Dirac adjoint
Template:Short description 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 "Template:Math-bar".
Definition
Let <math>\psi</math> be a Dirac spinor. Then its Dirac adjoint is defined as
- <math>\bar\psi \equiv \psi^\dagger \gamma^0</math>
where <math>\psi^\dagger</math> denotes the Hermitian adjoint of the spinor <math>\psi</math>, and <math>\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>\lambda</math> is a projective representation of some Lorentz transformation,
- <math>\psi \mapsto \lambda \psi,</math>
then, in general,
- <math>\lambda^\dagger \ne \lambda^{-1}.</math>
The Hermitian adjoint of a spinor transforms according to
- <math>\psi^\dagger \mapsto \psi^\dagger \lambda^\dagger.</math>
Therefore, <math>\psi^\dagger\psi</math> is not a Lorentz scalar and <math>\psi^\dagger\gamma^\mu\psi</math> is not even Hermitian.
Dirac adjoints, in contrast, transform according to
- <math>\bar\psi \mapsto \left(\lambda \psi\right)^\dagger \gamma^0.</math>
Using the identity <math>\gamma^0 \lambda^\dagger \gamma^0 = \lambda^{-1}</math>, the transformation reduces to
- <math>\bar\psi \mapsto \bar\psi \lambda^{-1},</math>
Thus, <math>\bar\psi\psi</math> transforms as a Lorentz scalar and <math>\bar\psi\gamma^\mu\psi</math> as a four-vector.
Usage
Using the Dirac adjoint, the probability four-current Template:Math for a spin-1/2 particle field can be written as
- <math>J^\mu = c \bar\psi \gamma^\mu \psi</math>
where Template:Math is the speed of light and the components of Template:Math represent the probability density Template:Math and the probability 3-current Template:Math:
- <math>\boldsymbol J = (c \rho, \boldsymbol j).</math>
Taking Template:Nowrap and using the relation for gamma matrices
- <math>\left(\gamma^0\right)^2 = I,</math>
the probability density becomes
- <math>\rho = \psi^\dagger \psi.</math>