REVIEW

Jeffrey Bub, Interpreting the Quantum World. Cambridge:

Cambridge University Press, 1997. xiv+298 pp.

Jeffrey Bub himself once mentioned that whenever someone

arrives at a position in the philosophy of quantum mechanics s/he

seems to lose momentum. Here we have a clear counterexample:

since his first book on the subject in 1974, which pioneered many

of the subsequent development, Bub has gained a great deal of

momentum. His position today has the additional virtue of

providing sufficient conditions for the tenability of a whole

range of alternative interpretations, thus locating quite clearly

many of the diverse ideas found in recent literature. Moreover,

he can demonstrate optimal desirable properties for his own

favorite interpretation within that range.

For physics students and for much of the general public,

interpretation of quantum theory begins with certain mystifying

pronouncements by the great scientists of Copenhagen.

Familiarity and authority have dulled the sense of mystery, and

left us with remnants delicately balanced between vacuity and

inconsistency. For the specialist, however, the story of

interpretation begins a few years after the Copenhagen

breakthrough with von Neumann's unification of matrix and wave

mechanics in the Hilbert space formalism. Not that this had its

meaning written on its face. Von Neumann strove mightily to add

some, through his discussion of measurement ("collapse of the

wave packet", "projection postulate") and of propositions

representable by projection operators. Fundamental to his effort

was what is now commonly called the "eigenstate-eigenvalue link".

That is the semantic rule which says that an observable

pertaining to a given system has a value if and only if the

system is in the corresponding eigenstate of that observable. If

that is so then not only do incompatible observables never have

simultaneous values -- but in fact most of time most observables

have no definite value at all. Accordingly, to assure that at

least the outcome of a measurement (and the measured observable

at that time) have a definite value, that "collapse" postulate

had to be added.

That is the beginning of the story; the latest developments

and perhaps the denouement (but that is debatable) come to us in

Jeffrey Bub's new book Interpreting the Quantum World.

To begin Bub shows clearly how the (in)famous measurement

problem and its unsolvability derive directly from the

eigenstate-eigenvalue link. (Let us abbreviate to "EE-link".)

The options are: either to change the theory or to reject that

interpretative principle. Such versions as von Neumann's (or the

more recent GRW) which postulate collapse, as interruption to the

reign of the Schroedinger equation, are changes to quantum

theory, in Bub's view. His book is devoted to the second option,

originally mooted by Henry Margenau in response to the

Einstein-Podolsky-Rosen paradox, but only recently much explored.

Bub presents a fundamental result (the Bub-Clifton uniqueness

theorem) which selects a precise spectrum of interpretations that

escape the unsolvable measurement problem under certain 'natural'

constraints.

Quick aside: what seem like a few natural constraints to one

person may seem intolerable to another. In the last part of this

review I'll discuss how much is left out from Bub's range of

admissible interpretations.

So what is Bub's approach? Bub follows von Neumann in the

representions of such propositions as "Observable m has value k"

by the k-eigenspace of m (the set of pure eigenstates of m

corresponding to value k, a subspace of the relevant Hilbert

space). Undoubtedly this identification was originally linked to

the EE-link. But there is separate motivation for this. The

following more basic principle was implicit in much of the early

discussion:

Identity of Observables:

If the probabilities for measurement outcomes for

observables m and m' are the same for every QM state of any

system to which m and m' both pertain [if m, m' are

'statistically equivalent'] then m = m'.

Equivalently: If observables m and m' are represented

by the same (Hermitean) operator then m = m'.

Given this conviction there is a one-one relation between

observables and their representing Hermitean operators. Hence it

is possible to represent the propositions which assign values to

observables by the eigenspaces of the corresponding operators.

There are also interpretations which run counter to this

principle, but Bub accepts it.

So there we have the propositions whose truthvalue was

settled by the quantum state itself as long as the EE-link was

assumed. At this point (after rejecting the EE-link) all we can

say is that some of these propositions are true, some are false

(=contraries to true propositions) and some are not truth-valued

at all (when he observable has no definite value).

Without the EE-link we need to lay down new conditions on

which propositions can be true or false when the quantum state is

given. What is the set of truth-valued propositions like? Logic

requires us to say that if one proposition entails another then

the second is true if the first is. Moreover if sets of

propositions have greatest lower and least upper bounds -- as is

indeed the case for subspaces -- then other familiar bits of

logic appear. These correspond to the familiar rules governing

"and" and "or". If we assume this (and not all interpretations

go along with it), then the truth-valued propositions form a

lattice, a sublattice of the lattice of subspaces of Hilbert

space. But they cannot form the entire lattice of subspaces --

it is not possible to assign truthvalues to all the propositions,

i. e. definite values to all the observables, while respecting

logic in this way. That follows from the famous 'no-go' theorems

about hidden variables.

So, how large can the lattice of truth-valued propositions

be? It is certainly possible to assign a value to one

observable; hence to assign truth values to all the propositions

"R has value k" for a given observable R. These propositions

form a Boolean sublattice. Can such a sublattice be expanded;

and if so, how far?

Here is Bub's idea. Suppose that one observable R has

privileged status, in that the lattice of truth-valued

propositions is a function D(e,R) of the quantum state e and the

observable R. Presumably at least some of the eigenspaces of R

belong to it -- or all of them perhaps. What more would we like?

Why not impose a few more desiderata, and then see if we can

identify what that function must be like?

Crucial to the empirical content of quantum theory are the

answers to questions of form: if observable S is measured, what

is the probability that value k is found? These answers come

from Born's rule for calculating probabilities. But they are

notoriously hard to interpret, and we cannot think of them all

together as simply constituting a measure of our ignorance. But

what if we look only at D(e,R), the truth-valued propositions?

Can we construe those questions as just asking "what is the

probability that 'S has value k' is true" (when the k-eigenspace

of observable S is in D(e,R)), and interpret the answer as a mere

measure of ignorance? After all, if we fix what the lattice of

truth-valued propositions is, we are still ignorant of how truth

and falsity are distributed in it. That ignorance can have a

measure, and it seems natural to identify it with the Born

probability.

How natural? Well, it would certainly follow at once if we

assume that measurement of S will reveal the truth if S already

has a definite value. That is not so easy an assumption to

motivate when the EE-link has been given up. True: a good

measurement of S will not change the quantum state if that state

was already an eigenstate of S. But in the present context we

are not assuming that S had a definite value only if it was in an

eigenstate! In the absence of a motivating argument we should

expect to see interpretations that do not agree. Nevertheless,

it would be a 'nice' feature of the interpretation if Born's

calculations yield can be so simply construed.

Now we come to the Bub-Clifton theorem. Its proof is by a

beautiful symmetry argument, on the additional premise that

D(e,R) is preserved under the automorphisms that preserve e and

R. (To put it another way: the identity of D(e,R) is a assumed

to be a function of e and R alone, no other factors are relevant

to the question of which propositions have a truthvalue.) The

result proved is then that the above desiderata determine D(e,R)

uniquely.

So now all we need to ask is: what is D(e,R)like then; i. e.

what is the set of propositions with definite truth values like,

on the suppositions of this theorem? That is also really quite

simple.

Let me first state the answer in precise, technical form,

and then give it a more intuitive gloss. D(e,R) is generated from

a set of rays (1-dimensional subspaces). First take all the

projections of state e onto the eigenspaces of R to which e is

not orthogonal. Let's call the set of these D1. Next take the

set of all rays that are orthogonal to all the members of D1.

That second set (call it D2) is a subspace, namely the

orthocomplement of the subspace spanned by D1. Obviously it

includes all the eigenspaces of R to which e is orthogonal. Now

D(e,R) is the lattice generated by D1 and D2 together (i.e.

D(e,R) is the smallest lattice of subspaces that contains both D1

and D2.)

Let us put this in more intuitive dress. Those rays are

very informative propositions. Each member f of D1 is an

eigenstate of R corresponding to some eigenvalue Ef. Clearly the

Ef-eigenspace belongs to D(e,R) in that case. So then the

proposition "R has value Ef" is truth-valued. Notice that these

are precisely the cases which receive a positive Born probability

in state e (i.e. there is a positive probability that if R is

measured on a system in state e, the value Ef will be found).

Let us call such propositions about R allowed by e. If the Born

probability for eigenvalue r of R is 0 in state e, let's call the

proposition that R has value r disallowed by e. Since all the r-

eigenstates of R for which that is the case belong to D2, all

those propositions about R that are disallowed by e are also

truthvalued.

So far so good; but of course this lattice D(e,R) contains

many more similar propositions about other observables. For

example, the projection of e onto the 1-eigenspace of R may also

be an eigenstate of a quite different observable S, corresponding

to eigenvalue 2 of S, say. In that case the proposition that S

has value 2 will also be truth-valued. We can divide the truth-

valued propositions about any observable S into those allowed by

e and those disallowed by e in a similar way (but not apply these

labels to propositions which are not truth-valued).

Thinking of it this way, it is understandable how the Born

probabilities can be recovered as a possible measure of ignorance

about how the truth values True and False are distributed in

D(e,R). Given that the system is in state e, we may take it that

all disallowed propositions are definitely false. Thus the whole

of D2 should be treated as a region with zero probability. We

can then add that one of the rays in D1 is a true proposition,

though we don't know which, and we can propose (or interpret) the

Born probability for the eigenvalue Ef of R as the measure of our

ignorance about whether member f of D1 is true. Starting in this

way, the probability assignment can be extended to the whole of

D(e,R) by additivity, without running into trouble with such'no-

go' theorems as the Kochen-Specker result.

Bas C. van Fraassen

Princeton University

Princeton, NJ

An interpretation along these lines results when a

particular observable is taken to have privileged status. This

specification might be once and for all or it might depend on the

state. Bohm chose position for this role, once and for all. Von

Neumann's EE-link chose a privileged observable very tightly

linked to the state, namely the projection on the ray containing

that state. Bub has a different suggestion: given a proper

selection of observables for such privilege, the measurement

problem will disappear.

Here we find the most important conceptual gain of this

approach. To explain how the values of the privileged observable

can change with time, Bub adapts a proposal by John Bell (as

elaborated by Vink). This account of the dynamics of values,

together with the above construal of the Born probabilities,

provides a very satisfactory corollary about what happens in a

measurement. The measurement outcome and the value of the

measured observable are both definite, and correspond to each

other, at the end of measurement.

I am passing rapidly over some difficult passages here. In

a measurement we deal with a composite system of measured object

and measuring apparatus. In the above discussion I focused on

the total system, assumed to be in a pure state. The two

component parts will be in mixed states, and the observables

pertaining to the components will be functions of those which

pertain to the total system. Obviously the nice corollary about

measurement is forthcoming only when the 'right' observable is

privileged. The 'pointer observable' pertaining to the apparatus

and the measured observable must be properly related to the

privileged observable. Because of the entanglement of the two

systems during the measurement interaction, it will suffice if

either the measured observable or the pointer observable receives

the privileged status. If we assume that either will take on a

definite value at the end of measurement then so will the other.

How compliant will nature be? Is it a question of fact,

which observables are privileged in nature in this way? Can we

assume that nature will privilege just those observables that we

have a special interest in? But perhaps that is the wrong

question to ask here. Quantum mechanics works; the task is to

propose an interpretation that makes sense of how it works. That

proposal can be precisely that nature privileges certain

observables under certain circumstances, so that the measurements

in which we have an interest have definite outcomes. Bub has

therefore presented us with no mean achievement: an explanation

of how a certain range of interpretations have the conceptual

resources to change the measurement problem from unsolvable

riddle to solved conundrum.

We have seen therefore that we have here a major

contribution to the debates concerning the interpretation of

quantum mechanics. In addition, the wide scope of these results

is illuminating. They give us a new insight into a number of

earlier interpretations which Bub is able to locate with respect

to his own. But I do want to enter a demurral here to the claims

of universality made in the book's early pages (e.g. 1, 4, 5).

The uniqueness interpretation does not characterize the entire

range of 'no collapse' interpretations, nor that of all modal

interpretations, and certainly not of all tenable

interpretations.

Of course, Bub is quite clear on this. His claims of

universality should be taken as relative to the underlying

program. That point is made quite clearly in the Coda,

particularly pages 240 and 241 around the quote

from von Neumann. It is especially clear if those passages are

read with the parable about the 19th century math student (pp.

115 - 117) in mind.

But it may still be illuminating for us if we take stock of

how other interpretations can and do differ from the range that

Bub characterizes here. To begin, the representation of

"Observable A has value k" by the k-eigenspace of the operator

that represents A rules out quite a bit. First of all it rules

out that an observable can have values other than its eigenvalues

(as happens in the Bohm-Hiley way of dealing with observables

other than position). Secondly it rules out 'de-occamized' and

'contextual' interpretations in which a single observable

ambiguously represents several distinct though statistically

equivalent observables. Thirdly, by taking all the subspaces as

propositions and respecting their lattice structure, Bub rejects

doubts about the meaningfulness of conjoining value attributions

to incompatible observables. (For example, in Healey's 1988

interpretation the set of truth-valued propositions is not closed

under logical conjunction.)

Some of these limitations of the uniqueness theorem are

pointed out in a paper by J. L. Bell and Clifton (Int. J. Theor.

Phys. 34 (1995), 2409-21). As they also point out, the

interpretation this reviewer proposed ("Copenhagen Variant of the

Modal Interpretation") is not covered. There the EE-link is

violated only when the state is not pure. The divergence from

orthodoxy will in general appear for the components of many-body

systems. For those components will typically be in mixed states

even when the total systems are themselves in pure states.

Measurement and Schroedinger's cat are typically represented as

dynamic evolution of composite systems (possibly with the

environment as one component). The desire for values concerns

observables that pertain to individual components (such as the

cat, or the pointer on the measuring apparatus). Thus the shared

goal to do justice to our intuitions in those cases can perhaps

be met without abandoning the EE-link for pure states. In other

respects there are significant similarities between this and

Bub's interpretations, such as adherence to the Identity of

Observables principle and hence the representability of

propositions by subspaces.

We may note that one intuitive reading of the original Bohm

interpretation also cannot belong to the range characterized by

Bub. I mean this: that position is the only observable that ever

has a value. A quick look at the lattice D(e,R) shows that there

will in general be many observables other than the privileged

observable itself which receive true and false value

attributions. The reason is that the propositions concerning

R's value and do not exhaust the lattice of truth-valued

propositions.

These points are not drawbacks to Bub's interpretation, nor

do they diminish its virtues or the importance of his general

results. They only mean that still more general results are in

the offing (and indeed, the results of the Bell and Clifton paper

are more general). The virtues Bub can claim are indisputable,

and his achievement gives us a truly successful and illuminating

way to understand quantum theory. The book is a must for

everyone in the field.