The quantized Hamiltonian of radiation could have been obtained replacing the classical expressions alpha_ell times alpha_ell conjugate by a_dagger_ell times a_ell in the classical expressions of the energy. The term one-half would then be derived from the fact that a and a_dagger do not commute, and the classical term alpha times alpha conjugate must be symmetrized. The case of the linear momentum of radiation can be treated with such method. Starting from the classical expression of the electromagnetic linear momentum per unit volume, integration over the quantization volume leads to an expression that is a sum of terms associated with each mode, without any crossed term. I recommend that you do the calculation yourself, in analogy to what we have done for the energy. You must expand the field over modes ell and use a periodic boundary condition on the quantization volume so that most of the terms do cancel, except the squares of the modulus of alpha_ell. When you replace the squared modulus of alpha_ell by the quantum expression, you find an expression analogous to the one of the energy with the one-half term for each mode. But for each mode ell, there is a counter propagating mode minus ell, with the opposite k vector, so that the one-half terms do cancel. So the final expression has no one-half term. A Fock state is an eigenstate of any operator a_dagger_ell a_ell. It is thus en eigenstate of the total linear momentum observable with the eigenvalue sum of n_ell, hbar k_ell. At this point, you may remember that I had not mentioned one-half factor in the linear momentum observable of a single mode. Well, it is embarrassing, I have to admit that I had swept that detail under the carpet. But I knew that I would come back to it. Today, it gives me the opportunity to point out that even if only a single mode is excited, you should not forget that other modes do exist even if they are in the vacuum state. So, when I consider the mode ell, I am not allowed to forget the mode minus ell. In the case of the linear momentum, it leads to the cancellation of the 1/2 terms. In quantum optics, you should never forget the vacuum. You have learned that a photon is an elementary excitation of a mode of the electromagnetic field, and that its properties are linked to the properties of that mode. Here the modes we have considered are monochromatic traveling waves, each with a well-defined frequency and k vector and the corresponding photons have well-defined energy and linear momentum. A classical traveling wave is also characterized by a polarization. There is a quantum observable associated with this classical property provided that one considers circular polarizations rather than linear polarizations. I've introduced, earlier in this lesson, the two circular polarizations characterized by the complex unity vectors epsilon_ell plus and epsilon_ell minus rotating in the direct or reciprocal sense around k_ell. The circular polarizations are also called sigma plus and sigma minus relative to the direction k_l. The quantum observable associated with circular polarization is the intrinsic angular momentum of the electromagnetic field, also called the spin of the photon. Its eigenvalues are plus or minus hbar, and they correspond to rotations around the direction of propagation in the direct, or reciprocal sense. But what can we say about a linearly polarized photon? Inverting the formulae for epsilon plus and epsilon minus we see that the linearly polarized mode is a superposition of counter rotating circularly polarized modes, and a one photon state with a linear polarization can be considered a superposition of a sigma plus and a sigma minus photon. This is another example of a one-photon state, which can be considered a superposition state. Once again we find that the photon is not necessarily an eigenstate of all the observables that we have considered. You find this difficult to understand? Alfred Kastler, who received the Nobel prize for the invention of optical pumping, based on the angular momentum of photons, said that when he was young, in the 1930s, he had a hard time understanding how a linearly polarized photon can be a superposition of circularly polarized photons. So, if you think it is a difficult notion, you are in good company.