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Prior to 1947, Julian Schwinger had not worked in quantum electrodynamics (QED), apart from his first unpublished paper ‘On the interaction of several electrons’.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.

It was apparent from its beginning that special relativity developed as the invariance theory of electrodynamics would require a modification of Newton’s three laws of motion. This chapter discusses that modified theory. The relativistically modified mechanics is presented and then recast into a fourvector form that demonstrates its consistency with special relativity. Traditional Lagrangian and Hamiltonian mechanics can incorporate these modifications. This chapter also discusses the momentum fourvector, fourvector form of Newton’s second law, conservation of fourvector momentum, particles of zero mass, traditional Lagrangian and traditional Hamiltonian, invariant Lagrangian, manifestly covariant Lagrange equations, momentum fourvectors and canonical momenta, extended and invariant Hamiltonian, manifestly covariant Hamilton equations, the Klein-Gordon equation, the Dirac equation, the manifestly covariant N-body problem, covariant Serret-Frenet theory, Fermi-Walker transport, and example of Fermi-Walker transport.

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.

This chapter introduces the concept of linear vector functions of vectors and the related dyadic notation, a concept that is particularly important in the study of rigid body motion and the covariant formulations of relativistic mechanics. Linear vector functions of vectors have a rich structure, with up to nine independent parameters needed to characterise them, and vector outputs that need not even have the same directions as the vector inputs. The subject of linear vector operators merits a chapter to itself, not only for its importance in analytical mechanics, but also because study of it will help the reader to master the operator formalism of quantum mechanics. The chapter begins with a definition of operators and matrices, and concludes with the Dirac Notation, used by quantum mechanics for complex vectors.