There is no relativistic Hamiltonian for many particles systems except for free particles and this has been accepted since the 1960s from the work of Currie, Jordan and Sudarshan, Cannon and Jordan, and Leutwyler. This is the problem we will address in this paper by coupling relativistic particles with a scalar field. The Hamiltonian we derive is explicitly relativistic and in the non-relativistic limit gives a Hamiltonian with two body interaction. However, we find that the Poincare algebra is still not satisfied because of the inherent time delayed interaction in relativistic particle systems.
This paper studies a system of
n∈N:n≥2 non-relativistic, spinless quantum particles moving on the real line and interacting via a two-body delta potential. The Hamiltonian of such a system is proved to be affiliated to the resolvent algebra of the case,
R(R2n,σ); it is further shown the existence of a
C∗−dynamical system and of a subalgebra
πS(S0)−1⊂R(R2n,σ), stable under time evolution, where
πS is the Schr{ö}dinger representation of the algebra.
The process algebra has been used successfully to provide a novel formulation of quantum mechanics in which non-relativistic quantum mechanics (NRQM) emerges as an effective theory asymptotically. The process algebra is applied here to the formulation of quantum field theory. The resulting QFT is intuitive, free from divergences and eliminates the distinction between particle, field and wave. There is a finite, discrete emergent space-time on which arise emergent entities which transfer information like discrete waves and interact with measurement processes like particles. The need for second quantization is eliminated and the particle and field theories rest on a common foundation, clarifying and simplifying the relationship between the two.