Chapter 2 - Introduction of a Quantum of Time (“chronon”), and its Consequences for the Electron in Quantum and Classical Physics

Abstract

In this chapter we discuss the consequences of the introduction of a quantum of time τ0 in the formalism of nonrelativistic quantum mechanics, by referring, in particular, to the theory of the chronon as proposed by Caldirola (1956; 1979a). Such an interesting “finite-difference” theory, forwards—at the classical level—a self-consistent solution for the motion in an external electromagnetic field of a charged particle like an electron, when its charge cannot be regarded as negligible, overcoming all the known difficulties met by the Abraham–Lorentz and Dirac approaches (and even allowing a clear answer to the question whether a free-falling electron does or does not emit radiation), and—at the quantum level—yields a remarkable mass spectrum for leptons. After briefly reviewing Caldirola's approach, our first aim is to discuss and compare the new formulations of quantum mechanics (QM) that can be drawn from it in the Schrodinger, Heisenberg and density-operator (Liouville-von Neumann) pictures, respectively. For each picture, three (retarded, symmetric and advanced) formulations are possible, which refer either to times t and t − τ0, or times t − τ0/2 and t + τ0/2, or times t and t + τ0, respectively. We shall see that, when the chronon tends to zero, the ordinary QM is obtained as the limiting case of the “symmetric” formulation only; whereas the “retarded” scenario does naturally appear to describe QM with friction, that is, it describes dissipative quantum systems (like a particle moving in an absorbing medium). In this sense, discretized QM is much richer than the ordinary one. We also obtain the (retarded) finite-difference Schrodinger equation within the Feynman path integral approach and study some of its relevant solutions. We then derive the time-evolution operators of this discrete theory, and use them to obtain the finite-difference Heisenberg equations. In discussing the mutual compatibility of the various pictures of QM that emerge from this procedure, we find that they can be written in a form so that they are all equivalent (as it happens in the “continuous” case of ordinary QM), even though the Heisenberg case cannot be derived by direct discretization of the ordinary Heisenberg representation. Latter, some typical applications and examples are studied, such as the case of a free particle (electron), the harmonic oscillator and the hydrogen atom; and various cases are noted for which the predictions of discrete QM differ from those expected from continuous QM. Finally, the density matrix formalism is applied as a possible solution of the measurement problem in QM with interesting results. For instance, a natural explication of “decoherence” which reveals the power of discretized (in particular, retarded) QM.