"I know that when I was in my late teens and early twenties the world was just a Roman candle - rockets all the time ... You lose that sort of thing as time goes on ... physics is an otherworld thing. It requires a taste for things unseen, even unheard of - a high degree of abstraction ... These faculties die off somehow when you grow up ... profound curiosity happens when children are young. I think physicists are the Peter Pans of the human race ... Once you are sophisticated, you know too much - far too much. Pauli once said to me 'I know a great deal. I know too much. I am a quantum ancient.'" - Isidor Rabi
Quantum Reality
Part 5 of "An Introduction to Quantum Reality"
The previous pages of this "Introduction to Quantum Reality" have presented the groundwork for the following discussion. We are now in a position to review what we have discovered, and hopefully reveal a more accurate picture of "reality" at the quantum scale.
Hopefully by this stage you will have read the previous four pages:
1) Quantum Mechanics: An Introduction
A Review of Previous Pages
In Quantum Mechanics: An Introduction we were introduced to the double-slit experiment which reveals the strange duality between particles and waves. Even though single electrons were shot at the double-slits, an interference pattern was produced. This result shows that until it is observed, a particle has a wave-like nature which is extended in space.
In The Quantum Casino it was revealed how nature - at its most basic level - is fundamentally random. The process of measuring a particle's property value was compared to a ball racing around a roulette wheel: the final state is randomly selected, and only certain values are allowed (the slots on the roulette wheel). Before observation, the particle's property value must be considered as being in a superposition of all possible states (equivalent to the ball racing around the roulette wheel).
In Quantum Entanglement it was revealed that when two particles interact they form a single "entangled" state. You can no longer talk of the wavefunction for just a single, localised particle, you have to talk of one single wavefunction for the entire system. The discussion on Bell's Inequality showed that particles are not localised in space, and do not possess fixed characteristics ("hidden variables"). There is a strange connection between particles which instantaneously informs an entangled particle of any type of measurement performed on its partner particle (this may be counter-intuitive, but it is not as surprising as it might seem as we have already shown that duality between a particle and a spatially-extended wavefunction).
In Quantum Decoherence a solution was presented for the apparent "collapse" of the wavefunction to a single eigenstate. If the components of a superposition state are to produce interference effects (as seen in the double-slit experiment) they must be in phase (must be coherent). What happens to a quantum particle in the real-world - a particle which is no longer isolated in space as it is in the double-slit experiment - is that each of its component states gets entangled with different aspects of its environment. As seen in the page on Quantum Entanglement, when particles become entangled you have to consider them as one single, entangled state. The phases of these states will be altered. This destroys the coherent phase relationships between the components. The components are said to decohere and the interference effects no longer appear.
The Copenhagen Interpretation
Now we have a good picture of behaviour at the quantum scale, we can start to ask more philosophical questions about the nature of quantum reality ("ontology").
Because the theory of quantum mechanics is only concerned with the accurate prediction of the results of experiments, it says nothing about any underlying mechanism which produces those results, i.e., it says nothing about the deeper quantum reality. This has left the door open for several competing "interpretations" of what those results say about the underlying reality. But one thing all the different interpretations must have in common is that they all predict exactly the same observable phenomena.
The most common interpretation of the results of quantum mechanics is the Copenhagen Interpretation. It was developed around 1927 by Niels Bohr, Werner Heisenberg and other collaborators, and named after the city where they often met. According to this interpretation, when a particle is not being observed all we have is a mathematical formula (the wavefunction) representing the knowledge we have of the probability of finding the particle after we take our observation. When we observe the particle, the wavefunction "collapses" and we perceive a localised particle. But the crucial thing is that this interpretation says nothing about what the particle is doing when we are not observing it. Indeed, such a question would be considered "unscientific", and the responsibility of philosophy. (For an hilarious - but surprisingly accurate - comparison of the Copenhagen Interpretation and Deal Or No Deal, see here).
The authors of the Copenhagen Interpretation were heavily influenced by the philosophy of positivism which was highly fashionable in Europe at that time. According to positivism, "the true laws of nature should involve only such quantities that can be observed". One should not waste one's time seeking the answers to unanswerable, deeper questions. Results such as the Heisenberg Uncertainty Principle seemed to suggest fundamental, insurmountable limitations on what we could ever determine about underlying quantum reality (see the roulette wheel analogy in The Quantum Casino for more about limitations on our knowledge about deeper mechanisms), so we should not ask the questions about reality if there was no way we could ever find the answer.
In taking this approach the Copenhagen Interpretation neatly sidesteps any philosophical questions such as "Does the electron exist before we observe it?". But there is a price to be paid for this apparently very neat tying together of the loose ends. In fact, there are a couple of nasty implications of this interpretation:
- Firstly, it requires this peculiar "collapse of the wavefunction" which apparently relies on observation. This might be taken to be a real physical process, or an "increase in our knowledge", but these precise details were left unexplained.
- Secondly, the interpretation effectively denies the existence of an deeper, objective reality that exists in the absence of observation.
This should all have been highly contentious. Nick Herbert considers the objections in his book "Quantum Reality": "From the results of experiments carried out in the twenties, how could Bohr conclude that no future technology would ever reveal a deeper truth?". However, it could be said that the advocates of the interpretation did such a good sales job that the Copenhagen Interpretation went largely unchallenged for many years, reprinted in text books as science fact, and is still the most widely-accepted interpretation among physicists. But still not everyone was satisfied. The mathematician James R. Newman said: "In this century the professional philosophers have let the physicists get away with murder. It is a safe bet that no other group of scientists could have passed off and gained acceptance for such an extraordinary principle." And Murray Gell-Mann said: "Niels Bohr brainwashed a whole generation of physicists into believing that the problem had been solved".
So let's dig a bit deeper ...
Beyond Copenhagen
We're now considerably wiser about the first problem with the Copenhagen Interpretation: as explained in detail in the page on Quantum Decoherence, different components of the quantum superposition form entanglements with the environment. Components which are out-of-phase coherence just dissipate into the wider environment. That seems a reasonable explanation for the apparent "collapse of the wavefunction". So let's concentrate on the second nasty implication of the Copenhagen Interpretation. Especially as this website is designed to consider questions about the nature of reality, let's concentrate on this principle of the Copenhagen Interpretation which denies the existence of an objective independent reality that exists in the absence of observation (or measurement).
Firstly, we've got to define what we mean by "observation". Do we limit this to mean "conscious human observation"? Surely not. For example, a radioactive uranium nucleus buried in rock on a distant planet will decay to emit an alpha particle. It does not matter if a human observer looks at the rock or not. Clearly, "measurements" must somehow be taking place all the time and do not require conscious observers. Instead, let us describe a "measurement" or "observation" as the process which produces a single property value from a state which was previously in quantum superposition, i.e., we now define a measurement to be the process of quantum decoherence which reduces the superposition state. In this case, any connection with the environment could produce a measurement. However, for all interference terms to disappear, i.e., for decoherence to be complete with the object no longer in a superposition state, the particle must make some macroscopic effect. This is described in the book Quantum Enigma: "Whenever any property of a microscopic object affects a macroscopic object, that property is 'observed' and becomes a physical reality". For example, if we use a macroscopic photon detector to detect the photon in the double slit experiment then that will destroy the interference pattern. So as long as there is a macroscopic effect from a quantum entity, that object can be considered to be "observed" or "measured" - no need for a conscious human observer.
OK, so now we're getting somewhere, digging a bit deeper than the Copenhagen Interpretation. We've got a better understanding of the "collapse of the wavefunction", we've defined what constitutes an "observation", so all we now have left to consider is what we can say about the state of the system before observation; specifically, do we have an independent reality that exists in the absence of observation? And, if so, what form would that reality take?
Here's what Nick Herbert says on this subject in his book "Quantum Reality": "Little has been said about the character of the unmeasured state. Since most of reality most of the time dwells in this unmeasured condition, which quantum theory represents by an uncollapsed superposition of possibilities, the lack of such a description leaves the majority of the universe (everything that's not currently being measured) shrouded in mystery".
We've got a few clues as to whether reality exists before observation. For a start, not all particle properties are in superpositions, subject to the quantum formulation. For example, an electron's mass and electric charge (so-called static attributes) have well-defined properties - they are not in a quantum superposition. Here, then, is objective reality without observation. The particle is there, and its properties have values. It's only the properties which can alter (the so-called dynamic attributes) such as position and momentum do not behave in the same manner - all dynamic attributes are subject to the quantum formulation. These quantities are in a superposition state before measurement.
So if there is any reality before observation, it would have to embrace the possibility of multi-valued properties. We now are beginning to get hard experimental evidence of this real multi-valued nature of reality. A team from the State University of New York and the Technical University of Delft have delayed quantum decoherence by isolating a quantum system from the environment and, as a result, they created a mesoscopic (in-between microscopic and macroscopic) electric current flowing through a superconducting ring in opposite directions at the same time (see here).
So now we're saying that quantum superposition might, in fact, be a superposition of many real particle properties. In which case maybe there's more to this wavefunction thing (which describes the superposition) that meets the eye ...
In the Copenhagen Interpretation, the wavefunction is treated as nothing more than a useful mathematical measuring tool, used for calculating the probabilities of finding a particle. The wavefunction is not considered to have any reality, i.e., it does not exist in this universe in the same way that a light wave, say, exists. It is purely a mathematical entity. But maybe the wavefunction is more than that. On the basis of this discussion so far, maybe we should now understand the form of the wavefunction as describing "reality before observation". Certainly, it would appear that the wavefunction is more just a simple probability wave giving the probability of finding a particle in a particular position. The wavefunction can introduce interference effects (seen in the double-slit experiment) but probabilities can never be negative (and so can never produce interference). Refer back to the form of the wavefunction derived in the page on The Quantum Casino:
It can be seen from this result that the wavefunction has the form of a wave in complex space, and it has a phase value in the complex plane. It is this phase value which introduces the interference effects. The wavefunction appears to have a structure - it's certainly more than a simple probability. So now we're considering the wavefunction as describing "reality before observation". This makes a lot of sense: if it is a mathematical model then it must be a mathematical model of an underlying mechanism. You do not have a functional model without an underlying mechanism. Surely a mathematical model does not exist on its own - it models something real. Even if we do not understand that mechanism, the wavefunction can provide us with a good understanding of its structure.
It's interesting to compare the wavefunction to television signals for many channels, which are permeating our everyday environment. Just because we can't see the multitude of signals does not mean they are not there, are not real. When we tune-in our televisions we can display just one of the channels, "bringing it into our physical reality" (the equivalent of selecting a single quantum measurement from a superposition). But the key thing is that the television signals were always real.
So now we have an answer to the last nasty implication of the Copenhagen Interpretation: there could, indeed, be an objective reality which exists in the absence of observation, and that reality would be described by the wavefunction. Which leaves only one problem: if the wavefunction is the true description of reality, then that reality is described in complex space. Clearly what we're dealing with here is a conception of reality which is very far removed from our everyday conception of what constitutes reality. This idea of there being a discrepancy between our human conception of reality and what, in fact, appears to be the true picture of reality is a recurring theme (see this New Scientist article).
Things are getting weird ...
Hidden Variables Theories
As discussed in detail on the page on Quantum Entanglement, Albert Einstein was one of the physicists who rejected the Copenhagen Interpretation and its insistence that there was no deeper quantum reality. Einstein considered quantum entanglement of unmeasured pairs of particles separated by a large distance. As soon as one particle was measured, quantum theory required an instantaneous transmission of particle property values to the other entangled particle (Einstein called it "spooky action-at-a-distance"). Such faster-than-light (superluminal) communication violated Special Relativity. Einstein considered this to be unacceptable, and argued that the particles must have possessed the property values already, albeit hidden from our view in some deeper level of reality. These property values (referred-to as hidden variables) only emerged into physical reality when a measurement was taken.
However, this would mean the particles possessing more information than quantum theory said they should have. If particles had hidden variables then quantum theory was wrong.
In 1932, John von Neumann published his definitive analysis of quantum theory, Mathematical Foundations of Quantum Theory. In that influential book, von Neumann showed that if electrons are ordinary objects with inherent properties (which would include any hidden variables) then the behaviour of those objects must contradict the predictions of quantum theory. At the time, Von Neumann was considered the greatest mathematician in the world and his result effectively killed-off hidden variables research for two decades.
However, in 1952 David Bohm showed that hidden variables theories were plausible if the restriction of locality was relaxed - essentially allowing faster-than-light communication (see here for a technical explanation of Bohm's findings).
Bohm illustrated his theory with the example of an undetectable, non-localised pilot wave which could explain the double-slit experiment. The pilot wave guides a single, real electron (i.e., the electron is never in a quantum superposition, it is always an "ordinary object"). The pilot wave guides the electron through just one of the two slits in the double slit experiment (see here) - none of this "going through both slits at once".
The guidance of the pilot wave is the cause of the interference effect when the electron reaches the screen. This is achievable because the pilot wave which is a special entity - it is in touch with every other particle in the universe (hence it is completely non-local). While this is peculiar, it does mean the electron is never in a quantum superposition state - it is always a real, ordinary object. The pilot wave is also a "real" wave (i.e., more than just a mathematical tool like the wavefunction, it is as real as a light wave - albeit undetectable!).
So both the pilot wave and the electron are real objects which together can describe quantum behaviour with no need for quantum superpositions.
Bohm generalised this pilot wave example to propose an holistic hidden variables theory. Bohm suggested the universe was a form of "hologram" called the Implicate Order, each point in the Implicate Order containing all the information about the universe (just as in a hologram - see here). This idea of an holistic quantum connectedness was continued in Fritjof Capra's bestselling book The Tao of Physics.
Veiled Reality
For the final part of our discussion of quantum mechanics, let's consider what would be involved in constructing a generalised hidden variables theory to examine the principles involved.
We've just discussed how the wavefunction could be considered to represent "reality before observation", with particles having multi-valued properties. If that is the case, why do we not see this multi-valued properties in our physical reality? When we perform a measurement of a particle property we only obtain a single value. Therefore, a hidden variables theory would appear to require some sort of mechanism or structural feature for hiding this multi-valued nature of reality from our eyes. How can this be so? Well, it might be due to the way in which we define our reality.
Bernard d'Espagnat has described a "veiled reality" - a deeper layer of reality - whose form is quite different from our human notion of reality (which he refers to as "empirical reality"). Our empirical reality could be thought of as a "World of Particles" (my term - see the diagram below). Because our measurement apparatus are themselves composed of point-particles (particle accelerators are themselves composed of particles and used to detect particles), this would explain why the multi-valued nature of reality is hidden from our view. We can only detect particles. The problem is the "measurement equipment" of our reality (including our own eyes - everything in our reality is defined as being composed of particles).
This imposes a structural limitation on what we can know about the veiled reality layer. In fact, we cannot interact with the veiled reality directly, we can only infer some clues as to its structure (I have denoted the structure of the veiled reality on the diagram below as some form of black network - don't think I know the real structure!). If we inhabit a "World of Particles" then the deeper veiled reality could be thought of as the "World of Wavefunctions". We can only infer the general structure of the veiled reality (wavefunction) - we are fundamentally incapable of interacting with it directly or knowing its details.
(Note on the diagram I have defined a barrier I have named the Firewall (based on the computing "firewall"). Like the computing firewall, it is not a physical barrier, rather it designed to fundamentally restrict the flow of knowledge in one direction. It is an inevitable consequence of the veiled reality layer.)
(I can't help seeing an analogy between this "World of Particles" and group theory: in group theory, we can apply operations to produce other (different) elements of the group, but we can never produce elements outside the group. In our world of particles we can perform experiments on particles with equipment made of particles, but they will be limited to telling us only information about particles. We will never be able to generate information about any reality beyond our empirical reality using these methods. Here's what Austrian physicist Anton Zeilinger has to say about the similar notion of quantum information being defined in terms of itself, and the resultant "firewall": "This world is not directly ascertainable or describable. Because every description must be done in terms of the information, and so you inevitably get into circular reasoning. There's a limit we can't cross. And even a civilisation on Alpha Centauri can't cross it. For me that's something almost mystical." (See here).
We do not really understand the real nature of wave/particle duality. However, even if we do not yet have a full understanding, we at least know what particles are not: they are not localised point particles in space. So why does the "observed particle" approach dominate physics? Clearly, it's because the scientific method places such importance on the outcome of experiment and, while we can detect particles with ease, we are unable to detect the flip-side (wavefunctions) directly.
So maybe we should turn our attention to the type of experiments we are performing. We do seem to place a great deal of emphasis (and money) on particle accelerators. Machines to detect particles. Machine which are themselves made of particles. Is that wise? Or is it just the easy way out. Isn't it easier to spend millions of dollars on creating ever larger accelerators than to admit that we're not dealing with localised particles after all but - as Bell's Inequality proved - we're dealing with spatially extended wavefunctions which we don't understand and cannot directly perceive. (In the book Quantum Reality it is considered if other creatures might be able to "see" wavefunctions: "Is it conceivable that other beings could experience reality differently? Could they possibly directly experience the superposition states whose existence we can only infer?").
In Thomas Kuhn's landmark 1962 book The Structure of Scientific Revolutions, it was argued that scientific thinking goes through revolutions, instead of gradual theory development through testing and experimentation. Maybe the current fixation with particles is merely fashion? Kuhn also questioned whether scientific experimentation is truly unbiased and neutral since the experimenter had previous theories and preconceptions - influenced by his fellow scientists - which could affect what experiments are chosen and the way in which the results are interpreted: "The existence of the paradigm sets the problem to be solved; often the paradigm theory is implicated directly in the design of apparatus able to solve the problem". Surely by using a particle detectors we are biasing our investigation towards a localised view of particles, and away from other approaches? Christoph Schiller agrees: "Observers have to be made of matter; an observer cannot be made of radiation. Our description of nature is thus severely biased: we describe it from the standpoint of matter." (see page 834 of Motion Mountain).
What other clues do we have about the true nature of reality? Well, we know that there is an apparent spooky connection between entangled particles separated by some distance. This "non-locality" can be explained by communication in the veiled reality layer. Remember, this deeper veiled reality layer could be thought of as being outside the jurisdiction of our human laws of physics which only apply to our "World of Particles". There's no problem achieving non-locality (faster-than-light communication) in the veiled reality layer as it the layer below that in which light exists (light (photons) being a constituent of the "World of Particles"). The "hidden variables" mechanism in the veiled reality layer transmit the characteristics between the particles instantaneously. However, we are always fundamentally incapable of accessing those hidden variables in that veiled reality.
At the end of this New Scientist article, David Deutsch considers our experimental limitations in exploring deeper reality: "All Deutsch will say about a theory of everything is that it is likely to come from uniting quantum theory and relativity at a more fundamental level than current entanglement experiments allow" (this refers to entanglement experiments such as Bell's Inequality and Leggett's Inequality - see back to the page on Quantum Entanglement). The interviewer then goes on to consider if we should be searching for deeper, hidden reality or just continuing with particle-centred experiments: "Maybe it is time to revisit Einstein's lost quest, if we are serious about uncovering the basic laws of the universe; the money spent on particle smashers such as the Large Hadron Collider certainly suggests we are. Perhaps we need to move quantum entanglement and the nature of reality to the centre of the quest to find a theory of everything. What was once a quirky sideshow may yet prove to be the main event."
There's no such thing as "nothing"
We all have a mental image of what constitutes "absolutely nothing". You might imagine that if you pump all the air out of a sealed box so that it contains no particles, you could form a perfect vacuum. However, the equations of quantum mechanics tell us that particles can appear in a vacuum, apparently "out of nothing". According to the Heisenberg uncertainty principle, there can be enough uncertainty about the energy at a point,
, to allow the creation of a particle/antiparticle pair, but only for a very short time,
:
This is surely the best possible example of how our particle-oriented conception of physical reality is flawed. The apparently empty space is still capable of producing particles; it is a system which is still described by a quantum state (actually, it is the quantum state with the lowest possible energy: zero-point energy). There is an underlying "veiled reality" layer present determining the quantum state of the system even when we have removed all particles from our "World of Particles". However, that layer is completely undetectable to our eyes and measuring apparatus - all of which are made of particles.
This shows that there is, in fact, no such thing as our intuitive concept of "absolutely nothing" in nature. The concept of "absolutely nothing" is purely a human invention.
Finally, a return to determinism?
Now we have defined the concept of a "veiled reality" we might even be able to cast some (extremely speculative) light on the probabilistic nature of the wavefunction. It has been suggested that at the very small Planck Scale (10-35m), quantum fluctuations become powerful enough to twist and turn the geometry of the Universe. Space and time break down into random "foam":
Maybe the random selection of a quantum state during wavefunction collapse is a result of this apparently random structure. Or maybe the selection process occurs in the veiled reality, hidden from us. What appears fundamentally random to us might actually be pseudo-random (i.e., deterministic!) at the Planck Scale or in the veiled reality "processing" (as suggested by Gerard 't Hooft - see this Nature article or this New Scientist article). Either way, the process would be completely hidden from our eyes. According to Jim Al-Khalili in his book "Quantum": "The uncertainty principle becomes now just a statement about our inescapable ignorance."
"God does not play dice", said Einstein, believing that quantum physics was incomplete, some deeper theory being able to predict outcomes with certainty. Maybe one day Einstein will be proved right.
Comments
Here you can add your comments about this
article.
Your comments will appear instantly on this page (underneath this
form):
Interesting article; although I
get a bit lost in the mathematics, you make it very clear what 'veiled reality'
is. I always like to find metaphors to more easily understand a concept; your
veiled reality picture reminded me of The Matrix, which contains an interesting
analogy to the 'veiled reality' concept. For Neo, when still plugged in the
matrix, his 'reality' is, for him, the world of particles. He is initially
unaware that his world is actually being run by an underlying system (the veiled
reality, hidden by a firewall). Since in Neo's simulated reality he only has
simulated equipment, which obeys the same rules imposed by the matrix, he is
unable to know or analyse the veiled reality; further more any random event in
Neo's life may be determined by logic in the veiled reality.
In the
movie, at one point Neo knows and understands the veiled reality, visualized by
seeing everything in 'matrix code' (a visualization of 'understanding the
quantum foam'), which gives him 'superpowers' (the fast 'downloading' of
knowledge portrayed in the movie is more or less a metaphor to the 'faster than
light' communication that you attribute to the veiled reality.) - Ivo Jansch, 5th January 2007
Hi Ivo, you've picked up on
something there as I (maybe subconciously) seemed to portray the "veiled
reality" layer as some sort of computing hardware (I see I portrayed it that way
on the diagram). Maybe that was a weakness in my presentation as we really don't
understand it yet, but, like you, the metaphor seemed to work for me.
I
think David Bohm's "Implicate Order" is also very similar to the Veiled
Reality/Matrix. Bohm's "Implicate Order/Explicate Order" would correspond to the
"World of Wavefunctions/World of Particles" described in my article. - Andrew Thomas, 5th January 2007
I think the only way to get a grip on the veiled reality would be by experiment; make assumptions on how it could work (the 'your guess is as good as mine' part will yield several different possibilities), and devise experiments that either confirm or disprove the idea. Basically this is what happens in the double slit experiment as well; it's an experiment designed to uncover things about the underlying principles, without being able to actually measure those principles. - Ivo Jansch, 6th January 2007
Absolutely, unfortunately we
simply don't have the technology to probe down to the "micro-quantum" level -
hence the room for the different interpretations.
On the subject of the
double-slit experiment, John Wheeler's Delayed Choice Experiment - see http://en.wikipedia.org/wiki/Wheeler's_delayed_choice_experiment
is a useful modification to the double slit experiment because it introduces so
many different aspects of quantum behaviour.
Firstly, the experiment
shows that idea of particles as isolated points in space is inaccurate: as
explained in the main article above, "reality before observation" is described
by a wavefunction. Wavefunctions, unlike particles, are extended in space and so
can pass through both slits at once.
When an observation is made, the
wavefunctions of the particle and the observing equipment become entangled
(i.e., form an entangled state - see http://www.ipod.org.uk/reality/reality_entangled.asp
). The behaviour and appearance of the particle (is this a particle or a wave?)
is determined by this entangled state (i.e., the wave/particle behaviour of the
particle is not independent of the observing equipment, rather the type of
observing equpiment determines behaviour). For example, if a screen is used to
detect particles, the wavefunction of the screen (which is, in itself, an
entangled state of all the particles which comprise the screen) allows the
interference pattern to appear. But if a single particle detector is used, the
wavefunction of the detector only allows the registering of single particles. In
that case, the interference superposition effects do not appear. Instead they
dissipate into the environment through decoherence - see http://www.ipod.org.uk/reality/reality_decoherence.asp
Some have suggested that the Delayed Choice experiment implies some sort
of backwards causality in time (see http://www.ipod.org.uk/reality/reality_backwards_causality.asp
): if the particle is detected by the screen or a single detector then some sort
of signal has to travel backwards in time to tell the particle to act as a
particle or wave when it travels through the double slits. But I believe the
process I have just described shows no backwards causality is required. -
Andrew Thomas, 7th January 2007
Everyone Can Understand Quantum
http://www.quantummotion.org/ -
Shan Gao, 21st March 2007
Brilliantly written Andrew. I just sat and digested your 5 part series over the past hour. Enjoyed it immensely. I am researching for a short film script and this hit the nail on the head. Without doubt the best overview of the components of quantum mechanics I have found on the web. Congratulations and keep writing. - brett, 22nd March 2007
Thanks so much! It took a while to do. I'm glad people are enjoying it. - Andrew Thomas, 22nd March 2007
Outstanding presentation! I'm strictly an amateur, but it could be that the Big Bang produced only a single particle with many dimensions, allowing the particle to exist everywhere at the same time. If its position and momentum could be pinned down at the same moment, the Universe would collapse. This is what would happen if one were allowed to travel at the speed of light. Time comes to a complete halt and the Universe collapses. Great articles, very clearly written. I've read other descriptions of Bell's inequality theorem which were totally confusing. You've cleared all that up. - Jim Carlson, 16th April 2007