So if spin isn't really spin

[nhmllr]

So if spin isn't really "spin"...

Then how do magnets work?

I really don't know much about spin, except that it's not really the particles spinning. I've read a few articles on them but I haven't yet learned the math necessary to understand what's going on there (although I've heard that nobody really gets it).

I know that magnetism arrives from the movement of electrical charges. I've been told that magnetism arrive from many electrons all "spinning" in the same direction, amplifying the magnetic effect, thus making a magnet as we know it.

But the electrons aren't REALLY spinning! So why should magnetism arise from them at all?

I hope I haven't opened too big a can of worms...

 

[Mordred]

Rather than go through lengthy explanation on the properties of angular momentum, magnetic dipoles etc YOU may find this link useful

http://hyperphysics.phy-astr.gsu.edu/hbase/spin.html

It explains the spin porperties in relatively simple terms and has hyper links on any term used to provide more detail Hope this helps.

 

[The_Duck]

Just as electrons possesses intrinsic angular momentum ("spin"), electrons also possesses an intrinsic magnetic dipole (which is aligned with their spin).

 

[nhmllr]

I've looked at the link, but I'm confused about one thing: "intrinsic angular momentum."
When I think "momentum," I think things with velocity vectors moving around. But if there is some momentum that is intrinsic... Where does that leave us? I'm just having a hard time thinking about what it means for such a thing to exist.

 

[Mordred]

All particles can be thought of as a particle or as a wave, as thy both have characteristics that can be described as a particle and a wave. The Heisenburg uncertainty principle taught us that one cannot visualize a particle as both a wave or particle at the same time. It also taught us that when we examine a particle we alter its position and sometimes energy state. When you try to describe an electron by its wave function you need to determine its instrinsic angular motion. Most of quantum mechanics deal with the wave function of a particle, where you seldom see that discussed in macro world sciences. It is discussed but they usually describe particles as particles rather than waves.
Electrons are both a particle and a wave therefore it always has angular momentum
particles also never stay still for that matter.
In quantum mathematics instrinsic angular momentum can also be termed as "Spin quantum number" http://en.wikipedia.org/wiki/Intrinsic_angular_momentum

List of angular momentum quantum numbers


Intrinsic (or spin) angular momentum quantum number, or simply spin quantum number
orbital angular momentum quantum number (azimuthal quantum number)
magnetic quantum number, related to the orbital momentum quantum number
total angular momentum quantum number


these are all needed to to describe the various wave, motion behavior characteristics of particles. The energy state of a particles also effect these characteristics.

Keep in mind I am still studying quantum mechanics myself so others may have corrections to my answer.

 

[Naty1]

So why should magnetism arise from them at all?

well they exhibit that characteristic...but WHY they do is unknown...as is why they have the charge or the mass they exhibit...


Then how do magnets work?

decent discussion here:
http://en.wikipedia.org/wiki/Magnet

A magnetic material is one where the dipoles are mostly aligned in the same direction.
But an electromagnet magnetism arise from the current flow and resulting electromagnmetic field without any particular alignment of any core material.

 

[Runner 1]

I have a question about "spin" as well. The terminology confuses me.

The way I see it -- "spin" implies that some property of the electron is a function of time, and this function has a period (returns to its original state at a certain frequency). To me, this is the semantic meaning of the word rotation, which is implied by "spin".

Is this correct?

And if not, then why the heck is it called spin? If it's just an intrinsic property of which two different states give two different measurements, then why not call it moose, or happiness, or color?

 

[nhmllr]

I have a question about "spin" as well. The terminology confuses me.

The way I see it -- "spin" implies that some property of the electron is a function of time, and this function has a period (returns to its original state at a certain frequency). To me, this is the semantic meaning of the word rotation, which is implied by "spin".

Is this correct?

And if not, then why the heck is it called spin? If it's just an intrinsic property of which two different states give two different measurements, then why not call it moose, or happiness, or color?

I think historically people used to think that the particles actually were spinning, but then they wised up and they were stuck with the name "spin." From how I'm understanding it, it helps to think of it as if it is spinning, although there's no reason to think that that's ACTUALLY happening.

 

[nhmllr]

Just as electrons possesses intrinsic angular momentum ("spin"), electrons also possesses an intrinsic magnetic dipole (which is aligned with their spin).

So then does EVERYTHING with spin have a magnetic property? They don't interact electromagnetically but they still have spin.

If spin is about magnetism, then what does it even MEAN for neutrinos to have it?

 

[jtbell]

why the heck is it called spin? If it's just an intrinsic property of which two different states give two different measurements, then why not call it moose, or happiness, or color?

Because we physicists like to mess with people's minds.

Remember, we're the folks who also use the word "color" to represent the property of quarks that is associated with the strong interaction in a similar way that charge is associated with the elecromagnetic interaction; and the word "flavor" to distinguish between different types of quarks (up, down, etc.) or leptons (electron, mu, tau). (I've been waiting for years for Ben & Jerry to pick up on that one. )

Also, the intrinsic angular momentum of an electron really is angular momentum, in the sense that it contributes to the total angular momentum of a macroscopic system, and can affect the macroscopic rotation of an object under the right circumstances. Look up the Einstein - de Haas effect.

 

[Polyrhythmic]

I have a question about "spin" as well. The terminology confuses me.

The way I see it -- "spin" implies that some property of the electron is a function of time, and this function has a period (returns to its original state at a certain frequency). To me, this is the semantic meaning of the word rotation, which is implied by "spin".

Is this correct?

And if not, then why the heck is it called spin? If it's just an intrinsic property of which two different states give two different measurements, then why not call it moose, or happiness, or color?

It is called spin because the generators of the group of spin transformations obey the same algebra as those for angular momentum (SU(2)). So, the analogy to something spinning around is purely mathematical.

 

[dydxforsn]

And if not, then why the heck is it called spin? If it's just an intrinsic property of which two different states give two different measurements, then why not call it moose, or happiness, or color?

It's called a "spin" because it follows the same form of mathematics that the angular momentum of an electron has orbiting an atom. (this is completely analogous to a planent like Earth "orbiting" the sun and possessing some resultant angular momentum, but then also having angular momentum due to its spin. Both forms of angular momentum follow the same mathematics in classic mechanics yet are interpreted differently depending on naming conventions. Well, in quantum mechanics spin and angular momentum differ more than just their names, however, they follow the same mathematical pattern as each other in much the same way orbits and spins do in classical mechanics. Further, the "spin" number in QM also denotes the magnetic moment of an electron, much like a spin could create in classic mechanics/electromagnetic theory, so using the naming convention "spin" makes sense for those two reasons and possibly more I don't know about.)Unfortuantely nhmllr, I can not answer your question. It's possible that nobody knows the answer to this, but I don't know (I don't know what relativistic QM says about this or other upper level theories like quantum electrodynamics, etc. After all, the spin is only derivable from relativistic quantum mechanics which is usually saved for graduate school.) I certainly know I didn't come across it as an undergraduate. However, I simply posted to make it clear to you that nobody has answered your question yet, just to confirm that you didn't miss anything in anybody's explanations. Somebody who knows more might come and post something soon though.

Edit** I typed this up before the last two posters posted.

 

[Polyrhythmic]

The Heisenburg uncertainty principle taught us that one cannot visualize a particle as both a wave or particle at the same time.

Where did you learn that? That's definitely not the meaning of the Heisenberg uncertainty principle.

 

[Runner 1]

It is called spin because the generators of the group of spin transformations obey the same algebra as those for angular momentum (SU(2)). So, the analogy to something spinning around is purely mathematical.

This is great! This is exactly what I've wanted to know for years. Why don't schools just say that instead of leaving everyone to wonder why electrons are just randomly rotating like tops?

 

[Runner 1]

Because we physicists like to mess with people's minds.

Remember, we're the folks who also use the word "color" to represent the property of quarks that is associated with the strong interaction in a similar way that charge is associated with the elecromagnetic interaction; and the word "flavor" to distinguish between different types of quarks (up, down, etc.) or leptons (electron, mu, tau). (I've been waiting for years for Ben & Jerry to pick up on that one. )

Oh wow, color's already taken huh? I guess "moose" will have to suffice ;)

EDIT: If I ever become some great particle physicist, I swear I'm naming the first undiscovered property "moose".

 

[Polyrhythmic]

This is great! This is exactly what I've wanted to know for years. Why don't schools just say that instead of leaving everyone to wonder why electrons are just randomly rotating like tops?

I don't know your level of education, but I guess one usually learns this in introductory quantum mechanics courses.

 

[Polyrhythmic]

Oh wow, color's already taken huh? I guess "moose" will have to suffice ;)

EDIT: If I ever become some great particle physicist, I swear I'm naming the first undiscovered property "moose".

I hate to disappoint you, but there is already something called "moose model" in particle physics! ;)

 

[Runner 1]

I don't know your level of education, but I guess one usually learns this in introductory quantum mechanics courses.

4th year chemical engineer. I've been taking QM for three weeks now. We just finished the time independent Schrodinger equation and have now begun the derivation of the Uncertainty Principle (well, at least for a particle in a box).

I should really say its Quantum Chemistry, not QM. We use https://www.amazon.com/dp/1891389505/?tag=pfamazon01-20.

It really doesn't go into a lot of the deep math about it -- which is why I'm trying to learn more on here, because I find the subject interesting.

(Btw, "I don't know your level of education, but..." usually comes across as "You seem kind of ignorant, but..." even if you didn't mean it that way).

 

[Runner 1]

I hate to disappoint you, but there is already something called "moose model" in particle physics! ;)

Holy crap! I thought you were kidding, but I just Googled it, and indeed there is a moose model!

 

[Polyrhythmic]

4th year chemical engineer. I've been taking QM for three weeks now. We just finished the time independent Schrodinger equation and have now begun the derivation of the Uncertainty Principle (well, at least for a particle in a box).

Ah, I see. If it is a decent course, they should teach you more about spin eventually!

(Btw, "I don't know your level of education, but..." usually comes across as "You seem kind of ignorant, but..." even if you didn't mean it that way).

Sorry, it was definitely not meant that way! I just wondered what kind of education you had, because it would've been odd if you were for example a physics major who never heard about the mathematical aspects of spin ;)

 

[Mordred]

Re: So if spin isn't really "spin"...

--------------------------------------------------------------------------------

Originally Posted by Mordred
The Heisenburg uncertainty principle taught us that one cannot visualize a particle as both a wave or particle at the same time.

Where did you learn that? That's definitely not the meaning of the Heisenberg uncertainty principle.

above should have been on quote

Actaully I'm not sure where I read that, may have been a Modern physics book (title of the book published in 1989 ) I borrowed from a friend when I first started looking at Quantum physics. Where ever it was I know that only that one literature stated that. Most others I've read refer to it

uncertainty principle (W. Heisenberg; 1927)
A principle, central to quantum mechanics, which states that two complementary parameters (such as position and momentum, or angular momentum and angular displacement) cannot both be known to infinite accuracy; the more you know about one, the less you know about the other.
It can be illustrated in a fairly clear way as it relates to position vs. momentum: To see something (let's say an electron), we have to fire photons at it; they bounce off and come back to us, so we can "see" it. If you choose low-frequency photons, with a low energy, they do not impart much momentum to the electron, but they give you a very fuzzy picture, so you have a higher uncertainty in position so that you can have a higher certainty in momentum. On the other hand, if you were to fire very high-energy photons (x-rays or gammas) at the electron, they would give you a very clear picture of where the electron is (higher certainty in position), but would impart a great deal of momentum to the electron (higher uncertainty in momentum).

Either way if its wrong then I'm happier knowing that come to think of it, its most likely a misinterpretation on my part from wave particle duality explanation I garnered from wiki

http://en.wikipedia.org/wiki/Wave–particle_duality

Although this describes particle and wave as complementary, now I'm not sure if Heisenburg uncertainty principle includes that as two viable complemenatary properties described by the above definition. Even though wiki includes Heisenburg in its definition page later on in the article.

 

[Polyrhythmic]

First of all, the physicist's name was Heisenberg, not Heisenburg.

Now I see where your confusion comes from: It's semantics. "Complementary parameters" is just a wrong wording: they are actually hermitian operators that obey the uncertainty principle for certain reasons. If you don't know that, you should study a little more before giving answers here.
Complementary in the case of wave-particle duality means something else: it adresses the fact that waves and particles are different descriptions of a physical entity that is actually the same. In this sense, they are "complementary".

 

[unusualname]

Just to correct the wide held belief that "spin" is a relativistic effect, we should note the Walter Greiner claims in one his influential textbooks Quantum Mechanics: an Introduction 4th Edition that "spin" arises from linearization of the (non-relativistic) Schroedinger Equation

p 365:

...Thus a completely nonrelativistic linearized theory predicts the correct intrinsic magnetic moment of a spin-1/2 particle

In contrast to this, almost all textbooks falsely claim that the anomalous magnetic moment is due to relativistic properties. The existence of spin is therefore not a relativistic effect, as is often asserted, but is a consequence of the linearization of the wave equations.

 

[qsa]

First of all, the physicist's name was Heisenberg, not Heisenburg.

Now I see where your confusion comes from: It's semantics. "Complementary parameters" is just a wrong wording: they are actually hermitian operators that obey the uncertainty principle for certain reasons. If you don't know that, you should study a little more before giving answers here.
Complementary in the case of wave-particle duality means something else: it adresses the fact that waves and particles are different descriptions of a physical entity that is actually the same. In this sense, they are "complementary".

from http://plato.stanford.edu/entries/qt-uncertainty/


closer account as regards the balance of momentum and energy. (Bohr, 1949, p. 210)
A causal description of the process cannot be attained; we have to content ourselves with complementary descriptions. "The viewpoint of complementarity may be regarded", according to Bohr, "as a rational generalization of the very ideal of causality".

In addition to complementary descriptions Bohr also talks about complementary phenomena and complementary quantities. Position and momentum, as well as time and energy, are complementary quantities.

 

[Mordred]

First of all, the physicist's name was Heisenberg, not Heisenburg.

Now I see where your confusion comes from: It's semantics. "Complementary parameters" is just a wrong wording: they are actually hermitian operators that obey the uncertainty principle for certain reasons. If you don't know that, you should study a little more before giving answers here.
Complementary in the case of wave-particle duality means something else: it adresses the fact that waves and particles are different descriptions of a physical entity that is actually the same. In this sense, they are "complementary".

Yeah thanks for the clarification, defining HUP using the term Hermitian operators instead of complementary parameter is far less confusing. I'm glad for that clarification. Still not sure why I posted it the way I did lol.

 

[vanhees71]

The "linearization" of either the Klein-Gordon or the Schroedinger equation to obtain the Dirac or Pauli equation, respectively, is only a handwaving argument, leading to the correct description of particles with spin (in that cases spin 1/2) by chance.

A more convincing argument is the group-theoretical method used to systematically derive the single-particle observables from the (continuous) symmetries of space-time. For the non-relativistic case, the symmetry group of space-time is the full group of inhomogeneous Galileo transformations, which are decomposed as temporal and spatial translations, spatial rotations, and boosts, reflecting homogeneity of time and space, isotropy of space, and the principle of inertia, which states that the physical laws do not change for observers that are in uniform motion with respect to each other.

The next step is to analyze, how these symmetries are realized in quantum theory. First of all one considers one single symmetry transformation. As has been proven by Wigner (and later simplified by Bargmann), such a symmetry transformation can be represented on the Hilbert-space vectors as either a unitary or an antiunitary transformation. If one has a transformation that is continuously deformable to the identity the transformation must be unitary, and since we consider only transformations which are continuously connected to the identity, we have to look for unitary representations of the Galileo group.

Now, there's one subtlety in this. In fact the (pure) states are not really represented by the Hilbert-space vectors, but only by these vectors modulo an arbitrary phase factor. That means that one needs not have unitary representations but only unitary ray representations, which are representations up to phase factors.

This has two very important consequences for physics: First of all the most general transformation is not necessarily the classical Galilei group but its covering group. That means that we are allowed to use the SU(2) to represent the rotations (making the group, SO(3)) within the Galilei group.

Second the Galilei group is such that it admits the introduction of a socalled nontrivial central charge, which is an observable that commutes with the generators of the one-parameter subgroups of the Galilei groups. The latter make the energy (Hamiltonian) and momentum (generating temporal and spatial translations), angular momentum (rotations), and boosts (center-of-mass position). The central charge turns out to be the mass of a particle, if the irreducible ray representations are interepreted as defining elementary (non-relavistic) particles. As it turns out, the representation without central charge, i.e., particles of zero mass doesn't give physically meaningful representations of the Galilei group (a famous paper by Wigner and Inönü).

The physically meaningful representations of the quantum-Galilei group, lead to representations for a particle, which has two intrinsic quantum numbers, namely its mass $m \in \mathbb{R}_{>0}$ and its spin $s \in \{0,1/2,1,3/2,\ldots\}.$ The spin determines the behavior of the one-particle state for particles at rest (zero momentum), i.e., for $s=1/2$, the zero-momentum states span a two-dimensional spinor space. Since in non-relstivistic physics, the spin commutes with momentum as well as with position operators, one can build a basis as either the direct product of momenum-eigenstates and spin-eigenstates or of position-eigenstates and spin-eigenstates.

In terms of wave functions this leads to spinor-valued wavefunctions $\psi_{\sigma}(t,\vec{x})$ or $\tilde{\psi}_{\sigma}(t,\vec{p})$.

The Pauli equation, including the correct gyrofactor of 2 (!), can be derived by using a specific form of minimal substitution to couple the electromagnetic field to the matter field in such a way as to make the invariance of quantum theory under changes of the wave function by a phase factor local. Thus, indeed one doesn't need relativity to make sense out of the spin observable, and also the gyro factor of 2 pops out of gauge invariance and the principle of minimal substitution rather than relativistic (Poincare) covariance.

However, in relativity the same analysis of the Poincare group as done above for the Galilei group (with the important difference that for the Poincare group there don't exist non-trivial central charges, and mass is thus not a central charge but a Casimir operator of the Poincare group) leads to the possibility of zero-mass particles, and if one considers massless particles with spin 1, these must be gauge fields, if one doesn't want continuous intrinsic quantum numbers, which has never been observed to exist. Thus, relativity leads necessarily to the principle of (Abelian) gauge invariance.

 

[Dali]

But the electrons aren't REALLY spinning! So why should magnetism arise from them at all?

To understand "why" something is the way it is, it is usually best to go back to the experiments that motivated the theory in the first place. In my humble opinion theory alone can never tell you "why" something happens, even though it can give you a deeper understanding how different experimental effects are related to each other. Because there are always experiments that motivates one or the other physics theory in the first place.
Quantum mechanics is a strange theory with many concepts that just have no analogies in classical physics, or is impossible for us humans to visualize. But the only reason we have QM theory at all is that the real world just turned out to behave that way and experimental results forced us to develop and adopt that theory. Without those experiments, no one would have anticipated or guessed those strange effects.

In this case, the reason why "spin" had to be introduced into physics theories in the first place is the Stern-Gerlach experiment(s). It just turns out that if you send a beam of electrons over the tip of a normal magnet, the beam is split in two distinct beams. Some electrons are bent upwards by the magnet, and some are bent downwards. This is the expected behavior if electrons were small permanent magnets. Now, what's "strange" is that they only bend up or down, never to the left or right. And also they always bend the same amount - the beam is not smeared out in the up-down direction but split in two distinct beams. The unavoidable conclusion of these experiments is that electrons do have a quantized magnetic moment, just like they where tiny magnets with a fixed strength.

The name "spin" comes from the classical analogy of a small charged ball, which would act like a little magnet if it were spinning. This classical analogy clearly doesn't fit the whole picture with electrons, since a (classical) spinning ball would give an arbitrary strenght of the magnetic moment depending on how fast it was rotating. Also, it would be able to rotate around any axis, not just around a fixed axis giving it only an "up" or "down" magnetic moment.

In other situations it also turns out that we need to include an angular momentum from the electrons spin to have total angular momentum conserved. So there are more similarities with classical rotation than just the magnetic moment. But is is still clear that the classical analogy of a small rotating sphere is just an analogy and can't be taken literally. Firstly, no experiment so far have ever seen any size of electrons, they appear to be point-like. And electron spin also survives in for example interference experiments where the picture of electrons as small localized balls clearly is completely wrong.

So the simple answer is that there is no real answer to "why", it just turns out that electrons are that way!

 

[unusualname]

The "linearization" of either the Klein-Gordon or the Schroedinger equation to obtain the Dirac or Pauli equation, respectively, is only a handwaving argument, leading to the correct description of particles with spin (in that cases spin 1/2) by chance.

A more convincing argument is the group-theoretical method used to systematically derive the single-particle observables from the (continuous) symmetries of space-time. For the non-relativistic case, the symmetry group of space-time is the full group of inhomogeneous Galileo transformations, which are decomposed as temporal and spatial translations, spatial rotations, and boosts, reflecting homogeneity of time and space, isotropy of space, and the principle of inertia, which states that the physical laws do not change for observers that are in uniform motion with respect to each other.

The next step is to analyze, how these symmetries are realized in quantum theory. First of all one considers one single symmetry transformation. As has been proven by Wigner (and later simplified by Bargmann), such a symmetry transformation can be represented on the Hilbert-space vectors as either a unitary or an antiunitary transformation. If one has a transformation that is continuously deformable to the identity the transformation must be unitary, and since we consider only transformations which are continuously connected to the identity, we have to look for unitary representations of the Galileo group.

Now, there's one subtlety in this. In fact the (pure) states are not really represented by the Hilbert-space vectors, but only by these vectors modulo an arbitrary phase factor. That means that one needs not have unitary representations but only unitary ray representations, which are representations up to phase factors.

This has two very important consequences for physics: First of all the most general transformation is not necessarily the classical Galilei group but its covering group. That means that we are allowed to use the SU(2) to represent the rotations (making the group, SO(3)) within the Galilei group.

Second the Galilei group is such that it admits the introduction of a socalled nontrivial central charge, which is an observable that commutes with the generators of the one-parameter subgroups of the Galilei groups. The latter make the energy (Hamiltonian) and momentum (generating temporal and spatial translations), angular momentum (rotations), and boosts (center-of-mass position). The central charge turns out to be the mass of a particle, if the irreducible ray representations are interepreted as defining elementary (non-relavistic) particles. As it turns out, the representation without central charge, i.e., particles of zero mass doesn't give physically meaningful representations of the Galilei group (a famous paper by Wigner and Inönü).

The physically meaningful representations of the quantum-Galilei group, lead to representations for a particle, which has two intrinsic quantum numbers, namely its mass $m \in \mathbb{R}_{>0}$ and its spin $s \in \{0,1/2,1,3/2,\ldots\}.$ The spin determines the behavior of the one-particle state for particles at rest (zero momentum), i.e., for $s=1/2$, the zero-momentum states span a two-dimensional spinor space. Since in non-relstivistic physics, the spin commutes with momentum as well as with position operators, one can build a basis as either the direct product of momenum-eigenstates and spin-eigenstates or of position-eigenstates and spin-eigenstates.

In terms of wave functions this leads to spinor-valued wavefunctions $\psi_{\sigma}(t,\vec{x})$ or $\tilde{\psi}_{\sigma}(t,\vec{p})$.

The Pauli equation, including the correct gyrofactor of 2 (!), can be derived by using a specific form of minimal substitution to couple the electromagnetic field to the matter field in such a way as to make the invariance of quantum theory under changes of the wave function by a phase factor local. Thus, indeed one doesn't need relativity to make sense out of the spin observable, and also the gyro factor of 2 pops out of gauge invariance and the principle of minimal substitution rather than relativistic (Poincare) covariance.

However, in relativity the same analysis of the Poincare group as done above for the Galilei group (with the important difference that for the Poincare group there don't exist non-trivial central charges, and mass is thus not a central charge but a Casimir operator of the Poincare group) leads to the possibility of zero-mass particles, and if one considers massless particles with spin 1, these must be gauge fields, if one doesn't want continuous intrinsic quantum numbers, which has never been observed to exist. Thus, relativity leads necessarily to the principle of (Abelian) gauge invariance.

Thanks vanhees71, that's certainly a deeper explanation. I guess what Greiner is trying to get at is that the concept of 'spin' can arise from fairly elementary algebraic manipulations of a non-relativistic equation, and while this might be "hand-waving" it is nonetheless surprising that such an easy algebraic manipulation should give a "prediction" of spin (and essentially, this all Dirac did to get his equation)

(incidentally, the relevant chapter (12) is not completely viewable on google books link I posted above, see the amazon "look inside!" link - you may need to create a (free) account. Search for '365' to go to page 365)

 

[lugita15]

If you can derive the notion of spin non-relativistically, can you also prove the spin-statistics theorem without relativity?

 

[Chronos]

I hope you all know, dark is a spin too.

 

[Demystifier]

If you can derive the notion of spin non-relativistically, can you also prove the spin-statistics theorem without relativity?

Unfortunately, no.

 

[unusualname]

If you can derive the notion of spin non-relativistically, can you also prove the spin-statistics theorem without relativity?

The derivation might not appear very "natural" but this paper constructs a non-relativistic argument:

Connecting spin and statistics in quantum mechanics - Arthur Jabs Foundations of Physics 40: 776-792, 2010

 

[strangerep]

If you can derive the notion of spin non-relativistically, can you also prove the spin-statistics theorem without relativity?

The derivation might not appear very "natural" but this paper constructs a non-relativistic argument:

Connecting spin and statistics in quantum mechanics - Arthur Jabs Foundations of Physics 40: 776-792, 2010

Thanks for mentioning that paper. I had seen a vaguely related argument many years ago but didn't understand it at that time.

For those who haven't read the above paper by Jabs, here's a quick summary of (what I perceive to be) the crucial point therein.

(The following in the context of non-relativistic QM.)
A spinor wave function is usually expressed wrt an arbitrary spin-quantization axis. However, there's an additional ambiguity arising from rotations around the spin-quantization axis. For many purposes, such rotations give rise only to a boring phase factor and don't affect physical predictions. Hence they're usually ignored. However, for superpositions of indistinguishable particles, such phase factors must be correctly taken into account. Jabs then shows that exchanging an angle parameter (for rotations around the spin-quantization axis) of the wave functions associated with the two particles potentially suffers the usual spin ambiguity (double-valuedness). But if this angle parameter is exchanged using a consistent convention (i.e., a consistent rotation direction to achieve the exchange), the factor $(-1)^{2s}$ pops out automatically without needing extra assumptions. (Here, s is the spin magnitude, and the factor is enough to demonstrate the spin-statistics connection.)

As yet, no other authors on the arXiv cite Jabs' paper. So I'd be interested to hear what others think of Jabs' proof. (Demystifier?)

 

[Demystifier]

So I'd be interested to hear what others think of Jabs' proof. (Demystifier?)

In the Introduction, the author says that his goal is to give a SIMPLE derivation of the spin-statistics theorem. Apparently, for him "simple" means - without using QFT and relativity. However, even though he does not use QFT and relativity, his proof seems to me much more complicated than the usual textbook proof based on relativistic constructive QFT. (Here by constructive QFT I mean not axiomatic QFT. The proof based on axiomatic QFT is indeed very complicated.) Owing to the complicated appearance of his proof, I have problems with motivating myself to study his proof in detail. And without studying it in detail, I cannot make any further comments on it.

What is perhaps even more interesting is the fact that in the Introduction he mentions a LARGE number of other derivations which do not use relativity and QFT. All these other derivations have some restrictions, but the fact that many such derivations exist suggests to me that it could be ultimately true that the relation between spin and statistics does not really depend on relativity and QFT, even if a completely satisfying proof does not exist yet.

 

[strangerep]

[...] suggests to me that it could be ultimately true that the relation between spin and statistics does not really depend on relativity and QFT, even if a completely satisfying proof does not exist yet.

A similar thought occurred to me also. After all, the fermion/boson distinction does not disappear at low speeds, hence ought to be convincingly explainable by nonrelativistic methods.

In the Introduction, the author says that his goal is to give a SIMPLE derivation of the spin-statistics theorem. Apparently, for him "simple" means - without using QFT and relativity. However, even though he does not use QFT and relativity, his proof seems to me much more complicated than the usual textbook proof based on relativistic constructive QFT. (Here by constructive QFT I mean not axiomatic QFT. The proof based on axiomatic QFT is indeed very complicated.) Owing to the complicated appearance of his proof, I have problems with motivating myself to study his proof in detail. And without studying it in detail, I cannot make any further comments on it.

In my first reading, I also had trouble understanding Jabs' use of the parameter $\chi$. But on a second reading during my lunch break, I was able to follow it up to and including section 5, which is enough to see his basic idea. I hope you might reconsider and try to study the proof at least up to that point. (The messy stuffy with permutations amongst multiple particles only comes later.)

To others: has anyone else around here studied Jabs' proof yet?
(Unusualname: I presume you've studied it?)