# Relativistic Off-Mass-Shell and Off-Energy-Shell Scattering Theories: An Application of the Rotation and Lorentz Groups

**Abstract**

A relativistic, phenomenological scattering theory for particles with arbitrary spin is presented, and the relation between off-mass-shell and off-energy-shell theories is discussed.

The theory is formulated from the Hilbert-space representation of particles with spin in relativistic quantum mechanics. This topic is reviewed in a basis independent manner by appealing to the properties of the rotation and Lorentz groups and their representations.

Spin is discussed and a set of basis state vectors for the single-particle Hilbert space is derived from this perspective. Two- and three-particle Hilbert-space bases are then constructed and angular momentum is discussed in complete analogy with the single-particle states. The z and helicity bases are presented as examples of the general procedure. These foundations permit a theoretically meaningful discussion of relativistic scattering theory and allow the on-shell scattering amplitude to be defined.

The space-inversion and time-reversal properties of this amplitude suggest that a new scattering function be defined such that a continuation of that function to negative energies can be considered. Antiparticle scattering events are associated with the continued function, and the CPT theorem arises as a natural consequence of this association. Moreover, these considerations lead to the definition of an off-mass-shell scattering function.

The resulting off-mass-shell scattering theory, which is shown to be essentially equivalent to perturbative field theory, has a number of very appealing properties.

- First, only 2s+l spin-projection indices are required to describe a particle with spin s in the off-mass-shell scattering function, and particles with any spin are described by exactly the same formalism.
- Second, it is an off-mass-shell theory, and the Lorentz transformation properties of the off-mass-shell scattering function are well defined.
- Third, the theory is easily subjected to reduction techniques that allow for the development of an equivalent off-energy-shell scattering theory.

The off-energy-shell theory is dependent on fewer variables than the off-mass-shell theory and is more susceptible to a phenomenological treatment.

Thus, phenomenology is discussed and dispersion-theoretic techniques of obtaining off-shell two-body scattering functions are described within this framework. Spin and angular momentum are treated in a covariant, kinematic-singularity free manner throughout these discussions.

Finally, an example of the theory, which is preliminary to a more complete theory of the relativistic three-body problem, is presented. A single-particle exchange diagram is described, and an off-mass-shell three-body recoupling coefficient, which is similar to the on-shell three-body recoupling coefficient that was defined by Wick, is derived.