Mathematically equivalent = physically equivalent?

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In summary, the conversation discusses the relationship between different mathematical formulations of quantum mechanics and whether they are also physically equivalent. There is a debate on whether these formulations offer different points of view or ultimately describe the same physical behavior of reality. It is argued that a theory is only used to make predictions and does not necessarily reflect reality. It is also noted that mathematical models are approximate and may not exactly fit a physical theory. Additionally, the idea of causality is brought up and its relation to mathematics and scientific understanding is discussed.
  • #1
kvantti
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Something has been bothering me for a while and I hope to get many productive answers. :smile:

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?
 
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  • #2
kvantti said:
Something has been bothering me for a while and I hope to get many productive answers. :smile:

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.
 
  • #3
kvantti said:
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
 
  • #4
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.
 
  • #5
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.
 
  • #6
kvantti said:
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.

kvantti said:
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.
 
  • #7
HallsofIvy said:
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.
 

1. What does it mean for two quantities to be mathematically equivalent?

Two quantities are mathematically equivalent when they have the same numerical value or can be expressed using the same mathematical equation. This means that they are equal in terms of numbers or equations, but they may represent different physical properties or quantities.

2. How is mathematical equivalence related to physical equivalence?

Mathematical equivalence is related to physical equivalence in that it describes the relationship between two quantities in terms of numbers and equations. Physical equivalence, on the other hand, describes the relationship between two quantities in terms of their physical properties or effects. In some cases, two quantities may be mathematically equivalent but physically different, or vice versa.

3. Can two quantities be physically equivalent but not mathematically equivalent?

Yes, it is possible for two quantities to be physically equivalent but not mathematically equivalent. This means that they may have the same physical effects or properties, but they cannot be expressed using the same mathematical equations or have the same numerical value. This often occurs in complex systems where multiple factors contribute to a physical property.

4. How do we determine if two quantities are mathematically equivalent?

To determine if two quantities are mathematically equivalent, we can perform mathematical operations on both quantities and see if they result in the same value. If they do, then the two quantities are mathematically equivalent. We can also use mathematical equations to represent both quantities and see if they are identical or can be manipulated to become identical.

5. Can mathematical equivalence be used to simplify complex physical systems?

Yes, mathematical equivalence can be used to simplify complex physical systems by reducing them to simpler mathematical equations. This allows us to better understand the relationship between different physical quantities and how they affect each other. By identifying mathematical equivalence, we can also make predictions and calculations that would be difficult or impossible to do with complex physical systems.

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