Foundations of Quantum Field Theory. Klaus D Rothe

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      reminiscent of the rotation group in four dimensions (except for the minus sign). This is not surprising, since the Lorentz transformations are connected to the four-dimensional rotation group by an analytic continuation in the parameters parametrizing the boosts, to pure imaginary values. The underlying Minkowski character of the space-time manifold is hidden in the anti-hermiticity of the operators

      where we have made the identifications

      with θ the angle labelling the boost (2.16). For a pure rotation in the ij-plane,

      where Jk is the generator of rotations around the k-axis (i, j, k taken cyclicly), with the explicit realization

      Next, it will be our aim to obtain higher dimensional representations of the generators of the Lorentz group.

2.4Finite irreducible representation of

      In order to obtain a characterization of the irreducible representation

, we define the new operators

      Note that

and

(see (2.22)). It then follows from the commutation relations (2.23) that

      The operators

. In the irreducible basis, the matrix elements of the generators

and

are evidently given by the well-known expressions known from the rotation group:

      where a and b take the values

      In particular, we have

      For the operators

and i

these matrix elements read from (2.27)

      From these matrix elements we then obtain the corresponding representations of finite Lorentz transformations

by simple exponentiation, in a way analogous to (2.24),

      or compactly

      with

      where

. They are non-unitary except for the trivial representation D^(0,0)

       Example 1

      Consider the representation (A, B) = (j, 0). In that representation the operator

is realized by zero or

. This possibility is allowed by the commutation relations (2.28). One has

      with

       Example 2

      Consider the representation (A, B) = (0, j). In that case

= 0 or

, and the Lorentz transformations are realized by

      with the corresponding matrix elements for

:

      The above representations will play a central role in the chapters to follow. We observe that

      We introduce the following notation⁸

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