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I am reading this article discussing quantum field theory. The author mentioned that 'a particle has a certain spin because its underlying field has a respective spin'. I think intuitively this makes sense because particles are excitations of the underlying quantum field. However, I was thinking if I can come up with a specific example to illustrate this. Assume we have the Dirac lagrangian in chiral representation: $$ \mathcal L = \bar\Psi(i\not\partial-m)\Psi $$

Since we know that the electron is described by a Dirac field, why this field has a half-integer spin? How can we say the mathematical structures of $\Psi$ make electrons spin 1/2?

Qmechanic
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IGY
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    The Lorentz representation on fields is very closely related to the unitaty Poincaré representation on the space of quantum states. This connection is behind the quantum field's spin and the associated particle's spin. In fact, quantum fields are in a sense constructed from this relation. See Weinberg's The Quantum Theory of Fields Chapter 5 for the complete details. – Gold Apr 24 '23 at 00:56
  • @Gold Thanks so much, that's quite helpful! – IGY Apr 24 '23 at 01:09
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    The verb/noun combination of "spinning electrons" is one of the more unfortunate language constructs of physics. It would be much better if we would say that "an electron" is an excitation of the physical vacuum by half a unit of angular momentum, one (negative) unit of electric charge and one unit of lepton number. Even that is not enough for "foundational" questions. The only thing that withstands rational analysis is that physics is (and always was) entirely about system properties. Objects are merely emergent classical phenomena. That's what that article says, isn't it? – FlatterMann Apr 24 '23 at 01:51
  • Related: https://physics.stackexchange.com/q/270357/50583 – ACuriousMind Apr 25 '23 at 13:15

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