Quantum spaces without classical counterparts?

In summary: The classical counterpart would be the spin of the electron in a classical magnetic field, which does not experience this quantization. In summary, there are finite-dimensional spaces that are quantum spaces but do not have a classical counterpart in classical mechanics. One example is the spin of an electron in a magnetic field, which can be seen as a two-state quantum system without a classical counterpart. However, classical Hamiltonian models for particles with spin do exist, but upon quantization, the spin parameter becomes discrete.
  • #1
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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  • #2
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.
 
  • #3
@hilbert2 thanks for your example!
Ssnow
 
  • #4
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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  • #5
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
 
  • #6
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?
 

1. What are quantum spaces without classical counterparts?

Quantum spaces without classical counterparts are mathematical spaces that cannot be described by classical physics. They are characterized by the principles of quantum mechanics, which govern the behavior of particles at the subatomic level.

2. How are quantum spaces without classical counterparts different from classical spaces?

Quantum spaces without classical counterparts exhibit behaviors that are not possible in classical spaces. This includes phenomena such as quantum entanglement, superposition, and uncertainty. These spaces also have discrete energy levels and probabilistic outcomes, unlike classical spaces which have continuous energy and deterministic outcomes.

3. What are some examples of quantum spaces without classical counterparts?

Some examples of quantum spaces without classical counterparts include the quantum harmonic oscillator, the hydrogen atom, and the quantum spin system. These systems cannot be fully described using classical physics and require the use of quantum mechanics to understand their behavior.

4. What is the significance of studying quantum spaces without classical counterparts?

Studying quantum spaces without classical counterparts is crucial for understanding the fundamental principles of quantum mechanics and for developing new technologies such as quantum computing and quantum communication. It also allows us to explore the boundaries of our current understanding of the universe and potentially discover new physical phenomena.

5. Are quantum spaces without classical counterparts purely theoretical or do they have practical applications?

While the study of quantum spaces without classical counterparts is primarily theoretical, it has already led to practical applications in various fields such as cryptography, quantum information processing, and quantum sensing. As our understanding of these spaces continues to improve, we can expect to see even more practical applications in the future.

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