Foundations of Quantum Field Theory. Klaus D Rothe

Foundations of Quantum Field Theory

(boost in the +z direction)

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

⁰ⁱ with the generators for pure velocity transformations along the z-axis. Notice that these matrices are anti-hermitian. On the other hand, the hermitian matrices

^ij generate rotations in the (ij)-plane:

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 relations^6,7

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

ⁱ⁰ generating the boosts. Correspondingly we have for a finite transformation

in space-time

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

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