Physics:Dirac adjoint
In quantum field theory, the Dirac adjoint defines the dual operation of a Dirac spinor. The Dirac adjoint is motivated by the need to form well-behaved, measurable quantities out of Dirac spinors, replacing the usual role of the Hermitian adjoint.
Possibly to avoid confusion with the usual Hermitian adjoint, some textbooks do not provide a name for the Dirac adjoint but simply call it "ψ-bar".
Definition
Let [math]\displaystyle{ \psi }[/math] be a Dirac spinor. Then its Dirac adjoint is defined as
- [math]\displaystyle{ \bar\psi \equiv \psi^\dagger \gamma^0 }[/math]
where [math]\displaystyle{ \psi^\dagger }[/math] denotes the Hermitian adjoint of the spinor [math]\displaystyle{ \psi }[/math], and [math]\displaystyle{ \gamma^0 }[/math] is the time-like gamma matrix.
Spinors under Lorentz transformations
The Lorentz group of special relativity is not compact, therefore spinor representations of Lorentz transformations are generally not unitary. That is, if [math]\displaystyle{ \lambda }[/math] is a projective representation of some Lorentz transformation,
- [math]\displaystyle{ \psi \mapsto \lambda \psi }[/math],
then, in general,
- [math]\displaystyle{ \lambda^\dagger \ne \lambda^{-1} }[/math].
The Hermitian adjoint of a spinor transforms according to
- [math]\displaystyle{ \psi^\dagger \mapsto \psi^\dagger \lambda^\dagger }[/math].
Therefore, [math]\displaystyle{ \psi^\dagger\psi }[/math] is not a Lorentz scalar and [math]\displaystyle{ \psi^\dagger\gamma^\mu\psi }[/math] is not even Hermitian.
Dirac adjoints, in contrast, transform according to
- [math]\displaystyle{ \bar\psi \mapsto \left(\lambda \psi\right)^\dagger \gamma^0 }[/math].
Using the identity [math]\displaystyle{ \gamma^0 \lambda^\dagger \gamma^0 = \lambda^{-1} }[/math], the transformation reduces to
- [math]\displaystyle{ \bar\psi \mapsto \bar\psi \lambda^{-1} }[/math],
Thus, [math]\displaystyle{ \bar\psi\psi }[/math] transforms as a Lorentz scalar and [math]\displaystyle{ \bar\psi\gamma^\mu\psi }[/math] as a four-vector.
Usage
Using the Dirac adjoint, the probability four-current J for a spin-1/2 particle field can be written as
- [math]\displaystyle{ J^\mu = c \bar\psi \gamma^\mu \psi }[/math]
where c is the speed of light and the components of J represent the probability density ρ and the probability 3-current j:
- [math]\displaystyle{ \boldsymbol J = (c \rho, \boldsymbol j) }[/math].
Taking μ = 0 and using the relation for gamma matrices
- [math]\displaystyle{ \left(\gamma^0\right)^2 = I }[/math],
the probability density becomes
- [math]\displaystyle{ \rho = \psi^\dagger \psi }[/math].
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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