Quantum theory was developed during the first part of the last century by studying the
behaviour of small things like atoms and photons
(particles of light). The word "quantum" refers to the fact that
light is apparently released and absorbed in lumps or quanta. But this is not
the main point of quantum theory. There are other features of the theory that
mark it out as representing a radical point of departure from classical
theories (theories predating quantum theory are often collectively referred to
as "classical") and force us to reconsider our picture of reality. In
classical theories the mathematical symbols in the theory relate in a simple
way to a picture of the world that is not far removed from our everyday
experience. For example, a ball flying through the air might be described in
Newtonian mechanics by mathematical symbols representing its speed, its
position, the direction in which it is moving, and the speed and direction in
which it is spinning. These quantities relate directly to the picture (which we
can quite easily visualize in our minds) of a spinning ball moving through the
air. Before quantum theory came along, it was taken for granted that the
quantities we use in physics relate in a simple way to some such picture of the
world and it came as something of a shock that quantum theory is not obviously
like that. What quantum theory provides us with is a neat and simple
mathematical formalism for calculating probabilities relating to measurements
we might make on quantum systems.
In
fact, once stripped of all its inessential structure, quantum theory can be
regarded as a generalization of classical probability theory - that theory we use to calculate the odds when
rolling dice. The quantum formalism consists of various mathematical symbols.
The great thing about quantum theory is that it works! So far no one has seen a
deviation from the rules of quantum theory. However, unlike in the classical
case (for example with the spinning ball), the symbols in the theory do not
relate in any obvious way to any simple intuitive picture of reality. Knowing
how to use quantum theory is a little like knowing how to perform magic but
without understanding why the magic works. This
leads us to ask three
types of question:
- Exactly what is it
that is odd about quantum theory - where does it contradict our usual intuitions about
reality?
- How should we
interpret quantum theory - what kind of
picture of reality, if any, does it give us?
- Why is nature described
by quantum theory (rather than some other possibly more sensible theory)?
Attempts
to answer these questions represent the main research efforts in the
foundations of quantum theory.
What
Is Odd About Quantum Theory?
Let us
start by considering a ball (a baseball for example) which can be in one of two
boxes - box A or box B. Imagine we do
not know which box it is in. In this case we are still inclined to believe it
is actually in one of the two boxes while nothing is in the other box. The fact
that we do not know which one is interpreted as ignorance on our part, having
nothing to do with the real world. However, now imagine that rather than a
ball, we have a quantum object like an atom. In this case it would be wrong, in
general, to suppose that the atom was actually in one box and not the other. In
quantum theory the atom can behave like it is, in some sense, in both boxes at
once. Of course, if we look into the boxes to find out where the atom is (we
could do this by shining bright laser light into the boxes and look for the
light scattered by the atom) then we will only find it in one of the two boxes,
not both, since there is only one atom. Why, then, say it can behave like it is
in both boxes at once? Well, physics is ultimately a discipline which relates
to experiments, so to answer this question let us consider an experiment which,
as we will see, leads us to the strange conclusion that something can be in two
places at the same time.
Consider
a quantum particle (it could be an electron, an atom, a photon, or any other
type of quantum particle). Let the particle impinge on a beam splitter. This is
a device which may either transmit the particle or reflect it.
If we
now place one detector in the transmitted path and another in the reflected
path then only one will fire. There is only one particle and it can only be
detected at one place. However, instead of placing detectors in these two
paths, we can bring the two paths together at a second beam splitter, placing
detectors in the rightward and upward paths of this second beam splitter. It is
a consequence of quantum theory, and demonstrated many times in laboratories,
that:
- If we ensure that the
two paths between the two beam splitters are equal in length, then the
particle will always be detected in the rightward path of the second beam
splitter.
- If one path is longer
than the other by a certain amount (equal to half the wavelength
associated with the particle) then the particle will always be detected in
the upward path of the second beam splitter.
Now
imagine trying to explain this behaviour as if the
particle went along only one of the two paths, though without us knowing which
one. Imagine that, in actual fact, the particle went along the lower path while
nothing went along the upper path. When the particle arrives at the second beam
splitter, it has to choose whether to go up or to the right. This particle has
only transversed the lower path whilst nothing has transversed the upper path and hence, the particle has no information
as to whether the two paths are equal in which case it should go right, or
differ by half a wavelength in which case it should go up, and hence there is
no way the particle can behave in the way we actually observe in experiments.
The same dilemma would apply if the particle took the upper path between the
beam splitters. This means that we simply cannot think in this way. We cannot
think in terms of one path being empty. There is some sense in which the
particle is in two places at once.
This device,
called a quantum interferometer, illustrates the central property of quantum
theory that distinguishes it from classical theories. We have two possibilities - "the particle is in path A" and "the
particle is in path B". In a classical theory only one of these two
statements could be true at any given time, while in quantum theory, both
statements can be partially true at once. This is called the superposition
principle. The state of reality is given by superposing two states which are,
from our usual classical way of thinking, mutually exclusive. Mathematically,
we represent this by taking the two mutually exclusive statements and adding
them together:
|
"the
atom is in box A" + "the atom is in box B"
|
(Actually,
the formalism is a little more complicated than this since we multiply each
statement by a number which represents "how much" the atom is in one
or the other box.) The superposition principle can be applied in many
situations. For example, we can have an atom in two boxes at once, a photon
over here and, at the same time, over there, a particle having clockwise spin,
and at the same time having anticlockwise spin. We are forced to think in this
way because all quantum objects are subject to interference experiments like
the one we just described.
Another
feature of quantum theory is that it is inherently probabilistic. Before we
look to see whether the atom is in one box or the other, there is no way of
knowing which box it will be found in. This fact is problematic for people who
believe that every effect must be fully determined by some cause. In fact, one
can attempt to remove this inherent indeterminism by supposing that quantum
theory is a statistical theory derivable from a deeper deterministic theory in
which the true state of affairs is given by a more complete description of the
world than that provided by the quantum state. Such theories are often called
"hidden variable theories". These hidden variables determine, in the
above example, exactly which box the atom will actually be found in. The
variables are called "hidden" since, before making a measurement, we
do not know what values they take.
A big
problem in quantum theory arises because we do not know when to stop using the
superposition principle. Imagine that rather than an atom in box A or box B we
have a molecule. This is a bit bigger than an atom but still very small and so
definitely quantum. What about a slightly bigger object like a single cell?
What about a really big object like a baseball or a cat or even a whole galaxy?
In principle, there is nothing that tells us when the superposition principle
does not apply. Thus, in principle we really could imagine a cat, for example,
being in a superposition of being alive and dead at the same time. In practise it would be very difficult to actually see quantum
interference with very big objects like baseballs or cats (though interference
has been seen with large molecules) and hence we can, for all practical
purposes, ignore the possibility that there is a superposition. However, if we
are interested in the "in principle" question, we cannot ignore this
possibility. This is a problem because we have to explain why we do not
actually see superpositions of large objects in two
places at once - our everyday experience is of
a macroscopic world with well localized objects. There are various approaches
to solving this problem. One is to say that, for sufficiently big systems, the
state "collapses" onto one or the other of the two possibilities.
Thus, it goes from being in the superposed state "A" + "B"
to being in one of the states "A" or "B". Such theories of
quantum theory are called "collapse theories".
One of
the strangest features of quantum theory is quantum entanglement. This arises
when we have two physical systems which have interacted with each other. After
such an interaction, the total system becomes describable only as a single
entity. There is no way of describing it by completely describing its
components. Entanglement is a fascinating subject from an interpretational
point of view, from a mathematical point of view and from the point of view of
applications. The most striking effect that arises from quantum entanglement is
quantum nonlocality - first discovered by John S Bell in 1964. Bell's
theorem is one of the most important results in physics and has played a large
role in motivating much recent work in quantum theory. Imagine that two quantum
systems interact and then separate to a great distance where measurements are
made. These measurements will reveal certain correlations between the two
systems. So far, this is what we expect since the systems have interacted in
the past. Bell showed, however,
that we cannot explain these correlations purely in terms of the previous
interaction between the systems. It appears that, even though the two systems
are at a great distance, they continue to talk with one another. However, this
effect is not so strong that it can actually be used to send signals, so the
principle that information cannot travel faster than the speed of light is safe
for the time being.
How
Should We Interpret Quantum Theory?
There
are many interpretations of quantum theory. At the risk of committing a gross
over-simplification, we can say that they fall into two basic categories. There
are realist/ontological interpretations and positivist/instrumentalist
interpretations. Realism is the idea that the real world exists independently
of us. To provide an ontology is to provide a picture
of the world. So, in realist/ontological interpretations, an attempt is made to
come up with a picture of an independently existing world which goes beyond
what we actually see (that is to say it goes beyond the detector clicks and
meter readings we record in our laboratory notebooks). Positivism is the
doctrine that there is nothing beyond what we directly observe. An
instrumentalist emphasizes the role of instruments (in our case, the laboratory
equipment) in the description of any phenomena. Positivist/instrumentalist
interpretations are not so concerned with developing a picture of an
independently existing world. Rather, an attempt is made to develop a
consistent way of thinking about the various detector clicks, meter readings,
and other effects.
The
realist/ontological interpretations can best be summarized by where they fall
on the following table:
|
|
hidden variables
|
no hidden variables
|
|
collapse
|
(no
major interpretations here)
|
collapse models
(GRW, ....)
|
|
no
collapse
|
the pilot-wave model (de Broglie, Bohm)
|
the many worlds
interpretation
|
In
collapse models the evolution of the state is chosen such that the state will
collapse whenever a superposition of a sufficiently large object occurs. This
means that the macroscopic world does consist of objects with definite properties
and explains why, when we look, we do not have the experience of seeing a
baseball in two places at once. The quantum state can be regarded as a real
existing object which, in the limit of the macroscopic world, accords with our
usual experience. There have been various attempts to build explicit collapse
models with some degree of success, though without guidance from some
overriding principle, such models are ad hoc. It has been suggested by Penrose
and others that gravitational mechanisms should induce collapse, but this idea
has yet to lead to a fully worked out model. If collapse really happens then
this should be detectable in experiments and there is currently some effort
being made to do this. Unfortunately, it may be a long time before experimental
techniques are sufficiently refined to detect collapse - if it happens.
In the
pilot wave model, the usual quantum description of the state (as a state vector
or wave function) is supplemented by an additional variable actually specifying
the positions of the particles. In the interferometer we can say that there is
a wave that goes along both paths whilst the particle only goes along one path.
The wave collects information about both paths and the particle uses this
information to decide which way to go at the second interferometer (hence the
"pilot-wave"). This interpretation of quantum theory has various
advantages. For example, it explains why the particle is only detected in one
place at once and it provides an almost classical picture of what is happening
at the microscopic level. However, to do this it is necessary to introduce
variables which go beyond quantum theory and detract from much of its
simplicity and elegance.
The
third option on the table is when we have no collapses and no hidden variables.
In this case, we really do get superpositions of
macroscopic objects. We need a way to explain why we do not actually see this.
The solution proposed by what is variously called the many worlds
interpretation, the Everett
interpretation (after Hugh Everett who invented it), and the relative state
interpretation (this was Everett's
name for it and, despite being less popular is a more accurate description of Everett's
idea) is that we can regard the world as splitting when a macroscopic superposition
occurs. In each branch one of the macroscopic possibilities happens. In an
individual world, macroscopic objects have well defined locations but when all
the worlds are taken together, it is still the case that there are superpositions of macroscopic objects. One way of thinking
about the splitting is that it is merely an artefact
introduced to help us understand the whole picture. Consider the analogy of
drawing a grid on a map so we can describe features on the map relative to the
grid. The grid does not actually exist. Likewise we can argue that there is, in
Everett's
interpretation, only one universe but by introducing a splitting we can explain
the appearance of the world relative to a particular observer. There are many
problems with this interpretation. The most obvious is that it represents an
extremely radical new view of the universe that many of us feel uncomfortable
with. But, even if we can swallow this, there remain technical problems. One is
that there are many ways to introduce the splitting into many worlds and yet we
need to pick out some preferred way of doing this. Another more serious problem
is probability. Each world in the splitting has a weighting attached to it
which is supposed to represent the probability of that world. A probability is
usually regarded as a probability of some particular possible outcome actually
happening. However, in the Everett
picture all outcomes are realized and so it is not clear what this probability
is a probability of.
The
main positivist/instrumentalist interpretation is the Copenhagen interpretation which was developed
by Niels Bohr and colleagues working in Copenhagen.
Bohr emphasizes that all experiments must be described in classical language.
Thus, we have to make statements like "the pointer is pointing at the
number 5" or "the ball was found in box A". Any laboratory
notebook consists of many statements just like this. Bohr then argues that
quantum theory is simply a consistent way of dealing with such statements.
Often one can tell a story about what happened in an experiment such as
"the photon was reflected at the beam splitter and travelled
along path A, finally being detected at detector
A". In fact, we did not see the particle being reflected nor travelling along path A. All we know is that the detector
in path A fired. In an interference experiment we would not be so keen to tell
a story about which path the particle travelled
along. In this case we might tell a different type of story in which a wave travelled along both paths, these two waves being brought
back together at the second beam splitter. The type of story we are likely to
tell depends on the experimental context. Bohr describes such descriptions as
complementary and he describes his interpretation of quantum theory as complementarity. It is not clear whether these
"stories" should be regarded as anything more than just stories, or
whether we should think of them as corresponding to what is actually happening
in the world. The advantage of the Copenhagen
interpretation is that it provides a consistent way of talking about quantum
experiments. However, since it requires all physical statements to be related
to concrete classically described apparatuses, it is difficult to arrive at a
picture of what is happening in the world independent of any measurements that
may be made.
Why
Quantum Theory?
We
know from experimental evidence that quantum theory works very well. But we
also know that it is a very strange physical theory. This motivates the obvious
question: why is nature described by quantum theory? How do we begin to answer
a question like this? Usually explanation in physics is by means of appealing
to some set of deep, simple and intuitive principles. The principles of quantum
theory, whilst mathematically simple, are rather abstract and cannot be
described as intuitive. It is instructive to compare this with the situation in
Einstein's theory of special relativity which follows from two principles: the
laws of physics are the same in all frames of reference and the speed of light
is a constant for all observers. Is it possible to find a similar set of simple
and intuitively reasonable principles for quantum theory? This question has
been around for a while. One approach has been to reconsider logic. It can be
argued that quantum theory forces us to modify the rules of logic. In this
case, we could attempt to derive quantum theory from a modified set of logic
rules. There has been a tremendous amount of work on quantum logic (which is,
by now, a field in its own right). However, it is fair to say that, so far, no
simple approach has emerged. Another approach stems from the observation that
quantum theory is, when stripped of inessential structure, simply a new type of
probability theory. Hence, we can consider what a reasonable theory of
probability might look like. One of our researchers, Lucien Hardy, has shown
that we can obtain quantum theory from five principles in this probability
context. Four of the five principles are true in classical probability theory
and hence it is the remaining principle that gives rise to quantum theory. To
understand this remaining principle, consider a system which can be in the
state 0 or the state 1 (a bit). In classical physics the only way to go from
the 0 state to the 1 state is to jump - one moment the
state is 0, the next it is 1. But in physics we prefer things to change in a
gradual fashion. The remaining principle (which gives rise to quantum theory)
demands that there exist a way of gradually transforming the state. This
implies that there exist an infinity of states between 0 and 1. These "in
between" states are just the usual quantum superpositions.
The future of quantum theory?
Quantum
theory remains a deeply mysterious subject. Progress in understanding it will
come from theoretical, philosophical and experimental developments. On the
experimental side we might, one day, expect to see some deviation from the laws
of quantum theory in the laboratory. Perhaps we will observe collapse for
macroscopic objects. Perhaps we will see a violation of quantum theory at a
much more fundamental level. At the theoretical end we need new tools for
gaining insight into the theory. By understanding more deeply the reasons why
nature is described by quantum theory we might be able to go beyond the theory.
At a philosophical level, we might come to a better understanding of the world
and our place in it if we can decode what the equations of quantum theory are
trying to tell us.
© Lucien
Hardy is an associate of the Perimeter Institute
for Theoretical Physics
