Relativistic Invariance of Quantum-Mechanical Equations

Kenneth G. Dyall, Knut Fægri

The purpose of this chapter is to introduce the Dirac equation, which will provide us with a basis for developing the relativistic quantum mechanics of electronic systems. Thus far we have reviewed some basic features of the classical relativistic theory, which is the foundation of relativistic quantum theory. As in the nonrelativistic case, quantum mechanical equations may be obtained from the classical relativistic particle equations by use of the correspondence principle, where we replace classical variables by operators. Of particular interest are the substitutions In terms of the momentum four-vector introduced earlier, this yields In going from a classical relativistic description to relativistic quantum mechanics, we require that the equations obtained are invariant under Lorentz transformations. Other basic requirements, such as gauge invariance, must also apply to the equations of relativistic quantum mechanics. We start this chapter by reexamining the quantization of the nonrelativistic Hamiltonian and draw out some features that will be useful in the quantization of the relativistic Hamiltonian. We then turn to the Dirac equation and sketch its derivation. We discuss some properties of the equation and its solutions, and show how going to the nonrelativistic limit reduces it to a Schrödinger-type equation containing spin.

Eugene Paul Wigner

Eugene Paul Wigner

Jussi Lindgren, Jukka Liukkonen

The main contribution of this paper is to explain where the imaginary structure comes from in quantum mechanics. It is shown how the demand of relativistic invariance is key and how the geometric structure of the spacetime together with the demand of linearity are fundamental in understanding the foundations of quantum mechanics. We derive the Stueckelberg covariant wave equation from first principles via a stochastic control scheme. From the Stueckelberg wave equation a Telegrapher's equation is deduced, from which the classical relativistic and nonrelativistic equations of quantum mechanics can be derived in a straightforward manner. We therefore provide meaningful insight into quantum mechanics by deriving the concepts from a coordinate invariant stochastic optimization problem, instead of just stating postulates.

A. O. Barut, Arthur S. Wightman

Albert A. Jackson

This thesis presents a development of classical canonical formalism and the usual transition schema to quantum dynamics. The question of transition from relativistic mechanics to relativistic quantum dynamics is answered by developing a homogeneous formalism which is relativistically invariant. Using this formalism the Klein-Gordon equation is derived as the relativistic analog of the Schroedinger equation. Using this formalism further, a method of generating other relativistic equations (with spin) is presented.