Hermitian conjugation and conserved current in the Dirac equation

In summary, the Dirac equation in the ordinary form uses a and \beta matrices which are hermitian, denoted by a^\dag and \beta^\dag respectively. In order to prove probability current conservation, a conjugate equation must be derived from the Dirac equation. However, the hermitian conjugation operation affects Dirac matrices and their corresponding columns and rows, but not differential operators. This leads to the need for two different definitions of hermitian conjugation, one for matrices and one for differential operators. The first one is used to derive the continuity equation, while the second one is needed to ensure hermiticity of the Dirac Hamiltonian. This is different from the case of the nonrelativistic
  • #1
gheremond
7
0
Consider the Dirac equation in the ordinary form in terms of [tex] a [/tex] and [tex]\beta [/tex] matrices
[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.
 
Physics news on Phys.org
  • #2
You are mixing up two different notions of hermitian conjugation. The first acts in the space of components of the Dirac wave function; it takes the column vector [itex]\psi_i(x)[/itex] and turns it into the row vector [itex]\psi^*_i(x)[/itex], and replaces the Dirac matrices with their hermitian conjugates. Differential operators such as [itex]\nabla[/itex] are unaffected by this conjugation. The analog for the nonrelativistic Schrodinger equation is just complex conjugation.

The second sort is hermtian conjugation in the space of square-integrable functions. In this space, [itex]\nabla[/itex] is an antihermitian operator: [itex]\nabla^\dagger=-\nabla[/itex].

It is the first sort that you want to use to derive the continuity equation.
 
  • #3
Thanks for the reply! The parallel with complex conjugation in the Schroedinger equation indeed makes some sense. Yet it is not fully formally equivalent. It would be, provided you could define a current and density for the Dirac equation that is valid "component-wise", not as a sum of terms involving all four components. This summation is in fact the same as forming a partial inner product in the Dirac Hilbert space, summing over the spinor components only. This makes the comparison with the Schroedinger case less obvious (since there you did not have to interfere at all with manipulations comparable to taking an inner product).
 

FAQ: Hermitian conjugation and conserved current in the Dirac equation

1. What is Hermitian conjugation in the context of the Dirac equation?

Hermitian conjugation, also known as adjoint or conjugate transpose, is an operation that involves taking the complex conjugate of a matrix or operator and then transposing it. In the Dirac equation, Hermitian conjugation is used to find the adjoint operator of the Dirac Hamiltonian, which is important for determining the conserved current.

2. What is the significance of Hermitian conjugation in the Dirac equation?

Hermitian conjugation plays a crucial role in the Dirac equation as it allows us to define a conserved current, which is a quantity that remains constant over time. This current is important in understanding the behavior and properties of particles described by the Dirac equation.

3. How is Hermitian conjugation related to the concept of symmetry?

In physics, symmetries refer to invariances or transformations that leave a physical system unchanged. Hermitian conjugation is closely related to the concept of symmetry, as it allows us to define a conserved current that is invariant under certain symmetry transformations. This is known as Noether's theorem and has important implications in the study of fundamental particles.

4. Can you provide an example of Hermitian conjugation in the Dirac equation?

Sure, let's consider the Dirac Hamiltonian operator, which is Hermitian and given by H = cα · p + mc^2β. The Hermitian conjugate of this operator is given by H† = cα · p - mc^2β. This Hermitian conjugate operator is used to define the conserved current in the Dirac equation.

5. How does Hermitian conjugation relate to the concept of charge conservation?

Hermitian conjugation is closely related to the concept of charge conservation. In the Dirac equation, the conserved current defined using Hermitian conjugation is closely related to the charge density and charge current. This allows us to understand how charges are conserved in physical systems described by the Dirac equation.

Similar threads

Back
Top