- #1
pellman
- 684
- 5
The conserved current for a field [tex]\phi[/tex] obeying the Klein-Gordon equation is (neglecting operator ordering) proportional to [tex]i\phi^{\dag}\partial_\mu \phi-i\phi\partial_\mu \phi^{\dag}[/tex].
The conserved current for a four component field [tex]\psi[/tex] obeying the Dirac equation is [tex]\psi^{\dag}\gamma^0\gamma^\mu \psi[/tex], with no derivatives.
The Klein-Gordon conserved current cannot be interpreted as conservation of probability because the 0-component is not non-negative and so cannot be a pdf. On the other hand, it does make sense as a conservation of charge (or particles vs anti-particles).
The Dirac current does lend itself to an interpretation as conservation of probability.
So far, so good? I am simplifying for the sake of brevity, but let me know if I said anything wrong.
Now, what do we make of the fact that each of the individual components [tex]\psi_j[/tex], j=1,2,3,4 of the Dirac field separately obey the Klein-Gordon equation, and therefore also satisfy
[tex]\partial^\mu\{i\psi^{\dag}_j\partial_\mu \psi_j-i\psi_j\partial_\mu \psi^{\dag}_j\}=0[/tex] ?
That is, both currents are conserved--the Dirac current in terms of all four components, coupled, and the Klein-Gordon current, each component independently. I presume that the two conservations are not equivalent, since one involves derivatives and the other doesn't.
What is the physical significance of each of these conserved currents?
The conserved current for a four component field [tex]\psi[/tex] obeying the Dirac equation is [tex]\psi^{\dag}\gamma^0\gamma^\mu \psi[/tex], with no derivatives.
The Klein-Gordon conserved current cannot be interpreted as conservation of probability because the 0-component is not non-negative and so cannot be a pdf. On the other hand, it does make sense as a conservation of charge (or particles vs anti-particles).
The Dirac current does lend itself to an interpretation as conservation of probability.
So far, so good? I am simplifying for the sake of brevity, but let me know if I said anything wrong.
Now, what do we make of the fact that each of the individual components [tex]\psi_j[/tex], j=1,2,3,4 of the Dirac field separately obey the Klein-Gordon equation, and therefore also satisfy
[tex]\partial^\mu\{i\psi^{\dag}_j\partial_\mu \psi_j-i\psi_j\partial_\mu \psi^{\dag}_j\}=0[/tex] ?
That is, both currents are conserved--the Dirac current in terms of all four components, coupled, and the Klein-Gordon current, each component independently. I presume that the two conservations are not equivalent, since one involves derivatives and the other doesn't.
What is the physical significance of each of these conserved currents?
Last edited: