Dirac adjoint
In quantum field theory, the Dirac adjoint defines the dual operation of a Dirac spinor. The Dirac adjoint is motivated by the need to form well-behaved, measurable quantities out of Dirac spinors, replacing the usual role of the Hermitian adjoint.
Possibly to avoid confusion with the usual Hermitian adjoint, some textbooks do not provide a name for the Dirac adjoint but simply call it "ψ-bar".
Definition
[edit]Let be a Dirac spinor. Then its Dirac adjoint is defined as
where denotes the Hermitian adjoint of the spinor , and is the time-like gamma matrix.
Spinors under Lorentz transformations
[edit]The Lorentz group of special relativity is not compact, therefore spinor representations of Lorentz transformations are generally not unitary. That is, if is a projective representation of some Lorentz transformation,
- ,
then, in general,
- .
The Hermitian adjoint of a spinor transforms according to
- .
Therefore, is not a Lorentz scalar and is not even Hermitian.
Dirac adjoints, in contrast, transform according to
- .
Using the identity , the transformation reduces to
- ,
Thus, transforms as a Lorentz scalar and as a four-vector.
Usage
[edit]Using the Dirac adjoint, the probability four-current J for a spin-1/2 particle field can be written as
where c is the speed of light and the components of J represent the probability density ρ and the probability 3-current j:
- .
Taking μ = 0 and using the relation for gamma matrices
- ,
the probability density becomes
- .
See also
[edit]References
[edit]- B. Bransden and C. Joachain (2000). Quantum Mechanics, 2e, Pearson. ISBN 0-582-35691-1.
- M. Peskin and D. Schroeder (1995). An Introduction to Quantum Field Theory, Westview Press. ISBN 0-201-50397-2.
- A. Zee (2003). Quantum Field Theory in a Nutshell, Princeton University Press. ISBN 0-691-01019-6.