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 $\psi$ be a Dirac spinor. Then its Dirac adjoint is defined as

:$\bar\psi \equiv \psi^\dagger \gamma^0$

where $\psi^\dagger$ denotes the Hermitian adjoint of the spinor $\psi$, and $\gamma^0$ 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 $\lambda$ is a projective representation of some Lorentz transformation,

:$\psi \mapsto \lambda \psi$,

then, in general,

:$\lambda^\dagger \ne \lambda^$.

The Hermitian adjoint of a spinor transforms according to

:$\psi^\dagger \mapsto \psi^\dagger \lambda^\dagger$.

Therefore, $\psi^\dagger\psi$ is not a Lorentz scalar and $\psi^\dagger\gamma^\mu\psi$ is not even Hermitian.

Dirac adjoints, in contrast, transform according to

:$\bar\psi \mapsto \left(\lambda \psi\right)^\dagger \gamma^0$.

Using the identity $\gamma^0 \lambda^\dagger \gamma^0 = \lambda^$, the transformation reduces to

:$\bar\psi \mapsto \bar\psi \lambda^$,

Thus, $\bar\psi\psi$ transforms as a Lorentz scalar and $\bar\psi\gamma^\mu\psi$ 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

:$J^\mu = c \bar\psi \gamma^\mu \psi$

where c is the speed of light and the components of J represent the probability density ρ and the probability 3-current j:

:$\boldsymbol J = (c \rho, \boldsymbol j)$.

Taking 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. .
*M. Peskin and D. Schroeder (1995). ''An Introduction to Quantum Field Theory'', Westview Press. .
*A. Zee (2003). ''Quantum Field Theory in a Nutshell'', Princeton University Press. {{ISBN, 0-691-01019-6.

Quantum field theory
Spinors
Mathematical notation
Paul Dirac