Quantum spaces without classical counterparts?

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Discussion Overview

The discussion centers on the existence of finite-dimensional quantum spaces that lack classical counterparts in classical mechanics. Participants explore specific examples, particularly focusing on quantum systems like spin and their relationship to classical models.

Discussion Character

  • Exploratory
  • Debate/contested
  • Technical explanation

Main Points Raised

  • Some participants propose that finite-dimensional quantum spaces may not have classical counterparts, questioning the nature of examples like entanglement.
  • One participant mentions the spin of an electron in a magnetic field as a two-state quantum system that lacks a classical counterpart.
  • Another participant argues that classical Hamiltonian models for particles with spin exist, suggesting that their phase space can be described by Poisson manifolds, which upon quantization yield discrete spin parameters.
  • There is a discussion about the interpretation of the effects of a strong magnetic field on energy eigenvalues, with some participants agreeing on the classical treatment while others question the necessity of a classical interpretation for quantum effects.

Areas of Agreement / Disagreement

Participants express differing views on whether certain quantum systems can be interpreted classically. Some agree on the existence of classical models for spin, while others challenge the need for classical interpretations of quantum phenomena.

Contextual Notes

Participants acknowledge the complexity of transitioning from quantum to classical descriptions, particularly in the context of specific examples like spin in magnetic fields. There are unresolved questions regarding the interpretation of quantum effects and their classical analogs.

Ssnow
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Hi to all members! From few days that I am thinking on this question: there are finite-dimensional spaces that are quantum spaces (models for QM) but without the classical counterpart in classical mechanics ? For example I know that entanglement have not a ''clearly'' classical counterpart ... but I ask for an example in finite dimension ...
I was thinking about the sphere with spin but it is not a good example because we have a quantization (not a quantization in term of rigorous geometric quantization but in term of deformation quantization ...) and the sphere is a Kahler manifold so a model for classical mechanics ... I have the suspect that the aswer is no because with the projectivization we can always pass from quantum world to the classical world in the finite dimensional case ... but I am not sure...

Thanks,
Ssnow
 
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The spin of an electron in a magnetic field can be seen as a two-state quantum system without a classical counterpart. The energy eigenvalues of the spin up and spin down states differ more when the magnetic field is stronger.
 
@hilbert2 thanks for your example!
Ssnow
 
hilbert2 said:
The spin of an electron in a magnetic field can be seen as a two-state quantum system without a classical counterpart.
Classical Hamiltonian models for particles with spin are known for a long time. Their phase space is the Poisson manifold defined by suitable coadjoint orbits of the Poincare group. These are parameterized for positive mass by two continuous parameters, mass and spin.

Upon quantization, the Poisson manifolds turn into unitary representations, and the spin parameter becomes discrete - in a similar way as angular momentum for the rotation group in place of the Poincare group.
 
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In fact the example of a charge in the magnetic field has been also treated in Guillemin Sternberg in ''Symplectic thecniques in physics'' in a classical way, so I am agree with @A. Neumaier ... on the other side @hilbert2 has reason on the effects of a strong magnetic field on the eigenvalues, this must be interpreted classically, how ? Ssnow
 
Ssnow said:
on the other side @hilbert2 has reason on the effects of a strong magnetic field on the eigenvalues, this must be interpreted classically, how ? Ssnow
It is a quantum effect, so why should it have a classical interpretation?
 

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