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.
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.
Time evolution of relativistic particle systems can be accomplished by means of a Schrödinger wave equation, provided the Hamiltonian of the N particle system satisfies some commutator equations involving the other generators of a suitable representation of the Poincaré group on the initial data space of the Schrödinger equation. This set of operator equations is solved in the N free particle case. Further, the structure of N interacting particle Hamiltonians is worked out to include second-order relativistic corrections, i.e., (1/c2)-terms. It is shown that Breit and Barker–O’Connell Hamiltonians (nonzero spin particles) and the Bažański Hamiltonian (zero-spin particles) are particular cases of this more general Hamiltonian.
It is well known that the analysis of a relativistic n -body problem invariant under the transformations of the Poincaré group and involving one time has only been done for n =1. For n >1 one uses the second quantization formalism of field theory. In this paper we state it in the ordinary space time coordinates associated with the n-bodies as Dirac did it in the one body case. We apply the formalism first to the two body problem if the interaction is of an harmonic oscillator form and then extend it to n particles. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
For an arbitrary number of particles an explicit construction is given for direct interactions satisfying cluster separability (macrolocality). The Hamiltonian is in many ways similar to the nonrelativistic many-body Hamiltonian. Under conditions of physical interest the strength of the N-body interaction decreases rapidly with increasing N.
Abstract Avoiding the assumption that relativistic scattering be describable by interacting fields we find in the Schrödinger picture relativistic scattering closely analogue to the non-relativistic case. On the space of scattering states the invariant mass operator M′ M ′ of the interacting time evolution has to be unitarily equivalent to the invariant mass M=P2 M = P 2 where P, acting on many-particle states, is the sum of the one-particle four-momenta. For an observer at rest P0 P 0 generates the free time evolution. Poincaré symmetry requires the interacting Hamiltonian H′ H ′ to Lorentz transform as 0-component of a four-vector and to commute with the four-velocity U=P/M U = P / M but not with P, else there is no scattering. Even though H′ H ′ does not commute with P, the scattering matrix does. The four-velocity U generates translations of states as they are seen by shifted observers. Superpositions of nearly mass degenerate particles such as a Klong K long are seen by an inversely shifted observer as a shifted Klong K long with an unchanged relative phase. In contrast, the four-momentum P generates oscillated superpositions e.g. a shifted Kshort K short with a changed relative phase. The probability of scattering of massive particles is shown to be approximately proportional to the spacetime overlap of their position wave functions. This is basic to macroscopic locality and justifies to represent the machinery of actual scattering experiments by the vacuum. In suitable variables the relativistic Hamiltonian of many-particle states is not the sum of a Hamiltonian for the motion of the center and a commuting Hamiltonian for the relative motion but factorizes as their product. They act on different variables of the wave functions.
In this paper I develop a relativistic Hamiltonian and Schr\"odinger equation for three particles using a generalization of the assumptions that give rise to Todorov's relativistic quasipotential equation for two particles. This formalism is treated by a method that leads to a generalization of the separable Lagrange equilateral-triangle solution from gravitational nonrelativistic to relativistic nongravitational bound-state problems in quantum as well as classical mechanics. As an example I derive an analytic solution for the spectrum of a three-body system bound by scalar Coulomb forces.
A review is given of a new relativistic classical mechanics for N-particle systems compatible with relativistic bound states. It is formulated in global non-inertial frames in Minkowski space-time by means of parametrized Minkowski theories and then restricted to the intrinsic inertial rest frame of the isolated system. There is a complete control on the relativistic collective variables (Newton-Wigner center of mass, Fokker-Pryce center of inertia, M/oller center of energy) and on the realization of the Poincare' algebra (with the explicit form of the interaction-dependent Lorentz boosts). The particle world-lines are found to correspond to the ones of predictive mechanics and localization problems are clarified. The model can be consistently quantized avoiding the instantaneous spreading of the center-of-mass wave packets (Hegerfeldt theorem), because the non-local non-covariant center of mass is a non-measurable quantity. This implies that the Hilbert space must be the tensor product of the center-of-mass Hilbert space with the one of relative motions. This spatial non-separability (due to the Lorentz signature of space-time) makes relativistic entanglement much more involved than the non-relativistic one.