Is it the case that the Born probabilities are necessarily explained—can only be explained—by some hidden component of our brain which says that we care about the alternatives in proportion to their squared modulus?
I have been thinking about quite a bit in the last few days and I have to say, I find this close to impossible.
The solution must be much more fundamental: Assumptions like the above ignore that the Born rule is also necessary for almost everything to work: For example the working of our most basic building blocks are tied to this rule. It is much more then just our psychological “caring”. Everything in our “hardware” and environment would immediately cease to exist if the rule would be different
Based on this, I think that attempts (like that of David Wallace, even if it would be correct) trying to prove the Born rule based on rationality and decision theory have no chance to be conclusive or convincing. A good theory to explain the rule should also explain why we see the reality as we see it, even if never really make conscious measurements on particles.
In our lives, we (may) see different type of apparent randomnesses:
incomplete information
inherent (quantum) randomness
To some extent these two type of randomness are connected and look isomorphic on surface (in the macro-world).
The real question is: “Why are they connected?”
Or more specifically: “Why does the amplitude of the wave function result in (measured) probabilities that resembles to those of random geometric perturbations of the wave function?”
If you flip a real coin, for you it does not look very different from flipping a quantum coin. However the 50⁄50 chance of heads and tails can be explained purely by considering the geometric symmetry of the object. If you assume that the random perturbing events are distributed in a geometrically uniform way, you will immediately deduce the necessity of even chance. I think the clue of the Born rule will be to relate similar geometric considerations to relate perturbation based probability to quantum probability.
Quantum probability is only “inherent” because by default you are looking at it from the system that only includes one world. With a coin, the probability is merely “epistemic” because there is a definite answer (heads or tails) in the system that includes one world, but this same probability is as inherent for the system that only includes you, the person who is uncertain, and doesn’t include the coin. The difference between epistemic and inherent randomness is mainly in the choice of the system for which the statement is made, with epistemic probability meaning the same thing as inherent probability with respect to the system that doesn’t include the fact in question. (Of course, this doesn’t take into account the specifics of QM, but is right for the way “quantum randomness” is usually used in thought experiments.)
I don’t dispute this. Still, my posting implicitly assumed the MWI.
My argument is that the brain as an information processing unit has a generic way of estimating probabilities based on a single-worldline of the Multiverse. This world both contains randomness stemming from missing information and quantum branching, but our brain does not differentiate between these two kind of randomnesses.
The question is how to calibrate our brain’s expectation of the quantum branch it will end up. What I speculate is that the quantum randomness to some extent approximates an “incomplete information” type of randomness on the large scale. I don’t know the math (if I’d knew I’d be writing a paper :)), but I have a very specific intuitive idea, that could be turned into a concrete mathematical argument:
I expect the calibration to be performed based on geometric symmetries of our 3 dimensional space: if we construct a sufficiently symmetric but unstable physical process (e.g. throwing a coin) than we can deduce a probability for the outcome to be 50⁄50 assuming a uniform geometric distribution of possible perturbations. Such a process must somehow be related to the magnitudes of wave function and has to be shown to behave similarly on the macro level.
Admitted, this is just a speculation, but it is not really philosophical in nature, rather an intuitive starting point on what I think has a fair chance ending up in a concrete mathematical explanation of the Born probabilities in a formal setting.
Does your notion of “incomplete information” take into account Bell’s Theorem? It seems pretty hard to make the Born probabilities represent some other form of uncertainty than indexical uncertainty.
I don’t suggest hidden variables. The idea is that quantum randomness should resemble incomplete information type of randomness on the large scale and the reason that we perceive the world according to the Born rule is that our brain can’t distinguish between the two kind of randomnesses.
I have been thinking about quite a bit in the last few days and I have to say, I find this close to impossible.
The solution must be much more fundamental: Assumptions like the above ignore that the Born rule is also necessary for almost everything to work: For example the working of our most basic building blocks are tied to this rule. It is much more then just our psychological “caring”. Everything in our “hardware” and environment would immediately cease to exist if the rule would be different
Based on this, I think that attempts (like that of David Wallace, even if it would be correct) trying to prove the Born rule based on rationality and decision theory have no chance to be conclusive or convincing. A good theory to explain the rule should also explain why we see the reality as we see it, even if never really make conscious measurements on particles.
In our lives, we (may) see different type of apparent randomnesses:
incomplete information
inherent (quantum) randomness
To some extent these two type of randomness are connected and look isomorphic on surface (in the macro-world).
The real question is: “Why are they connected?”
Or more specifically: “Why does the amplitude of the wave function result in (measured) probabilities that resembles to those of random geometric perturbations of the wave function?”
If you flip a real coin, for you it does not look very different from flipping a quantum coin. However the 50⁄50 chance of heads and tails can be explained purely by considering the geometric symmetry of the object. If you assume that the random perturbing events are distributed in a geometrically uniform way, you will immediately deduce the necessity of even chance. I think the clue of the Born rule will be to relate similar geometric considerations to relate perturbation based probability to quantum probability.
Quantum probability is only “inherent” because by default you are looking at it from the system that only includes one world. With a coin, the probability is merely “epistemic” because there is a definite answer (heads or tails) in the system that includes one world, but this same probability is as inherent for the system that only includes you, the person who is uncertain, and doesn’t include the coin. The difference between epistemic and inherent randomness is mainly in the choice of the system for which the statement is made, with epistemic probability meaning the same thing as inherent probability with respect to the system that doesn’t include the fact in question. (Of course, this doesn’t take into account the specifics of QM, but is right for the way “quantum randomness” is usually used in thought experiments.)
I don’t dispute this. Still, my posting implicitly assumed the MWI.
My argument is that the brain as an information processing unit has a generic way of estimating probabilities based on a single-worldline of the Multiverse. This world both contains randomness stemming from missing information and quantum branching, but our brain does not differentiate between these two kind of randomnesses.
The question is how to calibrate our brain’s expectation of the quantum branch it will end up. What I speculate is that the quantum randomness to some extent approximates an “incomplete information” type of randomness on the large scale. I don’t know the math (if I’d knew I’d be writing a paper :)), but I have a very specific intuitive idea, that could be turned into a concrete mathematical argument:
I expect the calibration to be performed based on geometric symmetries of our 3 dimensional space: if we construct a sufficiently symmetric but unstable physical process (e.g. throwing a coin) than we can deduce a probability for the outcome to be 50⁄50 assuming a uniform geometric distribution of possible perturbations. Such a process must somehow be related to the magnitudes of wave function and has to be shown to behave similarly on the macro level.
Admitted, this is just a speculation, but it is not really philosophical in nature, rather an intuitive starting point on what I think has a fair chance ending up in a concrete mathematical explanation of the Born probabilities in a formal setting.
Does your notion of “incomplete information” take into account Bell’s Theorem? It seems pretty hard to make the Born probabilities represent some other form of uncertainty than indexical uncertainty.
I don’t suggest hidden variables. The idea is that quantum randomness should resemble incomplete information type of randomness on the large scale and the reason that we perceive the world according to the Born rule is that our brain can’t distinguish between the two kind of randomnesses.