Pointer Momentum of a Measuring Apparatus

[SheikYerbouti]

I am writing a paper on the problem of measurement and I am stuck on what the pointer momentum corresponds to physically. The coupling of the pointer momentum with the observable being measured is used to define the interaction Hamiltonian between the measuring apparatus and the system being measured. I understand that the observable conjugate to the pointer momentum is the pointer coordinate/read-out and that the pointer momentum generates the translations in for this coordinate. Is this all that there is to the pointer momentum, or does it correspond to something else as well?

 

[bhobba]

I am writing a paper on the problem of measurement and I am stuck on what the pointer momentum corresponds to physically. The coupling of the pointer momentum with the observable being measured is used to define the interaction Hamiltonian between the measuring apparatus and the system being measured.

Yes - that's how generalised measurements are arrived at:
http://www.quantum.umb.edu/Jacobs/QMT/QMT_Chapter1.pdf

I understand that the observable conjugate to the pointer momentum is the pointer coordinate/read-out and that the pointer momentum generates the translations in for this coordinate. Is this all that there is to the pointer momentum, or does it correspond to something else as well?

Well that's all there really is to the momentum operator - but what exactly you mean by 'pointer momentum' in this context I am not sure.

Thanks
Bill

 

[SheikYerbouti]

By the pointer momentum, I am referring to the observable of the measuring apparatus that is coupled to the observable that is being measured in the system. I don't know if that helped at all, so here is a link that explains the setup: http://www.theory.caltech.edu/people/preskill/ph229/notes/chap3.pdf. It goes into the model on the second page and I am referring to the P operator. When we discuss the momentum of a particle, there is more to it than simply being the generator of coordinate translations. I was just wondering if there was something similar going on for the pointer momentum in that there was a deeper, manifestly physical interpretation or if it simply is a generator of translation.

 

[bhobba]

When we discuss the momentum of a particle, there is more to it than simply being the generator of coordinate translations.

Not so sure about that.

See chapter 3 - Ballentine - Quantum Mechanics - A Modern Development.

It's constrained by the fact probabilities are frame invariant.

Symmetry is its fundamental basis - symmetries implied by probability invariance in QM - symmetries implied by Lagrangian invariance in classical mechanics.

IMHO there is nothing deeper going on - its the magic and mystery of symmetry in physics.

Symmetry tells us what the momentum operator is. But its relation to the measurent problem is another matter - as it is for any observable in QM.

I know that paper you linked to and it is good.

However for understanding the measurement problem in light of modern developments in decoherence I think the following is better:
http://philsci-archive.pitt.edu/5439/1/Decoherence_Essay_arXiv_version.pdf

It emphasises the crucial difference between improper and proper mixed states.

An even better reference is Decoherence and the Quantum to Classical Transition by Schlosshauer
https://www.amazon.com/dp/3540357734/?tag=pfamazon01-20

Its my go-to book on such things.

Thanks
Bill