Dirac adjoint ( $\overset{ˉ}{ψ} = ψ^{†} γ^{0}$ )

The Dirac adjoint of a spinor is

\overset{ˉ}{ψ} = ψ^{†} γ^{0}

. It transforms contragrediently, making

\overset{ˉ}{ψ} ψ

a Lorentz scalar and

\overset{ˉ}{ψ} γ^{μ} ψ

a Lorentz vector.

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- Dirac scalar bilinear Dirac adjoint

Dirac scalar bilinear ( $\overset{ˉ}{Ψ} Ψ$ )

The Dirac scalar bilinear is invariant under the simultaneous spinor and adjoint transformations because

\overset{ˉ}{Ψ}^{'} Ψ^{'} = \overset{ˉ}{Ψ} S^{- 1} S Ψ

. It is the mass term of the Dirac action. At fixed coordinates it transforms as a scalar field, with an orbital transport term and no spin term.

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The Dirac adjoint is a mathematical concept used in quantum mechanics and quantum field theory, specifically in the context of Dirac spinors and the formulation of the Dirac equation, which describes the behavior of fermions such as electrons. In the context of Dirac spinors, we have a Dirac spinor \(\psi\), which is a four-component complex vector.

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Ciro Santilli  40 Updated 2025-07-16

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