The proper homogeneous Lorentz transformation operator e(L)=e(-omega center dot S-xi center dot K): Where's it going, what's the twist?

A discussion of the proper homogeneous Lorentz transformation operator e(L) =exp[-omega .S- xi .K] is given where e(L) transforms coordinates of an obs erver O to those of an observer O '. Two methods of evaluation are presente d. The first is based on a dynamical analog. It is shown that the transform ation 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 f ield E and magnetic field B, where E is parallel to xi and B is antiparalle l to omega, and E/B = xi/omega). The principal difference in the two proble ms is that in the dynamics problem, the initial conditions for the four-vel ocity it must satisfy the constraint, uu = 1, whereas the inner product of the coordinates acted on by e(L) can have any real value. In order to evalu ate e(L),one can then apply the simplifying techniques of transforming to t he frame where E is parallel or antiparallel to B, whereupon the transforma tion e(L) in this special frame is trivially evaluated. Then we transform b ack to the original frame. We determine the beta and the rotation Omega tha t results from a successive boost and rotation that the operator e(L) produ ces. A second method is based on a direct summation of the power series of the matrix elements of e(L) that has been used in relativistic quantum theo ry. The summation is facilitated by observing that the operators J(+/-) dro p 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 th e product of spinor operators to give a compact way to compute the elements of the Lorentz operator e(L). (9 2001 American Association of Physics Teac hers.