I recently came across Solèr's theorem which seems to state that for quantum mechanics to make sense, we have to use a continuous vector space.

But how can that possibly be?

I believe we can, in principle, simulate any quantum mechanical system with the discrete numbers in our classical computers to predict any measurement which we could make. This seems like an obvious counter-example. Instead of the perfect real valued math, quantum mechanics seems to work fine with just the discrete set of values a computer memory can hold.

Is that not the case?

Does Soler somehow prove that using rational numbers instead of real valued components are incapable of describing quantum phenomena?

Does this have any bearing on people attempting discrete spacetime approaches to quantum gravity?

This theorem sounds amazing, but perplexing. I wonder why it isn't mentioned in intro quantum classes along with Bell's theorem and invalidating local hidden variable theories.

  • $\begingroup$ Related: physics.stackexchange.com/q/8062/2451 , physics.stackexchange.com/q/32422/2451 and links therein. $\endgroup$ – Qmechanic Mar 4 '18 at 10:39
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    $\begingroup$ I'm not familiar with Solèr's theorem, but I don't see any contradiction here. You cannot exactly simulate a quantum mechanical system using only discrete numbers, because discrete numbers translate to only finite digit precision in the simulation. Note that this has nothing to do with QM predicting discrete things, such as discrete energy levels and such. Even in those cases you still need continuous numbers to properly characterise the probability amplitudes. $\endgroup$ – glS Mar 5 '18 at 18:38

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