Download ELEMENTARY QUANTUM METAPHYSICS Once upon a

276
ROBIN COLLINS
DAVID Z ALBERT
NOTES
1
Of course, even among scientific realists it will be a controversial matter in many cases
whether a virtue, such as simplicity or some aesthetic virtue , is really a truth-indicating
theoretical virtue.
2 For the purpose of this paper, the word "probability" is taken to denote any measure
over a space of outcomes that obeys what is known as Miller's rule, a rule according to
which one's degree of expectation (that is, one 's subjective probability) that a sequence of
events E has occurred, conditioned on the supposition that the objective probability of E is
x, is x. (See van Fraassen 1989, 82.) I shall leave open the question as to whether the probability in Bohmian mechanics is to be further interpreted as an epistemic, logical, theoretical ,
or statistical probability, or as a measure of typicality as Diirr eta/. do, or as something
else. (For various other interpretations of probability, and how they apply to probability as
used in statistical mechanics, see Sklar (1993, Chapter 3).)
3 P(q , t)- fP([q , q ], t) dq , where q- [qi> q ] and q represents the position coordinates
2
1
2
2
1
1
of the particles in the subsystem and the integral is taken over all values of q 2•
4
For example , if one interprets the probability in Bohmian mechanics as an epistemic
probability, then one should interpret the probability in this account as an epistemic probability (see footnote (2)).
5 This reversibility objection is the same sort of objection raised against Bolztmann 's original
H theorem by Loschmidt and Zermelo, an objection that was finally answered only by
assuming an equiprobability measure over phase space. (Davies 1-974, 56-61.) I shall show
below that in order to justify P(q, t) ~ '11 2, one cannot avoid introducing an explicit probability measure.
6 See Courant and Hilbert (1989, 28-32).
7
See ibid.
ELEMENTARY QUANTUM METAPHYSICS
Once upon a time, the twentieth-century investigations of the behaviors
of sub-atomic particles were thought to have established that there can be
no such thing as an objective, observer-independent, scientifically realist,
empirically adequate picture of the physical world.
And it was part and parcel of thinking things like that, it was (you
might even say) the essence of thinking things like that, that one looked
at quantum-mechanical wave functions not as representing physical objects
directly, but (say) as representing what observers know of such objects,
or as representing imaginary ensembles of such objects, or as representing
the probabilities of the outcomes of measurements on such objects, or something like that.
And it has consequently been essential to the project of digging one 's
way out of those sorts of confusions, it has been essential (that is) to the
project of quantum-mechanical realism (in whatever particular form it takes
- Bohm's theory, or modal theories, or Everettish theories, or theories of
spontaneous localization), to learn to think of wave functions as physical
objects in and of themselves. 1
And of course the space those sorts of objects live in, and (therefore)
the space we live in, the space in which any realistic understanding of
quantum mechanics is necessarily going to depict the history of the world
as playing itself out (if space is the right name for it - of which more
later) is configuration-space. And whatever impression we have to the
contrary (whatever impression we have, say, of living in a three-dimensional
space, or in a four-dimensional space-time) is somehow flatly illusory.
I learned all this (insofar as I can reconstruct it now) from a few casual
remarks scattered here and there in varigus papers and private communications of John Bell. And it has seemed so straightforward and so ineluctable
to me since then as not to merit any further discussion.
But it turns out not to have seemed that way to everybody. It turns out
(as a matter of fact) that this sort of talk still frequently manages to surprise
people, even to appal them.
Maybe there's a point, then, in writing it all down in some detail -just
to see how it looks when we're done, and to give its critics something
convenient to shoot at.
278 '
DAVID Z ALBERT
I.
REALITIES
The sorts of physical objects that wave functions are, on this way of
thinking, are (plainly) fields - which is to say that they are the sorts of
objects whose States one specifies by specifying the values of some set of
numbers at every point in the space where they live, the sorts of objects
whose states one specifies (in this case) by specifying the values of two
numbers (one of which is usually referred to as an amplitude, and the
other as a phase) at every point in the universe's so-called configuration
1
space.The values of the amplitude and the phase are thought of (as with all
fields) as intrinsic properties of the points in the configuration space with
which they are associated. And so (for example) the fact that the integral
over the entirety of the configuration of the square of the amplitude of
the universe's wave function is invariably equal to one is going to have
to be thought of not as following analytically from the sorts of physical
objects wave function are (which it certainly can not), but as a physical law,
or perhaps as an initial condition.
What physical role this object plays in the world, precisely, will depend
on precisely how the measurement problem gets solved.
On Bohm 's theory, for example, the world will consist of exactly two
physical objects. One of those is the universal wave function and the other
is the universal particle.
And the story of the world consists, in its entirety, of a continuous succession of changes of the shape of the former and a continuous succession
of changes in the position of the latter.
And the dynamical laws that govern all those changes - that is: the
Schrodinger equation and the Bohmian guidance condition- are completely
deterministic, and (in the high-dimensional space in which these objects
live) completely local.
The correspondence with our ordinary, three-dimensional, multi-particle
language is trivial. Every particle invariably has a perfectly determinate
position, and (more particularly) the x, y, and z co-ordinates of particle m
are, respectively, the (3m - 2)th, the (3m - I )th, and the (3m)th coordinates of the world-particle.
On the GRW theory (or for that matter on any theory of collapse), the
world will consist of exactly one physical object - the universal wave
function. What happens, all that happens, is that that function changes its
shape in accord with the theory's dynamical law s. And those changes are
not entirely continuous, and the Jaw s which govern them are not entirely
deterministic, 3 and (even in the high-dimensional configuration space) not
entirely Jocal. 4
The correspondence with the three-dimensional multi-particle language
is a bit more complicated. Talking about particles, in the context of these
ELEMENTARY QUANTUM METAPHYSICS
279
sorts of theories, is a way of talking about the wave function's shape. In
particular, locutions like "particle m is located in the (finite) region bounded
by x 1 and x 2 and y 1 and y2 and z 1 and z2" mean nothing more or Jess than
that the universal wave function is (at the moment in question) bunched
up in such a way that almost the entirety of its squared amplitude is confined
to the (infinite) region between the x 1 and x 2 values of the (3m - 2)th coordinate, and between the y 1 and y2 values of the (3m - l)th co-ordinate,
and between the z 1 and z2 values of the (3m)th co-ordinate.
Note, by the way, that the above understanding of locutions like "particle
m is located in the region bounded by x 1 and x 2 and y 1 and Y2 and z 1 and
z2 " counts such locutions as ineluctably vague. And there is no point in
denying that this comes as something of a surprise. And it is surely worth
recalling, at this juncture, that the meanings of locutions like that are perfectly precise on Bohm's theory. But there's nothing enormously mysterious
here either, and (more particularly) none of this represents any fundamental impediment to the project of describing the world exactly. The world
invariably has a perfectly exact description, on these theories, in the
language of the wave function; and particle-talk is inexact purely and simply
because it amounts to an inexact description of that. 5
One more example will suffice, I think.
Consider, then, what you might call a 'maximally atomistic' version of
a modal interpretation of quantum mechanics, in which a determinate
property is assigned, at every instant, to every separate co-ordinate-space
degree of freedom, by means of a bi-orthonormal decomposition. 6
On that sort of a theory, the physical world will consist, in its entirety,
of two wave functions. One of those evolves in accord with the Schrodinger
· equation, and the other (which will have its own, separate, laws of evolution) is at every moment a simultaneous eigenfunction of all of the
(commuting) Hermetian operators picked out by the above full set of decompositions.
Particle-location talk will supervene, here, on the second of these wave
functions; and the rules of that supervenience will be precisely the ones
we have just now discussed, precisely the same (that is) as the rules whereby
particle-location talk supervenes on the unique universal wave function in
the GRW theory. 7
2.
APPEARANCES
Those, then, are the rules whereby the three-dimensional multiple-particle
language of our everyday lives supervenes on the exact and complete and
fundamental language of the world, which is the language of wave functions (and whatever else) in configuration space.
But what is it about the world that can have suggested that everyday
language to us - that false language, that mirage - in the first place?
What it is (I think) is the world's Hamiltonian; what it is is that the
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DAVID Z ALBERT
ELEMENTARY Q U ANTUM METAPHYSICS
world has the sort of potential-energy operator whose expectation-value
gets big exactly when (on that three-dimensional language; but not on any
two-dimensional language, and not on any one-dimensional language, and
not on any five-dimensional language) the expectation values of the "distances between particles" gets small.
material particles which are actually floating around in it, can be distinguished as uniquely natural and reasonable and elegant and whatever else
you might like: the hypothesis that we are looking at a three-dimensional
space, in which N/3 distinct material particles are floating around; the
hypothesis (that is) is that the potential terms in this Hamiltonian represent an interparticle force whose intensity depends on the distan ce between
the particles in question.
Now (to put all this slightly differently) there is suddenly an available
conception of inter-particle distance - the three-dimensional one - which
is capable of playing a meaningful physical role; now there is an available conception of inter-particle distance which (other things being equal)
reliably measures the degree to which the particles in question can dynamically affect one another, can alter one another's trajectories.
Now we're dealing with a world that might reasonably be expected to
look to its inhabitants (if there are any) as if it's three dim ensional; now
(that is) we are dealing with a world whose laws might well entail, over
some reasonably broad range of initial conditions, that the world-particle
will eventually make its way into regions of the configuration-space which
correspond- in the three-dimensional language- to the existence of physics
textbooks, written by reputable experts, which contain inscriptions like "the
dimensionality of our world appears to be three".
Moreover, different sorts of modifications of the Hamiltonian can patently
generate different such appearances. If, for example, we had replaced (I)
not by (2) but by:
Let's think that through in some detail.
Let's think first (because it's a little simpler) about the classical case.
Let's think about how it happens that a Newtonian-mechanical universe
acquires the look of having some particular number of ordinary spatial
dimensions.
Consider, then, an N-dimensi onal classical-mechanical configuration
space, in which a single world-particle is floating around.
And suppose, to begin with, that that world-particle is floating around
freely; suppose (that is) that its Hamiltonian is:
N
(I)
H
=
L
(p;) 2 .
i - I
And note that the trajectory of a world-particle like this one can patently
contain no suggestion whatever as to whether we are dealing here with a
single material particle moving freely in anN-dimensional physical space,
or (say) N/3 distinct material particles moving freely in a three-dimensional physical space, or N/2 distinct material particles moving freely in a
two-dimensional physical space, or N distinct material particles moving
freely in a one-dimensional physical space. Nothing about a trajectory like
that (to put it slightly differently) can make it natural or make it plausible
or make it reasonable or make it simple or make it elegant or make it any
other desirable thing to suppose that any particular one of those possibilities, as opposed to any one of the others I mentioned, or any one of the
others I didn't mention, actually obtains.
The very question (in this context) seems pointless.
But try adding something to the dynamics - something along the lines
of an interaction.
Imagine, for example, an N-dimensional classical configuration-space
in which a world-particle is floating around not under the influence of the
free Hamiltonian in (l ), but under the influence of a more complicated
Hamiltonian of the form :
N
(2)
H
=
I
i - I
N/3
(py +
I
k,j -
I
V1k([(x3j- 2 - x 3k_2)2
k" j
Now, all of a sudden, one particular hypothesis about the number of
physical dimensions this space actually has, and about the number of distinct
N/2
(3)
H = i~l (py + j,kl:
j
~
I
vjk([(x 2j-l -
x2k- 1) 2
+ (x2j -
x 2k) 2]1 /2)
k
we would have made a world which appears to its inhabitants not to have
three dimensions, but two .
Good. Let's get back (with all this in mind) to quantum mechanics.
Consider, to begin with, a quantum-mechanical world (any quantummechanical world you like: a GRW world, a Bohm world, a modal world ,
a many-minds world, whatever) whose configuration space is N-dimensional, and whose Hamiltonian (considered now, of course, as an operator)
is the one in Equation ( 1).
Questions about the dimension of this world's classical counterpart
(remember) turned out to be silly; it turned out that nothing in any of the
possible trajectories of a world like that could imaginably tip the scales
in any particular direction. But the quantum mechanical case - as we discussed a few pages back - is going to be different: the set Q.f all possible
trajectories of a quantum-mechanical world with the Hamiltonian in (1)
is simply not going to be representable on a space whose dimension is
smaller than N. 8 And precisely the same thing will be true of a quantum-
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ELEMENTARY QUANTUM METAPHYSICS
DAVID Z ALBERT
mechanical world with the kind of Hamiltonian in Eq. (2) (notwithstanding
the fact that a classical world with a Hamiltonian like that turns out to
have three dimensions); and precisely the same thing will be true of a
quantum-mechanical world with the kind of Hamiltonian in Equation (3)
(notwithstanding the fact that a classical world with a Hamiltonian like
that turns out to have two dimensions).
But appearances are patently going to be another matter. How any particular quantum-mechanical world appears to its inhabitants, or at any
rate how it appears to them if they don't look too closely, is going to be
part and parcel of all of the usual arguments that insofar as so-called
' familiar macroscopic objects' under so-called 'familiar macroscopic circumstances' are concerned, the predictions of classical mechanics and the
predictions of quantum mechanics are going to correspond with one another.
Part of what has always followed from arguments like that (although we
have perhaps not been in the habit of thinking about them in these terms)
is that quantum-mechanical worlds are going to appear (falsely!) to their
inhabitants, if they don't look too closely, to have the same number of spatial
dimensions as their classical counterparts do. Part of what has always
followed from them is (for example) that a quantum-mechanical world
with a Hamiltonian like the one in (3) will appear to its not-too-closelylooking inhabitants to have two dimensions, and that a quantum-mechanical
world with a Hamiltonian like the one in (2) will appear to its not-tooclosely-looking inhabitants Uust as our own does) to have three.
That's what (I think) I think. But now that it's all on the table, maybe there's
a point in trying to put it a bit more diplomatically.
There are (you might say) two ideas we're accustomed to having in
mind when we think of 'physical space'.
There is, to begin with, the space of possible interactive distances, the
space (if there is one) that one reads off of the formal Pythagorean relations among the individual terms in the world's Hamiltonian - irrespective
of whether those terms are considered as classical variables or quantummechanical operators. The space of interactive distances of any classical
or quantum-mechanical world with a Hamiltonian like the one in (2), then,
is three-dimensional; and the space of interactive distances of any classical or quantum-mechanical world with a Hamiltonian like the one in (3)
is two-dimensional, and the space of interactive distances of any classical
or quantum-mechanical world with a Hamiltonian like the one in ( 1) has
no particular dimension.
And then there's an altogether different idea (and an altogether more
fundamental one, it seems to me; but let's not squabble about that for the
moment) of an arena within which the dynamics does its work, a stage
on which whatever theory we happen to be entertaining at the moment
depicts the world as unfolding: a space (that is) in which a specification
of the local conditions at every address at some particular time (but not
283
at any proper subset of them) amounts to a complete specification of the
physical situation of the world, on that theory, at that time.
And it just so happens that in the context of classical physics those two
spaces invariably coincide; 9 and it just so happens that in the context of
quantum physics they don't. 10 And so the habit of thinking of the two of
those spaces together, and of calling them by a single name, turns out to
be one more of the things that we are apparently going to have to learn
to give up.
Columbia University
NOTES
1
That wave functions have the ontological status of physical objects is often claimed as a
recent scientific discovery, as something that follows (more particularly) from the work of
Aharonov and Vi adman ( 1993) on so-called "weak" measurements.
But that can't be right. Aharonov and Viadman's very beautiful discoveries can (after
all) perfectly well be accommodated within any anti-realist interpretation of quantum
mechanics; and insofar as we are committed to realism, there was simply never anything other
than physical objects that wave functions could have been.
2
Or at any rate that's how things would work for a universe of elementary particles whose
spins are all zero. In more realistic cases, we shall of course have to specify the values of
some larger set of quantities at every point, but you get the idea.
3
Note, however, that to say that the laws of the evolutions of wave functions are probabilistic (which is perfectly true , on theories like GRW), is not at all to say that those wave
functions are somehow probabilities themselves, or that quantum mechanics somehow confronts us with a new and utterly mysterious modality of 'potentia' or ' possibilia' (which is
gibberish). On any realistic understanding of quantum theory, wave functions are never
anything more or less than perfectly actual, perfectly low-brow, field-configurations.
4
The probability that some particular GRW collapse will multiply the value of the world's
wave function, evaluated at some particular point in the world's configuration space, by
some particular number, will depend (after all) on the wave function of the world throughout
the entirety of that configuration space (and not just at the single point in question) at the
instant just before the collapse occurs.
5
Barry Loewer and I have gone into these matters in some detail (Albert and Loewer
1996).
6
This is a theory which perhaps nobody will be tempted to take very seriously (on thi s
theory, for example, the laws governing which Hermitian operators are assigned determinate values, at any particular moment, will not be invariant under rotations in ordinary
three-dimensional space) but it will do perfectly well for our purposes here - it will leave
the reader in no doubt whatever, I think, about how this sort of talk is to be extended to
any modal theory she likes.
7
Note that all of the pictures of the world I have been discussing here are what Paul Teller
(1989) calls 'particularist' ones: ones in which (as Teller puts it) the world is composed of
individuals, and the individuals have non-relational properties, and all relations between
individuals supervene on the non-relational properties of the relata. The individuals of which
the world consists on the GRW theory are the wave-function field s at all points in configuration space, and the individuals of which it consists on Bohm 's theory are those fields
together with the world-particle, and the individuals of which it consists on Modal theories
are two fields like that; and pinning the conditions of all of those individuals down, on any
of those theories, pins down everything.
TIM MAUDLIN
DAVID Z ALBERT
I
I
I
What Teller thinks, of course, is that quantum-mechanical worlds can't be particularist
ones; but (on the picture of the world we've been trying out here) that's because he misunderstands what the individuals of worlds like that are, that's because he supposes that
the individuals of worlds like that have got to be the same sorts of things that they are in
classical physics, things like elementary particles, or fields at points in 3-space, or something like that.
g
That is: for any co-ordinatization whatever of the configuration space in question, that
set will include trajectories which pass through states which are completely non-separable
in those co-ordinates, states which cannot be expressed as products of functions of any
proper subset of those co-ordinates.
9
It just so happens (that is) that insofar as classical-mechanical theories of the world are
concerned (and this is perhaps a way of putting one 's finger on what it is that 's distinctive
about such theories) whatever amounts to a space of possible interactive distances amounts
to a stage-space too.
10
The geometries of the interactive-distance-spaces of quantum theories depend (by definition - and precisely as they do in the classical case) on those theories' Hamiltoniwis , but
the geometries of their srage-spaces don't. The stage-space appropriate to a quantum theory
can (alas) never be anything other than that theory's configuration-space.
J'hat things can appear otherwise, if one doesn't look too closely, is a peculiarity of the
wave functions of macroscopic objects. What happens is that the wave function of the world
(on theories like GRW) or the effective wave function of the world (on theories like Bohm 's)
tends to be approximately separable in the centers-of-masses of objects like that, that the
wave function of the world , or its effective wave function, tends to be approximately of
the sort that can in fact be represented in the theory's interactive-distance space. And what
makes that happen, in the GRW case, is the collapse part of the dynamics; and what makes
it happen in the Bohm case is the part that governs the interactions of such objects· with
their environments.
SPACE-TIME IN THE QUANTUM WORLD
I.
NON-LOCALITY
Consider pairs of electrons produced in the singlet state and allowed to
separate to a very great distance. We know from the work of Bell that no
theory can predict violations of Bell's inequality for spin measurements
on those pairs if it satisfied two conditions. The first (condition a) is that
the theory ascribe either a single initial state or a convex sum of states to
the ensemble of pairs such that the initial states are statistically independent
of the spin measurements later carried out on the electrons. The second
(condition b) is most easily understood if expressed differently for deterministic and for stochastic theories. For a deterministic theory we require
(condition b') that the theory determine the results of each measurement
solely on the basis of the initial state and the details of the measurement
carried out on that particle. For a stochastic theory we demand (condition
b*) that the theory assign probabilities for measurement results based solely
on the initial state and the measurement carried out on a single electron,
which probabilities are unchanged when one conditionalizes on the measurements and results obtained on the other particle in the pair. (Conditions
b' and b* are really the same condition, expressed in the one case appropriately to a deterministic theory, in the other for an indeterministic one.
A theory which violates b' also violates b* since conditionalizing on information about the measurement carried out on the second electron can render
the result of the first measurement certain.) Einstein et al. ( 1935) had already
pointed out the impossibility of a non-deterministic theory which obeys a
and b* to recover the predictions of quantum theory, and so argued in
favor of a deterministic theory. Bell showed that no deterministic theory
obeying a and b' could recover all of the predictions of quantum mechanics.
The later work of Greenberger et al. (1989) showed that the reference to
ensembles is otiose: there can be individual triples of particles such that
no initial state of type a can recover all of the predictions of quantum
mechanics if the theory is of type b' or b*.
These results take on a slightly different cast when embedded into spacetime theory. Since the settings of the measurement devices can be performed
in the absolute future of the creation of the electrons, and since the settings
can be determined independently of the process which creates the electrons (e.g., the setting could be determined by a computer program running
a complex algorithm), any theory which denies condition a must posit backwards causation: the ineliminable asymmetric dependency of absolutely
earlier states on absolutely later events. And since the measurement events