Название: Foundations of Quantum Field Theory
Автор: Klaus D Rothe
Издательство: Ingram
Жанр: Физика
Серия: World Scientific Lecture Notes In Physics
isbn: 9789811221941
isbn:
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-timewhere 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 operatorsNote that
and (see (2.22)). It then follows from the commutation relations (2.23) thatThe 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 haswith
Example 2
Consider the representation (A, B) = (0, j). In that case
= 0 or , and the Lorentz transformations are realized bywith 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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