• Kubo formula
• nonperturbative
• superconductivity

The Kubo formula can be used to discuss the Meissner effect for quantum systems of particles and electromagnetic field, within the framework of a semiclassical approximation (the Kubo analysis).
We discuss the general properties of the Kubo formula, for an infinitely extended system of particles in external field. Using the continuity equation, together with euclidean covariance and gauge invariance of the Kubo response function, we show that superconductivity is reduced to the study of the response to a magnetic field localized on the boundary of some large region. Thus, superconductors are characterized by long range correlations, which correspond to a singularity at low momenta of the Kubo response function. For relativistic systems, the additional use of the spectral condition allows us to demonstrate that the presence of the singularity (a nonlocal term) is equivalent to the presence, in the mass spectrum of matter, of a massless Goldstone mode. We discuss this result, also in comparison with the interpretations of superconductivity as a broken symmetry, especially with S. Weinberg's analysis.
Next we discuss the foundation and consistency of the Kubo analysis. We show that it can be derived by combining two approximations: a variational ansatz, which approximates the vacuum as a product state of matter and electromagnetic field, and the linear response theory, to further approximate the particle current to first order in the electromagnetic field. This approach turns out to be compatible with the nonperturbative characters of superconductivity with respect to the e.m. coupling, which is essential for the generation of a photon mass.