Consider the Dirac equation in the ordinary form in terms of [tex] a [/tex] and [tex]\beta [/tex] matrices(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

i\frac{{\partial \psi }}

{{\partial t}} = - i\vec a \cdot \vec \nabla \psi + m\beta \psi

[/tex]

The matrices are hermitian, [tex]

\vec a^\dag = \vec a,\beta ^\dag = \beta [/tex]. Daggers denote hermitian conjugation, i.e. complex conjugation followed by transposition for matrix operators. In order to prove the probability current conservation one has to derive from the Dirac equation the conjugate equation according to

[tex]

- i\frac{{\partial \psi ^\dag }}

{{\partial t}} = i\vec \nabla \psi ^\dag \cdot \vec a^\dag + m\psi ^\dag \beta ^\dag

[/tex]

Using these two equations one can easily show that the probability 4-vector is conserved and [tex] \rho = \psi ^\dag \psi [/tex] is the correct definition for the probability density. However, the previous hermitian conjugation operation, as employed, only affects Dirac matrices and the corresponding columns and rows. Notice that

[tex] \left( { - i\vec a \cdot \vec \nabla \psi } \right)^\dag = i\vec \nabla \psi ^\dag \cdot \vec a^\dag [/tex]

under this operation, so there is no notion of hermiticity of the momentum operator,

[tex]

\vec p^\dag = \left( { - i\vec \nabla } \right)^\dag = i\vec \nabla = - \vec p

[/tex]

as one would expect. In the meantime, the Dirac Hamiltonian is (again from the Dirac equation)

[tex]

H_D = - i\vec a \cdot \vec \nabla + m\beta = \vec a \cdot \vec p + m\beta

[/tex]

We would expect this to be Hermitian and indeed, using the last expression involving the momentum operator, this seems to be the case, provided though that we now use a hermitian conjugation operation that also affects differential operators, so that we recover

[tex] \vec p^\dag = \vec p [/tex]

and hermiticity is recovered. Why is this need for two different definitions of the hermitian conjugation in this case? One would expect that the operation affecting both matrix and differential operators should be used throughout, as hermitian conjugation should be defined in terms of the inner product in the full Hilbert space of the theory, involving both summation over spinor components and integration over the position variable. But this would not yield the correct expression for the current conservation. Note that this question applies to the one-particle theory (no QFT arguments) where the momentum is still an operator and a valid observable.

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# Hermitian conjugation and conserved current in the Dirac equation

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