Download How Theory Meets the World

Tim Maudlin
NYU
Physics & Philosophy of Time
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Physics is supposed to be an empirical
theory, i.e. a theory that can be tested against
empirical data.
Fundamental physics is supposed to provide
a complete physical ontology for the world.
Therefore, either the data against which a
fundamental physical theory is tested is not
itself physical, or else it is somehow to be
understood as contained in the ontology of
the theory itself.
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Saying that a theory is tested against
empirical data suggests that the data is
somehow “empirical”.
“Empirical” comes from the Greek “ἐμπειρία”,
which is usually translated as “experience”.
One might then conclude that either
experience is itself a physical thing, and
hence part of the fundamental physical
ontology, or else some extra principle is
required to connect claims about the physical
ontology to the experiential data.
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Both options here seem unpalatable. One
requires giving a physical analysis of
experience itself, and hence solving the
mind-body problem. The other requires some
sort of “bridge principle” from physical claims
to experiential claims, and raises the question
of both the metaphysical status and
justification of the bridge principle.
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In the early 20th century, the school of logical
empiricists tried to dress up the old
empiricist account of meaning found in Hume
and Locke with the modern garb of predicate
calculus. Many (but not all) followed the
principle that all meaningful claims must
either be analytic (and hence “empty”) or
reducible to claims about experience, such as
a particular sort of visual experience (“redspot-here-for-me-now”).
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Since claims about conscious experience do
not follow directly from claims about, e.g.,
the behavior of atoms, some logical
empiricists who wanted to connect physical
claims to data needed bridge principles.
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Consider the following characterization of a “theory”
due to Ernst Nagel, in The Structure of Science:
“For the purposes of analysis, it will be useful to
distinguish three components in a theory: (1) an
abstract calculus that is the logical skeleton of the
explanatory system, and that ‘implicitly defines’ the
basic notions of the system; (2) a set of rules that in
effect assign an empirical content to the abstract
calculus by relating it to the concrete materials of
observation and experiment; and (3) an interpretation
or model for the abstract calculus, which supplies
some flesh for the skeletal structure in terms of more
or less visualizable materials.” (p. 90)
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Nagel’s presentation avoids any direct discussion
of the ontology of a theory. The “skeleton” of the
theory is not an ontological postulate about what
exists, but an “abstract calculus”, presumably a
piece of the mathematics (otherwise, why
“abstract”?). But such an “abstract calculus”
cannot “implicitly define” any physical notion: it
can only implicitly define other abstract,
mathematical notions. So the “skeleton”, on this
picture, contains no physics at all: it is just
mathematics.
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In Nagel’s picture, the abstract calculus comes to
have empirical content, and hence comes to be
empirically testable, only via the second
component, the “set of rules” that “assign
empirical content to the abstract calculus”. These
are sometimes called “bridge laws”, which are
supposed to connect the “unobservable” or
“theoretical” parts of the theory to empirical
phenomena.
With this sort of picture in place, it is easy to
make connections to, e.g., textbook quantum
theory.
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The standard presentation of the quantummechanical calculus in terms of probabilities for
measurement outcomes Nagel’s schema, but it
completely fails to provide any recognizable physical
account of anything. There is no explanation of why
any matrix at all, much less some particular one,
should be associated with a certain laboratory
procedure. There is no explanation of why another
operator should be associated with another
experimental condition. There is no explanation of
how the electron interacts with the experimental
apparatus to produce any particular outcome. There
even fails to be any clear notion of how to apply the
abstract calculus to anything except rather
constrained experimental conditions.
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Let’s consider a completely different approach,
one that does not fit Nagel’s schema well:
Newtonian gravitational theory.
Newton does not begin with any abstract
calculus. He rather begins by describing the
structure of Absolute Space, Time, and Motion
and postulating two sorts of physical magnitude:
the mass of an object and the notion of a force.
With these in hand, Newton can specify his laws
of motion, which we would render as Fnet = mA.
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With these postulates in place, Newton can
now derive what the motions of bodies will be
in certain circumstances. For example, having
postulated the gravitational force law Fg =
GmM/r2, Newton can derive the following
claim about the motion of bodies: if two
bodies of different mass are released
together at the same time above the surface
of the earth, with the only non-negligible
force being that of gravity, then they should
hit the ground at the same time.
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If one understands “empirical content” as having
to do with the content of experience, then
Newton’s theory has not yet been shown to have
any empirical content at all. Nothing has been
derived about anyone’s experience of anything:
all that has been derived is how, according to
Newton’s theory, bodies in certain physical
circumstances will move.
Indeed, it is obvious that no claims about
anyone’s experience of anything follow from the
whole of Newtonian mechanics, supplemented by
whatever force laws one cares to postulate.
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Given appropriate initial conditions, specifying
the masses, positions and velocities of the
planets, together with the assumption that only
gravity produces a non-negligible force,
Newtonian theory predicts the orbits of the
planets. Adding the laws of geometrical optics
(how light moves through empty space), the
theory predicts the exact timing of eclipses
(given initial data). But again, none of this makes
any direct claim about anyone’s experience of
anything.
So how is Newtonian mechanics an empirical
theory?
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The answer to the last question is obvious. Galileo, for
example, reported the result of some experiments he did
in exactly the terms that Newton’s theory predicts: two
objects of different mass, let fall off a tall tower at the
same time, hit the ground (nearly) together. Or again, one
can compare the predictions of Newtonian gravitation to
the observed periods of planetary orbits, times of eclipses,
etc. The language of observational data is the same as that
found in the theory: it reports the (relative) motions and
positions of bodies.
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Note: no “rules” connecting an abstract calculus to
observation are needed. Rather, the theory postulates laws
for items moving in space-time, and the predictions of the
theory are checked against data reporting the same.
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A complete physical theory (as opposed to an
effective practical tool for making predictions)
must contain within it a physical account of the
facts that are taken to constitute the data against
which the theory is tested.
One need not treat an effective device for
calculating predictions as a serious proposal for a
physical theory in this sense. Indeed, an effective
tool for making predictions may not be of the
form of a physical theory with a clear ontology.
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One approach to forging a connection
between the mathematical quantum
formalism of wavefunctions, operators, etc.,
and data was pioneered by Bohr, and is the
foundation of one understanding of the
Copenhagen interpretation.
On the data side, Bohr insisted that the data
of the theory, typically laboratory operations,
be described using “classical language”.
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“For this purpose, it is decisive to recognize that,
however far the phenomena transcend the scope
of classical physical explanation, the account of
all evidence must be expressed in classical terms.
The argument is simply that by the word
“experiment” we refer to a situation where we can
tell others what we have done and what we have
learned, and that, therefore, the account of the
experimental arrangement and of the results of
the observations must be expressed in
unambiguous language with suitable application
of the terminology of classical physics.”
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“As stressed in the [1927 Como] lecture, an
adequate tool for a complementary way of
description is offered precisely by the quantummechanical formalism which represents a purely
symbolic scheme permitting only predictions, on
lines of the correspondence principle, as to
results obtainable under conditions specified by
means of classical concepts. It must be
remembered that even in the indeterminacy
relation [uncertainty relation] we are dealing with
an implication of the formalism which defies
unambiguous expression in words suited to
describe classical physical pictures.”
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“Thus, a sentence like ‘we cannot know both the
momentum and the position of an atomic object’
raises at once questions as to the physical reality of
two such attributes of the object, which can be
answered only by referring to the conditions for the
unambiguous use of space-time concepts, on the one
hand, and dynamical conservation laws, on the other.
While the combination of these concepts into a single
picture of a causal chain of events is the essence of
classical mechanics, room for regularities beyond the
grasp of such a descriptions is afforded by the
circumstance that the study of the complementary
phenomena demands mutually exclusive
experimental arrangements.”
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Bohr’s insistence that classical concepts such
as space-time trajectory or energy-andcausation can only be applied to quantum
systems in certain classically described
circumstances implies that the classically
described circumstances themselves can’t be
the consequences of a quantum-mechanical
description. It is this that yields the
“measurement problem” in its most virulent
form.
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To work our way out of Bohr’s trap we need
for the physics to attribute some features to
systems that obtain independently of their
environment.
These features constitute the “experimental
arrangements” that in turn may play a role in
defining some of the properties of the
“quantum system”.
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It is important to distinguish the two senses of
“classical” that Bohr uses. One sense is tied to
classical dynamics, and describes a system that
has both “space-time” and a “energymomentum” properties that obey classical
equations. Bohr insists that “quantum
mechanical” systems are not “classical” in this
sense.
The other sense is that in which the experimental
conditions must be described in “classical” terms.
This does not and cannot require that the
experimental apparatus obeys classical dynamics.
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Bohr describes the “classical terms” in which
the phenomena (data) are reported as
communicable in unambiguous language.
This, of course, does not demand that the
phenomena are even approximately
“classical” in terms of dynamics. It is also
hard to see what the issue of
“communicability” is: is there a fear that the
phenomena cannot even be described in
natural language?
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Obviously, the language in which the data is
described must be comprehensible natural
language. That is a very weak constraint. The
more difficult and important requirement on
the language of the data is that it be
unambiguously connected to the description
given by the theory. It is only in this way that
the data can be derived from the theory in a
principled way.
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The most obvious way that theory can connect to
data in a principled way has been clearly
articulated by John Bell.
Bell notes that any complete physical theory must
postulate some beables, i.e. physical elements
that are simply there. “In particular we will
exclude the notion of ‘observable’ in favor of that
of ‘beable’. The beables of the theory are those
elements which might correspond to elements of
reality, to things which exist. Their existence
does not depend on ‘observation’. Indeed
observation and observers must be made out of
beables.”
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Beables are needed to end Bohr’s regress. If
some “properties” that a quantum system has
depend on the environmental conditions,
then at least some environmental conditions
must hold of their own accord. These will be
beables in Bell’s sense. And if the
environment is describable in the language of
quantum theory, then there must be some
beables in quantum systems.
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Among the beables, Bell further distinguishes the
local beables.
“These are the mathematical counterparts in the
theory to real events at definite places and times
in the real world (as distinct from the many
purely mathematical constructions that occur in
the working out of physical theories, as distinct
from things that may be real but not localized,
and as distinct from the ‘observables’ of other
formulations of quantum mechanics, for which
we have no use here.”
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“The concept of ‘observable’ lends itself to very precise
mathematics when identified with ‘self-adjoint operator’.
But physically, it is a rather wooly concept. It is not easy to
identify precisely which physical processes are to be given
the status of ‘observations’ and which are to be relegated
to the limbo between one observation and another. So it
could be hoped that some increase in precision might be
possible by concentration on the beables, which can be
described in ‘classical terms’, because they are there. The
beables must include the settings of switches and knobs
on experimental equipment, the currents in coils, and the
readings of instruments. ‘Observables’ must be made,
somehow, out of beables. The theory of local beables
should contain, and give precise physical meaning to, the
algebra of local observables.”
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Note Bell’s use of “classical” in the last quote.
It has nothing to do with classical dynamics.
Rather, the “classical” terms allow one to
describe the experimental conditions at
macroscopic scale in terms of the disposition
of objects in space and time.
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Bell’s point is that the data and description of
experimental conditions is always given in
terms of local beables at macroscopic scale
(“the pointer moved to the right”). If this is to
be connected in a principled way to the
language of the theory, the theory must have
some local beables from which the
macroscopic objects are composed.
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Bell further notes that there is a simple,
transparent, comprehensible way for
macroscopic objects to have locations in
space-time at macroscopic scale: postulate
that they are composed of local beables at
microscopic scale.
The bizarre position defended by Bohr seems
to be that there must be macroscopic
localized objects in a familiar space-time but
they can’t be composed of microscopic
localized parts described by quantum theory.
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Bell explicates Bohr’s theory in terms of a dual
ontology: macroscopic “classical” objects that
define the experimental condition and a
“quantum world” the is associated only with a
symbolic wavefunction.
The significance of the wavefunction is
exhausted by its role in calculating predictions
for the behavior of the classical objects.
The “quantum system” never can be described by
both a space-time and a “causal” vocabulary, but
each individually may be appropriate given
experimental conditions.
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“The kinematics of the world, in this orthodox
picture, is given by a wavefunction (maybe more
than one?) for the quantum part, and classical
variables—variables which have values—for the
classical part: (Ψ(t,q…),X(t)…). The Xs are
somehow macroscopic. This is not spelled out
very explicitly. The dynamics is not very precisely
formulated either. It includes a Schrödinger
equation for the quantum part, and some sort of
classical dynamics for the classical part, and
‘collapse’ recipes for their interaction.”
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“It seems to me that the only hope of precision
with the dual (Ψ,x) kinematics is to omit
completely the shifty split [between classical and
quantum], and let both Ψ and x refer to the world
as a whole. Then the xs must not be confined to
some vague macroscopic scale, but must extend
to all scales. In the picture of de Broglie and
Bohm, every particle is attributed a position x(t).
Then instrument pointers—assemblies of
particles—have positions, and experiments have
results.”
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“The dynamics is given by the world
Schrödinger equation plus precise ‘guiding’
equations prescribing how the x(t)s move
under the influence of Ψ. Particles are not
attributed angular momenta, energies, etc.,
but only positions as functions of time.
Peculiar ‘measurement’ results for angular
momenta, energies, and so on, emerge as
pointer positions in appropriate experimental
setups.”
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If all of this works, then one gets from the
description of the physical world offered by the
theory to the description of the data provided by
experiment without any mention of observers,
observables, systems, apparatuses,
measurements, or anything like that.
The work is done by local beables that are
postulated to constitute the macroscopic objects,
the dynamics (including other beables) provided
by the physical theory, and coarse-graining.
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The methodological role of the local beables
in this picture can be fulfilled by many
different sorts of things at microscopic scale,
including:
Particles
Continuous matter density
Flashes
Fermion Number Density
Strings
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Bell’s local beables are local in space-time, but
he does not specify exactly what sort of spacetime structure he has in mind. We can get a clear
constraint on the space-time structure by
reflecting on the methodology: if we want to
derive the language of the data from a
description in the language of the theory by
coarse-graining, then the theoretical description
of the local beables and the space-time should,
in the right circumstances, coarse-grain to a
familiar, everyday description at macroscopic
scale.
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1: Newtonian Absolute Space and Time
2: Neo-Newtonian Space-Time
3: Newton-Cartan Space-Time
4: Minkowski Space-Time
5: Globally Hyperbolic Solution to the Einstein
Field Equations on a 4-D Manifold
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6: Solutions to the Field Equations with
Closed Timelike Curves, or other nonglobally hyperbolic solutions
7: 10- or 11-Dimensional Space-Time with 6
or 7 Compactified Dimensions Forming a
Calabi-Yau Space
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8: The Holographic Principle, together with
the claim that space-time is “really” just the
“boundary space” and that the “mass” is an
illusion (parallel to how an actual hologram is
really only 2-dimensional but nonetheless
gives the appearance of being 3-dimensional)
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9: The fundamental physical space (for nonrelativistic
quantum
mechanics)
is
geometrically isomorphic to a classical
configuration space
10: “We live in Hilbert space” (Sean Carroll)
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The last two suggestions are qualitatively
completely unlike the first 7. (Let’s leave
aside the Holographic Principle.) According to
them, the fundamental “space” is extremely
high-dimensional, with no four dimensions
picked out at macroscopic scale. In the first
7, the macroscopic structure of space-time is
four-dimensional. So the weird suggestions
cannot be used for our purposes.
Why should weird structures be suggested?
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The inspiration of the last two suggestions is
evidently the quantum state. In nonrelativistic quantum mechanics, the
wavefunction, which represents the quantum
state, is a complex function on a classical
configuration space. And since the
wavefunction must be square-integrable, it
lives in (i.e. is an element of) a highdimensional Hilbert space.
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These last two, radical, positions begin with the
wavefunction (the mathematical representation of
the quantum state) and then retrofit an account
of the fundamental structure of physical space or
space-time that is inspired by the wavefunction.
I want to argue that this strategy, which tries to
derive a physical ontology by focusing on the
mathematical structure of the wavefunction, is a
bad hangover from the historical avoidance of
the “measurement problem”.
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The most straightforward example of this
sort of inference is in “Elementary Quantum
Metaphysics”:
“..it has been essential (that is) to the project
of quantum-mechanical realism…to learn to
think of wave functions as physical objects in
and of themselves.”
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“And of course the space those sorts of
objects live in, the space 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 threedimensional space, or a four-dimensional
space-time) is somehow flatly illusory.”
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“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 state 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 space”.
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If we accept Bell’s approach, which
completely solves the ‘measurement
problem’, we arrive at an entirely different
understanding of the quantum state (the
thing represented by the wavefunction) and
its role in the theory, and therefore an
entirely different understanding from Albert’s
of the significance of the mathematical form
of the wavefunction.
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If the data of the theory are a function of the
distribution of local beables (via simple
coarse-graining to get macroscopic
variables), then the quantum state has
empirical consequences only insofar as it
helps determine the behavior of the local
beables.
That is, the quantum state manifests itself in
the data exactly to the extent that it
influences the configuration of local beables.
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The same conclusion would hold, according
to this analysis, for any non-local beable,
because the non-local beables are not
directly given in the data.
The mathematical form of the representative
of the quantum state must therefore be of a
kind to generate (by some equation) a
specification of how configurations of local
beables (will or might) change. An obvious
mathematical object that can play this role is
a function on configuration space.
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It is essential, in this picture, that the
mathematical space represent the configuration
of local beables in a space-time of the form 1-7,
at least in the present state of the universe. (The
space-time, and hence configuration space,
could be quite different at different times and
places in the history of the universe.)
Note that Albert’s naïve interpretation of the
wavefunction as indicating the fundamentality of
“configuration space” has entirely evaporated.
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If we follow Bell’s suggestions, a clear,
complete physical theory that makes
transparent connections to data will have
three main components:
1) a space-time structure that (now) coarsegrains to a familiar 4-dimensional form
2) a set of postulated physical entities, or
beables, divided into two types: local and
non-local
3) a dynamics for the beables and (ultimately)
the space-time structure
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The beables divide into local and non-local
because, on the one hand, familiar
macroscopic objects, as reported in the data,
should be made of microscopic local beables
and, on the other hand, we want to
implement
the
non-locality
without
superluminal local entities (tachyons).
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The division of beables suggests a parallel
division of dynamics: one for the non-local
beable (quantum state) and another for the local
beables (ultimately including space-time
structure).
In Bohm’s theory, this yields, e.g., Schrödinger’s
equation for the quantum state and the guidance
equation for the local beables.
In a GRW-type theory, it yields a GRW dynamics
for the quantum state and a second equation
specifying the distribution of local beables in
terms of the quantum state.
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GRW-type theories therefore have only one
fundamental dynamical law: the dynamics of
the local beables piggy-backs on that of the
quantum state via the relation between
quantum state and local beables.
Bohm-type theories have the potential of
having only one fundamental dynamical law if
the universal quantum state turns out to be
static. The observable changes that constitute
the data are then generated by the guidance
equation.
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I have not argued that it is impossible to
create a clear, complete physical theory with
a transparent connection between physics
and data in some other way.
But Bell’s basic approach leave a wide scope
for variation in terms of local beables, nonlocal beables, space-time structure, and
dynamical law. Both Bohmian mechanics and
GRW fit easily in the scheme.
Any other sort of approach ought to be able
to explain its basic architecture as precisely.