|
|
Quantum Weirdness III:
Bell's Inequality
Randomness and Hidden Variables
The Copenhagen Interpretation views the randomness of
quantum phenomena as an irreducible aspect of nature. In the
double-slit experiment, for example, we know that there is
high probability of an electron being detected at certain
locations, and a low or zero probability of it being
detected at other locations. Since one cannot
observationally "track" the electron before it strikes
the detector (without destroying the interference pattern),
it is unscientific speculation to ask why this particular
electron struck the screen at this particular
location. The observation is simply that it did; end of
story.
Other examples of randomness in nature, however, are
explainable in terms of deterministic processes, which are
simply impractical to track in sufficient detail to predict
exact outcomes. For example, a hand of cards delt from a
shuffled deck can be treated as a random sample, and
discussed probabilistically. But if one knew the minute
details of each step in the suffling process, then the
position of every card would be known, and the contents of
the draw predictable in advance. The most notable example of
this in the history of science is the kinetic theory of
gases and the field of statistical mechanics, which explains
the thermodynamic properties of macroscopic objects in terms
of the deterministic motions of individual molecules.
Could it be that quantum randomness also reflects
deterministic behavior at a level too difficult to observe
and track, but still thoroughly real? We've already
encountered this suggestion in David Bohm's interpretation
of the wave function as a field that guides the motion of
the particles, such as electrons, casuing them to strike the
detector at locations consistent with the probabilities
given by quantum mechanics. The class of theories that
attempt to account for quantum probabilities in this way are
called hidden variable theories, the idea being that
the outcomes of particular experiments are not fundamentally
random, but determined by the values of variables that are
unknown to the experimenter. Hidden variable theories are
aimed at accounting for the interference seen in
superposition states without abandoning the philosophical
orientation of metaphysical realism.
The prospect of devising a satisfactory hidden variable
theory was prematurely dismissed in the early days of
quantum mechanics. John von Neumann published a proof that
no hidden variable theory could be made consistent with the
prediction of quantum mechanics. This proof was accepted for
some time, but was eventually found to be in error.
The current status of hidden variable theories is framed
by an influential paper by John Bell, which acknowledges
that hidden variable theories are possible, but only if the
theories are nonlocal, meaning that changes in the
quantum system are conveyed from place to place faster than
the speed of light, in violation of Einstein's special
relativity theory.
Coupled Particles
Bell's work addressed the case of coupled particles,
first used by Einstein, Podolsky, and Rosen to argue that
quantum mechanics must be an incomplete theory. Consider an
atomic process that produces two photons of light, emitted
in opposite directions, but with the same
polarization. (Polarization is the direction of the
electic field associated with a light wave. You might
picture the photon as a frisbee, that can sail along
horizontally, as most frisbees do, or tipped up at any
arbitrary angle, including vertical.)
Polarization is measured by passing the photon through a
polarizer. If the polarizer is oriented parallel to the
polarization of the photon, the photon passes through
unimpeded. If it is oriented perpendicular to the
polarization of the photon, the photon is absorbed. At an
intermediate angle, the photon will have a certain
probability of being transmitted.
The interesting thing is that there are two
photons with the same polarization, moving apart from
one another at the speed of light. If the coherence of the
system can be maintained, a measurement made at one location
will assign a single definite state to the system, which
should be verified by any subsequent measurements, no matter
where they occur. Whether the first photon passes through
the polarizer or is absorbed completely determines the
polarization of both photons from that point onward. The
polarizer can be thought of as asking the system "are you
polarized at angle x?" and accepting only a yes/no
answer, which the system is then obliged to repeat
consistently thereafter.
In the Copenhagen interpretation, nothing can be said
about the angle of polarization of the photons before the
measurement. The angle of polarization has no objective
reality until it is measured, and the measurement result is
not predetermined by anything, it is intrinsically random.
After it is measured, then the polarization state of both
photons is completely known and will not change.
In a hidden variable theory, there is a variable or
variables that determine the real polarization angle of the
photons; these variables have definite values from the
moment the photons are formed, and they determine what the
result of the polarization measurement will be.
An interesting thing happens if we make the polarization
measurement on the second particle at a different angle from
our measurement on the first. Now the two photons will have
different probabilities of passing through their respective
polarizers. We can expect that sometimes photon A will
pass through its polarizer, but photon B will be absorbed by
its polarizer. We can call such events errors, not in
the sense of mistakes, but just in the sense that there is a
disagreement between the measurements of the two
polarizers.
 Suppose
the polarizer at A is rotated by an angle x relative
to the polarizer at B. A certain number of errors are
produced, say E. Now what happens if we rotate the B
polarizer through an angle x in the opposite
direction, so that the total angle between them is
2x? We expect more errors, but how much
more? Twice as many? Bell demonstrated that the error
rate with an angle 2x must be less than or equal to
twice the error rate at angle
x: E(2x)<=2E(x),
provided that the photons have a definite polarization (as
postulated by the hidden variable theories) and that the A
photon cannot affect the B photon's state
instantaneously (this is the requirement of
locality). Although it may not be immediately obvious
why this inequality must hold true for all local hidden
variable theories, you can get some intuitive sense of its
correctness by imagining both polarizers being rotated while
the photons are in flight. Before they are rotated, the
error rate will be a minimum. Rotating polarizer A causes
some of the photons striking it to be absorbed when they
would have been transmitted, or vice versa. This increases
the error rate to E. Because of symmetry, we would
expect exactly the same thing to happen if we left
A fixed and rotated B through the same angle in the
opposite direction. With both polarizers rotated, we should
get both contributions to the total error rate, but
reduced by a certain number of cases where the errors
cancel (for example, a pair of photons that were both
transmitted when the polarizers were parallel, but are both
absorbed when both polarizers are rotated; since this is two
absorptions, it would not count as an error). The
self-cancelling errors might be very few, even 0, but cannot
be negative, so you could never get more than twice
the error rate by doubling the angle between polarizers.
Although this inequality is rather subtle, it has a
profound implication, because quantum mechanics does
predict that the error rate may more than double
when the angle between the polarizers is doubled.
Why? Because when the first measurement is made at
polarizer A, the entire system enters a definite state
corresponding to that measurement. If the photon at
A passes through the polarizer, then the photon at B
must have exactly the same polarization as the one at A.
Depending on the angle, this can greatly reduce the
probability of transmission for photon B, and increase the
error rate above the limit given by Bell's inequality.
The Experiment and Its Implications
Bell's Inequality has been tested experimentally numerous
times, to an astounding degree of precision. Photon pairs
have been separated by distances as large as 15 km, and the
polarizers rotated with such precision timing that there is
insufficient time for any physical signal to be transmitted
between the first photon and the second. The predictions of
quantum mechanics are confirmed; Bell's Inequality is
violated. What does this mean for the status of hidden
variable theories? It means that for any hidden variable
theory to be consistent with experiment, the principle of
locality must be violated. The theory must allow the
instantaneous transfer of information across arbitrarily
large distances. One might say that subatomic particles are
required to "telephathically" communicate with other
particles kilometers away.
We seem to be left with a choice of interpretations,
neither of which accords well with the naive realism of
classical physics. Either subatomic particles are capable of
a mysterious form of instantaneous communication, requiring
no exchange of energy and no transmission time, or else we
may follow the Copenhagen interpretation and regard the
properties of the particles as having no objective reality
until they are measured. Until hidden variable theories are
developed with testable predictions that differ from
standard quantum mechanics, there is no scientific basis for
favoring one interpretation over another. It is a matter of
esthetics.
To many physicists, however, the hidden variable approach
has the look of an ad hoc complication to a theory
that doesn't need it. The addition of hidden variables and
guiding waves help solve no scientific problem nor do they
simplify any calculation. They might be compared with
Ptolemaic epicycles, which complicate the theory of the
solar system but maintain consistency with a philosophical
doctrine from outside of science, that of the centrality of
the Earth. Likewise hidden variable theories allow us to
maintain a position of metaphysical realism and reject the
epistemological view of reality offered by the Copenhagen
interpretation. On the other hand, the Copenhagen
interpretation can be rightly criticized as antithetical to
the inquisitive spirit of science, by rejecting out of hand
any speculation about lower-level deterministic process
behind the phenomenon of quantum randomness.
For myself, I do not see any reason to think that quantum
mechanics is the final word on scientific understanding of
the microscopic world, and hidden variable ideas should
certainly be pursued as long as there are interested
researchers to do so; we cannot know in advance whether new
important insights will emerge from such work or not.
However, I also think the burden is on the hidden
variable theorists to show that their approach has actual
scientific value, and is not just an elaborate
complication introduced for the sole purpose of saving a
philosophical belief that is not itself a part of
science.
In the meantime, though, I think it is fair to regard the
Copenhagen interpretation as the cleanest and most natural
framework for understanding and using the equations of
quantum mechanics. If one thinks that the Copernican system
is a good reason for abandoning geocentrism, and that
relativity theory is a good reason for abandoning concepts
of absolute space and time, then quantum mechanics would
likewise seem to be a good reason for abandoning
metaphysical realism in favor of a more epistemological view
of the world.
Quantum Mechanics and Mysticism in Popular Culture
If quantum mechanics challenges naive metaphysical
realism, does it consequently validate every mystical,
magical, or spiritual speculation we can think of? A
visit to the new age section of your local bookstore might
lead you to think so. Quantum mechanics actually has very
little to offer in providing a scientific mechanism for
puported phenomena such as ESP, divination, telepathy,
synchronicity, channelling, out-of-body experiences, and so
on. As we have seen, quantum weirdness and the
"interconnectedness" it seems to imply requires coherent
superposition states, which are really not to be encountered
outside the carefully controlled conditions of a physics
laboratory. Our day-to-day experience is in the world of
macroscopic objects with definite single states.
Furthermore, it can be demonstrated that the quantum
interconnectedness exhibited in Bell's experiment, for
example, actually cannot be exploited to send messages from
place to place faster than light. The correlation between
the results of the two measurements is something that can
only be recognized with records of the two observations are
brought together and compared. Until this is done, the
observer at B remains clueless as to what the observer at
A is doing.
Despite the inadequacy of the pop-culture attempts to
link quantum mechanics with mystical and magical phenomena,
I do think quantum weirdness has some relevance to matters
of religion, philosophy, and spiritual experience. For many
centuries, western thinking has been dominated by the idea
of an objective reality, completely laid out in advance,
just waiting for us to learn about it. Quantum mechanics
tells us there is something wrong with this
picture--although there are different opinions as to what
exactly is wrong about it. It appears that when something is
truly left to itself--a system small and isolated
enough to preclude even the possibility of observation--its
properties lose their objective reality, leaving us with no
way to make an adequate mental picture of what it is doing.
It is as if Nature doesn't bother to keep track of details
that aren't needed; she sometimes doesn't have an answer
ready until the question is asked. Sometimes it is human
beings who ask the questions; more often it is some
macroscopic amplification process that does it for us.
This picture undermines metaphysical reductionism, which
would have us believe that macroscopic objects depend on
microscopic ones, but microscopic objects do not depend on
the macroscopic world. On the contrary, it seems that the
two levels depend on each other. From our human perspective,
at least, things like Geiger counters and photographic
plates seem to be necessary to get electrons to decide where
they are and what they're doing there. Neils Bohr and some
of his colleagues always kept a clear focus on the
macroscopic world of meter readings and other phenomena
available to the human senses as constituting the primary
reality. For them, the microworld was an extrapolation of
human understanding into a realm of uncertain metaphysical
status.
There are also profound implications to the Copenhagen
Interpretation's elevation of epistemology to take priority
over metaphysics. Experience is primary; from those
experiences we collectively construct an image, a story, a
model of what the world must be like in order to provide the
experiences we have. We then place such trust in the story
that it becomes primary for us, and our experiences are seen
as nothing more than its byproducts. Quantum mechanics
shakes that trust, and calls us back to experience itself,
the immediacy of consciousness of the here and now. This is,
of course, what world's great mystical and meditative
traditions have been urging for centuries.
So I do think quantum mechanics offers some support
for mystical approaches to understanding the world, not
because it provides a physical mechanism to account for
paranormal experiences, but rather because it puts the
brakes on our unexamined trust in "objective
reality" and alerts us to the possibility of radically
different philosophical orientations, including some that
are much more congenial to a reality in which consciousness
has some primacy over matter, or at least some autonomy from
it.
Return to Quantum Weirdness menu
|