When are isomorphic Hilbert spaces physically different?

In summary, the classical limit of a Hilbert space is a concept that only applies to linear operators, not states. In quantum mechanics, a Hilbert space is defined as a Hilbert space with a distinguished unitary representation of a given Lie algebra of preferred observables. Two Hilbert spaces are considered physically different if their representations are non-isomorphic. The classical limit is achieved when the representation depends on Planck's constant in a way that the commutator of operators tends to a finite limit, representing a Poisson bracket. The Wigner phase space formulation of classical mechanics leads to the Koopman-von Neumann formulation, where linear operators are associated with dynamical variables. However, the relation between classical and quantum Hilbert spaces
  • #1
A. Neumaier
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martinbn said:
What is a classical limit of a Hilbert space? And these Hilbert spaces, for one or two or many particles, are all isomorphic.

In quantum mechanics, a Hilbert space always means (in mathematical terms) a Hilbert space together with a distinguished irreducible unitary representation of a given Lie algebra of preferred observables on a common dense domain. Two Hilbert spaces are considered (physically) different if this representation is different (in the sense of non-isomorphic). The Lie algebra defines the kinematics of the system of interest. The semidirect product of ##(2dN+1)##-dimensional Heisenberg algebra with ##N## copies of ##so(d)## has a unique irreducible unitary representation, which defines the Hilbert space of ##N## particles in ##d##-dimensional Euclidean space.

Hilbert spaces don't have a classical limit. The latter is restricted to linear operators, which may have one. To have a classical limit, the above representation must depend on Planck's constant hbar in such a way that ##i[A,B]/\hbar## tends (at least for ##A## and ##B## in the Lie algebra of preferred observables) to a finite limit ##\{B,A\}##, which represents a Poisson bracket.
 
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  • #2
A. Neumaier said:
Hilbert spaces don't have a classical limit. The latter is restricted to linear operators, which may have one. To have a classical limit, the above representation must depend on Planck's constant hbar in such a way that i[A,B]/hbar tends (at least for A and B in the Lie algebra of preferred observables) to a finite limit {B,A}, which represents a Poisson bracket.

Uhm, I suppose you are right, but I'm not so sure.

The wigner phase space formulation of classical mechanics goes in the classical limits to the Koopman- von neumann formulation of classical mechanics, which is a Hilbert space formalism for classical mechanics where there are linear operators associated to the dynamical variables.

I don't know what could be the relation between the quantum Hilbert space and the classical Hilbert space though.
 
  • #3
“Quantum phenomena do not occur in a Hilbert space. They occur in a laboratory.”
Asher Peres, Quantum Theory: Concepts and Methods

In my own words, isomorphic Hilbert spaces may not be equivalent physically. Or even more directly, quantum physics is not only about states in Hilbert spaces.
 
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  • #4
andresB said:
The wigner phase space formulation of classical mechanics goes in the classical limits to the Koopman- von neumann formulation of classical mechanics, which is a Hilbert space formalism for classical mechanics where there are linear operators associated to the dynamical variables.

I don't know what could be the relation between the quantum Hilbert space and the classical Hilbert space though.

I described the Heisenberg picture, while the Wigner representation describes the Schroedinger picture. There the Hilbert space has no classical limit either. Instead, the classical limit again happens on the operator level. The states are the density matrices, and the classical limit that takes ##\hbar## to zero (essentially corresponding to infinitely fast decoherence) replaces these by diagonal operators. These are essentially the density functions of classical stochastic processes, corresponding to the Koopman formulation.
 
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FAQ: When are isomorphic Hilbert spaces physically different?

1. What is an isomorphic Hilbert space?

An isomorphic Hilbert space is a mathematical concept that refers to two Hilbert spaces that have the same structure and can be transformed into each other through a bijective linear isometric mapping. In other words, they have the same dimension and preserve the inner product and norm of vectors.

2. How can two isomorphic Hilbert spaces be physically different?

While two isomorphic Hilbert spaces may have the same mathematical structure, they can still have different physical interpretations. This means that while the vectors and operations in the two spaces may be the same, they may have different physical meanings or applications.

3. What is the significance of isomorphic Hilbert spaces in physics?

In physics, isomorphic Hilbert spaces are important because they allow us to study and analyze physical phenomena in different ways. By transforming a problem into a different isomorphic Hilbert space, we can gain new insights and perspectives that may not have been apparent in the original space.

4. Can isomorphic Hilbert spaces be used to describe the same physical system?

Yes, isomorphic Hilbert spaces can be used to describe the same physical system. While they may have different physical interpretations, they still represent the same underlying mathematical structure and therefore can be used to describe the same physical phenomenon.

5. Are there any limitations to using isomorphic Hilbert spaces in physics?

While isomorphic Hilbert spaces can be useful in providing new perspectives and insights, they are not always applicable in all physical situations. In some cases, the physical interpretation of a problem may be lost when transforming it into a different isomorphic Hilbert space. Therefore, it is important for scientists to carefully consider the physical implications of using isomorphic Hilbert spaces in their research.

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