Born rule as an axiom or as a theorem?

[nomadreid]

TL;DR Summary

Does the statement that the Born rule is an axiom change when one adds an axiom excluding hidden-variable theories? That is, doesn't the Born rule then become a theorem? But when it is an axiom, can one provide a mathematically consistent theory with the negation of the Born rule as an axiom, even if not empirically valid?

First, a warning: I do not know much about hidden-variable theories, so my apologies in advance when some of my questions seem rather obvious to people better versed in them than me.

Without the Born rule, much of quantum physics falls apart, and the Born rule does wonders at predicting data in our universe. However, in this question I want to regard it from a purely mathematical point of view.

I have occasionally read that the Born rule is an (independent) axiom (postulate) of quantum physics. I find this puzzling, as I have also read that one can derive Born's rule (Gleason's Theorem) using other reasonable axioms (e.g., the ones used for the Kochen-Specker theorem).

So I assume the statement that the Born rule is an axiom is based on a weakening of those other (reasonable) axioms. What would be, from a purely mathematical point of view, a non-trivial weakening to allow a theory which does not obey Born's rule (either (a) one that does not contradict empirical results, or, certainly easier, (b) one that contradicts empirical results but which would still allow the resulting theory to remain consistent mathematically). More precisely, what would a probability density function look like in this new theory?

That is, an independent axiom A in a theory T has the property that one could replace A by its negation ~A (while keeping all the other axioms fixed) to get a new theory T*, so that T is consistent iff T* is consistent. Obviously T and T* would have different interpretations, so that if we replace Born's rule by its negation we won't be describing our physical universe. But that is not the point here. The idea is that, if we assume our present theory to be mathematically consistent, and the Born rule to be an axiom, ,then a theory with a probability density function that does not obey Born's rule should also be mathematically consistent (regardless of whether it is empirically valid). (Even one which simply tweaks the usual probability density function by a little.)

Any comments such as that my question is unclear (if not downright silly) would also be appreciated, if one then has the patience to allow me to try to clarify. Or a comment to show that my quest is hopeless would also be interesting. In any case, my thanks in advance.

 

[gentzen]

What would be, from a purely mathematical point of view, a non-trivial weakening to allow a theory which does not obey Born's rule (either (a) one that does not contradict empirical results, or, certainly easier, (b) one that contradicts empirical results but which would still allow the resulting theory to remain consistent mathematically). More precisely, what would a probability density function look like in this new theory?

Bohmian mechanics without the quantum equilibrium hypothesis seems to me like the "canonical example" of what you are asking. From my POW, it is an example for (b), but Antony Valentini is a proponent that claims it would be an example of (a).

The idea is that instead of assuming a probability distribution for the particle positions, you just assume initial particle positions to be given without being associated with any probability distribution.

 

[nomadreid]

Thanks very much, gentzen! And thank you for the link.

 

[Demystifier]

Another important result, first understood by Bohm, but actually independent on the interpretation, is this: If Born rule is postulated (or assumed) for positions, then Born rule for other observables can be explained from considerations of macroscopic apparatuses. See e.g. my https://arxiv.org/abs/1811.11643

 

[nomadreid]

Thank you very much for the interesting response and the corresponding link, Demystifier.

 

[bhobba]

I have occasionally read that the Born rule is an (independent) axiom (postulate) of quantum physics. I find this puzzling, as I have also read that one can derive Born's rule (Gleason's Theorem) using other reasonable axioms (e.g., the ones used for the Kochen-Specker theorem).

Gleason is a mathematical theorem, so it is rock solid. The issue is the explicit and implicit assumptions.

Explicably, non-contextuality and the principle of strong superposition are assumed. I think it is an important theory, but that is just me

Von Neumann mistakenly assumed additivity of expectations in proving his no hidden variable theorem:
The assumption is quite intuitive; the expectation of the observable A+B, E (A+B), is E (A) + E (B) but is not necessarily true:

https://arxiv.org/pdf/1801.09305

However, assuming it, it is reasonably easy to prove Born's Rule.

First, it's easy to check <bi|O|bj> = Trace (O |bj><bi|).

O = ∑ <bi|O|bj> |bi><bj| = ∑ Trace (O |bj><bi|) |bi><bj|

Now we use the linearity we have assumed.

E(O) = ∑ Trace (O |bj><bi|) E(|bi><bj|) = Trace (O ∑ E(|bi><bj|)|bj><bi|)

Define P as ∑ E(|bi><bj|)|bj><bi| and we have E(O) = Trace (OP).

P, by definition, is called the state of the quantum system. The following are easily seen. Since E(I) = 1, Trace (P) = 1. Thus, P has a unit trace. Note because the |bi> is an orthonormal basis, that E(|bi><bj|) is positive. A nice challenge is to make that more rigorous (look at the eigenvalues of |bi><bj|). Hence, P is a positive operator of unit trace.

Thanks
Bill

 

[nomadreid]

Thanks very much, Bill, aka bhobba. Very clear explanation, and I have downloaded the ArXiv article, which looks interesting.

 

[A. Neumaier]

What would be, from a purely mathematical point of view, a non-trivial weakening to allow a theory which does not obey Born's rule (either (a) one that does not contradict empirical results, or, certainly easier,

I get Born's rule (in the cases where it applies) as a consequence of another axiom, the Detector Response Principle (DRP). See
https://www.physicsforums.com/threa...-a-new-approach-to-quantum-mechanics.1011069/