Conformal group of transformations of the quantum field operators in the momentum space and the five dimensional Lagrangian approach

TL;DR

This paper explores the conformal group of transformations in momentum space and develops a five-dimensional Lagrangian framework for equations of motion of interacting massive particles, demonstrating invariance of the S-matrix under certain transformations.

Contribution

It introduces a five-dimensional generalization of equations of motion for massive particles using conformal transformations in momentum space, with explicit separation of equations based on inversion symmetry.

Findings

S-matrix invariance under momentum translations

Separation of equations over the fifth coordinate

Explicit inversion symmetry in five-dimensional equations

Abstract

Conformal group of transformations in the momentum space, consisting of translations $p'_{\mu}=p_{\mu}+h_{\mu}$, rotations $p'_{\mu}=\Lambda^{\nu}_{\mu}p_{\nu}$, dilatation $p'_{\mu}=\lambda p_{\mu}$ and inversion $p'_{\mu}= -M^2p_{\mu}/p^2$ of the four-momentum $p_{\mu}$, is used for the five dimensional generalization of the equations of motion for the interacting massive particles. It is shown, that the ${\cal S}$-matrix of the charged and the neutral particles scattering is invariant under translations in a four-dimensional momentum space $p'_{\mu}=p_{\mu}+h_{\mu}$. In the suggested system of equations of motion, the one-dimensional equations over the fifth coordinate $x_5$ are separated and these one dimensional equations have the form of the evaluation equations with $x_5=\sqrt{x_o^2-{\bf x}^2}$. The important property of the derived five dimensional equations of motion is the explicit separation of the parts of these equations according to the inversion $p'_{\mu}=-M^2 p_{\mu}/p^{2}$, where $M$ is a scale constant.