The Problem of Probability

The Problem of Probability

The Everett approach does away with the objectively indeterminate universe suggested by both the collapse and Bohm approaches to quantum mechanics. There is no objective uncertainty because every physical possibility is actualised. This may be seen as a virtue by some, Einstein was critical of the collapse approach because of its reliance on objectively indeterminate probabilities, probabilities which relate to an uncertainty that cannot be explained by our ignorance of the physical system. In a letter to German physicist Max Born, written in 1926, Einstein stated that; "quantum mechanics is certainly imposing. But an inner voice tells me that it is not yet the real thing. The theory says a lot, but does not really bring us any closer to the secret of the 'old one'. I, at any rate, am convinced that He does not throw dice" (Einstein, 1926).

Everett did not see his approach as being any better or worse than traditional approaches in this regard. In a letter to DeWitt, written shortly after his theory was published in 1957, Everett claimed that; "from my point of view there is no preference for deterministic or indeterministic theories. That my theory is fundamentally deterministic is not due to any deep conviction on my part that determinism holds any sacred position...I only object to mixed systems where the character changes with mystical acts of observation" (Everett, 1957a).

The Incoherence Problem

Determinism may be favoured by some, but it appears to cause problems for the Everett approach because the concept of objective uncertainty has become an accepted part of quantum theory, our understanding of radioactive decay provides a good example of this. The argument that the Everett approach fails because it cannot assign probabilities to events is known as the incoherence problem.

Everett attempted to resolve the incoherence problem by arguing that, from a subjective point of view, an observer will not be aware of every possibility, probabilities represent their chances of observing a specific result (Everett, 1957b, pp.70). Everett argued that "the subjective experience...is precisely in accordance with the predictions of the usual probabilistic interpretation of quantum mechanics" (Everett, 1957b, pp.63).

The Quantitative Problem

Given that it makes sense to talk of probabilities within the Everett approach, a more serious problem arises. What good is it in saying that an atom has a 1% chance of decaying in the next twenty four hours when there are only two possibilities; a world where it decays and a world where it does not? The quantitative problem asks why Everett is justified in using the Born rule to assign probabilities, rather than assigning an equal probability to each branch. Everett suggested that "in order to establish quantitative results, we must put some sort of measure (weighting) on the elements of a final superposition" (Everett, 1957b, pp.70). In the above example, the universe can be thought of as branching into 100 copies, the atom decays in one but not in the 99 others. Because these 99 worlds remain identical until new quantum interactions force them to diverge, they can be thought of as a one world with a weight of 99. The meaning of the word 'weight' is still being debated.

The quantitative problem is analogous to the problems raised by classical probabilities. When we throw a weighted dice, for example, we know that there are only six possible outcomes, and so this raises the question of why are we entitled to give them unequal probabilities. Classically, we understand probabilities in terms of decision theory (Savage, 1972 and Jeffrey, 1983). The decision-theoretic link states that it is rational for a person to use their objective knowledge of a system in order to determine how to act.

Objectively, we know that regular dice have a 1/6 chance of landing on any particular number and coins have a 1/2 chance of landing either heads or tails. A weighed dice will have different probabilities associated with each side. The decision-theoretic link entails that a rational person should try to bet on the number that has the highest objective probability. If the stakes are high then they should use their knowledge of these objective probabilities in order to decide whether to bet at all.

The problem with this approach is that we do not know how to derive probabilities without knowing the symmetry of the system. If we actually throw dice and count how many times each number occurs then a set of objective probabilities is expected to emerge, but there is no certainty to this assumption. No matter how many frequency trails we run, we can never know for certain if the dice is weighed, there is always the possibility that we have just been unlucky. This raises the question of why we should be rationally compelled to use our objective knowledge of probabilities when placing bets. There are three solutions to this problem; functionalism, primitivism and eliminativism.

Functionalism, Primitivism and Eliminativism

Functionalism suggests that we will one day have a better understanding of objective probability and will be able to define it as a physical property. This property will be defined independently of the decision-theoretic link but will come to the same conclusions as it does. The frequency approach is an example of a failed attempt at a functional definition.

Our inability to find a functional definition of objective probability has led some to adopt the second approach, primitivism. Primitivism is the view that we should accept the decision-theoretic link as a fundamental law of nature and not look for a deeper explanation. This is a popular approach but it does not seem entirely satisfactory, the decision-theoretic link is not similar to other mathematical concepts that we are willing to accept as fundamental.

The only option left is eliminativism. This is the view that there is no such thing as objective probabilities and so nothing can explain the decision-theoretic link. This is not acceptable as the concept of objective probability is used in all branches of science as well as ordinary life.

There is no satisfactory explanation for what classical probabilities really are, but primitivism, the denial that a functional definition exists beyond the decision-theoretic link, and eliminativism, the denial of objective probabilities all together, should only be accepted once we have given up on finding a functional definition. Cautious functionalism is the view that we will one day find a functional definition and, in the meantime, we can use the decision-theoretic link as such. This allows scientists to continue to use decision theory when considering objective probabilities.

There is no further justification for the use of classical probabilities and so proponents of the Everett approach can defend their use of the Born rule in the same way that proponents of the Bohm and collapse approaches do; using the decision-theoretic link. If Everett's use of the Born rule is correct, then we should still be rationally compelled to take the probability of each possibility into account when we decide whether a bet is worth taking. We should also be compelled to bet on an event which has the highest objective probability when we do.

Proponents of the Everett approach can simply state that their definition of the Born rule meets the conditions set by the decision-theoretic link and defend its use on the basis of cautious functionalism. However, British physicist David Deutsch argued that they can go further than this and prove that their concept of weight fits the functional definition of objective probability (Deutsch, pp.3129-3137). This is because it defines objective probability as a physical property which is independent of the decision-theoretic link but comes to the same conclusions about how it is rational to act when faced with uncertainty. If we accept this claim then proponents of the Everett approach can defend their use of the Born rule more fully than proponents of the Bohm or collapse approaches can. But even if we do not accept Deutsch's proof, the Everett approach is no worse off than the collapse or Bohm approaches in this regard.

References

Deutsch, D., 1999, 'Quantum Theory of Probability and Decisions', Proceedings of the Royal Society of London, A455

Einstein, A., 1926, 'Letter to Max Born (4 December 1926)', The Born-Einstein Letters, Born, I. (trans.), Walker and Company, New York

Everett, H., III, 1957a, 'Letter from Everett to DeWitt'

Everett, H., III, 1957b, 'The Theory of the Universal Wavefunction'

Jeffrey, R. C., 1983, 'The Logic of Decision', University of Chicago Press, Chicago

Savage, L. J., 1972, 'The foundations of statistics', Dover, New York