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. Since the usual Hermitian adjoint lacks the Lorentz symmetry of the system, the Dirac adjoint must be used instead.
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".
Let ψ be a Dirac spinor. Then its Dirac adjoint is defined as
where ψ† denotes the Hermitian adjoint of the spinor ψ and γ0 is the time-like gamma matrix.
The Lorentz group of special relativity is not compact, therefore representations of Lorentz transformations in the Dirac spinor space are not unitary. That is, in general,
where λ is the corresponding Lorentz transformation that maps spinors:
The Hermitian adjoint of spinors transforms according to
Therefore, using only the Hermitian adjoint, one finds that ψ†ψ is not a Lorentz scalar and ψ†γμψ is not even Hermitian.
Using the definition, one finds that the Dirac adjoint of spinors transforms according to
Using the identity γ0λ†γ0 = λ−1, the transformation reduces to
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