# QM derived or axiomatic ?

1. Jul 28, 2007

### Rade

In another thread on the forum, a member (Doctordick) made the following statement:

....What you should take note of is the fact that modern quantum mechanics, as seen by the academy (the religious authority of modern physics) is not derived from fundamental concepts; but is rather put forth in axiomatic form and that derivation of the relationships from more fundamental analysis is really of no interest to them...

I am interested in knowing if this is in fact the modern view of those that work in quantum mechanics.

2. Jul 28, 2007

### jostpuur

That sounds a strange claim. Aren't these axioms the same thing as the fundamental concepts? If they have different meaning, perhaps their meaning should be explained in more detail then by the one who is claiming this.

I my opinion the idea that we have some coordinate set $\{q\}$, whose members represent some classical states, and the wave mapping $\Psi(t,q):\mathbb{R}\times\{q\}\to\mathbb{C}$, whose time evolution is defined from a classical action by

$$\Psi(t+T,q) = \textrm{sum}_{q'} e^{iS(q'\to q)/\hbar} \Psi(t,q') + O(T^2)$$

is the most fundamental idea behind the quantum mechanics. Or I mean, the most fundamental that I have encountered so far. I agree with the original claim to the extent that this is often not emphasized in the education of QM. Usually people are interested to know about QM precisely what is needed to know so that we could calculate stuff.

Now when I actually took a glance on the Doctordick's posts, I think this has nothing to do with his ideas. I'll put it this way: "Physicists are usually not interested in philosophy, they are interested in calculating." That is something that many will probably agree with, and if Doctordick is criticizing it, it is understandable, although I'm not convinced that he himself would be improving anything.

Last edited: Jul 28, 2007
3. Jul 28, 2007

### jostpuur

No! In fact, for example the axiom that says the we can make a substitution

$$\boldsymbol{p} \mapsto -i\hbar\nabla$$

is way too abstract to be considered something fundamental. So on this matter I agree fully with Doctordick.

Rade, as you can see, now I'm already too much guessing what Doctordick has meant with this, and it results my responses getting confusing. Now I'll shut up, and continue only when the original claims have been made clearer.

4. Jul 28, 2007

### country boy

This sounds a bit overly angry. Quantum Mechanics was originally worked out (derived?) to be self-consistent, to agree with Classical Mechanics, and to produce correct results. Once the rules were established they could be applied without thinking about where they came from. But every physicist I know is interested in the possibility that QM and other aspects of modern physics might be derivable from more fundamental, as yet unrecognized, principles.

5. Jul 28, 2007

### Hurkyl

Staff Emeritus
Well, what do you mean by "fundamental"? If you mean those things that provide a starting point for further study -- e.g. axioms -- then being fundamental is not an inherent property of anything; it depends on your choice of presentation.

I'm not sure upon what grounds you label this particular statement as "abstract"; it's an explicit, constructive definition of the momentum operator! You can't get much more concrete than that. :tongue: While I certainly agree that there is merit to defining the momentum operator in terms of a list of properties that it should have, you run the risk of losing some of your audience if they have to do a lot of theoretical work before they can actually compute anything.

6. Jul 28, 2007

### Llewlyn

Please note that all physics is put in axiomatic form. Those principles are the fundamental concepts because we could analyze any phenomena just starting from it.
If you are thinking at something specific as "fundamental", explain what is.

Ll.

7. Jul 28, 2007

### jostpuur

The operator approach is intended to give humans tools to calculate. Perhaps it is fundamental in such mathematical sense. The path integral approach instead looks like fundamental in a sense, that nature could be working by some principles very similar to these path integrals.

At least my instinct tells me so.

8. Jul 28, 2007

### meopemuk

Google for "quantum logic", "Mackey axioms", "Piron theorem" and you will find a beautiful axiomatic approach from which all rules of quantum mechanics would follow, including the rule $\mathbf{p} \to -i \hbar \nabla$

Eugene.

9. Jul 28, 2007

### country boy

Thanks for the Google directions. Mackey's ideas are certainly valid and have apparently led to a search for the logical underpinning of QM. If the fundamental principles were discovered, however, it would be nice if they produced an understanding beyond just explaining QM. Something that included gravity, for instance, and maybe things we hadn't even thought about.

10. Jul 28, 2007

### meopemuk

As far as I know, Birkhoff - von Neumann - Mackey - Piron quantum logic didn't lead to any deeper insights. It simply provided a transparent and simple axiomatic foundation for the laws of quantum mechanics. This is a major thing, by itself!

It is very important to know that quantum mechanics lies on a very solid foundation, and if one decided to modify QM rules one should be aware that he would violate some fundamental rules of logic when doing that. For example, there are numerous proposals (e.g., Penrose) to change QM rules to make them "compatible" with general relativity. Quantum logic is a big roadblock on this path.

Eugene.

11. Jul 29, 2007

### Fra

I agree with this.

My major incentive to find a better line of reasoning behind QM, is so that the methodology can be more readily extrapolate to the new, harder, more general problems in reality.

If you do not have the next step in mind, the driving measure will be different. To rethink a working model, just for the fun of it, with no intention that this rethinking can provide a better stance is not my motivation.

Learning how to apply and do calculations, is not the same as finding a deeper understanding that give you a more fit stance to attack the unknown, and maybe even extend the theory to more general cases and do away with idealizations.

My impression is that different peoples views often boils down to different views of realism, and different views of the "scientific method". The way I was presented quantum mechanics at the university when I took those course was certainly enough to teach me a tool, but they didn't convince me with their handwaving argumentaion. It's clearly alot of guessing and ad hoc reasoning, that is, formalized by axiomatising, and ultimately justified by noting that no matter how we arrived at this, it seems to work. But there are still fuzzy ends all over the place IMO.

/Fredrik

12. Jul 29, 2007

### Hurkyl

Staff Emeritus
Of course! That's essentially the definition of science: the ultimate arbiter of truth is empirical evidence. Pure reason, no matter how clever, cannot justify a scientific theory.

Last edited: Jul 29, 2007
13. Jul 29, 2007

### Mike2

That seems to suggest that at some level reality (i.e. QM) is not reasonable. I think that the basic premise of the scientific method is that everything is reasonable at EVERY level of existence. We just have to find out how it all stands to reason.

Whatever our theories about reality might be, they ultimately end up making statements about reality that we either decide are true or false. Therefore, the algebra of true and false, namely propositional logic, is relavant at every level of existence.

If science is only based on the contingencies of what we can measure, then we will always be asking how those contingencies are possible. Questions about reality will not end until we can show that existence is derivable from pure reason. The questions will stop when we answer that it is required by reason.

14. Jul 29, 2007

### Anonym

It is not clear what you/ Doctordick/ jostpuur/ country boy/ Hurkyl/ Llewlyn/ meopemuk/ Fra/ Mike2 mean using term axiomatic. As usual each one tries to fit it according to his personal knowledge/understanding/experience/purpose. For example, one well known physicist wrote:

“Our streams of thought are constantly stimulated and redirected by interactions with the external world, and by internal drives, in ways that don’t seem to resemble at all the unfolding of mathematical algorithms.

Another argument derives from our experience with modern digital computers. For these are, in a sense, ideal mathematicians. They follow precise rules (axioms) with relentlessness, speed, and freedom from error that far surpasses what is possible for humans.”

No doubt that the axioms are the precise rules. However, I consider them as constraints that allow any structure that do not contradict these constraints.

So, the point is what we agree to use as a standard reference?

I suggest: J.S.Hadamard “Elementary Geometry” (Lecons de Geometrie Elementaire).

Regards, Dany.

P.S. By the way my quotation was selective and manipulative. The writer finally stated the opposite.

15. Jul 30, 2007

### country boy

Thank you for clarifying/elucidating/defining/explaining the meaning of axiomatic/fundamental/basic/essential.

This is getting to be more of a philosophical thread, but here's one more thought. Scientific understanding involves an interplay between evidence and context (experiment and theory). A fact has no meaning without a framework that attempts to link it with other facts. Theory has no validity if it doesn't fit the evidence. Experiment drives theory and theory drives experiment. They develop together.

In the end, however, evidence is the arbiter. If the experimental result is valid and the theory is not consistent with it, then the theory must be changed. We are allowed to discard theory, but not experiment.

16. Jul 30, 2007

### Anonym

If you treat my post so, I was not successful. I do not consider fundamental /basic/ essential close synonyms of axioms.

Then why discussion should be in QP of PF?
And you aren’t tired of philosophy?

Regards, Dany.

17. Jul 30, 2007

### Fra

I don't suggest that "pure reason" can replace real interactions and real observations.

However, human reason is bound to be a physical process, and can be thought of as self observation/interaction, and my self-interaction we can increase the effiency in our scientific method by improving it, and thus giving us better theories in shorter time. After all the human brain is one of the more fascinating information processing structures in nature IMO, so I still wouldn't underestimate reason, because it is in an indirect sense also "observations".

My own subjective conclusion has drawn my attention to the method itself, and the dynamics of evolving theories, and how this is related to the communication between observer and environment.

From this viewpoint, there is alot to wish from most standard formulations. The lacking part is the line of reasoning that you'd expect if a more intelligent evolutionary strategy was used. Still, everything may have started with "random" trials, but I think it's sooner or later time for a review of our methods, for the benefit of increased fitness. This is my vision at least.

/Fredrik

18. Jul 30, 2007

### Fra

Dany, I know I come out as fuzzy and philosophical, and I don't get tired of philosophy, but I often get tired of "talking about it" because there is rarely any progress. These things are often best solved by self-communication.

Those who think that physics is _just_ math, is from another school of thinking than me, so I can't comment on what they are doing. To me I try to understand reality. Not mathematical models. In this quest mathematics is a significant and almsot dominating component but it's not, IMO, the only component. It also contains realy fuzzy problems, that requires fuzzy thinking at times. That's my experience at least.

/Fredrik

19. Jul 30, 2007

### Rade

To anyone who has responded to this thread I started--if you have any interest in dialog with Doctordick about his comment that I presented in the OP--please see and respond at the end of this thread link (post #480):

https://www.physicsforums.com/showthread.php?p=1389412#post1389412

My motivation here is to link professional physicists/mathematicians with out-of-box and very creative philosophic ideas being presented by Doctordick.

As he often closes, have fun.

20. Jul 31, 2007

### country boy

Whoops!, I picked this thread up on the QP Forum and didn't notice that it had been moved.

Philosophy is fine, but it is best when it leads to a practical outcome. In this case, we should hope that there is a way to deal with axiomatic physical theory, but still be able to look beyond the axioms for more fundamental principles. The method for doing this probably can't be procedural, since that would be too confining. But are there at least guidelines that help us make progress past prevailing understanding?

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