Adjoint Dirac equation

Reference: References: Robert D. Klauber, Student Friendly Quantum Field Theory, (Sandtrove Press, 2013) – Chapter 4, Problem 4.11.

The Dirac equation in condensed form is

\displaystyle i\gamma^{\mu}\partial_{\mu}\left|\psi\right\rangle =m\left|\psi\right\rangle \ \ \ \ \ (1)

 

where the gamma matrices have been defined earlier. The Hermitian conjugate of the gamma matrix {\gamma^{\mu}} is given by the Hermiticity condition

\displaystyle \gamma^{\mu\dagger}=\gamma^{0}\gamma^{\mu}\gamma^{0} \ \ \ \ \ (2)

 

To get the adjoint form of the Dirac equation, we use the adjoint solutions

\displaystyle \left\langle \bar{\psi}^{\left(n\right)}\right|\equiv\left\langle \psi^{\left(n\right)}\right|\gamma^{0} \ \ \ \ \ (3)

 

The Hermitian conjugate of 1 is

\displaystyle -i\partial_{\mu}\left\langle \psi\right|\gamma^{\mu\dagger}=m\left\langle \psi\right| \ \ \ \ \ (4)

Remember that when we take the Hermitian conjugate of a matrix equation we must take the Hermitian conjugate of each matrix and also reverse the order of matrix multiplication, which is why the {\gamma^{\mu\dagger}} term appears at the end on the LHS. The {\partial_{\mu}} is a differential operator, not a matrix, so it retains its position in front of the {\left\langle \psi\right|}.

Using 2, we can write this as

\displaystyle -i\partial_{\mu}\left\langle \psi\right|\gamma^{0}\gamma^{\mu}\gamma^{0}=m\left\langle \psi\right| \ \ \ \ \ (5)

Then, by post-multiplying by {\gamma^{0}} and using {\left(\gamma^{0}\right)^{2}=I}, the identity matrix, we get, using the definition 3

\displaystyle -i\partial_{\mu}\left\langle \psi\right|\gamma^{0}\gamma^{\mu}\left(\gamma^{0}\right)^{2} \displaystyle = \displaystyle m\left\langle \psi\right|\gamma^{0}\ \ \ \ \ (6)
\displaystyle -i\partial_{\mu}\left\langle \bar{\psi}\right|\gamma^{\mu} \displaystyle = \displaystyle m\left\langle \bar{\psi}\right|\ \ \ \ \ (7)
\displaystyle i\partial_{\mu}\left\langle \bar{\psi}\right|\gamma^{\mu}+m\left\langle \bar{\psi}\right| \displaystyle = \displaystyle 0 \ \ \ \ \ (8)

Note that although the terms {\left\langle \bar{\psi}\right|} in this adjoint equation are adjoint solutions, the gamma matrices {\gamma^{\mu}} are the original (that is, not the Hermitian conjugate) gamma matrices.

4 thoughts on “Adjoint Dirac equation

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