Dirac adjoint
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In quantum field theory, the Dirac adjoint <math> \bar\psi </math> of a Dirac spinor <math>\ \psi </math> is defined to be the dual spinor <math>\ \psi^{\dagger} \gamma^0 </math>, where <math>\ \gamma^0 </math> is the time-like gamma matrix. Possibly to avoid confusion with the usual Hermitian adjoint <math>\psi^\dagger</math>, some textbooks do not give a name to the Dirac adjoint, simply calling it "psi-bar".
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Motivation
The Dirac adjoint is motivated by the need to form well-behaved, measurable quantities out of Dirac spinors. For example, <math>\psi^\dagger\psi</math> is not a Lorentz scalar, and <math>\psi^\dagger\gamma^\mu\psi</math> is not even Hermitian. One source of trouble is that if <math>\lambda</math> is the spinor representation of a Lorentz transformation, so that
- <math>\psi\to\lambda\psi,</math>
then
- <math>\psi^\dagger\to\psi^\dagger\lambda^\dagger.</math>
Since the Lorentz group of special relativity is not compact, generally <math>\lambda</math> will not be unitary, so <math>\lambda^\dagger\neq\lambda^{-1}</math>. Using <math>\bar\psi</math> fixes this problem, in that it transforms as
- <math>\bar\psi\to\bar\psi\lambda^{-1}.</math>
Usage
Using the Dirac adjoint, the probability density for a spin-1/2 particle field can be written as
- <math> \rho = \bar\psi \psi</math>
and the current can be written as
- <math> j^\mu = \bar\psi\gamma^\mu\psi.</math>
See also
References
- 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 0201503972.
- A. Zee (2003). Quantum Field Theory in a Nutshell, Princeton University Press. ISBN 0-691-01019-6.