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  Is the Born rule trivially encoded in the structure of Hilbert space?

+ 5 like - 0 dislike

I am working my way through Everett's long thesis (1956) and got to the section where he "derives" the Born rule. On working it through myself, I found that his derivation was essentially a restatement of the normalization condition, ie that the norm of a state as the square root of the sum of squares of the coefficients of its decomposition in some basis must be equal to one. The abbreviated derivation would be essentially that if we have a generic QM state $\sum a_i \phi_i$ that is normalized $\sum a_i^*a_i = 1$ and we want to find a measure M($a_i$) for each possible outcome $\phi_i$ then by definition we want $\sum M(a_i)=1$, and so by comparison with the normalization condition it must be that $M(a_i)=a_i^*a_i $. Again this isn't exactly what Everett does, but as far as I can tell his derivation is morally equivalent to the abbreviated argument above.

At this point I had two competing realizations: 

  1) That I understand why Everett's derivation of the Born rule does not stand up to scrutiny, in that it is tautologically dependent on the inner product structure of Hilbert space.

  2) That there is really no other option if you are working with a Hilbert space, it is basically just an example of Gleason's theorem in action. The only way to have a more satisfying derivation of the Born rule would be to somehow derive the inner product structure of Hilbert space from some form of branch counting, but that presents a chicken/egg problem. 

My question is whether my above reasoning is correct. Is the Born rule trivially encoded in the structure of Hilbert space (at some level it must be given Gleason's theorem)? If so, isn't all the fuss about the Born rule in the MWI completely missing the point?

asked Apr 5, 2015 in Theoretical Physics by user1247 (540 points) [ no revision ]

As per Arnold Neumaier's suggestion, and with the consent of user1247 and Vladimir Kalitvianski, I've moved a tangential (yet nevertheless perhaps useful) discussion to chat.

1 Answer

+ 3 like - 0 dislike

I agree with your interpretation (1) that his derivation of the Born rule is spurious; see the entry Circularity in Everett's measurement theory from my theoretical physics FAQ.

For how I see the relation between Born's rule and the mathematical structure of quantum mechanics see Section 10.5 Quantum probability of my free online book Classical and Quantum Mechanics via Lie algebras.

answered Apr 5, 2015 by Arnold Neumaier (15,787 points) [ no revision ]
But your FAQ entry has a different complaint entirely about Everett's Born rule derivation. Your complaint there seems to me to be more about state reduction than the Born rule, something that has been since been addressed via decoherence and anthropics, and is separate from the issue I raise, which is about the derivation of the branch counting measure.

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