EvC Forum: Water under the Bridge

On the standard collapse formulation of quantum mechanics, somehow during the measurement interaction the state would collapse to either the first term of this expression (with probability equal to a squared) or to the second term of this expression (with probability equal to b squared). In the former case, J ends up with the determinate measurement record "spin up", and in the later case J ends up with the determinate measurement record "spin down". But on Everett's proposal no collapse occurs. Rather, the post-measurement state is simply this entangled superposition of J recording the result "spin up" and S being x-spin up and J recording "spin down" and S being x-spin down. Call this state E for Everett. On the standard eigenvalue-eigenstate link (Rule 3) E is not a state where J determinately records "spin up", neither is it a state where J determinately records "spin down". So the puzzle for an interpretation of Everett is to explain the sense in which J's entangled superposition of mutually incompatible records is supposed to agree with the empirical prediction made by the standard collapse formulation of quantum mechanics. The standard collapse theory, again, predicts that J either ends up with the fully determinate measurement record "spin up" or the fully determinate record "spin down", with probabilities equal to a-squared and b-squared respectively.One might hope that the selection of a preferred basis would work the other way around: that the biological evolution of observers would select for observers who record their measurement results in whatever physical observable is in fact determinate. The idea is that observers would either start recording their measurement results in whatever physical observable is in fact determinate or face some sort of failure in action for not having determinate measurement records. The full explanation for why this does not work in a straightforward way is subtle, but the basic idea is simple: in a no-collapse theory, there is no decreased fitness for a good observer who fails to have determinate measurement records. Suppose that only the position of particles is in fact determinate, as in Bohm's theory, but that an observer nonetheless tries to record his measurements in terms of the x-spin of particles in his brain. Such an observer would typically not have any determinate measurement records, but it would be difficult for anyone to tell. The observer would himself not know because, for bare theory reasons, he would falsely believe that he had determinate measurement records. But neither would other observers typically know that he failed to have determinate records, for as soon as the observer's brain state becomes quantum-mechanically correlated with the position of anything, the observer would have an effectively determinate measurement record by dint of this correlation in the physically preferred quantity. The evolutionary upshot of this is that, as soon as there is the possibility of a determinate failure in action, a good observer would have precisely those determinate dispositions that would lead to successful action regardless of whether he started with a determinate measurement record. While such an observer eventually has something that serves the dispositional role of a measurement record, his belief that he had a determinate measurement record before he correlated his brain state with the state of the determinate preferred observable was simply false. (See Albert's 1992 discussion of measurement records in GRW for more details on the formation of effectively determinate records from an observer's dispositions to act.) = |A| |B| cos
Let |A1> and |A2> be vectors of length 1 ("unit vectors") such that = 0. (So the angle between these two unit vectors must be 90 degrees.) Then we can represent an arbitrary vector |B> in terms of our unit vectors as follows:
|B> = b1|A1> + b2|A2>
For example, here is a graph which shows how |B> can be represented as the sum of the two unit vectors |A1> and |A2>:

Figure 2: Representing |B> by Vector Addition of Unit Vectors
Now the definition of the inner product has to be modified to apply to complex spaces. Let c* be the complex conjugate of c. (When c is a complex number of the form a bi, then the complex conjugate c* of c is defined as follows:

[a + bi]* = a bi
[a bi]* = a + bi
So, for all complex numbers c, [c*]* = c, but c* = c just in case c is real.) Now definition of the inner product of |A> and |B> for complex spaces can be given in terms of the conjugates of complex coefficients as follows. Where |A1> and |A2> are the unit vectors described earlier, |A> = a1|A1> + a2|A2> and |B> = b1|A1> + b2|A2>, then
= (a1*)(b1) + (a2*)(b2)
The most general and abstract notion of an inner product, of which we've now defined two special cases, is as follows. is an inner product on a vector space V just in case

(i) = |A|2, and =0 if and only if A=0
(ii) = *

(iii) = + .

It follows from this that
(i) the length of |A> is the square root of inner product of |A> with itself, i.e.,
|A| = , When a pair of physical systems interact, they form a composite system, and, in quantum mechanics as in classical mechanics, there is a rule for constructing the state-space of a composite system from those of its components, a rule that tells us how to obtain, from the state-spaces, HA and HB for A and B, respectively, the state-space -- called the ‘tensor product’ of HA and HB, and written HAHB -- of the pair. There are two important things about the rule; first, so long as HA and HB are Hilbert spaces, HAHB will be as well, and second, there are some facts about the way HAHB relates to HA and HB, that have surprising consequences for the relations between the complex system and its parts. In particular, it turns out that the state of a composite system is not uniquely defined by those of its components. What this means, or at least what it appears to mean, is that there are, according to quantum mechanics, facts about composite systems (and not just facts about their spatial configuration) that don't supervene on facts about their components; it means that there are facts about systems as wholes that don't supervene on facts about their parts and the way those parts are arranged in space. The significance of this feature of the theory cannot be overplayed; it is, in one way or another, implicated in most of its most difficult problems.

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