# What exactly is spin? Does the standard model work without spin?

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DomDominate
I did some research online and found that "When certain elementary particles move through a magnetic field, they are deflected in a manner that suggests they have the properties of little magnets." To explain this phenomenon, physicists invented the concept of spin. So far so good.

What I don't understand is this, doesn't a charged particle moving through an electric field already generate a magnetic field? So, can't the deflection of the particle be explained by the interaction between the magnetic field the particle enters into and the magnetic field generated by the particle? Why do we even need to invent this concept of spin to explain this phenomenon?

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Quantum "spin" is not like classical spin and we didn't invent it, we observed it.

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And since we observed it, no the SM doesn't work without it.

topsquark and vanhees71
simonjech
You dont really need to use spin to explain this. Magnetic force can act on any charged object which has non zero velocity.

And we can define spin using Noethers theorem as a quantity which is conserved when lagrangian is symetric under the lorentz transformation (boost). If you analyse this more in field theory using the noethers charge you can obtain a general mathematical definition of spin for given lagrangian.
But we can understand spin more if we look on a very similar quantity angular momentum. These two have almost identicall properties but the angular momentum is connected with spacial rotations and spin is connected with space time rotations. You can not really imagine what a spin is for this exact reason.

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"physicists invented the concept of spin"
Spin was invented by two graduate students, Goudsmit and Uhlenbeck, to explain fine structure in atomic physics. It was later seen to come out of relativistic quantum mechanics. It cannot be avoided.

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Spin comes as well out of non-relativistic quantum mechanics or rather it's natural to occur in any field theory on any spacetime admitting rotations as a symmetry.

topsquark
simonjech
I think that you mean the Stern Gerlach experiment here where atoms of silver went through an inhomogenious magnetic field and the magnetic force acted on them even when these atoms were neutral. In classical physics this would not make any sence because classicaly, the magnetic moment of a particle depends only on the angular momentum. So it was a mystery for physicists back than.
But later they figered out that there should be another quantity, similar to angular momentum which also contributes to final magnetic moment of a particle. So even a neutral particle can interact if the spin of the particle is non zero. Later in 1927 Dirac showed how spin is connected with lorentz transformation.

topsquark
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Spin cannot be explained on B-level. It comes from the representation theory of the symmetry group of (Newtonian or special-relativistic) spacetime, particularly the representation of the rotation group. The spin of a massive particle is defined by the representation of the rotation group for the particle at rest, ##\vec{p}=0##.

topsquark, dextercioby, malawi_glenn and 1 other person
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Sure you can: it's something that looks put artificially by hand alongside orbital angular momentum to explain the Stern-Gerlach experiment or the Zeeman effect.

vela, topsquark and gentzen
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It's not artificial. It's totally natural given the mathematical structure of field theory and quantum theory.

topsquark and malawi_glenn
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Spin comes as well out of non-relativistic quantum mechanics or rather it's natural to occur in any field theory on any spacetime admitting rotations as a symmetry.
What I meant was that the relativistic Dirac equation produces and requires requires spin. Intrinsic spin has to be put in by hand for classical electromagnetism or non-relativistic quantum mechanics.

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No! The possibility of spin follows in QT based on Galilei spacetime in the same way as it follows in Q(F)T based on Minkowski spacetime. You just investigate the realization of the spacetime symmetry in terms of ray representations.

topsquark, simonjech and malawi_glenn
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No!? Of course, spin is "possible", but the relativistic Dirac equation REQUIRES spin.

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The non-relativistic Pauli equation requires spin too. It's simply that the representation theory of the rotation group tells us that there can be spin. Whether a given particle has spin, is of course a matter of observation, and electrons have spin 1/2. That's why they are described by the Pauli equation in non-relativistic and as a Dirac quantum field in the relativistic QT.

topsquark
What I don't understand is this, doesn't a charged particle moving through an electric field already generate a magnetic field? So, can't the deflection of the particle be explained by the interaction between the magnetic field the particle enters into and the magnetic field generated by the particle?

The consequence of spin is an extra magnetic deflection quite apart from the one you are referring to. That extra deflection is a function of the non-uniformity of the magnetic field, and is at right angles to the standard one that classical physics predicts. For example, an electron will move in a circle in a plane normal to a uniform magnetic field - that is basic classical physics. But if the field is non-uniform, the electron will have a velocity component either in the direction of the field, or directly opposite to it. And that's what spin helps to explain and calculate.

topsquark and vanhees71
andresB
A non-quantized (of course) version of spin is also allowed in classical physics, it is just that it is not very useful for describing elementary particles. It also arises as an extra degree of freedom in the representation of the Galilei/Minkowski algebra.

mattt, topsquark and vanhees71
Fra
Why do we even need to invent this concept of spin to explain this phenomenon?
The troublesome part to understand is fermionic spin right? If we can't understand it as a "spinning top" - is there another way to understand it beyond that "it works"?

Here is a simple personal reflection explain in simple terms, I made decades ago when examining the handwaving arguments going from KG to Dirac equation in class. A common argument is to just then postulate the linear dirac equation, rather than trying to related solutions in KG space with dirac space.

But suppose instead we have a spin 0 "particle" that we expect to comply to SR, so we naively expect it to comply to the klein gordon equation.

Then lets supposed we have an observer, we let processes inputs of hte KG solution space, and then we ask the agent to make a guess what's in there? Does anyone think the agent will come up with the idea that we have a system of spin 0 particles, that moreoever does not seem to nserve probability?? Possible yes, but not very likely.

It may be a more likely guess that we have a system of fermions in here.

But the two explanations seems mathematicall related, just like the KG and dirac equations and it's components are related via the pauli transformations. But do we postulate them, or can we motivate them?

Are these "apparent" fermions related to the initially naively assume spin 0 bosons? Could it simply be that, from the perspective of inference, a system of spin 0 bosons and a system of fermions are dual descriptions, but that the fermionic description simply is more stable? Is the reason we have not observed the supersymmetric partners, if they exist, maybe becase even if we did, the preferrable abduction would be the fermionic dual? After all, a model failing to preserve probability is NOT an "inconsistency" per see - it's just STRANGE and by definition a bad explanatory model. But I think when it comes to inference, one should not confuse bad or unfit with "forbidden"? This may be another loose way to consider a "spontanously broken symmetry" in terms of inference.

For me at least, this is my best conceptual intutitive picture of what a "fermion is", it can be seen as a "transformation of something else", fermion or boson is in the eye of the beholder. Works for me.

/Fredrik

DeBangis21
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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).

mattt
Fra
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.
I agree that this is maybe the best way to understand the standard model.

One uses given symmetries as a "constraint" then one essentially find the missing "terms" or components following from transformations and identify these with something phenomenological that we managed to fit with experiment. This is indeed a powerful method. But it does not exactly offer any deeper insight ontology of terms popping out, as we arrive at them by a top down approach.

And to undertand the fermion deeper than that (for those that are not satisfied) we need to look for a deeper explanation of spacetime itself (and thus the origin of the symmetries we use as constraints)...

/Fredrik