Noether currents for a complex scalar field and a Fermion field

[Sandglass]

For a complex scalar field, the lagrangian density and the associated conserved current are given by:

$$ \mathcal{L} = \partial^\mu \psi^\dagger \partial_\mu \psi -m^2 \psi^\dagger \psi $$
$$J^{\mu} = i \left[ (\partial^\mu \psi^\dagger ) \psi - (\partial^\mu \psi ) \psi^\dagger \right] $$
whereas for a fermion field, results are:
$$ \mathcal{L} = \bar \psi ( i \gamma^\mu \partial_\mu -m ) \psi $$
$$J^{\mu} = \bar \psi \gamma^\mu \psi $$

In the former case, a derivative of $\psi$ appears in the Noether current and not in the latter. Apart from the technical aspect, does this difference tell us anything about the physics of these situations ?

 

[malawi_glenn]

What situation?

Well the equation of motion would be different.

 

[Sandglass]

My question is only a formal comparison between the two currents: the presence of a derivative term in one case and its absence in the other intrigues me (although its demonstration from the Lagrangian is straightforward). But perhaps there is no lesson to be learned.

 

[div_grad]

For a scalar field, the equations of motion involve second-order derivatives with respect to time & spatial variables - hence the corresponding current contains the "one-order-less" derivatives (i.e., first-order differentiation).

Observe, for example, that the (spatial) current corresponding to the ordinary Schrödinger equation contains gradient operators while the equation itself involves a Laplacian. The important difference is that the Schrödinger equation is of first-order in the time derivatives, which is not the case for the equations of motion of the scalar field.

The Dirac equation on the other hand is of first-order in both the time & spatial derivatives, hence the corresponding current does not involve any additional differentiation. So as regards to the question

(...) does this difference tell us anything about the physics of these situations ?

I would say that one can infer about the character of the relevant equations of motion by looking at the expressions for the associated conserved currents. This is of some importance, since the dynamics of the fields are encoded in the solutions of the corresponding equations of motion.

Perhaps looking at it this way can help build some intuition for working with/teaching the Lagrangian formalism of field theories, or deriving/justifying the form of the equations of motion "the other way around".