# Do we really need the Hilbert space for Quantum Mechanics?

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jonjacson
TL;DR Summary
In other words, What formalism would Quantum Mechanics use if the Hilbert space were not allowed?
Let's play this game, let's assume the infinite Hilbert Space, the operators and all the modern machinery introduced by Von Neuman were not allowed.

How would be the formalism?

Thanks

Gold Member
2022 Award
You cannot do physics without using the appropriate language, which is math. Your question doesn't make sense, because quantum mechanics uses the math of Hilbert spaces (in fact of rigged Hilbert spaces). As well you could ask me, what would Shakespeare use to write a play when the use of the English Langauge were not allowed (maybe German, but then it weren't Shakespeare ;-)).

atyy, dextercioby, Vanadium 50 and 3 others
Kolmo
There are other formalisms like the Path Integral, C*-algebra formalisms or GPT formalism, but they all follow from or imply the Hilbert space formalism. So there's no real sense in which one could imagine the Hilbert space formalism "not being allowed" as vanhees71 has above.

vanhees71, Demystifier, atyy and 3 others
Van Hees is right, of course.

If you want to avoid functional analytic complications, maybe this can be done by considering spin-only systems. (##\mathbb{C}^n## is still a Hilbert space, but at least it is not infinite dimensional.) The question of how useful this is, is probably better left to physicists.

vanhees71 and bhobba
Milsomonk
I'm in total agreement with VanHees and others on this.

One alternative approach that can be taken in QM (and many other areas of physics) is to formulate quantum mechanics in terms of the spacetime algebra (STA). The STA is a real geometric algebra, in this approach one does away with complex hilbert spaces and all operators/state vectors are represented by real elements of the STA.

However if you were to simply "switch off" the mathematics underpinning Hilbert spaces you'd likely switch off the required mathematics for geometric algebra and most probably all other alternative mathematical approaches to QM not to mention the rest of physics.

vanhees71
I think we need a modified version of Brandolini's Law, i.e., "The amount of energy needed to refute BS is an order of magnitude larger than to produce it."

The (polite) variation of that law applicable to this thread is "The amount of energy needed to explain why a question is idle is an order of magnitude larger than to ask it".

A more equitable proportioning of effort would be as follows. Instead of asking the vague open-ended question "what if Hilbert space were outlawed", the OP should construct an alternative theory of physics (sans Hilbert space, etc), compatible with all empirical results, and get it published in a reputable peer-reviewed journal. (Good luck with that.)

physika, mattt, russ_watters and 4 others
Mentor
Let's play this game
The game is not played by you throwing a question out to others and expecting them to do all the work. It is played, as @strangerep has correctly pointed out, by you constructing an alternative model and publishing it in a peer-reviewed paper.