An Infinite Lie Algebra Associated with the Quantum Coulomb Field

Abstract

The theory of the quantum Coulomb field associates with each Lorentz frame, i. e., with each unit, future oriented time-like vector $u$, the operator of the number of transversal infrared photons $N(u)$ and the phase $S(u)$ which is the coordinate canonically conjugated with the total charge $Q: [Q, S(u)] = ie, e$ being the elementary charge. It is shown that the operators $N(u), Q/e S(u)$ and $Q^2$ form an infinite Lie algebra. One can conclude from this algebra that $\DeltaN(u) = (4/π) Q^2$, where $Δ$ is the Laplace operator in the Lobachevsky space of four-velocities $u$, thus relating the total charge $Q$ with the number of infrared photons.