Quantum Lorentz covariance

Apart from Lorentz covariance the quantum mechanical state equation must obey certain mathematical criteria (i) it must be homogeneous in order to fulfill Eq. (4.7) for all times, and (ii) it must be a linear equation so that linear combinations of solutions are also solutions. The latter requirement is often denoted as the superposition principle, which is required for the description of interference phenomena. However, it is equally well justified to regard these requirements as the consequences of the equation of motion in accordance with experiment if the equation of motion and the form of the Hamiltonian operator are postulated. 

In 1928, Dirac proposed a new quantum mechanical equation for the electron [99,100], which solved two problems at once, namely the Lorentz-covariance requirement and the duplexity of atomic states, which was accounted for by Goudsmit and Uhlenbeck s phenomenological introduction of spin. In fact, he showed how the dynamic spin variable is connected to Lorentz covariance — a connection that will become clear in the following. To derive this fundamental quantum mechanical equation for the electron, which features relativistic covariance, we set out with a basic ansatz for this equation based on the results of the preceding section. 

In 1948, techniques introduced by Schvttinger and Feynman enabled these difficulties to be avoided, without being removed. Their relativisti-cally covariant development of the theory allowed such infinite terms to be treated unambiguously, and in particular terms which are to be understood as electrodynamic contributions to the charge and mass of a particle were put in a form which is invariant under Lorentz transformations. The program of charge renormalization and renormalization of mass then enabled such terms to be related to the experimentally observed charge and mass of the particle. See also Quantum Mechanics. 

This part is concerned with variational theory prior to modem quantum mechanics. The exception, saved for Chapter 10, is electromagnetic theory as formulated by Maxwell, which was relativistic before Einstein, and remains as fundamental as it was a century ago, the first example of a Lorentz and gauge covariant field theory. Chapter lisa brief survey of the history of variational principles, from Greek philosophers and a religious faith in God as the perfect engineer to a set of mathematical principles that could solve practical problems of optimization and rationalize the laws of dynamics. Chapter 2 traces these ideas in classical mechanics, while Chapter 3 discusses selected topics in applied mathematics concerned with optimization and stationary principles. 

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