Dirac adjoint

Dual to the Dirac spinor

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

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.