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The Dirac equation is a first order 4×4-system of partial differential equations of the form

The present chapter will be irrelevant for the mathematical deployment in succeeding chapters. We offer this material only to provide a motivation for our claim that a Dirac observable should be a self-adjoint operator.

Spectral theory of the Dirac Hamiltonian of (1.0.2) has been vigorously pursued since the early 1930-s.

In this and the following chapter we will investigate time-dependence of ψdoobservables when physical states are kept constant in time. In particular we look for “smooth” dependence on in uniform operators norms (of our weighted Sobolev spaces). Clearly the exponential operator - for time-independent - or, more generally, the evolution operator (, ) of (), are not even continuous in strong operator topology, they are only (what functional analysts call) “strongly continuous”. We shall see that our smoothness translates into a powerful condition on the symbol a(, ) of a ψdo, which is not passed by many observables. In a sense, the rejected observables experience some kind of “Zitterbewegung”.

In this chapter we will start by discussing a precise theorem giving a necessary and sufficient condition for smoothness of the (inverse) Heisenberg transform, with some “framing conditions” added. Note, the symbol classes ψ carry a “topology” (in fact, a Frechet topology), defined by the sup norms

It is known and very essential to Dirac’s theory that it is compatible with a transformation of coordinates under the laws of special relativity. In other words, if we change the (space-time) coordinate system by a Lorentz transform then Dirac’s equation remains intact - except for the physically signifficant changes of potentials - for example, a moving electrostatic field will also generate a magnetic field, under Faraday’s laws.

In this chapter we will analyze the spectral theory of a few of our “precisely predictable approximations” of dynamical observables, which are not precisely predictable. Let us emphasize again: . The approximations only are good for calculating an “approximate expectation value” for predicting outcome of a measurement of the observable in question.

In this chapter we shall venture beyond the Dirac equation - so far our only object of study - and try reflecting on other wave equations in Quantum Mechanics. Perhaps we have fortified our opinion that - for the hydrogen atom - and, more generally, any “one-particle problem” considering a single charged particle in an electromagnetic field - the Dirac equation would be preferable - i.e., more accurate, and more to the point - to the Schrödinger equation, already introduced in (3.0.2).