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Quantum Physics
What exactly is spin? Does the standard model work without spin?
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[QUOTE="vanhees71, post: 6868769, member: 260864"] The "hand-waving" arguments are fine to get started with the relativistic wave equations, but it completely lacks the unerstanding, why these equations look as they look. The same holds for the non-relativistic wave equations (Schrödinger and Pauli). I think the best way to understand, why we get these equations and why they precisely look the way they do, is the analysis of symmetries. Since in our current understanding QT takes over the spacetime models from classical physics, the first step is to find wave functions which obey field equations that are invariant under the spacetime-symmetry-group transformations. For Newtonian physics that's the Galilei group, for special relativity it's the proper orthochronous Poincare group. It turns out that symmetries of continuous Lie groups are represented by unitary ray representations on a Hilbert space appropriate for that purpose. In addition you have the very general constraint of causality and stability, which leads in the relativistic case to the additional principle of locality or microcausality. You end up with the various wave equations. It's slightly different between non-relativistic and special-relativistic QT. In the non-relativistic case it turns out that the symmetry group of the classical Newtonian mechanics in QT is represented by unitary representations of a slightly different group. First the rotation subgroup is represented not necessarily by proper representations of SO(3) but there's also the possibility to represent it by unitary representations of its covering group, SU(2). This reflects the usual result that for angular momentum, which are the generators of rotations by definition, you have integer and half-integer spins, ##s \in \{0,1/2,1,\ldots \}##. In addition the Lie algebra of the Galilei group admits a non-trivial central charge, and this turns out to be the mass, which cannot be set to zero in non-relativistic physics, because that wouldn't lead to any physically interpretable dynamics of the resulting quantum theory. The final result then is that you get the Schrödinger equation for particles of any spin. For the corresponding fields with spin you have of course additional types of couplings, which you don't have for spin-0 particles, which are usually treated in the QM 1 lecture first. For the electromagnetic interaction you inherit the gauge principle from classical physics, which becomes a local symmetry in QT. Using the "principle of minimal coupling", you get a guess for the interaction between the quantum wave function of charged particles and the electromagnetic field (either keeping the em. field classical, leading to the semiclassical approximation, which has a wide range of applicability in atomic physics or the full quantized version, leading to "non-relativistic QED"). For the spin-1/2 case one is lead to the Pauli equation with the correct gyrofactor of 2 for the spin-magnetic moment. In the relativistic case, also the rotation subgroup is substituted by its covering group, SU(2), but there are no non-trivial central charges, i.e., everything is represented by unitary representations of the proper orthochronous Poincare group and microcausality leads to the concept of local quantum field theories, with quantized fields of all kinds of spin, and also the couplings are pretty much determined by the Poincare symmetry. It's a bit more restrictive than in the non-relativistic case, such that already on the non-interacting level you get non-trivial predictions: the existence of anti-particles in order to fulfill the microcausality constraint, the spin-statistics relation (half-integer-spinfields must be quantized as fermions, with integer spin they must be quantized as bosons), and additional symmetry under CPT (charge-conjugation, space reflection, time reversal). There are two types of spin-1/2 representations, which for P-symmetric theories (like electrodynamics) must be combined to a Dirac representation, and using the gauge principle for electromagnetism you get the Dirac equation for the quantum fields (of course with the correct gyrofactor of 2). [/QUOTE]
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What exactly is spin? Does the standard model work without spin?
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