A Recent papers refuting real-valued quantum mechanics

[Paul Colby]

So, in a rare instance I actually read APS News, I came across “New Experiment Suggests Imaginary Numbers Must be Part of Real Quantum Physics.” In November 2022, Volume 31, Number 10.

Since complex numbers are isomorphic to a real 2x2 matrix algebra, I was confused how such a claim can be made. I tried several times to find the PF threads on this with no success. A relevant peer review reference is, Phys Rev Let 129, 140401, (2022).

Reading only the introduction of the letter, it appears there are measurable consequences in how composite systems are formed by tensor products of real versus complex Hilbert spaces. Anyway, if there are PF discussions, a link would be appreciated. If not any comments would be.

 

[gentzen]

Here is a thread where the difference between real versus complex Hilbert spaces and tensor products is important: https://www.physicsforums.com/threa...-consistent-histories-interpretation.1044960/

But otherwise, it is completely unrelated to that paper. You might have to skip the initial CH (and QM interpretations more general) bashing, probably roughly to Morbert's first post (#44).

 

[Paul Colby]

My expectation was this paper would be a hotly discussed topic here about. Apparently this isn't the case so it's worth articulating my question better. We have two allegedly different theories of quantum mechanics, one based on real-Hilbert spaces (RQM) and one based on complex-Hilbert spaces (CQM). Clearly, I'm having trouble seeing exactly where differences arise in the mathematics and hence, the physics.

In the experiment one prepares independent quantum systems, combines (entangles) them, then performs statistical analysis. Somewhere in the QM modeling of this chain there is a substantive difference between RQM and CQM. Where is this difference? Since the algebra of complex numbers is isomorphic to a real algebra of two by two matrices I'm having a hard time seeing this. I suspect but don't know, this difference arises in the entanglement step. One must model this step by writing down an interaction hamiltonian, $H_e$. I assume these are somehow -necessarily- different operators in the two theories? If so, how?

 

[gentzen]

Somewhere in the QM modeling of this chain there is a substantive difference between RQM and CQM. Where is this difference?

I guess it is related to Tsirelson's Problem:

The situation of two independent observers conducting measurements on a joint quantum system is usually modelled using a Hilbert space of tensor product form, each factor associated to one observer. Correspondingly, the operators describing the observables are then acting non-trivially only on one of the tensor factors. However, the same situation can also be modelled by just using one joint Hilbert space, and requiring that all operators associated to different observers commute, i.e. are jointly measurable without causing disturbance.

The difference is in how independent observers are modelled. Intuitively, one would expect that all three different ways (real tensor product, complex tensor product, commuting observables) are equivalent. But whether they really are was known as Tsirelson's Problem:

The problem of Tsirelson is now to decide the question whether all quantum correlation functions between two independent observers derived from commuting observables can also be expressed using observables defined on a Hilbert space of tensor product form. Tsirelson showed already that the distinction is irrelevant in the case that the ambient Hilbert space is of finite dimension.

Surprisingly, the recent computer science breakthrough MIP*=RE showed

Using a known connection, undecidability of the entangled value implies a negative answer to Tsirelson's problem: we show, by providing an explicit example, that the closure $C_{qa}$ of the set of quantum tensor product correlations is strictly included in the set $C_{qc}$ of quantum commuting correlations.

Seeing that modelling independence via real tensor product is not equivalent to modelling it via complex tensor product is much easier. So it is known now that all three different ways are totally different. Hence it made sense to test experimentally which one is used by nature. The "real tensor product" is now shot down. Experimentally distinguishing between "complex tensor product" and "commuting observables" will be much harder, I guess.

 

[gentzen]

https://www.physicsforums.com/threa...standard-formalism-of-quantum-theory.1011800/

 

[Paul Colby]

Okay, this echoes what's been bothering me. RQM is a different theory than QM, one that experiment shows is wrong. I can live with this.