This follows from the construction of unitary (ray) representations of the rotation group in three or more dimensions, where one discovers the condition that 2s+1 must be integer.

Unitary ray representation because we describe spin by Quantum Mechanics.

Note: There is not a similar restriction in two dimensions, where quantum spin in principle may vary continuously (cf. the theory of anyons).

Hello Kare,

can you name an article that explains this somewhat more in detail.

Generally it makes no sense to explain general text book stuff (here textbooks of quantum mechanics together with Wikipedia articles) here in RG. In this special case, however, one may hold a different oppinion, since, in fact, the standard sources don't give a short precise description of the reasons that answer the question. So I feel invited to give such an description, essentially by adding some observations to the answer of Kare with which I agree 100%.

1. Informally speaking, spin results from rotational symmetry, and so we need to know what the rotation group SO(3) in physical 3 dimensional space is: After introduction of a system of reference with orthogonal axes in physical 3-space this is the group or real 3 by 3 matrices R which are orthogonal (i.e. R*transpose(R) = identity ) and satisfy det(R)=1.

2. We argue within quantum mechanics and so have a Hilbert space H which describes the states of the physical system under consideration. A unitary transformation U in H is a linear mapping U:H-->H such that (Ux|Uy) = (x|y) for all x,y in H, where (.|.) denotes the scalar product in H. The group (composition of mappings as product) of unitary mappings in H shall be denoted G(H) (instead of G most books use U in some special font). We write U ~ U' if U' = z U where z is some complex number (of course |z| =1). The reason to consider this equivalence ~ is that U and U' give indentical actions on operators A according to rule A |--> U A U* (where * indicates hermitean conjugation which, for unitary operators, coincides with inversion).

3. Rotational symmetry is described by a mapping D: SO(3) --> G(H) that is consistent with the composition of mappings, that is:

D(g) D(g') ~ D(g g') (*)

Such a D is called a unitary ray representation of SO(3) and, if '~' is replaces by '=', simply a unitary representation of SO(3).

The interesting thing is that by a heroic endevor of many mathematicians and also physicists (e.g. Hermann Weyl, Eugene Wigner) one knows all ray representations of SO(3). This is the state of affairs not only for the rotation group but also for all classical compact Lie groups. To put things into perspective, consider an extremely simplified problem: replace the rotatation group by the additive group of real numbers, consider (*) only with '=' instead of '~' and replace H by the complex numbers C (i.e. the one-dimensional Hilbert space). Then we also know all continuous solutions of (*): These are of the form r |--> exp(zr), where each z in C gives just one of the solutions. So, in a sense, group representations are generalized exponential functions.

For the rotation group the complete result is.

a) Each ray representation of SO(3) is a direct sum of irreducible ray representations of SO(3).

b) For each irreducible ray representation D of SO(3) there is j in {0,1/2,1,3/2,...}

such that D is unitarily equivalent to Dj , where Dj is a given by explicit formulas given more or less completely in all quantum mechanical text books.

The meaning of j results, in a sense, from the appearence as a parameter in these explicit formulas. A more direct meaning is that all all components of angular momentum operators (which are determined by Dj in a way for which I also refer to quantum mechanics textbooks) have the eigenvalues {j,j-1,...-j}.

The wave functions of single particle states can be written as a linear superposition of tensor products of spin parts and spatial parts which transform under rotations seperately giving rise to a representation of the angular momentum operator as a sum of a spin angular momentum and an orbital angular momentum. The orbital momentum always gives integer j's and in the spin angular momentum is allowed to give 'half-integer' eigen values. For the half-integer j's one has '~' in (*) which one expresses by saying that their spin components form a spinor instead of a vector. The particles having such spin operators behave as fermions in many-particle situations.

You will have noticed that this last paragraph used some undefined concepts for which I alo like to refer to textbooks. Also earlier, I did not define what an irreducible representation and a direct sum of representation is, see Wikipedia.