I Confusion About the Born Rule Postulation

Summary
Can the Born rule of quantum mechanics be derived from more fundamental principles?
Hey there!

I’ve been thinking about the Born rule recently and whether it can be derived from the other postulates from QM.

I’ve done a bunch of google searching, across PF, stackexchange and the arxiv, but most of it has felt a little opaque, particularly on whether anything has been ‘accepted’ as a derivation yet.

I decided to just try myself: using the assumptions that probability needs to be conserved under unitary transformations, that the rule holds for all quantum systems, and that the probabilities are functions of the respective amplitudes, I think I’ve showed that the Born rule must hold.

Do those assumptions go beyond the other postulates of QM, or are they perhaps in any sense less ‘fundamental’ than the Born rule itself?

Thanks in advance!
 

vanhees71

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That's very interesting. As far as I know so far nobody has been able to derive Born's rule from the other postulates without introducing any other assumption(s). So it would make and interesting research paper!

The issue is very nicely discussed in Weinberg's textbook on quantum mechanics:

S. Weinberg, Lectures on Quantum Mechanics, Cambridge University Press
 
Well, I'll write up about it properly and talk to a tutor or something, always a decent chance I've make a mistake or assumed something implicitly somewhere. I just so happen to have that book, so I'll have a read :)
 

Demystifier

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I’ve been thinking about the Born rule recently and whether it can be derived from the other postulates from QM.
The Born rule in the arbitrary basis can be derived from the Born rule in the position basis. See the paper linked in my signature below.
 

atyy

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As @vanhees71 mentions above, there is no known derivation of the Born rule "purely" from the other postulates of QM.

However, if one supplements the postulates or uses other postulates, the Born rule can be derived.

Gleason's theorem
Hardy, Quantum Theory from Five Reasonable Axioms
G. Chiribella, G. M. D'Ariano, P. Perinotti, Informational derivation of Quantum Theory
Lluís Masanes, Thomas D. Galley, Markus P. Müller, The measurement postulates of quantum mechanics are operationally redundant
 
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These are exactly the sorts of papers and ideas which I’ve found rather ‘opaque’ :P
Maybe I just don’t know enough maths to understand what exactly they’re doing, but perhaps I’ll go through them again. Just seems to me to be an awful lot more complicated than I imagine it could be...
 
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Summary: Can the Born rule of quantum mechanics be derived from more fundamental principles?

Hey there!

I’ve been thinking about the Born rule recently and whether it can be derived from the other postulates from QM.
Thanks in advance!
Possibly, but I'm not sure if there is any point. Quantum mechanics is essentially an an extension of classical probability theory, so I'm not sure the Born rule is really out of place as a fundamental axiom. In any axiomatic stystem, you have a choice of which statements take the role of axioms and which statements are left to be proved from those axioms. Since quantum mechanics is a probabilistic theory, the Born rule would seem to be a good choice to consider fundamental to the theory. I really don't see any reason for a lot of angst and hand wringing over this.
 

N88

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Summary: Can the Born rule of quantum mechanics be derived from more fundamental principles?

Hey there!

I’ve been thinking about the Born rule recently and whether it can be derived from the other postulates from QM.

I’ve done a bunch of google searching, across PF, stackexchange and the arxiv, but most of it has felt a little opaque, particularly on whether anything has been ‘accepted’ as a derivation yet.

I decided to just try myself: using the assumptions that probability needs to be conserved under unitary transformations, that the rule holds for all quantum systems, and that the probabilities are functions of the respective amplitudes, I think I’ve showed that the Born rule must hold.

Do those assumptions go beyond the other postulates of QM, or are they perhaps in any sense less ‘fundamental’ than the Born rule itself?

Thanks in advance!
Tom,

I suspect that you might like my friend Fritz Fröhner (1988):

“Missing link between probability theory and quantum mechanics: the Riesz-Fejér theorem."

Z. Naturforsch. 53a, 637-654. http://zfn.mpdl.mpg.de/data/Reihe_A/53/ZNA-1998-53a-0637.pdf

All the best; N88
 

Mentz114

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Last edited:
Summary: Can the Born rule of quantum mechanics be derived from more fundamental principles?

I decided to just try myself: using the assumptions that probability needs to be conserved under unitary transformations, that the rule holds for all quantum systems, and that the probabilities are functions of the respective amplitudes, I think I’ve showed that the Born rule must hold.
Haven't you assumed the born rule in order to derive the born rule?
 
Gleason's Theorem, proven in 1957, says that the Born rule is the only one that is unitary, the only one where all the probabilities add up to exactly 1. So if you want probabilities its got to be proportional to the square of the magnitude of the particle's wave-function and not the cube or something else. However why we must use probabilities at all and not certainties is unknown.

John K Clark
 

vanhees71

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atyy

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Gleason's Theorem, proven in 1957, says that the Born rule is the only one that is unitary, the only one where all the probabilities add up to exactly 1. So if you want probabilities its got to be proportional to the square of the magnitude of the particle's wave-function and not the cube or something else. However why we must use probabilities at all and not certainties is unknown.
As I learned from @bhobba, this is not correct. Probability does not preclude certainty. The crucial assumption in Gleason's Theorem is not the assumption of probability, but the assumption of contextuality.
 

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