Foundations of Quantum Field Theory. Klaus D Rothe

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alt="figure"/> then represent the probability of finding a particle in the interval at time t. (Notice the treatment of space and time on unequal footing.) In QFT we can have particle production, that is, we are dealing with “many-particle” physics. Hence notions linked to a one particle picture must be abandoned in the relativistic case.

      (2)Galilei transformations

      For a scalar function

and a Galilei transformation , t′ = t we must have

      Consider in particular a plane wave

in S as seen by an observer in S′ moving with a velocity

with respect to S:

      where

      We have

      with

      We seek an operator

with the property

      From

      and

      we conclude, by comparing with (1.1),

      where

is the position operator. Now

      Hence we may write

in the form

      where we have used

      Denoting by

the eigenstates of the momentum operator

      we have

      For the solution

      of the “free” Schrödinger equation

we obtain from (1.3),

      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

is a solution of the equations of motion, then

is also a solution.

       Group-property

      As Eq. (1.4) shows, a Galilei transformation is represented by the unitary operator

      with

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