Let $\hat{a}^{\dagger}_{\sigma}$ be the creation operator of a photon with the polarization $\sigma $ towards some reference. What are the commutator relations for the creation operators of a photon with different polarizations. Example let $\hat{a}^{\dagger}_{H}, \hat{a}^{\dagger}_{V}$ be the creation operator for a photon with horizontal and vertical polarization. Is $\hat{a}^{\dagger}_{H}\hat{a}^{\dagger}_{V}=\hat{a}^{\dagger}_{V}\hat{a}^{\dagger}_{H}$. I would say, if they do commute, then also $\hat{a}^{\dagger}_{H}\hat{a}^{\dagger}_{D}=\hat{a}^{\dagger}_{D} \hat{a}^{\dagger}_{H}$, where $\hat{a}^{\dagger}_{D}$ creates a photon with diagonal polarization and for any other polarization as well. Since we can express any polarization as a linear combination of $\hat{a}^{\dagger}_{H}$ and $\hat{a}^{\dagger}_{V}$. I am thankful for any physical or mathematical explanation as well.
2 Answers
Boson creation operators always commute, i.e. $[a_\alpha^\dagger, a_\beta^\dagger]=0$ for all $\alpha,\beta$. In fact, $[a_\alpha^\dagger,a_\beta]=0$ if $\alpha\ne \beta$ as well provided that $\alpha$ and $\beta$ refer to distinct modes, be these wavevector modes or polarization modes.
On way to understand this is to imagine writing these for a harmonic oscillator in terms of position and momenta, with momenta expressed as a derivative. Then clearly $[x_\alpha,\frac{\partial}{\partial x_\beta}]=0$ if $\alpha\ne \beta$. The more abstract photon creation and destruction operators are constructed to duplicate the commutation relation of harmonic oscillator operators.
The commutation relations for the quantized EM field are $$\left[a_\mathbf{p}^s, a_\mathbf{p'}^{s'\dagger}\right] = \delta^{ss'}\delta^{(3)}(\mathbf{p} - \mathbf{p}')\,, \quad \left[a_\mathbf{p}^s, a_\mathbf{p'}^{s'}\right] = \left[a_\mathbf{p}^{s\dagger}, a_\mathbf{p'}^{s'\dagger}\right] = 0\,.$$ Here $s$ and $s'$ depend on your basis. If you take $s, s' \in \{ \mathrm{H}, \mathrm{V} \}$ then they are orthonormal and a diagonally polarized photon would indeed be a superposition of the horizontally and vertically polarized photons.
The only non-commuting relation is between a creation and an annihilation operator with the same momentum and the same polarization.
