Foundations of Quantum Field Theory. Klaus D Rothe
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of the “free” Schrödinger equation
or
Furthermore, making use of
we have
or
with
Correspondingly we have from (1.7)
in accordance with expectations.
(3)Covariance of equations of motion
From
follows
which we rewrite as
Noting from (1.3) and (1.2) that
we obtain, using (1.6),
or, recalling (1.7),
This equation expresses on operator level the covariance of the free-particle equation of motion: If
Group-property
As Eq. (1.4) shows, a Galilei transformation is represented by the unitary operator
with
the generators of boosts. We have
so that different “boosts” commute with each other. The Galilei transformations thus correspond to an abelian Lie group. In particular
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
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
Then
Define
as well as
The kernel
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