In quantum field theory, the Dirac adjoint $\bar\psi$ of a Dirac spinor $\ \psi$ is defined to be the dual spinor $\ \psi^{\dagger} \gamma^0$, 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, $\psi^\dagger\psi$ is not a Lorentz scalar, and $\psi^\dagger\gamma^\mu\psi$ is not even Hermitian. One source of trouble is that if λ is the spinor representation of a Lorentz transformation, so that $\psi\to\lambda\psi,$

then $\psi^\dagger\to\psi^\dagger\lambda^\dagger.$

Since the Lorentz group of special relativity is not compact, λ will not be unitary, so $\lambda^\dagger\neq\lambda^{-1}$. Using $\bar\psi$ fixes this problem, in that it transforms as $\bar\psi\to\bar\psi\lambda^{-1}.$

## Usage

Using the Dirac adjoint, the conserved probability four-current density for a spin-1/2 particle field $j^\mu = (c\rho, j)\,$

where $\rho\,$ is the probability density and j the probability current 3-density can be written as $j^\mu = c\bar\psi\gamma^\mu\psi$

where c is the speed of light. Taking μ = 0 and using the relation for Gamma matrices $\left( \gamma^0 \right)^2 = I \,$

the probability density becomes $\rho = \psi^\dagger\psi\,$ .

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