The Copenhagen interpretation of Quantum Mechanics embraces the idea that there are
no deterministic hidden variables that give rise to the probabilities of the
quantum world. This means that it is not generally possible to predict the outcome
of any measurement with certainty and that there is no deeper reality hidden beneath
quantum mechanics which could predict the outcome of each measurement with certainty.
But there are other theories that embrace
determinism and that seek out these hidden variables.
In 1932 Hungarian-born polymath John von Neumann presented what he claimed to be proof that there
could be no 'hidden parameters'. The validity of his proof was questioned by Hans Reichenbach.
Albert Einstein argued persistently that quantum mechanics could not be a complete
theory. His preferred argument relied on the principle of locality. He together with Boris
Podolsky and Nathan Rosen proposed a thought experiment which is now referred to as the
Einstein-Podolsky-Rosen Paradox. The thought experiment involved a pair of particles in an
entangled state. If the position of the first particle were measured, the result of measuring
the position of the second particle could be predicted. If instead the momentum of the first
particle was measured, then the results of the second particle's momentum could be predicted.
They argued that no action taken on the first particle could instantaneously affect the other,
since this would involve information being transmitted faster than light, which is not
allowed by the theory of relativity. They argued that if without any way of distributing a system,
we can predict with certainty the value of a psychical quantity, then there exists an
element of reality corresponding to that quantity. From this, they inferred that the second particle
must have a definitive value of both position and momentum prior to either being measured.
This contradicts quantum mechanics where two observables are not possible. They postulated that
these elements of reality are local in the sense that each belongs to a certain point in spacetime.
Bohr's response to this was that he felt their reasoning was wrong
due to the fact that the measurement of position and momentum are complementary,
making the choice to measure one excludes the possibility of measuring the other.
For Einstein, the most important problem he saw was that of non-locality. The choice of
measurement done on particle A would lead to two different
quantum states for particle B. He argued that, because of locality,
the real state of particle B could not depend on which kind of measurement was done on A
and that the quantum states therefore cannot be related with real states.
In 1951 David Bohm proposed a variation on the Einstein-Podolsky-Rose thought experiment.
If we take a source that emits electron-positron pairs, where the electrons go to point A and
positrons to point B. At point A Alice will be our observer and at point B Bob. If we assume
the pairs are entangled. This can be viewed as a quantum superposition of two states. In the first,
the electron has its spin pointing upwards and the positron pointing downwards,
in the second the electron point down and the positron point up.
As the particles are in a superposition of states,
it is not possible without measurement to know which state of spin either particle has.
Let's assume that Alice now measures the spin. It can be either up or down. If
she measures an up spin this means the system collapsed into the first state.
If Bob were now to measure his spin we would be 100% confident that he would have a down spin.
Now it is equally possible that the spin is not up and down but instead along the
X or Y axis. Again here if Alice measures a +X spin Bob would subsequently measure a -X spin.
Whatever axis their spins are measured along,
they are always found to be opposite. In quantum mechanics, the x-spin and z-spin are
'incompatible observables'. This means that Heisenberg's uncertainty principle applies.
So if Alice measures z-spin and gets an +z measurement and that now Bob defies his order
and instead measures the x-spin. In this case, Bob has a 50% chance of measuring a +x value.
Bob's positron will have a definitive spin when measured along the same axis as Alice's electron,
but when measured on a different axis its spin will be uniformly random. This
would seem to imply that information had propagated almost instantaneously
from Alice's position to make Bob's positron assume a definitive spin.
In 1964 the Northern Irish physicist John Stewart Bell proved a seminal theorem that
many physicists today misinterpret as rendering hidden variables impossible.
He set out to investigate whether it was possible to solve the non-locality problem
with hidden variables. He showed that it was possible for both paradoxes to be explained
with hidden variables. He showed that when Bob and Alice make measurements on that same axis
or on perpendicular axes hidden variables could account for this but when any angle was allowed,
local hidden-variable theories became unable to reproduce the quantum mechanical correlations.
This is what is known as Bell's inequalities.
Bell then showed that quantum physics predicts correlations that violate this inequality. The
only way hidden variables could explain the predictions would be if they were non-local.
This would mean two particles were able to interact instantaneously
no matter how far apart the two particles are separated.
Is there only one way of looking at Bell's Theorem?
In the Copenhagen interpretation, the violation of Bell's inequalities is
taken as grounds to reject the assumption often called counterfactual definiteness.
This means that it is not possible to speak meaningfully of the definiteness of the
results that have not been performed or in other words unperformed experiments have no results
There are other theories that take a very different approach.
The Many-world interpretation This is also known as the Everett
interpretation. Hugh Everett was an American physicist who first proposed the many-worlds
interpretation of quantum physics. Unlike the Copenhagen interpretation the wave function never
collapses and it holds that all possibilities of a quantum superposition are objectively real.
It can generate correlations that violate a Bell inequality because
it violates an implicit assumption by Bell that measurements have a single outcome.
In Everett's view, it was our concept of reality that was at fault. We only
think that there is a single outcome from a measurement when in fact all of them occur.
We are only able to see one of those realities. The others have a separate physical existence.
So the entire universe can be thought of as a giant wave function within which are
all the possible realities for the Universe. It begins as a superposition of all possible states
of all its particles. As it then evolves, some of these superpositions break down,
making certain realities distinct and isolated from one another.
So the act of measuring does not really create new realities just separates it from the rest.
This theory should not be confused with the multiverse hypothesis which envisions other
universes, born independently, that have always been physically disconnected from our own.
The problem with this concept is that once you initiate a collapse it spawns
a different reality where the opposite happened. You avoid the complication of
wave function collapse, but at the expense of making another universe.
In the 1970's and 80's this theory was given new life when a concept called decoherence was
developed. This would provide a clear rationale for why a quantum world would split. Using this
concept, the splitting of a world is not an abrupt event. It evolves through the decoherence and is
only complete when decoherence has removed all possibility of interference between worlds.
The main scientific attraction of this concept is that it requires no changes or additions to the
standard mathematical representation of quantum mechanics. There is no mysterious collapse of the
wavefunction and it predicts experimental outcomes that are fully consistent with what we observe.
On big question is that of the interpretation of the probabilities. If all outcomes occur
with 100% probability, where does that leave the probabilistic character of quantum mechanics?
Attempts to explain this come down to the idea that the quantum probabilities we see are because
our consciousness is restricted to only one world. This then raises the question of how your
consciousness or selfhood works in a universe that is constantly splitting you across new universes.
So the Many-Worlds Interpretations avoid the collapse of the quantum wave. There
is currently no experimental evidence that shows that a collapse actually occurs leaving the door
open for this interpretation. Although the MWI removes the most bothersome aspect of nonlocality,
action at a distance, the other aspect of quantum nonlocality, the non-separability
of remote objects manifested in entanglement, is still there. A world is a non-local concept and
this explains why we observe nonlocal correlations in a particular world.
It also removes most of the quantum mechanics paradoxes, such as Schrodinger's cat.
Strong proponents of the MWI can be found among
some cosmologists (like Tipler, Aguirre and Tegmark). Using it as a framework
for quantum cosmology allows you to discuss the whole universe and avoids the difficulty
in the standard interpretation of requiring an external observer. Consideration of the
framework of string theory also naturally leads to the many world interpretation.
Stochastic mechanics
In 1966 Edward Nelson wrote a paper and opened it with the following statement:
"We shall attempt to show in this paper that the radical departure from classical physics
produced by the introduction of quantum mechanics forty years ago was unnecessary"
At first glance, this might seem like a very bold statement but the results of the
paper are impressive. He derives the Schrodinger equation, by assuming that particles are subject
to a rapidly fluctuating random force. Microscopic particles such as electrons
are therefore described as executing something similar to Brownian motion.
The field has steadily grown since his first paper attracting a large community of researchers. Its
successes include an explanation for quantized angular momentum, quantum statistics, and the
famous double-slit experiment. There are however some problems, for certain types of measurements
it gives incorrect predictions and it still has an instantaneous action at a distance.
Pilot Wave Theory This theory stems back to the very beginning.
In 1927 de Broglie published a paper outlining his own theory called the pilot wave model. In this
model matter waves were seen as real physical objects that guided the motion of particles.
This predated the Copenhagen interpretation but few took notice of it until in the 1950s
when David Bohm rediscovered it and developed it further.
Bohm's modification still kept Schrodinger's equation to govern the wave function but added
a guiding equation that directly influences the particle's motion.
This concept is capable of reproducing the behaviour of
the quantum world but at the same time is able to retain realism.
Rather than probability being an intrinsic part of nature as in the Copenhagen interpretation,
here it is put back in its traditional place as a description of our incomplete knowledge.
David Bohm has been able to produce a hidden-variable theory which up and until recently
has been ignored by physicists and philosophers. This is what Bell wrote when he read Bohm's paper:
"But in 1952 I saw the impossible done. It was in papers by David Bohm. Bohm showed explicitly how
parameters could indeed be introduced into non-relativistic wave mechanics with the
help of which the indeterministic description could be transformed into a deterministic one.
More importantly, in my opinion, the subjectivity of the orthodox version,
the necessity of reference to the 'observer', could be eliminated. Moreover, the essential
idea was one that had already been advanced by de Broglie in 1927 in his 'pilot wave' picture.
But why then had Born NOT told me of this 'pilot wave'? If only to point out what was wrong with
it? ... Should it not be taught ... to show that vagueness, subjectivity, and indeterminism,
are not forced on us by experimental facts, but by deliberate theoretical choice"
While he didn't endorse Bohm's theory as the last word on interpreting quantum mechanics,
he clearly thought it superior to the Copenhagen alternative and stated the following:
"conventional formulations of quantum theory, and of quantum field theory in particular,
are unprofessionally vague and ambiguous. Professional theoretical physicists
ought to be able to do better. Bohm has shown us a way."
Bohmian Quantum Mechanics makes the same predictions for usual quantum mechanical
experiments but it is the way it accounts for the underlying mechanisms which is strikingly
different. Electrons and other elementary particles always possess definite positions
and have determinate trajectories. The evolution of the quantum state is governed
by the Schrodinger equation and the evolution of particle positions by a guidance equation.
Both of which are deterministic. Quantum mechanical probabilities enter the theory by
way of a postulate to the effect that particle positions are distributed in conformity with
the usual quantum mechanical probabilities. These probabilities represent the ignorance
of the precise values of the quantum state and the particle positions.
The variables that determine this exist on a lower level and in a sense are hidden. This level is
defined by a guiding wave. This wave essentially pushes the particles around, in a way that makes
it impossible to measure certain attributes accurately, hence the uncertainty principle. And
by comparison to the Copenhagen interpretation, these parameters are defined before measurement.
From an experimental point of view,
Bohm's theory is indistinguishable from the Copenhagen interpretation.
When Bohm was formulating the mathematics for his theory, he discovered that the pilot wave,
when changed even slightly in one place, could push a particle instantly in a distant location.
The beauty of Bohm's theory is that it claims that the variables have real,
defined values all the time.
No wave collapse is required as all particles possess determinate positions
even when the quantum state fails to assign them determinate positions.
The main problems with this theory are that it introduces a new conceptual element the
pilot wave which is unobservable. It also keeps the electron as a point particle
More recently experiments conducted showed that vibrating a silicon oil
bath up and down at a particular frequency can induce a droplet to bounce along the surface.
They discovered that the droplet's path was guided by the slanted contours of
the liquids surface generated from the droplet's own bounces. A mutual
wave-particle interaction which is analogous to de Broglie's and Bohm's pilot wave concept.
They then extended this experiment to examine what would happen with a single and double slit.
When the droplet bounced towards a pair of opening it would pass through only one slit
but the pilot wave passed through both. The experiment clearly showed that the
pilot wave steered the droplets to certain places and never to locations in between.
It also showed that a disturbance of the pilot wave would destroy the interference pattern.
They also demonstrated that droplets could tunnel through barriers,
orbit each other in stable 'bound states' and exhibit properties which are analogous
to quantum spin and electromagnetic attraction. They can annihilate
with subsurface bubbles showing a similar concept to matter anti-matter annihilation.
When they introduce an external force like a magnet and used a magnetic ferrofluid droplet
they were able to observe the droplets adopt a discrete set of stable orbits around the magnet,
each characterised by a set of energy levels and angular moment, clearly showing a quantisation
effect normally associated with the quantum realm like electron orbits.
Every bounce of the droplet causes ripples, these
chaotically but deterministically influence the droplet's future bounces, guiding it.
Many fluid dynamicists who are familiar with the new research are starting to see
that there is a classical, fluid explanation of quantum mechanics.
Quantum physicists tend to consider the findings less significant.
Could these experiments hold the key to understanding and
uniting the world of the small with the world of the large?
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