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Russian Physics Journal

Relativistic quantum theory damping equation in the Hamiltonian formulation of quantum field theory

I. L. Solovtsov

Published Apr 1, 1983• 1 citation
Publisher
DOI
10.1007/bf01882971

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A novel Hamiltonian scheme for non–relativistic quantum electrodynamics in which the gauge arbitrariness of the field potential is kept explicit is used to study the gauge–dependence properties of various versions of perturbation theory when resonance and line–broadening effects are admitted. Time–dependent perturbation theory is shown to have severe gauge–dependence problems unless ad hoc modifications are made. A time–independent formulation of S–matrix theory is then studied. Far from resonance, the S–matrix is gauge invariant in all orders of perturbation theory due to a very precise cancellation of gauge–dependent terms which requires, among other things, complete sets of intermediate states; energy conservation also has a crucial role. However, an obvious separation of the S–matrix into a resonant (pole) and non–resonant background leads to incomplete cancellation of the gauge–dependent terms. The introduction of the Heitler damping matrix into an integral equation for the T–matrix leads to a gauge–invariant result. This provides the basis for a gauge invariant S–matrix theory of atoms and molecules interacting with electromagnetic radiation that encompasses resonance and damping effects.
In the pilot-wave theory of quantum mechanics particles have definite positions and velocities and the system evolves deterministically. The velocity of a particle is determined by the wave function of the system (the guidance equation) and the wave function evolves according to Schrödinger’s equation. In this paper I first construct a Hamiltonian that gives Schrödinger’s equation and the guidance equation for the particle. I then find the Hamiltonian for a relativistic particle in Dirac’s theory and for a quantum scalar field.
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.