Magnetic energy of a quantum current

– It is shown that the magnetic energy of a quantum current, contrary to the classical case, is essentially negative. Since this result allows to escape a famous theorem by Bloch, it can be expected that, under appropriate conditions, the ground state of a quantum conductor may be characterized by a spontaneous current. A similar idea was suggested, many years ago, by Heisenberg and by Born and Cheng as a possible mechanism for superconductivity. Introduction. – It is well-known, from the study of the infra-red divergence problem in relativistic QED, that a charged particle cannot be separated from its classical field [1] [2]. It is known also that the motion of a classical particle cannot be described correctly, if the interaction with the self-generated field is not taken into account properly [3]. In this paper we analyse the effects induced by the self-generated classical (i.e. coherent) field on a stationary current flowing in a macroscopic quantum conductor. In particular we will estimate the contribution to the total energy of the system due to the classical magnetic field and to its interaction with the current. The analysis will concern explicitly two geometrically different models. In both cases we will obtain the remarkable result that the magnetic energy associated to the current is essentially negative. A completely different result would be obtained for a current due to a system of classical charged particles. Such a difference descends from the fact that, in classical mechanics, the kinetic energy of a particle is expressed, in a natural way, in terms of the velocity. We remark that the velocity is essentially a classical concept. In quantum mechanics the velocity of an electron corresponds to the following operator v = m[−ih̄∇+ eAc(t, r)], (1) where Ac is the classical vector potential. Owing to the presence of Ac , which depends on the other particles of the system as well as on the external environment, the velocity v, given by eq.(1), is essentially a collective operator. On the other hand, the canonical momentum is connected directly to the particle wave-length, which in turn is involved by a boundary (∗) E-mail: [email protected]