- Dirac adjoint
In quantum field theory, the Dirac adjoint of a Dirac spinor is defined to be the dual spinor , where is the time-like gamma matrix. Possibly to avoid confusion with the usual Hermitian adjoint , some textbooks do not give a name to the Dirac adjoint, simply calling it "psi-bar".
The Dirac adjoint is motivated by the need to form well-behaved, measurable quantities out of Dirac spinors. For example, is not a Lorentz scalar, and is not even Hermitian. One source of trouble is that if λ is the spinor representation of a Lorentz transformation, so that
Since the Lorentz group of special relativity is not compact, λ will not be unitary, so . Using fixes this problem, in that it transforms as
Using the Dirac adjoint, the conserved probability four-current density for a spin-1/2 particle field
where is the probability density and j the probability current 3-density can be written as
where c is the speed of light. Taking μ = 0 and using the relation for Gamma matrices
the probability density becomes
- 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.
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