Let mathematical quantum theories be theories of Hilbert-space rays together with probability / amplitude / unitary evolution / Hermitian operator machineries. Observations essentially correspond to orthogonal bases, maybe with specific weights attached to certain eigenvectors - but the core structure is the orthogonal basis. Two different orthogonal bases transform into each other by a unitary transformation, so in a certain sense all theories are isomorphic to each other. The only thing making them "different", of course, is dimension. We can formulate this even in a more rigorous sense, such as every infinite dimensional, separable Hilbert space is isometrically isomorphic to $l^2$.

In physical quantum theories this is completely different, which I want to illustrate in two scenarios.

In the (1) infinite dimensional case it is clear that the generator of the evolution is given by the Hamiltonian and it is clear how the Hamiltonian has to be formulated in position and in momentum base. Position and momentum bases, although they can be transformed into each other by a unitary transformation, physically are different: Position and momentum can clearly be distinguished by experiments and carry a different semantics.

In the (2) finite dimensional case we have a more subtle but (maybe??) similar situation. Mathematically we can consider a two dimensional Hilbert space generated by two abstract vectors $\vec{u}$ and $\vec{d}$ (for up and down) and we can connect the notion of state with a three dimensional unit (Pauli) vector. We could also also use any other basis, $\vec{r}$ and $\vec{l}$ (for left and right), the difference just is a unitary isomorphism and the respective base transformation has to be applied to the generator of the unitary evolution as well as to the observable and everything is just the same. But as soon as we do physics we turn on some evolution, for example a magnetic field, which couples to the spin and some observation, for example spin measurement in a specific direction. We still are free with regard to choices of the coordinate system in our three dimensional real space (or our two dimensional complex projective space). But there is some physical property which ties the magnetic field vector to the chosen direction of spin measurement. The argument of the mathematician that, essentially all qubit systems are unitarily equivalent, is not helpful...

I am not sure if examples (1) and (2) are strictly comparable.

When I am asking What generates physical meaning? I am interested in understanding which elements of physics remove that possibility of just treating all Hilbert space models of the same dimension on an equal footing.

I am searching for some deeper and formal answer on the boundary between math and physics which goes beyond the obvious answers: "The experiment", "Shut up and calculate" or "This is not a philosophical Q&A site".

  • $\begingroup$ You are ignoring what physics is. Maybe this answer of mine might help physics.stackexchange.com/questions/44196/… $\endgroup$
    – anna v
    Aug 19, 2019 at 11:50
  • $\begingroup$ I enjoyed reading the cited thread. I hint physics is empirical, not rigorous. Still, my question remains. We can do better :-) As to (1) we get hints from gravity. As to (2) there are (yet unsuccessful) attempts towards models of space-time emerging as averaged-out quantum phenomenon. I am reluctant to discard all this as difference in the methods of two disciplines. Similar questions had been driving developments in both disciplines. (a) Can non-euclidean geometries exist. (b) Putnam: Is logic empirical. (c) Poincare: Is space conventional. The questioning is valid, not ignoring & not easy $\endgroup$ Aug 19, 2019 at 12:21
  • $\begingroup$ Probably a reference which should be mentioned here as well is physics.stackexchange.com/q/407116/139287 $\endgroup$ Aug 19, 2019 at 12:25
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    $\begingroup$ It boils down between the platonists, those who believe that mathematics creates reality/nature, and the realists , who believe nature exists and mathematics is used to model it. It was that way with the pythagorians more than two thousand years ago, betting on the music of the spheres, and it has still a great number of followers among theoretical physicists. At the moment mathematics is used to model reality. If in the future, string theories manage to describe all known observations, I keep an open mind to become a platonist. $\endgroup$
    – anna v
    Aug 19, 2019 at 12:54
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    $\begingroup$ "...remove that possibility of just treating all Hilbert space models of the same dimension on an equal footing." To specify a model (like QED or non-relativistic 1-particle QM), we need to specify its observables. This means specifying (1) which operators on the Hilbert space qualify as observables, and (2) the physical significance of those observables. In quantum field theory, observables are explicitly associated with regions of spacetime. Together with their algebraic relationships, this more or less dictates their physical significance (more so in more complete models). $\endgroup$ Aug 19, 2019 at 16:23


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