It is in principle possible to measure the electric field or the magnetic field at a point of space. So we can define the quantum observables associated with the corresponding classical observables. To do it, we expand the field as a function of the alpha_ell complex numbers proportional to the complex amplitude in each mode and we replace alpha_ell and its complex conjugate by a_ell and a_dagger_ell. Since there is no product of alpha_ell and its complex conjugate there is no extra term associated with commutation relations. One can obtain similarly the expression of the magnetic field observable. We can also define the operators E plus_hat and E minus_hat such that E_hat is their sum. E plus and E minus are not hermitian, so they do not correspond to observables. But they are useful as we will see soon. The problem of the infinities, which we have encountered with energy, also happens with fields, or more precisely with field fluctuations. We present this problem in the case of the vacuum, which is striking. The average of E in the vacuum is obviously null since each a_ell applied to the vacuum returns the number 0 and similarly for a_dagger_ell applied to the vacuum on the left. But if now you calculate the average of E squared in the vacuum taking care of the commutation relations, there remains a non null term in each mode. Since there is an infinite number of modes, the sum is an infinite quantity. You may think that it does not play any role, as in the case of the energy of the vacuum. It turns out that in fact, field fluctuations in the vacuum can be invoked as an explanation of several phenomena. I already mentioned one of these phenomena, the Lamb shift. It is a correction to the calculated value of the energy levels of the atoms which was measured by a team lead by Willis Lamb in 1947. Calculations of effects such as the Lamb shift demand methods to obtain finite quantities, starting from the infinite values of the vacuum fluctuations. Such methods called renormalization were developed by Tomonaga, Schwinger, and Feynman in the 1940s. The excellent agreement, with the value measured by Lamb, was considered a triumph of the new quantum electrodynamics, as emphasized by the Nobel committee. You can read this presentation by the Nobel committee of the year 1965. It is very interesting and accessible to non-specialists, as all Nobel presentations. You can also read the lectures by each laureate. But here, I cannot promise that you will understand everything. Reading these documents, you will realize that the importance of these methods goes well beyond the Lamb shift calculation and beyond quantum electrodynamics. They have been applied successfully to many other phenomena and the idea of renormalization, is still a very important concept, in theoretical physics. Measurements of the Lamb shift and other quantities are still going on, always more precise with development of laser and of cold atoms physics. The comparison with the results of calculations which progress at the same pace is still considered a benchmark of quantum electrodynamics.