Huyssteen, The Many-worlds hypothesis

One of the fundamental interpretive problems of quantum theory arises from the fact that from any two or more states for a system one can create another state, their superposition (mathematically, a linear combination). Let s and s′ be possible states for a system, corresponding to two different values, k and k′ respectively, for the observable, O. (That is, they are mutually orthogonal eigenstates of O.) Their superposition, which is another possible state for the system, is denoted by s + s′. According to the standard interpretation of quantum theory, a system in the state s + s′ does not have the value k for O, nor the value k′ for O, nor neither, but if O is measured on the system, the system will be found to have either the value of k or the value of k′ for O.

The standard interpretation works in practice, but many physicists and philosophers find it to be unsatisfactory for a variety of reasons, not least because it contains the unanalyzed notion of “measurement.” With minimal experience, it is easy to judge when to say that a measure of O has occurred, but upon what principle can such a judgment be made? No satisfactory principle has been offered. The other problematic feature of the standard interpretation is that it countenances physical systems that are literally indeterminate with respect to their values for observables such as “position.” In other words, a physical system (outside of a context in which its location is being measured) can literally have no location (though if its location is measured, it will be found to have a location).

The Many-worlds Hypothesis, which originally arises from work by Hugh Everett (1930–1982), is an alternative approach to interpretation that purports to dispense with the notion of measurement and to resolve the problem of indeterminacy.

The central idea behind the Many-worlds Hypothesis is that a state such as s + s′ in fact describes a multiplicity of distinct, independent, worlds, some in which our system is in the state s and others in which our system is in the state s′. In most versions of the Many-worlds interpretation, there are, in all, an uncountable infinity of worlds, divided amongst the various states appearing in the superposition (in our case, s and s′) according to the probabilities attached to the various states.

So if, according to the standard interpretation, a measurement of O on our system would reveal the value k with probability 1/3, and k′ with the probability 2/3, then according to the Many-worlds interpretations, in 1/3 of the worlds our system is in the state s, and so has the value k for O, and in 2/3 of the worlds our system is in the state s′, and so has the value k′ for O.

It is important to keep in mind that the “worlds” of the Many-worlds interpretation are not the same as the “possible worlds” of philosophy. This point is clear in light of the fact that the philosopher’s possible worlds need not obey the laws of quantum theory, while the single “universe” of the Many-worlds interpretation does obey the laws of quantum theory. In the Many-worlds interpretation, therefore, there is a single “actual” world in the philosopher’s sense, but it consists of many distinct independent “realms of reality.” However, in standard usage, these realms of reality are called worlds.

A variant of the Many-worlds Hypothesis, called the Many-minds Hypothesis, asserts that the multiplicity in question is not a multiplicity of worlds, but a multiplicity of distinct, independent, minds. Each observer in fact has many minds (in most versions, an uncountable infinity of them), and when the observer observes a system in a superposition (for example, s + s′) some of the minds observe the system to be in the state s, while others observe it to be in the states s′, the proportions again matching the quantum-mechanical probabilities.

In the case of the Many-minds interpretations, rather than a single actual world with many realms of reality, there is a single “person” with many minds. Other than that, there are many similarities between the two interpretations. The notion of a measurement is supposed to play no fundamental role in these interpretations. A measurement of an observable O on a system merely reveals the pre-existing value in “your” world that the system had for O. That is, if you witness the result k, then you are in a world in which the system already had the value k for O. Similar remarks hold for the Many-minds theories, mutatis mutandis.

Many-worlds interpretations face a number of interpretive difficulties. One is that any quantum state can be written as a superposition in many ways. In the terms stated earlier, s + s′ is equivalent to an infinity of other super-positions, t + t′, where s and s′ are different from t + t′. So given that the quantum state of the universe is V= s + s′ = t + t′, are the realms of reality (the “worlds”) defined by the states s and s′ or t and t′, or all of the above? If one of the former two, then the interpretation faces the obvious question why one (e.g., s and s′) rather than the other (e.g., t and t′). If the latter, then the interpretation faces the problem of how to define a probability measure over all of the components that can appear in any decomposition of the quantum state. Indeed, if such a measure is supposed to represent “ignorance” about which world one occupies (or which mind is “one’s own”) then it is far from clear that a satisfactory measure can be defined.

This issue is related to another severe problem facing these interpretations, namely, how to justify, or even to understand, the probabilities of standard quantum theory. The most obvious way to conceive of probabilities in these interpretations is as a measure of ignorance either about which world one occupies or about which mind is one’s own. The problem is that it is not at all clear why that ignorance should be measure by the quantum-theoretic measure (except by stipulation). But perhaps the most significant obstacle facing the Many-worlds and Many-minds interpretations is the sheer implausibility of the hypotheses. The central issue facing these interpretations is whether the difficulties we have understanding quantum theory really force us to such extreme measures.