polarization and frequency of a single photon

Question

I believe that we can take a single photon state as a tensor product of a frequency Hilbert space (infinite dimension) and a polarization Hilbert space (dim 2). Does this mean we can measure the polarization and the frequency at space-like separated regions a the same time?

My second question is about frequency and polarization entanglement. If we have an entangled state between the frequency and the polarization, but we model the system as having a classical distribution over frequencies, what kinds of mistakes will we make in understanding any joint measurements of polarization and frequency? I am thinking about a channel which entangles the polarization and the frequency which we imagine is a classical channel over frequencies.

Answer 1

The Helicity operator of a photon commutes with all of its momentum components thus with its energy, since from the commutation relations of the Poincare group, we have:

$[P_i, P_j] =0$

$[J_i, P_j] = \epsilon_{ij}^k P_k$

where $P^i$ and $J^i$ are respectively, the momentum and angular momentum generators. Thus the helicity

$h = \frac{\mathbf{J.P}}{E}$

where, $ E = \sqrt{\mathbf{P.P}}$ satisfies the commutation relation

$[h, P_j] = \frac{\epsilon_{ij}^k P_i P_k}{E} = 0$

(The result vanishes due to the complete antisymmmetry of $\epsilon$)

Consequently, we have: $[h, E] = 0$

Thus the helicity and energy (which is proportional to the frequency) can be simultaneously measured.