Is quantum mechanics formulated from 1st principles?

[wmikewells]

I was surprised recently to learn that one of the reasons both Newton and Einstein were so revolutionary was that they performed a neat mathematical trick. For Newton, it was the mathematical derivation of Kepler's laws from Newton's laws of gravitation and motion. For Einstein, it was the derivation of the Lorentz transformations from the principle of relativity and the constant speed of light.

I know that Dirac was able to show that Heisenberg's and Schrodinger's formulations were equivalent to his own, but was Dirac's formulation (motion in Hilbert space) based on first principles like Newton's and Einstein's? Or are these formulations just mathematical that happen to match quantum phenomenon?

Either way, it was powerful stuff. I was just wondering what those first principles were. Doing a little research online, it seems like first principles for quantum mechanics are still a work in progress.

 

[hilbert2]

It's possible to predict the observed spectrum of blackbody radiation, and several other things, using quantum mechanics.

 

[wmikewells]

I probably didn't set up and ask my question properly.

Predictions are important to a theory, and I know that quantum theory is exquisitely accurate and all-encompassing in its predictive power. However, predictions are not the same thing as first principles. If I use Einstein as a model, the Lorentz transformations for time can predict that cosmic rays will reach further down into the atmosphere because their decay is slowed down. This is in contrast to the principle of relativity which says that physics laws are the same in all frames of reference. Predictions have more to do with experimental verification, and first principles are more like working assumptions upon which theory can be built.

I am guessing that a set of empirical phenomenon (such as black body radiation) can serve as an acid test for a theory, but an acid test doesn't seem like the same thing as a first principle. I suppose experimentation could be used to help establish a first principle (such as the constant speed of light). However, to call the speed of light a law of nature requires a little something more than experimentation.

I suppose an example of first principles for quantum mechanics is in Lee Smolin's 2015 talk in the link below. Not that I am qualified to remark on his first principles.

https://perimeterinstitute.ca/videos/quantum-mechanics-first-principles

I hope this all makes sense.

Let me know if you were trying to make the point that quantum theory is considered its own first principle. If so, I will have more questions.

 

[PeterDonis]

If I use Einstein as a model, the Lorentz transformations for time can predict that cosmic rays will reach further down into the atmosphere because their decay is slowed down. This is in contrast to the principle of relativity which says that physics laws are the same in all frames of reference.

Huh? There is no contradiction here: the behavior of muons in cosmic rays is perfectly in accord with the principle of relativity.

 

[wmikewells]

Agreed. The purpose of my question was not to find anything wrong or contradictory with any theory. I was just trying to show how I thought prediction (muon behavior) and first principles (principle of relativity) play a different role in theory.

Originally, I was trying to ask if quantum theory is unique from prior accomplishments in that it is not mathematically derived from first principles. I kind of get the impression that it is not, but I wanted to confirm.

I suppose some would maintain that first principles in quantum mechanics are not possible, but that is outside the scope of my question. I don't want to go down that rabbit hole.

I hope that clears it up.

 

[PeterDonis]

I was trying to ask if quantum theory is unique from prior accomplishments in that it is not mathematically derived from first principles.

"First principles" is too vague. You gave Newton as an example: he showed how Kepler's laws could be derived from his own laws. But why should Newton's laws count as "first principles"?

 

[bhobba]

"First principles" is too vague. You gave Newton as an example: he showed how Kepler's laws could be derived from his own laws. But why should Newton's laws count as "first principles"?

Agreed. Using the approach those primarily interested in physics usually encounter, Ballentine provides the best development I have seen, requiring only two axioms, with the second one (often called the Born Rule) at least partially derivable from the first by Gleason's Theorem.

But the mathematicians have got into the act as well and rigorous developments of QM exist - the one I know the most about and actually have read is:
https://www.amazon.com/dp/0387493859/?tag=pfamazon01-20

Here everything is mathematically rigorously defined, even exactly what is an observation - but physicists normally don't use such as formal approach so its mostly of interest to mathematicians.

And we have studies into the deeper meaning in the formalism eg:
https://arxiv.org/abs/1402.6562

Note - this is different from interpretations - it's examining the actual formal structure of QM in a different way than Varadarajan.

Personally I would recommend starting with Ballentine - QM - A Modern Development and you will see things like how Schrodinger's equation comes from a symmetry so intuitively obvious nobody would really doubt it - specifically the probabilities of an observation does not depend on how fast you are travelling. Formally its invoking the Principle of Relativity - but is very intuitive by itself. Post if its a bit advanced for your level and me and others can suggest books to lead into it.

Thanks
Bill

 

[DarMM]

I know coming up with a first principles formulation of Quantum Mechanics in the manner you mean is one of the major goals of research in quamtum foundations.

The major question could be seen as "What is the physical reason for the non-commutative algebra of events?"

There are some information theoretic attempts at a first principles derivation, for example Hardy's axioms or D'Ariano et al's axioms or the Auffèves-Grangier axioms.

They wouldn't be completely first principles as the authors themselves admit. Sort of half mathematical and half physical.

 

[A. Neumaier]

both Newton and Einstein were so revolutionary was that they performed a neat mathematical trick. For Newton, it was the mathematical derivation of Kepler's laws from Newton's laws of gravitation and motion. For Einstein, it was the derivation of the Lorentz transformations from the principle of relativity and the constant speed of light.

And for Wolfgang Pauli, one of the fathers of modern quantum mechanics, it was the mathematical derivation of the hydrogen spectrum from the Hamiltonian of the hydrogen atom. This showed that quantum mechanics were better first principles than classical mechanics.

I was just wondering what those first principles were.

For the first principles in quantum mechanics itself, see the discussion in Postulates for the formal core of quantum mechanics.

 

[wmikewells]

"First principles" is too vague. You gave Newton as an example: he showed how Kepler's laws could be derived from his own laws. But why should Newton's laws count as "first principles"?

I agree. To paraphrase a line from the movie Notting Hill (1999), "Today's first principles (newspapers) will be lining tomorrow's wastepaper bin."

For example, from what I have read, Newton's 2nd Law (F=ma) can be derived from symmetries. So, I guess that would make Newton's 2nd Law a second principle. It had a good run as a first principle, though.

Also, from a historical perspective, first principles tend to incorporate metaphysics, whether explicitly or implicitly. I can give examples if you want, but I think that would be outside the scope of this forum.

From my standpoint, it was very interesting too see how first principles tinged with metaphysics can be used in such rigorous and mathematical ways. I am trying to explore how that applies to quantum mechanics. Hence, the reason for me asking about the first principles of quantum mechanics.

I agree that the term "first principle" is ambiguous. I was using it more in a historical perspective, whereas, the terms postulate or axiom are probably more accurate in how they are used in several of the examples in this thread and elsewhere.

I do have my work cut out for me. The postulates of quantum mechanics are not as accessible as I thought.

 

[wmikewells]

Thank you all for suggesting places to look for first principles of quantum mechanics.