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# Dirac adjoint

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

\( \rho = \psi^\dagger\psi\, .\)

See also

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.

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