No abstract

A physically relevant unitary irreducible non-projective representation of the Galilei group is possible in the Koopman–von Neumann formulation of classical mechanics. This classical representation is characterized by the vanishing of the central charge of the Galilei algebra. This is in contrast to the quantum case where the mass plays the role of the central charge. Here we show, by direct construction, that classical mechanics also allows for a projective representation of the Galilei group where the mass is the central charge of the algebra. We extend the result to certain kind of quantum–classical hybrid systems.

In calculating the energy corrections to the hydrogen levels we can identify two different types of modifications of the Coulomb potential V C , with one of them being the standard quantum electrodynamics corrections, δV , satisfying |δV | ≪ |V C | over the whole range of the radial variable r. The other possible addition to V C is a potential arising due to the finite size of the atomic nucleus and as a matter of fact, can be larger than V C in a very short range. We focus here on the latter and show that the electric potential of the proton displays some undesirable features.Among others, the energy content of the electric field associated with this potential is very close to the threshold of e + e − pair production. We contrast this large electric field of the Maxwell theory with one emerging from the non-linear Euler-Heisenberg theory and show how in this theory the short range electric field becomes smaller and is well below the pair production threshold.

We revisit quantum-classical hybrid systems of the Sudarshan type under the light of Galilean covariance. We show that these kind of hybrids cannot be given as a unitary representation of the Galilei group and at the same time conserve the total linear momentum unless the interaction term only depends on the relative canonical velocities.

We use the Schwinger action principle to obtain the equations of motion in the Koopman–von Neumann operational version of classical mechanics. We restrict our analysis to non-dissipative systems. We show that for velocity-independent forces the Schwinger action principle can be interpreted as a variational principle.