In this paper, as suggested by the classical Hamiltonian equations, a new set of corresponding relativistic equations is set up. Therefrom a relativistic form of Heisenberg's equation is obtained. The relativistic Hamiltonian system of a particle according to the formulation here established has been fully discussed and by following it, however, Dirac's equation appears only naturally as a necessary form of relativistic wave equation for electron in quantum mechanics. The process of taking square root in Dirac's theory is seen to have its classical analogy. Finally, some applications of the relativistic Heisenberg's equation to Dirac's theory have been discussed and it has thereby been pointed out that this equation brings some quantities to being more symmetrical in the relativity sense and also some more general than those the non-relativistic equation can give.
In this paper, as suggested by the classical Hamiltonian equations, a new set of corresponding relativistic equations is set up. Therefrom a relativistic form of Heisenberg's equation is obtained. The relativistic Hamiltonian system of a particle according to the formulation here established has been fully discussed and by following it, however, Dirac's equation appears only naturally as a necessary form of relativistic wave equation for electron in quantum mechanics. The process of taking square root in Dirac's theory is seen to have its classical analogy. Finally, some applications of the relativistic Heisenberg's equation to Dirac's theory have been discussed and it has thereby been pointed out that this equation brings some quantities to being more symmetrical in the relativity sense and also some more general than those the non-relativistic equation can give.
In this paper, as suggested by the classical canonical equations, a new set of the corresponding relativistic equations is set up. Therefrom a relativistic form of Heisenberg's equation is deduced. The rclalivistic Hamiltonian system of a particle according to the formulation here established has been fully discussed and by following it, however, Diracs equation appears naturally as a necessary form of relativistic wave equation tor electron in quantum mechanics. The process of taking square root in Dirac's theory is seen to have its classical analogy. Finally, some applications of the relativistic Heisenberg's equation to Dirac's theory have been discussed and it has thereby been pointed out that this equation brings some quantities to being more symmetrical in the relativity sense and also some more general than those the non-relativistic equation can introduce.
The quantization process was always tightly connected to the Hamiltonian formulation of classical mechanics. For non-Hamiltonian systems, traditional quantization algorithms turn out to be unsuitable. Numerous attempts to quantize non-Hamiltonian systems have shown that this problem is nontrivial and requires the development of new approaches. In this paper, we present the quantization methods that do not depend upon the Hamiltonian formulation of classical mechanics. Two approaches to the quantization of mechanical systems are considered: axiomatic and hydrodynamic. It is shown that the formal application of these approaches to the classical Hamilton-Jacobi theory allows obtaining the wave equation for the corresponding quantum system in natural way. Examples are considered that show the effectiveness of the proposed approaches, both for Hamiltonian and non-Hamiltonian systems. The spinor form of the relativistic Hamilton-Jacobi theory for classical particles is considered. It is shown that it naturally leads to the Dirac equation for the corresponding quantum particle and to its non-Hamiltonian generalization, the bispinor relativistic Kostin equation.
Abstract The steady development of the quantum theory that has taken place during the present century was made possible only by continual reference to the Correspondence Principle of Bohr, according to which, classical theory can give valuable information about quantum phenomena in spite of the essential differences in the fundamental ideas of the two theories. A masterful advance was made by Heisenberg in 1925, who showed how equations of classical physics could be taken over in a formal way and made to apply to quantities of importance in quantum theory, thereby establishing the Correspondence Principle on a quantitative basis and laying the foundations of the new Quantum Mechanics. Heisenberg’s scheme was found to fit wonderfully well with the Hamiltonian theory of classical mechanics and enabled one to apply to quantum theory all the information that classical theory supplies, in so far as this information is consistent with the Hamiltonian form. Thus one was able to build up a satisfactory quantum mechanics for dealing with any dynamical system composed of interacting particles, provided the interaction could be expressed by means of an energy term to be added to the Hamiltonian function. This does not exhaust the sphere of usefulness of the classical theory. Classical electrodynamics, in its accurate (restricted) relativistic form, teaches us that the idea of an interaction energy between particles is only an approximation and should be replaced by the idea of each particle emitting waves which travel outward with a finite velocity and influence the other particles in passing over them. We must find a way of taking over this new information into the quantum theory and must set up a relativistic quantum mechanics, before we can dispense with the Correspondence Principle.
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.
This paper re-examines the key aspects of Dirac's derivation of his relativistic equation for the electron in order advance our understanding of the nature of quantum field theory. Dirac's derivation, the paper argues, follows the key principles behind Heisenberg's discovery of quantum mechanics, which, the paper also argues, transformed the nature of both theoretical and experimental physics vis-a-vis classical physics and relativity. However, the limit theory (a crucial consideration for both Dirac and Heisenberg) in the case of Dirac's theory was quantum mechanics, specifically, Schrodinger's equation, while in the case of quantum mechanics, in Heisenberg's version, the limit theory was classical mechanics. Dirac had to find a new equation, Dirac's equation, along with a new type of quantum variables, while Heisenberg, to find new theory, was able to use the equations of classical physics, applied to different, quantum-mechanical variables. In this respect, Dirac's task was more similar to that of Schrodinger in his work on his version of quantum mechanics. Dirac's equation reflects a more complex character of quantum electrodynamics or quantum field theory in general and of the corresponding (high-energy) experimental quantum physics vis-a-vis that of quantum mechanics and the (low-energy) experimental quantum physics. The final section examines this greater complexity and its implications for fundamental physics.
The formal mathematical structure of the truly Hamiltonian formulations of nonrelativistic and relativistic quantum mechanics is critically examined for both spin 0 and spin 12 particles with nonzero mass. It is shown that the relativistic quantum mechanics can, in principle, at least be fitted into a truly covariant Hamiltonian procedure where the Hamiltonian is a Lorentz-invariant world scalar (and is the negative of 12 the rest energy of the particle, regardless of its spin). In particular, a truly covariant Hamiltonian theory of the electron is presented. It will be seen that the Lorentz-invariant Dirac theory of the electron does not fall within the framework of a truly covariant Hamiltonian procedure. Further, the dynamical variables associated with the translational degree of freedom of the particle (such as the position, the canonical momentum, and the velocity four-vectors of the relativistic quantum mechanics) correspond precisely to the same variables of the truly covariant formulation of relativistic classical mechanics. The mathematical structure of relativistic quantum mechanics is seen to be rather similar to that of nonrelativistic quantum mechanics for both spin 0 and spin 12 particles with nonzero mass.