Mathematically equivalent = physically equivalent?

[kvantti]

Something has been bothering me for a while and I hope to get many productive answers.

Heres the question:
if a physical theory is mathematically equivalent with another physical theory (eg. the different mathematical formulations of quantum mechanics), does it mean that the theory also physically equivalent with the other theory?

The obvious answer seems to be "yes", but then we seem to have another problem concerning reality itself:

All the interpretations of quantum mechanics are based on the same mathematics. But if they are mathematically equivalent and therefore physically equivalent, does it mean that they describe the same physical behaviour of reality?

In other words: could you consider it to be possible that all the interpretations of quantum mechanics are just different point of views of the same, "correct", interpretation?

 

[daso]

Something has been bothering me for a while and I hope to get many productive answers.

Heres the question:
if a physical theory is mathematically equivalent with another physical theory (eg. the different mathematical formulations of quantum mechanics), does it mean that the theory also physically equivalent with the other theory?

The obvious answer seems to be "yes", but then we seem to have another problem concerning reality itself:

All the interpretations of quantum mechanics are based on the same mathematics. But if they are mathematically equivalent and therefore physically equivalent, does it mean that they describe the same physical behaviour of reality?

In other words: could you consider it to be possible that all the interpretations of quantum mechanics are just different point of views of the same, "correct", interpretation?

My view is that a theory doesn't say anything about the reality. It's just used to make predictions. As long as the theory can predict some observed behaviour, it's fine. Hence, one can use eg. waves or particles to describe electrons.

 

[Office_Shredder]

All the interpretations of quantum mechanics are based on the same mathematics. But if they are mathematically equivalent and therefore physically equivalent, does it mean that they describe the same physical behaviour of reality?

No, because there are facts about quantum mechanics we don't yet know. It can be seen that different interpretations offer differing predictions on a number of questions, meaning they don't actually yield the same answer to all the questions

 

[selfAdjoint]

You have to be very careful about what you treat as a "formulation" of QM. For example "path integrals" is not a formulation; it's a heuristic calculation method.

But all the formulations I know of target the same QM. It is not the the case that they coincide mathematically therefore the theories they describe also concide, rather the other way around; the fact that they are all descriptions of the same theory guarantees that they will not disagree on the areas where they overlap.

 

[HallsofIvy]

There is no such thing as a "mathematical model" that exactly fits a physical theory. Mathematical models involve "ideal" object that are defined exactly. Physical theories involve measurements that are approximate. The best we can do is construct or choose a mathematical model that approximately fits the physical theory. It is quite possible that two different physical theories will approximately match the same mathematical model.

 

[LJM]

All the interpretations of quantum mechanics are based on the same mathematics. But if they are mathematically equivalent and therefore physically equivalent, does it mean that they describe the same physical behaviour of reality?

Reality is a whole, QM is an 'incomplete' description of reality. You've completely disregarded relativity.

In other words: could you consider it to be possible that all the interpretations of quantum mechanics are just different point of views of the same, "correct", interpretation?

Absolutely, QM throws out many implications, so it's a whittling down of questions that will correct this. However we're nowhere near this, but I'd look up 'topos' if it's any help.

 

[Mycroft7]

There is no such thing as a "mathematical model" that exactly fits a physical theory. Mathematical models involve "ideal" object that are defined exactly. Physical theories involve measurements that are approximate. The best we can do is construct or choose a mathematical model that approximately fits the physical theory. It is quite possible that two different physical theories will approximately match the same mathematical model.

Inasmuch as any mathematical description is going to be originally derived from observation, I agree with you. It is impossible to measure with infinite precision, therefore it is also impossible to be 100% certain that our mathematical descriptions are accurate. However, I don't think it means that they can't be accurate, only that their accuracy is unverifiable in an ultimate, metaphysical sense. I don't think it is necessarily true that there is some fundamental "imprecision of reality."

Also (unrelated), causality is unmathematical. To say, taking an example from an argument in classical mechanics, that gravity is some "real" force that pulls on objects (Newton), or that all objects contain within themselves the teleological power to exist in certain relationships (Leibniz) are mathematically indistinguishable. In other words, mathematics describes the "what" but not the "why." The latter is merely aid to understanding, it being helpful and in accordance with our intuitions to think in terms of causes, but ultimately is unmathematical and unscientific.