Dirac momentum density

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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:

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

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

[tex]\mathbf{G}=\frac{\hbar}{2i}[\psi^\dagger \nabla \psi -(\nabla \psi ^\dagger)\psi]+\frac{\hbar}{4}\nabla \times (\psi^\dagger \mathbf{\sigma} \psi)[/tex]
 
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  • #2
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Nobody has ever seen this before?
 
  • #3
dextercioby
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da_willem said:
How can I find the momentum density in the dirac field? Can someone show me, tell me how or give me a reference?
U use the real Lagrangian density of the DIrac field and the expression for the momentum density given by Noether's theorem.
 

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