Definitions

In quantum field theory, the Dirac adjoint $barpsi$ of a Dirac spinor $psi$ is defined to be the dual spinor $psi^\left\{dagger\right\} gamma^0$, where $gamma^0$ is the time-like gamma matrix. Possibly to avoid confusion with the usual Hermitian adjoint $psi^dagger$, some textbooks do not give a name to the Dirac adjoint, simply calling it "psi-bar".

## Motivation

The Dirac adjoint is motivated by the need to form well-behaved, measurable quantities out of Dirac spinors. For example, $psi^daggerpsi$ is not a Lorentz scalar, and $psi^daggergamma^mupsi$ is not even Hermitian. One source of trouble is that if $lambda$ is the spinor representation of a Lorentz transformation, so that
$psitolambdapsi,$
then
$psi^daggertopsi^daggerlambda^dagger.$
Since the Lorentz group of special relativity is not compact, $lambda$ will not be unitary, so $lambda^daggerneqlambda^\left\{-1\right\}$. Using $barpsi$ fixes this problem, in that it transforms as
$barpsitobarpsilambda^\left\{-1\right\}.$

## Usage

Using the Dirac adjoint, the conserved probability four-current density for a spin-1/2 particle field

$j^mu = \left(crho, j\right),$

where $rho,$ is the probability density and j the probability current 3-density can be written as

$j^mu = cbarpsigamma^mupsi$

where c is the speed of light. Taking $mu = 0$ and using the relation for Gamma matrices

$left\left(gamma^0 right\right)^2 = I ,$

the probability density becomes

$rho = psi^daggerpsi,$ .

## 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 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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