Foundations of Quantum Field Theory. Klaus D Rothe
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where we have set
Equation (2.20) just represents the first two leading terms of the expansion of the finite Lorentz boost (2.15) in the x3-direction around θ = 0. We thus identify
the generators of rotations about the z-, x- and y-axis respectively. One verifies from (2.18) that the generators of Lorentz transformations satisfy the Lie algebra
Explicitly
This algebra simplifies if we define the generators
which now satisfy the simple commutation relations6,7
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
Note that
The operators
where a and b take the values
In particular, we have
For the operators
From these matrix elements we then obtain the corresponding representations of finite Lorentz transformations
or compactly
with
where
Example 1
Consider the representation (A, B) = (j, 0). In that representation the operator