Dirac momentum density

1. Oct 31, 2004

da_willem

How can I find the momentum density in the dirac field? Can someone show me, tell me how or give me a reference?

Preferably not in relativistically covariant notation; I found this expression for the momentum density G, and want to know where it comes from:

$$\mathbf{G}=\frac{\hbar}{4i}[\psi^\dagger \nabla \psi + \psi^\dagger \mathbf{\alpha} (\mathbf{\alpha} \cdot \nabla)\psi]+hc$$

with hc the hermitian conjugate of teh expression. This can be written using the commutation relations for the matrices $\alpha_k$:

$$\mathbf{G}=\frac{\hbar}{2i}[\psi^\dagger \nabla \psi -(\nabla \psi ^\dagger)\psi]+\frac{\hbar}{4}\nabla \times (\psi^\dagger \mathbf{\sigma} \psi)$$

Last edited: Oct 31, 2004
2. Nov 1, 2004

da_willem

Nobody has ever seen this before?

3. Nov 3, 2004

dextercioby

U use the real Lagrangian density of the DIrac field and the expression for the momentum density given by Noether's theorem.