The Philosophy of Physics
Simon Saunders
‘Physics, and physics alone, has complete coverage’, according to Quine. Philosophers
of physics will mostly agree. But there is less consensus among physicists, many of
whom have a sneaking regard for philosophical questions, about the use of the word
‘reality’ for example.
Why be mealy-mouthed when it comes to what is real? The answer lies in quantum
mechanics. Very little happens in physics these days without quantum mechanics having
its say: never has a theory been so prolific in predicting new and astounding effects, with
so vast a scope. But for all its uncanny fecundity, there is a certain difficulty. After a
century of debate, there is very little agreement on how this difficulty should be resolved
- indeed, what consensus there was on it has slowly evaporated. The crucial point of
contention concerns the interface between macro and micro. Since experiments on the
micro-world involve measurements, and measurements involve observable changes in the
instrumentation, it is unsurprising how the difficulty found its name: the problem of
measurement. But really it is a problem of how, and whether, the theory describes any
actual events. As Werner Heisenberg put it, “it is the `factual' character of an event
describable in terms of the concepts of daily life which is not without further comment
contained in the mathematical formalism of quantum theory” (Heisenberg 1959, p.121).
The problem is so strange, so intractable, and so far-reaching, that, with the exception
of space-time philosophy, it has come to dominate the philosophy of physics. The
philosophy of space-time is the subject of a separate chapter: no apology, then, is needed,
for devoting this chapter to the problem of measurement alone.
1. Orthodoxy
Quantum mechanics was virtually completed in 1926. But it only compounded entrenched - a problem that had been obvious for years: wave-particle duality. For a
simple example, consider Young's two slit experiment, in which monochromatic light,
incident on two narrow, parallel slits, subsequently produces an interference pattern on a
distant screen (in this case closely spaced bands of light and dark parallel to the slits). If
either of the slits is closed, the pattern is lost. If one or other slit is closed sporadically,
and randomly, so that only one is open at any one time, the pattern is lost.
There is no difficulty in understanding this effect on the supposition that light consists
of waves; but on careful examination of low-intensity light, the interference pattern is
built up, one spot after another - as if light consists of particles (photons). The pattern
slowly emerges even if only one photon is in the apparatus at any one time; and yet it is
lost when only one slit is open at any one time. It appears, absurdly, as if the photon must
pass through both slits, and interfere with itself. As Richard Feynman observed in his
Lectures on Physics, this is ‘a phenomenon which is impossible, absolutely impossible, to
explain in any classical way, and which has in it the heart of quantum mechanics. In
reality, it contains the only mystery.’
Albert Einstein, in 1905, was the first to argue for this dual nature to light; Niels Bohr,
in 1924, was the last to accept it. For Einstein the equations discovered by Heisenberg
and Erwin Schrödinger did nothing to make it more understandable. On this point,
indeed, he and Bohr were in agreement (Bohr was interested in understanding
experiments, rather than equations); but Bohr, unlike Einstein, was prepared to see in the
wave-particle duality not a puzzle to be solved but a limitation to be lived with, forced
upon us by the very existence of the `quantum of action' (resulting from Planck's constant
h, defining in certain circumstances a minimal unit of action); what Bohr also called the
quantum postulate. The implication, he thought, was that a certain `ideal of explanation'
had to be given up, not that classical concepts were inadequate or incomplete or that new
concepts were needed. This ideal was the independence of a phenomenon of the means by
which it is observed.
With this ideal abandoned, the experimental context must enter into the very definition
of a phenomenon. But that meant classical concepts enter essentially too, if only because
the apparatus must be classically describable. In fact, Bohr held the more radical view
that these were the only real concepts available (they were unrevizable; in his later
writings, they were a condition on communicability, on the very use of ordinary
language).
Less obviously, the quantum postulate also implied limitations on the `mutual
definability' of classical concepts. But therein lay the key to what Bohr called the
`generalization' of classical mechanics: certain classical concepts, like `space-time
description', `causation', `particle', `wave', if given operational meaning in a given
experimental context, excluded the use of others. Thus the momentum and position of a
system could not both, in a single experimental context, be given a precise meaning:
momentum in the range Δp, and position in range Δx, must satisfy the inequality ΔpΔx≥h
(an example of the Heisenberg uncertainty relations).
As a result, phenomena were to be described and explained, in a given context, using
only a subset of the total set of classical concepts normally available - and to neither
require or permit of any dovetailing with those in another, mutually exclusive
experimental context, using a different subset of concepts. That in fact is how genuine
novelty was to arise, according to Bohr, despite the unrevizability of classical concepts:
thus light behaved like a wave in one context, like a particle in another, without
contradiction.
Concepts standing in this exclusionary relation he called complementary. Bohr's great
success was that he could show that indeed complementary concepts, at least those that
could be codified in uncertainty relationships, could not be operationally defined in a
single experimental context. Thus, in the case of the two-slit experiment, any attempt to
determine which slit the photon passes through (say by measuring the recoil, hence
momentum, of the slit) leads to an uncertainty in its position sufficient to destroy the
interference pattern. These were the highly publicized debates over foundations that Bohr
held with Einstein, in the critical years just after the discovery of the new equations; Bohr
won them all.
Bohr looked to the phenomena, not to the equations, surely a selling point of his
interpretation in the 1920s: the new formalism was after all mathematically challenging.
When he first presented his philosophy of complementarity, at the Como lecture of 1927,
he made clear it was based on ‘the general trend of the development of the theory from its
very beginning’ (Bohr 1934 p.52) - a reference to the so-called old quantum theory,
rather than to the new formalism. The later he acknowledged others in the audience
understood much better than he.
It is in the equations that the problem of measurement is most starkly seen. The state ψ
in non-relativistic quantum mechanics is a function on the configuration space of a
system (or one isomorphic to it, like momentum space). A point in this space specifies
the positions of all the particles comprising a system at each instant of time (respectively,
their momenta). This function must be square-integrable, and is normalized so that its
integral over configuration space (momentum space) is one. Its time development is
determined by the Schrödinger equation, which is linear - meaning, if ψ₁(t), ψ₂(t) are
solutions, then so is c₁ψ₁(t)+c₂ψ₂(t), for arbitrary complex numbers c₁, c₂.
Now for the killer question. In many cases the linear (1:1 and norm-preserving, hence
unitary) evolution of each state ψk admits of a perfectly respectable, deterministic and
indeed classical (or at least approximately classical) description, of a kind that can be
verified and is largely uncontentious. Thus the system in state ψ₁, having passed through
a semi-reflecting mirror, reliably triggers a detector. The system in state ψ₂, having been
reflected by the mirror, reliably passes it by. But by linearity if ψ₁ and ψ₂ are solutions to
the Schrödinger equation, so is c₁ψ₁+c₂ψ₂. What happens then?
The orthodox answer to this question is given by the measurement postulate: that in a
situation like this, the state c₁ψ₁+c₂ψ₂ only exists prior to measurement. When the
apparatus couples to the system, on measurement, the detector either fires or it doesn't,
with probability ‖c₁‖² and ‖c₂‖² respectively. Indeed, as is often the case, when the
measurement is repeatable - over sufficiently short times, the same measurement can be
performed on the same system yielding the same outcome - the state must have changed
on the first experiment, from the initial superposition, c₁ψ₁+c₂ψ₂, to either the state ψ₁,
or to the state ψ₂ (in which it thereafter persists on repeated measurements). This
transition is in contradiction with the unitary evolution of the state, prior to measurement.
It is wave-packet reduction (WPR).
What has this to do with the wave-particle duality? Just this: let the state of the photon
as it is incident on the screen on the far side of the slits be written as the superposition
c₁ψ₁+c₂ψ₂+c₃ψ₃+....+cnψn, where ψk is the state in which the photon is localized in the
kth region of the screen. Then by the measurement postulate, and supposing it is ‘photon
position’ which is measured by exposing and processing a photographic emulsion, the
photon is measured to be in region k with probability ‖ck‖². In this way the ‘wave’ (the
superposition, the wave extended over the whole screen) is converted to the ‘particle’ (a
localized spot on the screen). The appearance of a localized spot (and the disappearance
of the wave everywhere else across the screen) is WPR.
Might WPR (and in particular the apparent conflict between it and the unitary
evolution prior to measurement) be a consequence of the fact that the measurement
apparatus has not itself been included in the dynamical description? Then model the
apparatus explicitly, if only in the most schematic and idealized terms. Suppose as before
(as we require of a good measuring device) that the (unitary) dynamics is such that if the
microscopic system is initially in the state ψk, then the state of the joint system
(microscopic system together with the apparatus) after the measurement is reliably Ψk
(with the apparatus showing ‘the kth-outcome recorded'). It now follows from linearity
that if one has initially the superposition c₁ψ₁+c₂ψ₂+..., one obtains after measurement
(by nothing but unitarity) the final state c₁Ψ₁+c₂Ψ₂+..., and nothing has been gained.
Should one then model the human observer as well? It is a fool's errand. The `chain of
observation' has to stop somewhere - by applying the measurement postulate, not by
modeling further details of the measuring process explicitly, or the observers as physical
systems themselves.
These observations were first made in detail, and with great rigor, by the
mathematician John von Neumann in his Mathematical Foundations of Quantum
Mechanics in 1932. They were also made informally by Erwin Schrödinger, by means of
a famous thought experiment, in which a cat is treated as a physical system, and modeled
explicitly, as developing into a superposition of two macroscopic outcomes. It was
upsetting (and not only to cat-lovers) to consider the situation when detection of ψ₁
reliably causes not only a Geiger counter to fire but the release of a poison that causes the
death of the cat, described by Ψ₁. We, performing the experiment (if quantum mechanics
is to believed), will produce a superposition of a live and dead cat of the form
c₁Ψ₁+c₂Ψ₂. Is it only when we go on to observe which it is that we should apply the
measurement postulate, and conclude it is dead (with probability ‖c₁‖²) or alive (with
probability ‖c₂‖²)? Or has the cat got there before us, and already settled the question?
As Einstein inquired, ‘Is the Moon there when nobody looks?’. If so, then the state
c₁Ψ₁+c₂Ψ₂ is simply a wrong or (at best) an incomplete description of the cat and the
decaying atom, prior to observation.
The implication is obvious: why not look for a more detailed level of description? But
von Neumann and Schrödinger hinted at the idea that a limitation like this was inevitable;
that WPR was an expression of a certain limit to physical science; that it somehow
brokered the link between the objective and the subjective aspects of science, between the
object of knowledge, and the knowing subject; that…. Writings on this score trod a fine
line between science and mysticism - or idealism.
Thus John Wheeler's summary, that reads like Berklerian idealism: “In today's words
Bohr's point --- and the central point of quantum theory --- can be put into a single,
simple sentence. ‘No elementary phenomenon is a phenomenon until it is a registered
(observed) phenomenon.’” And Heisenberg’s: the ‘factual element’ missing from the
formalism “appears in the Copenhagen [orthodox] interpretation by the introduction of
the observer”. ‘The observer’ was already a ubiquitous term in writings on relativity, but
there it could be replaced by ‘inertial frame’, meaning a concrete system of rods and
clocks: no such easy translation was available in quantum mechanics.
Einstein had a simpler explanation. The quantum mechanical state is an incomplete
description. WPR is purely epistemic - the consequence of learning something new. His
argument (devised with Boris Podolsky and Nathan Rosen) was independent of micromacro correlations, resting rather on correlations between distant systems: they too could
be engineered so as to occur in a superposition. Thus Ψk might describe a particle A in
state ψk correlated with particle B in state φk, where A and B are spatially remote from
one another. In that case the observation that A is in state ψk would imply that B is in
state φk - and one will learn this (with probability ‖ck‖²) by applying the measurement
postulate to the total system, as given by the state c₁Ψ₁+c₂Ψ₂, on the basis only of
measurements on A. How can B ‘acquire’ a definite state (either φ₁ or φ₂) on the basis of
the observation of the distant particle A? - and correspondingly, how can the probabilities
of certain outcomes on measurements of B be changed? The implication, if there is to be
no spooky `action-at-a-distance', is that B was already in one or the other states φ₁ or φ₂
- in which case the initial description of the composite system c₁Ψ₁+c₂Ψ₂ was simply
wrong, or at best incomplete. This is the famous EPR argument.
It was through investigation of the statistical nature of such correlations in the 1960s
and '70s that foundational questions re-entered the mainstream of physics. They were
posed by the physicist John Bell, in terms of a theory - any theory - that gives additional
information about the systems A, B, over and above that defined by the quantum
mechanical state. He found that if such additional values to physical quantities (`hidden
variables') are local - unchanged by remote experiments - then their averages (that one
might hope will yield the quantum mechanically predicted statistics) must satisfy a
certain inequality. Schematically:
Hidden Variables + Locality ( + background assumptions) ⇒ Bell inequality.
But experiment, and the quantum mechanical predictions, went against the Bell
inequality. Experiment thus went against Einstein: if there is to be a hidden level of
description, not provided by the quantum mechanical state, and satisfying very general
background assumptions, it will have to be non-local.
But is this argument from non-locality, following on from Bell's work, really an
argument against hidden variables? Not if quantum mechanics is already judged nonlocal, as it appears, assuming the completeness of the state, and making use of the
measurement postulate. Bohr's reply to EPR in effect accepted this point: once the type of
experiment performed remotely is changed, and some outcome obtained, so too does the
state for a local event change; so too do the probabilities for local outcomes change. So,
were single-case probabilities measurable, one would be able to signal superluminally
(but of course neither they nor the state is directly measurable). Whether or not there are
hidden variables, it seems, there is non-locality.
2. Pilot-wave theory
The climate, by the mid 1980s, was altogether transformed. Not only had questions of
realism and non-locality been subject to experimental tests, but it was realized – again,
largely due to Bell’s writings, newly anthologized as Speakable and Unspeakable in
Quantum Mechanics - that something was amiss with Bohr's arguments for
complementarity. For a detailed solution to the problem of measurement - incorporating,
admittedly, a form of non-locality - was now clearly on the table, demonstrably
equivalent to standard quantum mechanics.
It is the pilot-wave theory (also called Bohmian mechanics). It is explicitly dualistic: the
wave function must satisfy Schrödinger's equation, as in the conventional theory, but is
taken as a physical field, albeit one that is defined on configuration space E3N (where N is
the number of particles); and in addition there is a unique trajectory in E3N - specifying,
instant by instant, the configuration of all the particles, as determined by the wave
function.
Any complex-valued function ψ on a space can be written as ψ=Aexp iS, where A and
S are real-valued functions on that space. In the simplest case of a single particle (N=1)
configuration space is ordinary Euclidean space E3. Let ψ(x,t) satisfy the Schrödinger
equation; the new postulate is that a particle of mass m at the point x at time t must have
the velocity:
v(x,t)=(h/m)∇S(x,t)
(the guidance equation). If, furthermore, the probability density ρ(x,t₀) on the
configuration space of the particle at time t₀ is given by
ρ(x,t₀)=A²(x,t₀)
(the Born rule) - that is, ρ(x,t₀)ΔV is the probability of finding the particle in volume ΔV
about the point x at time t₀ - then the probability of finding it in the region ΔV′ to which
ΔV is mapped by the guidance equation at time t, will be the same,
ρ′(x′,t)ΔV′=ρ(x,t₀)ΔV.
What does `probability' really mean here? Never mind: that is a can of worms in any
deterministic theory. Let us say it means whatever probability means in classical
statistical mechanics, which is likewise deterministic. Thus conclude: the probability of a
region of configuration space, as given by the Born rule, is preserved under the flow of
the velocity field.
It is a humble-enough claim, but it secures the empirical equivalence of the theory with
the standard formalism, equipped with the measurement postulate, so long as particle
positions are all that are directly measured. And it solves the measurement problem:
nothing particularly special occurs on measurement. Rather, one simply discovers what is
there, the particle positions at the instant they are observed.
The theory was in fact proposed very early, by Count Louis de Broglie, at the Fifth
Solvay conference of 1927. It found few supporters, and not even he was enthusiastic: it
was a flat-footed version of what he was really after, a theory in which particles were
singularities in fields. When it was rediscovered by David Bohm in 1952, the thought was
likewise that it was a step to something more (a solution, perhaps, to the mathematical
pathologies that plagued relativistic quantum theory). As such it languished: Bell was the
first to present the theory for what it was, a complete solution to the problem of
measurement in the non-relativistic arena.
Might orthodoxy have been different, had de Broglie's ideas been championed more
clearly or forcefully at Solvay and subsequently? Perhaps. But the window of opportunity
was small. Paul Dirac and others, architects of the relativistic quantum theory, were
rapidly led to a theory in which the particle number of a given species must dynamically
change. This appeared forced by relativity theory, for reasons internal to the structure of
the new mechanics. Since hugely successful, experimentally, by the mid 1930s there was
a rather decisive reason to reject the pilot-wave theory: for no guidance equation could be
found, then or subsequently, that described change in particle number. The empirical
equivalence of the theory with standard quantum mechanics only extended to nonrelativistic phenomena.
There was, however, another dimension to its neglect. For, if de Broglie's later
writings are to be believed (de Broglie 1990 p.178), what was never clear to him, even
following Bohm's revival of the theory (and, we must infer, what was unclear to everyone
else in this period) was how the pilot-wave theory accounted for WPR. It is that in certain
circumstances, the wave function can be written as a superposition of states, the vast
majority of which at time t can, given a specific particle trajectory at time t, be ignored,
both from the point of view of the guidance equation and for determining the probability
measure over configuration space. This dropping - pragmatically ignoring - components
of the state amounts to WPR. It is an effective process, reflecting a computational
convenience. The point is not difficult to grasp in simple cases - supposing the states
superposed are completely non-overlapping, for example - but it was only implicit in
Bohm's 1952 revival of the theory, and the generic understanding of this phenomenon,
named decoherence in the 70’s by Dieter Zeh, was slow in coming. So too was an
understanding of the true dimensions of the state.
This is the conception that de Broglie had failed to grasp and that not even Bohm had
made clear: that of the wave-function of the universe, a field on configuration space of
vast dimensionality, subject to a continuous process of branching, corresponding to the
countlessly large numbers of possible alternatives sanctioned by decoherence, including
among them all possible experimental outcomes. The branches are non-interfering, or
decohering, to just the extent that they exhibit no interference effects. This, the unitarily
evolving universal state in pilot-wave theory, extends, as it must, to the entire universe. It
is the same wave-function of the universe that one has in the Everett interpretation (§4).
3. State-reduction theories
The pilot-wave theory obviously has certain deficiencies, even setting to one side its
failure in particle physics. Chief of them is the whiff of epiphenomenalism: the
trajectories are controlled by the wave function, but the latter is the same whatever the
trajectory. Relatedly, it is the wave-function, the ‘effective’, local part of it, that is
explanatory of the dynamically properties and relations of quantum systems. Probability,
meanwhile, remains the enigma it has always been classically - but now tied to the Born
rule, a rule presumably to be applied to that first configuration of particles that (together
with the wave-function) made up the initial conditions of the universe.
Subtlety is one thing, malice is another, as Einstein said: the Born probability measure,
like Liouville measure in classical statistical mechanics, ought to admit of exceptions fluctuations away from equilibrium. The experimental implications of non-equilibrium
pilot-wave theory are far-reaching; to suppose they will be forever concealed in perfect
equilibrium smacks of conspiracy. They are so far-reaching, indeed, that they had better
be confined to length-scales so far unexplored: to the Planck length, for example, hence
to the very early universe. Still, there may be signatures of hidden variables written in the
heavens, and waiting to be found.
But what if no such evidence of hidden variables is uncovered? What if no progress is
made with relativistic guidance equations? De Broglie might have posed those questions
in 1927, and probably did: eighty years later, dispiritingly, we are posing them again.
But the alternative is scarcely welcoming. Given that Bohr did not rely on any
distinctively relativistic effects, the very existence of a fully realistic theory, involving
additional equations to the standard formalism and dispensing with the measurement
postulate, able to account for the appearance of WPR, and yielding the same probabilities
as ordinary quantum mechanics, undermines Bohr's arguments for complementarity.
Bohr argued for the impossibility of classical realism, not for its inferiority to idealism. If
pilot-wave theory is such a realism, those arguments cannot stand.
Furthermore, Bohr's positive claims for complementarity now seem implausible. One
of them, for the explanation of novelty even given the restriction to classical concepts,
was supposed to apply whenever the use of such concepts excluded, as a matter of
principle, certain others. Bohr gave as examples the life sciences and psychology, but
nothing came of either suggestion. And the restriction to classical concepts seems wrong,
in the light of decoherence theory and the approach to the classical limit that that
engenders. In term of theories it seems just the reverse: it is quantum theory that seems
better able to mimic the classical, not the other way round.
It is against this backdrop that the advent of dynamical WPR theories should be
assessed. The first theory of this sort, with a claim to genuinely foundational status, is due
to GianCarlo Ghirardi, Alberto Rimini, and Tullio Weber, in 1986. It introduced the idea
of a stochastic process - in the simplest case, of a `hitting' process, under which the wave
function ψ at random times t and at random points q is multiplied by a Gaussian (‘bellshaped’) function well-localized about q. The result (for a single particle) is the
transition:
ψ(x,t)→ψq(x,t)=Kexp(-(1/(2d²))(x-q)²)ψ(x,t)
in which d is a new fundamental physical constant (with the dimensions of length),
determining the degree of localization of the Gaussian, and K is a normalization constant.
A further fundamental constant f determines the mean frequency with which this hitting
occurs. Both are chosen so that for atomic systems the wave function is scarcely changed
(the hits are infrequent, say with mean frequency 10⁻¹⁶ sec, and d is large, say 10⁻⁵m, in
comparison to atomic dimensions).
Two further key ideas are, first, that the probability of a hit at point q is determined by
the norm of ψq (integral of the modulus square of the RHS with respect to x) - this has the
effect that a hit is more likely where the amplitude of the state prior to the hitting is large
- and, second, that when two or more particles are present, each particle is subject to a
hitting process. It follows that the state becomes well-localized at q if the wave function
of any one of the particles it describes is localized about q - that is to say, it is the sum of
the probabilities of any one of its constituents becoming localized at q that matter. For
very large numbers of particles (of the order of Avogadro's number, as comprise anything
like a macroscopic, observable object), even with f as small as 10⁻¹⁶sec, so an individual
atom is hit on average once in a hundred million years, the wave function of a
macroscopic system will become localized in a microsecond or less.
So much for the simplest model of this kind. There are various complications - for
example, it turns out that one constant f is not enough (there is one constant for each
species of particle, where the lighter the particle, the smaller the frequency) - and various
sophistications - the ‘continuous state reduction’ theory of Ghirardi, Rimini, Weber, and
Philip Pearle, which also accommodates particle indistinguishability and the concomitant
symmetrization of the state. But on a number of points the key ideas are the same. There
are of course no measurement postulates; the wave-function, at any instant, is perfectly
categorical - it is the distribution of ‘stuff’ at that time. In conventional quantum
mechanics (if we ask about position) only if the wave function vanishes outside ΔV is a
particle really (with certainty) in ΔV: all that goes out of the window. The distribution of
stuff determines the probabilities for subsequent ‘hits’, but it is not itself probabilistic.
This point tells against the criticism that Gaussians centered on any point q have ‘tails’,
suggesting that a particle thus localized at q is not really (not with probability one) at q.
Unless there is a genuine conceptual difficulty with the theory, the implication is this.
With the minimal of philosophical complications - without introducing anything
epiphenomenal, or a dualistic ontology, or things (trajectories) behaving in ways that
have no operational meaning - merely by changing the equations, the measurement
problem is solved. Therefore it cannot be a philosophical problem: genuinely
philosophical problems are never like that.
But is it true that dynamical state reduction theories are free of conceptual difficulties?
Here is a different difficulty, concerning the tails. Consider e.g., the Schrödinger cat
superposition c₁Ψ₁+c₂Ψ₂. Whilst the hitting mechanism will in a microsecond or less
greatly reduce the amplitude of one term (say Ψ₁, describing the dead cat), in comparison
to the other, it does not eliminate it - it is still there, as described by Ψ₁ (complete with
grieving or outraged pet-lovers, etc.). All that structure is still there, encoded in the state.
But the GRW theory simply denies that structure like this depicts anything - because its
amplitude is so much less than that of the other component.
Whether or not you find this a serious problem will probably depend on your viewpoint
on the Everett interpretation (see below). But unproblematically, uncontroversially,
dynamical state reduction theories face an overwhelming difficulty: there is no relativistic
GRW theory. Whether the problem is a principled one (whether dynamical WPR theories
are in outright conflict with relativity) is debatable, that there is a theoretical problem is
not: we are it seems to laboriously work out the equations of relativistic particle physics
all over again.
4. The Everett interpretation
If this were all there was to say about the foundations of physics, the conclusion would be
deeply troubling: the philosophy of physics would say of physics that it is seriously
confused, in need of revision. From a naturalistic point of view, one might better
conclude that it is the philosophy that is in trouble - specifically, that it is realism that is
in trouble, or if not this, then another fragment of our presuppositions.
Enter the Everett interpretation. Like GRW and pilot-wave theory, it involves wavefunction realism, and like them, it solves the measurement problem. Unlike them, it is
only an interpretation. Crucially, it does not rely on any aspects of non-relativistic
quantum mechanics not available in relativistic theory. So it applies smoothly to the
latter. It demands no revisions.
With so much going for it there had better be a terrible negative. It is that quantum
mechanics under the Everett interpretation is fantastic - too fantastic, perhaps, to take
seriously. For in the face of the unitary development of a superposition of states which, in
isolation, would each correspond to a distinct macroscopic state of affairs, it declares that
all of them are real. It does not look for a mechanism to enhance the amplitude of one of
them over all the others, or to otherwise put a marker on one other than all the others.
Welcome to many-worlds.
Is the approach at all believable? – but we should put this question to one side (how
believable, after all, is classical cosmology?). It was not in any case the usual question
(however much it may have weighed privately); the usual question was whether the
theory was even well-defined. Here some more history is needed.
The achievement of Hugh Everett III, in his seminal paper of 1957, was to show how
branching - the development of a single component of the wave-function into a
superposition - would as a consequence of the unitary evolution give rise to registrations
of sequences of states, as though punctuated by WPR. To this end he considered the
unitary dynamical description of a recording instrument - a device with memory - and the
question of what its memory would contain. What results after branching is a plurality of
recording instruments each with a record of a definite sequence of states (each the
‘relative state’ of the state of the recording instrument at that time). The Born rule defines
a measure over this plurality, much as it did in the pilot wave theory, thus recovering the
usual predictions of quantum mechanics.
The approach, however, has a drawback. It hinted that only registration, or memory, or
consciousness, need be involved in this notion of multiplicity; that, in fact, the theory was
ultimately a theory of consciousness, and to make good on its promise, that it had to
explain why consciousness of branching was impossible.
There is further the objection: what are the probabilities about? In pilot-wave terms
they were about the real trajectory of the universe in configuration space – of which is
actual or real. Uncertainty about chance events always reflects ignorance, it may be
thought. But if Everett is to be believed all such trajectories come about. There is nothing
to be ignorant of.
The interpretation was stillborn in another respect. Branching is basis-dependent meaning, the quantum state can be represented as a superposition with respect to any
orthogonal set of states; which one (which basis) is to be used? Normally this is fixed by
the measurement postulate: the states used represent the possible outcomes of the
experiment. In pilot-wave and GRW theory the multiplicity is, roughly speaking, the
possible particle configurations in E3N. But here Everett made no comment. As framed by
Bryce de Witt, in terms of a multiplicity of universes, the question is more urgent: what is
this plurality, the preferred basis, so called? De Witt along with Wheeler were the
founding fathers of the field of quantum gravity.
The three problems of probability, of consciousness, and of the preferred basis, can all
be linked. Thus, as conjectured by Michael Lockwood, a theory of consciousness (or
consciousness itself) might pick out a preferred basis, and even, according to David
Albert and Barry Loewer, a criterion of identity over time. The latter, Albert and Loewer
insisted, was needed to make sense of probability, of what one is ignorant of (of what will
happen to me). But if these are add-ons to the standard formalism, and idealistic to boot,
they are self-defeating. The selling point of the Everett interpretation is that it is a realist
interpretation, based on physics as is. No wonder it languished in this period.
But with the concept of decoherence, in the early 1990s, came a different solution to
the preferred basis problem. The key to it is that branching and classicality concern only
an effective dynamics, just as does WPR in the pilot-wave theory. Branching and the
emergence of a quasiclassical dynamics go together FAPP (`For All Practical Purposes').
If branching reflects decoherence, and nothing else, no wonder that there is no precise
definition of the preferred basis; no wonder, either, that there is no precise classical limit
to quantum mechanics (no limit of the form h→0), but only effective equations, FAPP,
more or less approximate, depending on the regime of energy, mass, and scale concerned.
This philosophy is moreover continuous with now-standard methodology in the
physical sciences. Thus Kenneth Wilson, winner of the 1982 Nobel prize for physics,
showed how renormalization was best viewed as a demonstrably stable scheme of
approximation, defined by a coarse-graining of an underlying physics that never needs to
be explicitly known. It is the same in condensed matter physics. In philosophy of science
quite generally, on this point there is wide consensus: from nuclear physics to the solid
state and biochemistry, the use of approximations and phenomenological equations is the
norm. Who today would demand that there exist a precise and axiomatic theory of
‘molecules’, for example, to legitimate the term?
But if the preferred basis problem can be answered, the probability problem remains.
Branch amplitudes had better be the quantity to which expectations should be tied, or we
make nonsense of basis for taking quantum mechanics seriously in the first place. Why
should they be? Why the particular function of the amplitudes used in the Born rule? And
the over-riding question: what is the appropriate epistemic attitude to take in the face of
branching? Does it make sense to speak of uncertainty? What are the probabilities
probabilities of?
Defenders of Everett have answers to these questions. For example, to take the last, that
they are the probabilities that things now are thus and so, given that they were such-andsuch then. But whether this is enough to ground an objective notion of uncertainty is hard
to say. If such a notion is available, they can also give reasons why it should take the
quantitative form that it does, in terms of the Born rule. Thus Deutsch, following Bruno
de Finetti's approach to probability, considered the constraints placed on rational agents
by the axioms of decision theory. Let them fix their utilities on the outcomes of quantum
experiments (‘games’) as they see fit; then, if subject to these constraints, their
preferences among games implicitly defines a probability measure over the outcomes of
each game (as that which yields the same ordering in terms of the expected utilities of
each game). Given quantum mechanics, the claim goes, then whatever the choice of
utilities, the only permitted measure is the Born rule.
5. Whither quantum mechanics?
And yet the Everett interpretation remains inherently fantastic. The prospects for a
relativistic pilot-wave theory or state-reduction theory are discouraging. Bohr's doctrine
of complementarity, as something forced by experiment, is no longer credible.
No wonder, in these circumstances, that many look to the frontiers of physics, and
especially to developments, whether theoretical or experimental, in quantum gravity.
There, all are agreed, key concepts of relativity theory or quantum theory or both will
have to give. Others look to frontiers in technology: whatever the deficiencies of
experiments to date to discriminate between the realist solutions on offer, discriminate
they eventually will (taking pilot-wave theory to include the concept of quantum
disequilibrium). Whether at the ultra-microscopic level, of at the boundary between
micro and macro, experiment will ultimately decide.
That, in the final analysis, is what is wrong with Bohr's quietism today. Grant that there
are realist alternatives, and it is reasonable to expect experiment to eventually decide
between them. Bohr could not so much as acknowledge them as genuine alternatives.
There are lessons for neo-Bohrians today, who propose to view quantum mechanics as a
generalization, not of classical mechanics, but of classical probability, or of information
theory: it is not enough to have as their intended outcome a form of quietism; they must
show there are no realist alternatives. There is nothing in their arguments to date to so
much as hint that they can.
References
Bohr, N. (1934), Atomtheorie und Naturbeschreibung (Springer, Berlin, 1931).
Translated as Atomic Theory and the Description of Nature (Cambridge University Press,
Cambridge).
de Broglie, L. (1990), Heisenberg's Uncertainties and the Probabilistic Interpretation of
Wave Mechanics (Kluwer, Dordrecht).
Heisenberg, W. (1959), Physics and Philosophy (Allen and Unwin, London).
Further Reading
Apart from the text already cited, Bohr’s most important writings on foundations is his
‘Is Can quantum-mechanical description of physical reality be considered
complete?’(1935), and ‘Discussion with Einstein on epistemological problems in atomic
physics’ (1949), both reprinted in J. Wheeler and W. Zurek (eds.), The Quantum Theory
of Measurement, Princeton 1983. This is a collection of almost all the most important
writings on the problem of measurement in the first half-century of quantum mechanics.
For a commentary on the debates between Einstein and Bohr, with special attention to
the EPR argument, see A. Fine, The Shaky Game, Chicago, 1988. For the charge that
orthodoxy in quantum mechanics amounted to ‘Copenhagen hegemony’, see J. Cushing,
Quantum Mechanics, Chicago, 1994; for a contrasting view, see S. Saunders,
‘Complementarity and scientific rationality’, Foundations of Physics, 35, 347-72 (2005),
available online at http://xxxx.arXiv.org/quant-ph/0412195. For detailed commentaries
and the proceedings of the 5th Solvay conference in English translation, including an
extensive discussion of the pilot-wave theory, see A. Valentini and G. Bacciagaluppi,
Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference,
Cambridge, 1007; available online at http://xxx.lanl.gov/ hep-th/0610032. For examples
of antirealist approaches today, see A. Zeilinger, ‘The message of the quantum’, Nature
438, 743 (2005), and J. Bub, ‘Quantum mechanics is about quantum information’,
availabe online at http://xxx.lanl.gov/quant-ph/0408020v2/.
For an overview of the measurement problem and much else in physics as it stands
today, see R. Penrose, Ch. 29, The Road to Reality, Jonathan Cape, 2004; for reasons to
think that the decisive solution to the problem (or ‘paradox’ as Penrose calls it) lies in the
realm of quantum gravity, see Ch.30. More introductory is A. Rae, Quantum physics:
illusion or reality, Cambridge, 1986, and, for mathematical beginners who yet want
rigor, R. Hughes, The Structure and Interpretation of Quantum Mechanics, Cambridge,
1989. The first chapter of M. Redhead’s Incompleteness, Non-locality, and Realism,
Oxford, 1987 is a self-contained introduction to the formalism, but at a much faster pace.
The rest of Redhead’s book is a systematic study of the constraints on hidden variables
posed by quantum mechanics, whether by violation of the Bell-inequalities, or by other,
algebraic constraints. Similar ground, but on a more general (geometric and logical)
plane, is covered by I. Pitowsky’s Quantum Probability-Quanum Logic, Springer-Verlag,
1989.
Neither Redhead nor Pitowsky discuss the pilot-wave theory (nor indeed special
relativity) explicitly. For an investigation of quantum non-locality in these contexts see T.
Maudlin’s Quantum Non-locality and Relativity, 2nd ed., Oxford, 2002. See Bell’s
anthology Speakable and Unspeakable in Quantum Mechanics, Cambridge, 1987, for the
dozen or more papers on similar themes that so reinvigorated the debate over foundations
in quantum mechanics. The penultimate chapter ‘Are there quantum jumps?’ remains one
of the clearest published outlines of the then just-discovered GRW theory.
For a recent review of state reduction theory, see C. Ghirardi, ‘Collapse theories’.
The Stanford Encyclopedia of Philosophy (Spring 2002 Edition), E. Zalta (ed.), available
online at http://plato.stanford.edu/archives/spr2002/entries/qm-collapse/. For pilot-wave
theory, see S. Goldstein, ‘Bohmian Mechanics’, ibid, available online at
http://plato.stanford.edu/entries/qm-bohm/. For ‘non-equilibrium’ pilot-wave theory, and
the clear hope of an experimental verdict on the existence of hidden-variables, see A.
Valentini, ‘Black holes, information loss, and hidden variables’, available online at
http://xxx.lanl.gov/hep-th/0407032v1.
For the decoherence-based Everett interpretation, see M. Gell-Mann and J. Hartle,
‘Quantum mechanics in the light of quantum cosmology’, in Complexity, Entropy, and
the Physics of Information, W. Zurek, ed., Reading, 1990. For reviews, see S. Saunders,
‘Time, quantum mechanics, and probability’, Synthese, 114, p.405-44 (1998), available
online at http://arxiv.org/abs/quant-ph/0111047, and D. Wallace, ‘Everett and structure’,
Studies in the History and Philosophy of Modern Physics 34, 87–105 (2002), available
online at http://arxiv.org/abs/quant-ph/0107144. For a review of decoherence theory that
is non-committal on many worlds, see W. Zurek, 1991. ‘Decoherence and the transition
from quantum to classical’, Physics Today 44 (10): 36 (1991); available online, with
added commentary, at http://arxiv.org/abs/quant-ph/0306072.
For Deutsch’s decision theory argument, see ‘Quantum theory of probability and
decisions’, in Proceedings of the Royal Society of London, A455, 3129–3137 (1999),
available online at: http://arxiv.org/abs/quant-ph/9906015, and its subsequent
strengthening by D. Wallace, ‘Epistemology quantized: circumstances in which we
should come to believe in the Everett interpretation’, forthcoming in The British Journal
for the Philosophy of Science, available online from http://philsci-archive.pitt.edu. For
criticism of the Everett interpretation on the ground of probability, see D. Lewis’s ‘How
many tails has Schrödinger’s cat?’, in Lewisian Themes, F. Jackson, and G. Priest, eds.,
Oxford: 2004.