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 ψ {\displaystyle \psi } be a Dirac spinor. Then its Dirac adjoint is defined as
ψ ¯ ≡ ψ † γ 0 {\displaystyle {\bar {\psi }}\equiv \psi ^{\dagger }\gamma ^{0}}
where ψ † {\displaystyle \psi ^{\dagger }} denotes the Hermitian adjoint of the spinor ψ {\displaystyle \psi } , and γ 0 {\displaystyle \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 λ {\displaystyle \lambda } is a projective representation of some Lorentz transformation,
ψ ↦ λ ψ , {\displaystyle \psi \mapsto \lambda \psi ,}
then, in general,
λ † ≠ λ − 1 . {\displaystyle \lambda ^{\dagger }\neq \lambda ^{-1}.}
The Hermitian adjoint of a spinor transforms according to
ψ † ↦ ψ † λ † . {\displaystyle \psi ^{\dagger }\mapsto \psi ^{\dagger }\lambda ^{\dagger }.}
Therefore, ψ † ψ {\displaystyle \psi ^{\dagger }\psi } is not a Lorentz scalar and ψ † γ μ ψ {\displaystyle \psi ^{\dagger }\gamma ^{\mu }\psi } is not even Hermitian. Dirac adjoints, in contrast, transform according to
ψ ¯ ↦ ( λ ψ ) † γ 0 . {\displaystyle {\bar {\psi }}\mapsto \left(\lambda \psi \right)^{\dagger }\gamma ^{0}.}
Using the identity γ 0 λ † γ 0 = λ − 1 {\displaystyle \gamma ^{0}\lambda ^{\dagger }\gamma ^{0}=\lambda ^{-1}} , the transformation reduces to
ψ ¯ ↦ ψ ¯ λ − 1 , {\displaystyle {\bar {\psi }}\mapsto {\bar {\psi }}\lambda ^{-1},}
Thus, ψ ¯ ψ {\displaystyle {\bar {\psi }}\psi } transforms as a Lorentz scalar and ψ ¯ γ μ ψ {\displaystyle {\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 μ = c ψ ¯ γ μ ψ {\displaystyle 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:
J = ( c ρ , j ) . {\displaystyle {\boldsymbol {J}}=(c\rho ,{\boldsymbol {j}}).}
Taking μ = 0 and using the relation for gamma matrices
( γ 0 ) 2 = I , {\displaystyle \left(\gamma ^{0}\right)^{2}=I,}
the probability density becomes
ρ = ψ † ψ . {\displaystyle \rho =\psi ^{\dagger }\psi .}
See also Dirac equation Rarita–Schwinger equation
References B. Bransden; C. Joachain (2000). Quantum Mechanics (2nd ed.). Pearson. ISBN 0-582-35691-1. M. Peskin; 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.