 Dirac adjoint

In quantum field theory, the Dirac adjoint of a Dirac spinor is defined to be the dual spinor , where is the timelike 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 "psibar".
Contents
Motivation
The Dirac adjoint is motivated by the need to form wellbehaved, 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
then
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
Usage
Using the Dirac adjoint, the conserved probability fourcurrent density for a spin1/2 particle field
where is the probability density and j the probability current 3density 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
 .
See also
References
 B. Bransden and C. Joachain (2000). Quantum Mechanics, 2e, Pearson. ISBN 0582356911.
 M. Peskin and D. Schroeder (1995). An Introduction to Quantum Field Theory, Westview Press. ISBN 0201503972.
 A. Zee (2003). Quantum Field Theory in a Nutshell, Princeton University Press. ISBN 0691010196.
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