# Photon carries spin angular momentum of $\hbar$

I know from numerous articles and also wiki (http://en.wikipedia.org/wiki/Spin_angular_momentum_of_light) that a photon carries spin angular momentum of $\hbar$. How to prove it mathematically? What is $S_z$ in operator form? What is going to be a circular equation look like?

• I am open to correction on this, but how can you derive a mathematical proof for a nature assigned value? Or do you want to produce a proof relating a photon spin/polarisation value to something already proven. My apologies if I am way off on this, but I ask to find out more. – user108787 Sep 9 '16 at 16:02
• I think this is more or less a duplicate of Where do the quantum fields encode the spin information? though I won't vote to close since it isn't an exact duplicate. – John Rennie Sep 9 '16 at 16:16
• You will need to learn a bit about spin more generally before you will gain a good understanding this. Spin is another degree of freedom, not related to position and momentum operators. For photons it is related to the fact that EM field is a pair of vector fields. – Andrew Steane Jul 22 at 7:57

You can't "prove it mathematically" - it's an observed aspect of nature. Indeed, all elementary particles carry spin - except for the recently discovered Higgs Boson. The fundamental unit of spin is 1/2 h bar and there are only two classes of elementary particles by spin..

1) Fermions have an odd number of basic units. 2) Bosons have an even number of basic units.

Nature keeps things simple - odd/even.

• For example, I have a linear momentum operator $-i\hbar \frac{\partial}{\partial x}$, I act with it on the photon wave function $\vec{A} \approx e^{-ikx}$ and I get a circular equation with an eigenvalue $\hbar k$ which is a linear momentum carried by a wave. The same must exist for spin operator. – MsTais Sep 9 '16 at 17:58

As stated by others, the fact that photons have spin 1 cannot be proven. I believe that is not what you are asking. You are asking about the mathematical formalism to describe photon spin.

The reason that this formalism does not exist is that conservation of photon spin is incompatible with gauge invariance. There exists no gauge invariant expression of photon spin. In the gauge theory the total angular momentum has the form of an orbital AM involving the Poynting vector. This is a consequence of Belinfante-Rosenfeld symmetrisation of the energy-momentum tensor.

The Noether theorem links spin conservation to invariance of the lagrangian under rotation of the EM four potential. The gauge invariant lagrangian does not have this symmetry. The Fermi lagrangian does. I worked out the consequences of using this lagrangian instead. It leads to a valid theory of electromagnetism with conserved spin. See https://arxiv.org/abs/physics/0106078.