Dirac adjoint

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".

Contents

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\, .

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 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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