Boosts

      Let S and S′ be two inertial frames whose clocks are synchronized in such a way, that their respective origins coincide at time t = 0. Then we have for an eigenstate of the momentum operator, as seen by observers O and O′ in S and S′

      respectively. In particular, for a particle at rest in system S we obtain, from the point of view of O′,

      Define

      where

stands for a Galilei transformation taking

and

. U[B(

)] is thus an operator which takes a particle at rest into a particle with momentum

. One refers to this as a “boost” (active point of view).

      We have the following property of Galilei transformations not shared by Lorentz transformations (compare with (2.23)): boosts and rotations separately form a group. Indeed one easily checks that

      (4)Causality

      We next want to show that NRQM violates the principle of causality. We have for any interacting theory,

      Let |En

be a complete set of eigenstates of H:³

      Then

      Define

      as well as

      The kernel

satisfies a heat-like equation:

      In terms of this kernel we have from above,

      We now specialize to the case of a free point-like particle. In that case

      and correspondingly we have with (1.8),

      Notice that the kernel K₀ satisfies the desired initial condition (1.9).

      From (1.10), for the initial condition Скачать книгу