The proper homogeneous Lorentz transformation operator eL=e−ω⋅S−ξ⋅K: Where’s it going, what’s the twist?

Abstract

A discussion of the proper homogeneous Lorentz transformation operator eL=exp[−ω⋅S−ξ⋅K] is given where eL transforms coordinates of an observer 𝒪 to those of an observer O′. Two methods of evaluation are presented. The first is based on a dynamical analog. It is shown that the transformation can be evaluated from the set of equations that are identical to the set of equations that determine the four-velocity of a charged particle in response to a combined spatially uniform and temporally constant electric field E and magnetic field B, where E is parallel to ξ and B is antiparallel to ω, and E/B=ξ/ω. The principal difference in the two problems is that in the dynamics problem, the initial conditions for the four-velocity u must satisfy the constraint, uu=1, whereas the inner product of the coordinates acted on by eL can have any real value. In order to evaluate eL, one can then apply the simplifying techniques of transforming to the frame where E is parallel or antiparallel to B, whereupon the transformation eL in this special frame is trivially evaluated. Then we transform back to the original frame. We determine the β and the rotation Ω that results from a successive boost and rotation that the operator eL produces. A second method is based on a direct summation of the power series of the matrix elements of eL that has been used in relativistic quantum theory. The summation is facilitated by observing that the operators J±≡K±iS commute with each other, and can be represented in terms of the Pauli spin matrices. Indeed, we can reduce the Lorentz transformation to the product of spinor operators to give a compact way to compute the elements of the Lorentz operator eL.