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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 \lambda 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, $$\lambda$$ 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 $$\mu = 0$$ and using the relation for Gamma matrices

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

the probability density becomes

$$\rho = \psi^\dagger\psi\, .$$

Dirac equation
Rarita-Schwinger equation

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.

Physics Encyclopedia