbital wave function, is it also the case that the intrinsic spin of a free electron (as one specific example) in QFT is associated with an asymmetry in the coherent excitation pattern of the electron matter field/EM field constituting the electron? Or, is intrinsic spin more fundamental, perhaps relating to the fact that the underlying matter field is a spinor field.
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1$\begingroup$ What do you mean by "an asymmetry in the coherent excitation pattern of the electron matter field/EM field constituting the electron"? $\endgroup$– my2ctsCommented Feb 11, 2019 at 21:29
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$\begingroup$ For example, the two conjugate phased nodes of an individual p-orbital. $\endgroup$– CSnowdenCommented Feb 13, 2019 at 3:42
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1 Answer
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No, the intrinsic spin of a particle is just a consequence of a "particle" being represented by an irreducible representation of the the Poincaré group.
The spin of the particle is determined by the dimension of the representation that it assumes under the double covering group $SL(2,\mathbb{C})$ of the Lorentz group $SO(1,3)$. For example, spin 1/2 particles live in the fundamental (or spin-1/2) representation of $SL(2, \mathbb{C})$. These come in either left handed reps (1/2,0) or in right handed reps (0,1/2), both of which are spin 1/2 representations.
answered Feb 11, 2019 at 22:40
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2$\begingroup$ There's a bit of terminological confusion in this answer: The unitary representations of the Poincaré group are all infinite-dimensional and not the finite-dimensional representations denoted by a pair of half-integers $(s_1,s_2)$ you talk about later. $\endgroup$– ACuriousMind ♦Commented Feb 12, 2019 at 19:29
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$\begingroup$ In what sense exactly are those reps infinite dimensional? My understanding was that a chiral spinor lives in a two dimensional complex space. Or are you saying that is the case, but that’s not a unitary representation of the PG? $\endgroup$ Commented Feb 12, 2019 at 22:17
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$\begingroup$ The latter. A chiral spinor field lives in two dimensions, its particle states live in an infinite-dimensional Hilbert space. It is the latter space upon which a unitary representation exists. $\endgroup$– ACuriousMind ♦Commented Feb 12, 2019 at 22:18
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$\begingroup$ I see.. but then how can we say that the (1/2,0) spinor is an irrep of the PG if it’s only dimension 2, or is that wrong too? $\endgroup$ Commented Feb 12, 2019 at 22:23
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$\begingroup$ It is irreducible. It is just not unitary, and it doesn't have to be - only the quantum space of states needs a unitary rep. $\endgroup$– ACuriousMind ♦Commented Feb 12, 2019 at 22:27