Comment on the equivalence of Bakamjian-Thomas mass operators in different forms of dynamics

TL;DR

This paper analyzes the equivalence of Bakamjian-Thomas mass operators across different forms of relativistic dynamics, showing that unitary transformations do not generate complex many-body forces and confirming the practical equivalence of various formulations.

Contribution

It demonstrates that different forms of Dirac's dynamics are physically equivalent without generating new many-body operators, clarifying the role of unitary transformations in relativistic quantum models.

Findings

Unitary scattering equivalences do not produce many-body forces.

Different forms of dynamics are mathematically equivalent.

Electromagnetic probes differ only in one-photon exchange approximation.

Abstract

We discuss the scattering equivalence of the generalized Bakamjian-Thomas construction of dynamical representations of the Poincar\'e group in all of Dirac's forms of dynamics. The equivalence was established by Sokolov in the context of proving that the equivalence holds for models that satisfy cluster separability. The generalized Bakamjian Thomas construction is used in most applications, even though it only satisfies cluster properties for systems of less than four particles. Different forms of dynamics are related by unitary transformations that remove interactions from some infinitesimal generators and introduce them to other generators. These unitary transformation must be interaction dependent, because they can be applied to a non-interacting generator and produce an interacting generator. This suggests that these transformations can generate complex many-body forces when used in many-body problems. It turns out that this is not the case. In all cases of interest the result of applying the unitary scattering equivalence results in representations that have simple relations, even though the unitary transformations are dynamical. This applies to many-body models as well as models with particle production. In all cases no new many-body operators are generated by the unitary scattering equivalences relating the different forms of dynamics. This makes it clear that the various calculations used in applications that emphasize one form of the dynamics over another are equivalent. Furthermore, explicit representations of the equivalent dynamical models in any form of dynamics are easily constructed. Where differences do appear is when electromagnetic probes are treated in the one-photon exchange approximation. This approximation is different in each of Dirac's forms of dynamics.