Some applications: The electron anomalous moment, invariant berry phases and the spacetime lattice

In this chapter we describe three important applications of the theory. In the first section, we discuss the application of the Stueckelberg theory to the calculation of the anomalous moment of the electron (Bennett 2012). The original work of Schwinger (1951), and many later treatments (Itzykson 1980) use the standard formalism of quantum field theory. We show here, following (Bennett 2012), that the results can be obtained, to lowest order, in the framework of the relativistic quantum mechanics that we have developed here, without the necessity of second quantization. In the second section we discuss the general formulation of Berry phases, the response of a wave function in the quantum theory to a cyclic adiabatic variation of parameters of the Hamiltonian, resulting in a phase when the parameters return to their original value. The basic theory was developed by Berry (1984) using the nonrelativistic quantum theory. Since the Stueckelberg quantum theory has the same structure as the nonrelativistic quantum theory, represented in a well-defined Hilbert space, one can calculate the Berry phases in a similar way (Bachar 2014). We show an example of a perturbed four dimensional harmonic oscillator, of the type considered by Feynman et al. (1971), and Kim and Noz (1977), discussed in some detail in Chap. 5 here, and show that the associated Berry phases are Lorentz invariant, and are therefore an intrinsic property of the relativistic dynamical system. In the third section, we introduce the idea of a spacetime lattice (Engelberg 2009) and the corresponding Bloch waves for a periodic potential distribution in space and time. The example that we treat is that of an electromagnetic standing wave in a cavity. The corresponding solution of the Schrödinger-Stueckelberg equation is that of Bloch type waves in space and time with associated mass (energy) gaps which appear to be observable in the laboratory.