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Prior to 1947, Julian Schwinger had not worked in quantum electrodynamics (QED), apart from his first unpublished paper ‘On the interaction of several electrons’. Before joining the City College of New York, he had already studied Paul Dirac's The principles of quantum mechanics, first published in 1930. As a freshman at CCNY, Schwinger studied the recently published papers on quantum field theory of Dirac, Werner Heisenberg, Wolfgang Pauli, Enrico Fermi, J. Robert Oppenheimer, and others; he absorbed all that was being done in this field. However, he maintained his interest in quantum field theory, and had more exposure to the subject when he went to the University of California at Berkeley to work with Oppenheimer for two years. This chapter deals with Schwinger's work on QED, Dirac's theory of radiation and relativistic theory, relativistic quantum mechanics, the infinities in QED, earlier attempts to overcome the infinities in QED, and earlier experimental evidence for the deviations from Dirac's theory of the electron.

Towards a relativistic quantum mechanics. Klein–Gordon and the problems of the probability current and the negative energy solutions. The Dirac equation and negative energies. P, C, and T symmetries. Positrons. The Schrödinger equation as the non-relativistic limit of relativistic equations. Majorana and Weyl equations. Relativistic corrections in hydrogen-like atoms. The Dirac equation as a quantum system with an infinite number of degrees of freedom.

Here an initial stab is made at constructing a relativistic quantum wave equation, the Klein–Gordon equal. This turns out to have some unsavoury characteristics that mean that it is not the right equation to describe electrons, but it is nevertheless illuminating and illustrates some of the issues we are going to come across later.

This chapter uses the traditional Hamilton equations as the basis for an extended Hamiltonian theory in which time is treated as a coordinate. The traditional Hamilton equations, including the Hamiltonian form of the generalised energy theorem, will be combined into one set of extended Hamilton equations. The extended Hamilton theory developed in the chapter is of fundamental importance for the more advanced topics in mechanics. It is used to write the relativistically covariant Hamiltonian, which is then used to derive the Klein-Gordon equation of relativistic quantum mechanics. The extended Hamilton equations also provide the basis for the discussion of canonical transformations. The objective of extended Hamiltonian theory is to write the equations of motion in terms of an extended set of phase-space variables.