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Oct
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comment Commutation relations for Dirac and Pauli electron
Dear Jeff: OK, I think I've figured out how to write down a position and momentum operator just on single particle states for the Dirac field. I'll try to actually calculate before too long and see what happens. But I didn't shut up just yet so as to tell you not to waste your time explaining anything yet.
Oct
17
comment Commutation relations for Dirac and Pauli electron
Maybe this is the point at which you're supposed to tell me 'shut up and calculate.'
Oct
17
comment Commutation relations for Dirac and Pauli electron
I'm quite worried about wasting your time with my remarks, but perhaps I will state my naive understanding anyway. Having quantized the Dirac field, even with a potential term, I can take a state in the corresponding representation space for the CAR with particle number one and let it evolve according to Schroedinger's equation w.r.t. the QFT Hamiltonian. But then, if I project the evolution to the single particle subspace again, I assumed I would get approximately Schroedinger time evolution in QM for an electron in a potential.
Oct
17
comment Commutation relations for Dirac and Pauli electron
Having thought about it, I still don't see that my confusion is so simple. (It's possibly even simpler.) I agree that the spacetime momentum will be obtained by integrating a density formed from the (local) spacetime positions and conjugate momenta. It still feels like the algebra of the latter will have a lot to say about the algebra of the momentum and position operators acting on the quantum state space. More precisely, it still doesn't appear completely silly to ask if one can derive CCR from CAR.
Oct
17
comment Commutation relations for Dirac and Pauli electron
Ah! Part of my silly confusion appears to be resolved by your simple remark. I see now that this was essentially Carlo Beenakker's comment as well. I'll think about it more and see if I'm still confused.
Oct
16
comment Commutation relations for Dirac and Pauli electron
Dear Jeff, Many thanks for the reference. I'll try to understand it. But I'm not sure my question depends on the existence of a position operator for quantum fields. It seems to me that $\phi(x)$ and $\pi(x)$ still give the field density and field momentum at spacetime points. I could (vaguely) argue that they should therefore determine algebraic relations between positions and momenta of various quantum mechanical particle states.
Oct
16
revised Commutation relations for Dirac and Pauli electron
added 14 characters in body
Oct
13
comment Commutation relations for Dirac and Pauli electron
Perhaps I should add that I'd be perfectly happy with a definitive answer that the two have nothing to do with each other.
Oct
13
comment Commutation relations for Dirac and Pauli electron
Abdelmalek Abdesselam: I agree with this as well. However, from a rather fundamentalist view, the representation of position and momentum you describe is a consequence of (CCR).
Oct
13
comment Commutation relations for Dirac and Pauli electron
Carlo Beenakker: Yes, I agree. But does this answer my question of deriving (CCR) for $x$ and $p$ from the (CAR) for the Dirac field?
Oct
13
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