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I know that spin gets a proper and complete definition in Quantum Field Theory, when we account for relativity in our quantum theory. This question is not about this.

I am instead interested in understanding how spin is defined in non relativistic quantum mechanics, assuming we know nothing about Dirac's equation and such.

Regarding this question I know that in "vanilla" QM spin is somewhat introduced by force, but I don't exactly know what this means: is spin introduced in the theory only on the base of experimental evidence, such as Stern-Gerlach experiment? Or maybe there is some other experimental evidence? Or other reasons to introduce it?

Anyway then, by some reasoning, we determine that it is a good idea to introduce spin, as an intrinsic property of particles, and to postulate that it has the algebric structure of angular momentum. This pratically means that the following is postulated to be true:

$$[S_i,S_j]=i\hbar \varepsilon _{ijk}S_k \tag{1}$$ $$[S^2,S_i]=0 \tag{2}$$

this should be it. No other postulate. But then something strange is stated, and it is stated not as a postulate but somehow as a consequence of what we have said so far:

Particles with spin 1/2 are associated with angular momentum in two dimensions and particle with spin 1 are associated with angular momentum in three dimensions.

Why? What does this mean exactly?

But the strangeness is not over. Then we are somehow able to show that spin $1/2$ particles have spin operators represented by Pauli Matricies:

$$S_i=\frac{\hbar}{2}\sigma _i$$

and we are also able to show that spin $1$ particles have spin operators represented by another set of matricies, these matricies as far as I know don't have a name, but they are 3x3, in accordance with what we have stated in the citation.

Is all this simply postulated? Or it is derived from (1),(2) as I understood? And if it is indeed derived from (1),(2): how exactly is it derived? How can we find out that the spin is represented by these matricies in particular? And also why spin $1/2$ is in 2D and spin $1$ is in 3D?
This bit in particular seems really strange also because in the case of spin $1/2$ spin can be measured either up or down so I can kinda understand why it is in 2D, but what about spin $1$ particles?

Noumeno
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    Are you aware of the correspondence between $\text{SU}(2)$ and $\text{SO}(3)$? – Nihar Karve Jan 10 '21 at 15:37
  • Maybe this answer of mine that describes how intrinsic spin was assigned, may help for part of your question https://physics.stackexchange.com/questions/586741/is-spin-necessary-for-electromagnetism/586743#586743 – anna v Jan 10 '21 at 15:39
  • I would like to understand this topic as much as possible without counting on knowledge regarding group theory and SU(2), SO(3). I haven't studied this part of math, and I suspect that there are many students that need to understand properly the definition of spin but they haven't properly studied group theory either, yet. An answer that does not make use of group theory would be best. – Noumeno Jan 10 '21 at 15:42
  • This semi-historical discussion of the (anomalous) Zeeman effect before spin was understood is worth considering, to explain the splitting and it's behavior in different situations one is forced to arbitrarily introduce extra quantum numbers. This begs for an explanation, hopefully without introducing anything new. As the discussion mentions, Heisenberg found introducing half-integers made sense of things, this could have spelt the end of quantization as it was known if it couldn't be made sense of. – bolbteppa Jan 10 '21 at 16:19
  • It turns out those extra quantum numbers and the possibility of half-integer angular momenta immediately fall out from the representation theory of the angular momentum algebra, clearly the usual differential operator representation of the angular momentum applied to wave functions has to be too strong a commitment and misses something, it motivates considering a more general group-theoretic perspective since those quantum numbers so directly fall out of the representation theory of $\mathrm{so}(3)$. – bolbteppa Jan 10 '21 at 16:21
  • (2) follows from (1) since $S^2:=S_jS_j$; it's not a separate postulate. – J.G. Jun 02 '21 at 20:58
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    Though you say that an answer not involving group theory would be best: its difficult, because the origin of spin comes directly from (projective) representations of SO(3), to actually gain an understanding of whats going on, I think a little group theory is unavoidable (which is also why, despite this protest, all of the other comments/answers will be explaining it in terms of the group theory involved) – QCD_IS_GOOD Jun 02 '21 at 20:58
  • It's essentially a tautology that you will end up learning group theory to understand the precise meaning of spin. But, that does not mean you need to take an entire course in group theory. You really need to understand a fairly narrow (but deep and beautiful) part of the subject. In a good book, you should find an argument which constructs the raising and lowering operators $S_{\pm} = S_x\pm i S_y$. Given these operators, you can construct the possible sets of states. Spin-1/2 and Spin-1 (including the matrices you mentioned) fall out of this argument. – Andrew Jun 02 '21 at 22:47

2 Answers2

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Particles with spin 1/2 are associated with angular momentum in two dimensions and particle with spin 1 are associated with angular momentum in three dimensions.

This is flat wrong and ambiguous nonsense, at best, and you should toss the sloppy text you saw it in. It is pig Latin group theory. A good introduction to Lie Group theory might well be in order. I do understand this is precisely what you aim to avoid, but it is a little bit like asking to bypass calculus and still utilize its techniques. The best you could ask for is a gentle introduction.

Your confusions arise from the newbie efforts of 1920s physicists to understand quantum angular momentum and spin, and how they enter the Lorentz group and their theories.

Both spin 1 and spin 1/2 (and higher integral or half-integer spins for that matter) are associated with the very same Lie algebra structure (1) you wrote down. ((2) is an easy consequence thereof.) The three ("generators") operators $S_i$ of this algebra, when suitably exponentiated describe the group of rotations and an associated Lie group with the same Lie algebra.

It turns out that, quite independently of physics, by ineluctable mathematical necessity, these operators may be irreducibly represented by 2×2, 3×3, 4×4, 5×5,... matrices, acting on spaces of 2d, 3d, 4d, 5d, ... vectors. The dimensionality of these vector spaces correspond to spin s = 1/2, 1, 3/2,2, etc. (D=2s+1). The operator $S^2$ in your (2) has characteristic different "eigenvalues" for each such irrep, namely numbers multiplying the identity matrix in each "space" dimension D=2s+1: $~~~S^2=s(s+1) 1\!\!1$. Do study these matrices, which certainly include the spin-1 ones you are asking about.

These spaces may represent peculiar internal symmetries such as isospin, etc, but, in spacetime, the 3d representation corresponds to our three space dimensions in which we rotate, and the 2d irrep to an abstract complex 2d "spinor space", much unlike our three space dimensions, so talking about it in the same breath as the three space dimensions as your pestiferous quote is bound to confuse you.

This marvelous group theory was invented/discovered out of pure reason in the 19th century, and, when QM emerged in the 20th, physicists had the ready tools to recognize it described the selection rules and Zeeman phenomena involved. Trying to "derive" it from physical "axioms" is as silly as trying to derive matrix calculus, or even geometry, out of physics. Because, at the time, physicists were not too familiar with Lie algebras, sophisticates like Wigner and Dirac (brothers in law) made it easy for them to apply these structures to QM without undue formalism; but, at the end of the day, you'd best start with the elegant and tight mathematical theory, and just apply it to physics, almost magically fitting it--the way geometry does.

Cosmas Zachos
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You are aware that spin arises naturally from Dirac's equations, but classically the justification for spin was historically more ad-hoc. The role that spin plays is well described by anna v.

What is spin? Whenever you ask this sort of question in QM you get into awkward territory, but here goes. It is the property that is measured by operators which obey $$[L_i, L_j] = i \hbar e_{ijk} L_k$$ It is one of the simplest non-trivial algebras, and has some beauty. For example, it does not rely on any co-ordinate system, and it has a natural scale. I.e. if you change $L_i \rightarrow \lambda.L_i$ is instantly detectable.

You don't need group theory to understand the representation (but it helps). For 1/2 spin, define 2 states \begin{bmatrix} 1 \\ 0 \end{bmatrix} and \begin{bmatrix} 1 \\ 0 \end{bmatrix} and look for 3 2x2 matrices that operator on them and obey the commutator relations, If you play around enough, the Pauli matrices pop out. Some algebra shows in this case the angular momentum is $\pm\hbar/2$ (spin 1/2).

Similarly, if you have a particle with 3 possible states, you get different matrices (obviously) and spin takes the values $-\hbar, 0, \hbar$.

The statement "Particles with spin 1/2 are associated with angular momentum in two dimensions and particle with spin 1 are associated with angular momentum in three dimensions" is confused. Spin 1/2 particles are associated with a spin space that is complex and 2D, they do not directly correspond to 2D space (or space-time).

shaunokane001
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