In QM the angular momentum algebra is introduced by considering differential operators acting on wave functions, leading to the angular momentum operators $L_x = i (y \partial_z - z \partial_y)$ etc... which satisfy the algebra $[L_x,L_y] = i L_z$ etc... , referred to as the $so(3)$ Lie algebra, and we want eigenvalues and eigenfunctions for these operators - another way to say this is that one is looking for complex irreducible representations of this algebra.
The $so(3)$ commutation relations imply that only one of the $L_x, L_y, L_z$ operators can have an eigenvalue at any one time, and from the angular momentum operators in spherical coordinates we know $L_x = - i \frac{\partial}{\partial \phi}$ which tells us that $L_z$ has integer eigenvalues, $L_z \psi = m \psi$, note this forces us to fix the axes.
Since $L^2$ commutes with $L_x,L_y,L_z$ it can also be diagonalized i.e. it also has eigenvalues on these eigenfunctions of $L_z$, and one can show $L^2 = l(l+1)$ for finite-dimensional representations. Note we showed that $m$ must be an integer by exploiting the $L_z$ differential operator, i.e. $-i \partial_{\phi} e^{im \phi} = m e^{im\phi}$, and one can easily show that $m$ can only take the $2l+1$ values $-l,..,0,..,l$ giving $2l+1$ possible wave functions.
However there is no reason why we can't rotate the coordinate system, wrecking the fact that our wave function is an eigenfunction of the $L_z$ operator. If we rotated the coordinate system the wave function with $m$ known has to become a linear combination of the $2l+1$ possible wave functions, each with different $m$'s.
This is just another $so(3)$ algebra rotating the coordinate system, however now we're not talking about differential operators acting on wave functions, we're talking about abstract rotations of a coordinate system, and the $2l+1$ wave functions rotated into one another have to live in an irreducible representation of the rotation algebra analyzed from the commutation relations alone.
In other words, we can no longer assume the $L_z$ eigenvalue has to be an integer, and it turns out analyzing the $so(3)$ commutation relations abstractly one can show that half-integer eigenvalues are possible, which we call spin representations.
the commutation relations are what matter, at the end of the day, not how they work in the specific case of position space (where only integer spin makes sense)
Right, even if you stubbornly just worked with angular momentum operators in position space, you would still unavoidably be led to considering abstract representations of the same algebra
well, you could insist that you should only worry about those which work in position space. it's just that, if you did, then you wouldn't describe nature :P
it's an empirical problem as much as a mathematical one.
please don't spam the starboard like that. if you want to hold on to a block of conversation, use the 'create new bookmark' feature under the room dropdown (right between the room description and the list of icons)
@MohamedObeidallah Photodisintegration in stars happens when the temperature is high because at such high temperatures there are many photons with sufficient energy to disrupt nuclei. But there are other ways to get high energy particles. If the particles are charged, like alpha particles, protons, or electrons, we can accelerate them electromagnetically. The particle energies involved in particle collisions in the LHC are far greater than what happens in a supernova.
Basically, rotations of the coordinate system force us to view wave functions as living in an irreducible representation of the (rotation) $so(3)$ Lie algebra, and the irreducible representations admit both integer and half-integer eigenvalues. The reason why this happens is that the $SO(3)$ Lie group is not simply connected, which means a wave function would not be the same wave function after a rotation, it would be sent to a different wave function,
but quantum theory doesn't care about wave functions so really we want to work with wave functions defined on the 'simply connected universal cover' of $SO(3)$ rather than $SO(3)$. This perspective is introduced here
