Название: Foundations of Quantum Field Theory
Автор: Klaus D Rothe
Издательство: Ingram
Жанр: Физика
Серия: World Scientific Lecture Notes In Physics
isbn: 9789811221941
isbn:
Tracelessness
Since
or
Dimensionality
Since
Minimal dimension
The Pauli matrices
together with the identity matrix 1 represent a complete basis for 2 × 2 hermitian matrices. Of these, the Pauli matrices satisfy the first of the conditions (4.3); however, the identity matrix cannot be identified with β, since trβ = 0. Since the dimension of the matrices must be even, we conclude that the dimension of these matrices must be at least four.
The following 4 × 4 matrices satisfy all the requirements (4.3):
The same applies of course to matrices obtained from the above ones via a unitary transformation (unitary, in order to preserve the hermiticity of the matrices). For the choice of basis (4.5), the equation reads
We can compactify the notation by introducing the definitions
where the subscript D stands for “Dirac representation”. Explicitly we have
We may collect these matrices into a 4-tuplet
This equation implies that ψ(r, t) is also a solution of the Klein–Gordon equation (3.8). We thus have the following Fourier decomposition into positive and negative energy solutions,
where
For ψ(x) to be a solution of the Dirac equation (4.8), the (positive and negative energy) Dirac spinors U(p, σ) and V(p, σ) must satisfy the equations
Recalling the explicit form (4.7) of the
where
with c the “charge conjugation” matrix defined by
Notice that
The matrix c has the fundamental property3
The following algebraic relations will turn out to be useful:
where
with
and
We further have
In the Dirac representation, γ5 is the off-diagonal 4 × 4 matrix
In order to fix the normalization constants in (4.11), we need to choose a scalar product. To this end we observe that
Hence the Dirac operator iγμ∂μ − m is hermitian with respect to the “Dirac” scalar product