Our everyday world exists in 3 dimensions, Length, Breadth and Height (+ Time). Maybe.

Quantum Entanglement

Simply put, Quantum Entanglement is the process of connecting two separate things, be they photons or nanoscale objects, so that they behave the same no matter their distance apart. What happens to one particle also happens instantly to the other, even if they are separated by the entire universe.

Does this violate the principles of light speed whereby instantaneous 'communication' between 2 distant points in time is effectively impossible??

Does "Quantum Atom Theory" go some way towards answering this question?

Dear Mr Hack, If you're around, could you please explain the Collapse of the Wave Function in lay-terms, please?

Also, I've read descriptions of the Double Slit Experiment many times before, but you seem to be pretty good with words so I'd like to hear your explanation. Thanks

"TONGUE-tied and twisted, just an earthbound misfit, I"

That's a tricky one, but it seems a reasonable Saturday morning challenge, so I'll give it a go.

The first thing to note is that the wavefunction may not be real, and certainly wavefunction collapse may not be real. There are formulations of QM that have no wavefunction collapse, although all the formulations of QM are essentially ontological in nature, and are therefore not really scientific.

Anyhoo, the simplest way to think of what the wavefunction is is to think about a particle, and specifically where it is and how quickly it's moving in a given direction (its velocity; velocity is a vector quantity, which means that it contains a directional component, as opposed to a scalar quantity, such as speed, which does not). Heisenberg's Uncertainty Principle tells us that these quantities are a pair of parameters, known as 'conjugate variables' whose values cannot both be known with arbitrary precision. The more precisely one is known, the more 'smeared out' the other becomes.

It should be noted that 'known' is a bit misleading here, because it implies that these values do actually exist, and that the problem lies in our ability to measure it, but in reality that's not how it works. I'll come back to this shortly.

So, we have a particle and, until we measure it, we can't know where it is or how fast it's moving. Until we start making measurements, we can only describe it in terms of the probability of finding it in a given location or with a given velocity. This is described by a probability distribution, thus:

So we have a wave describing a probability distribution, the height of which (actually, the square of the height) gives the probability of finding a particle at a given location. This is the wavefunction. The evolution of this wavefunction over time is given by the Schrödinger equation (yes, the Schrödinger with the cat in the box).

When we actually measure a particle's position, the probability of find the particle at that the position we measure it 'collapses' to a probability of 1 (probabilities are always given as ratios, or as an absolute value between 0 and 1; this particular probability can be given as the ratio 1:1), because there is zero probability that the particle will be found at any other location. This is, in a nutshell, wavefunction collapse.

Note that, the more precisely we measure the position of the particle, the less precisely can be known its velocity. The implication of this is pretty clear, namely that the more precisely we know where a particle is now, the less precisely can we know where it was a moment before or where it will be a moment later, because this would require that we know its velocity precisely, and this is exactly what the uncertainty principle prohibits.

Einstein hated this, insisting that the particle must have both a definite position and a definite velocity at all times, and that this uncertainty must be a feature of our inability to measure it, rather than being a fundamental feature of how the universe operates. He vehemently denied that the best we can do in describing the deepest principles governing the universe could be entirely probabilistic in nature, and this was the motivation for his famous quote 'I am convinced that He (God) does not play dice'. He was certain that those values really existed at all times, but were hidden from us in some way. Like many of his day, he was convinced that the universe was deterministic, but QM says otherwise (there's an extremely interesting digression about Darwin and his role in this here, but I'll save it for the moment).

One of the issues concerns something called 'entanglement'. Basically, when a pair of particles are entangled, measuring a given state for one forces the other into an opposing state. This occurs no matter how far apart they are, even to the extent that the particles could be on opposite sides of the universe. When we measure a particle to have a given 'spin' about a particular axis, the entangled particle is forced into having opposite spin. Einstein, along with Podolsky and Rosen, insisted that there were some 'local' hidden variables that hadn't been accounted for and that, therefore, the Copenhagen interpretation of quantum mechanics was incomplete.They were sure that it was possible to add parameters to QM that would be able to predict the exact outcome of a given experiment.

In 1964, John Bell, a physicist from Northern Ireland, developed a theorem showing that, if local hidden variables existed, the outcome of these experiments would show an inequality. Swathes of experiments have shown that these inequalities are not found, and thus that deterministic local hidden variables do not exist. Essentially, he ruled out a theory in which the outcome of a specific theory could be predicted with certainty, concluding:

John Bell wrote:In a theory in which parameters are added to quantum mechanics to determine the results of individual measurements, without changing the statistical predictions, there must be a mechanism whereby the setting of one measuring device can influence the reading of another instrument, however remote. Moreover, the signal involved must propagate instantaneously, so that a theory could not be Lorentz invariant.

In short, the universe is non-local.

There's a really good analogy dealing with this in Brian Greene's Fabric of the Cosmos, which features Mulder and Scully and some boxes sent to them by aliens.

As for the double-slit experiment, this is interesting stuff. The first iteration, by Thomas Young, was taken as a sure sign that light was wave-like in nature, because it always showed an interference pattern.

This result was taken to be the final nail in the coffin for what Newton called a 'corpuscular theory' of light, in which light came in corpuscles, or discrete packets. It wasn't seriously challenged until Einstein and his work on the photoelectric effect in 1905. It's interesting that not many people talk about this work now, especially given that it overturned the extant paradigm and won him the Nobel Prize. This was his annus mirabilis, in which he not only overturned the wave-theory of light, but also gave the first experimental evidence for atoms in his work detailing Brownian motion in pollen grains suspended in water, and also released his world-shattering paper On the Electrodynamics of Moving Bodies, the paper dealing with what we now call the Special Theory of Relativity.

Anyway, after Einstein's work on the photoelectric effect, in which he demonstrated categorically that light comes in corpuscles, or what we now call 'photons', the bosons of the electromagnetic force, the double-slit experiment was revisited, because it had categorically shown that light came in waves, or the interference pattern on the screen could not be explained. It should also be noted that the wave picture of light had already hit a slight snag, in the form of the Michelson-Morley experiment of 1887, because it had always been thought that waves required a medium to travel through, in this case the 'luminiferous aether'. The Michelson-Morley experiment demonstrated that no such beast existed.

Max Planck had, a few years before in 1900, been toying with a thought experiment dealing with the energy in an oven. Essentially, he worked out that, under a classical treatment, once all the frequencies of energy were added together, the energy in an oven should actually be infinite. This would be great for re-heating pizza very quickly, but it was clearly bunk, so he started thinking carefully about what was going on. What he worked out was that it could only be explained if most frequencies were excluded, so he set about finding out what frequencies were excluded and by what means. Eventually, he determined that the determining factor was actually the walls of the oven, and that any frequency that did not return exactly to the origin at the walls were excluded. In other words, only waves that completed a half or a full cycle could contribute to the energy of the oven. It's a lot like a guitar string, in which only standing waves contribute to the energy of the string. What this means is that we hear only the fundamental (one standing wave between the ends) and the various harmonics, which are all the frequencies that can complete a full cycle on that length of string. By this, he had realised that the energy in an oven was basically quantised, hence the name.

Anyhoo, the double-slit was revisited in light of Einstein's paper, and it was toyed with in terms of how light could exhibit this 'duality', the term for the two kinds of behaviour it exhibited, wave-like and particulate. Many iterations were tried, and many interpretations for the results, including that the waves were simply the behaviour exhibited by many photons at once interfering with each other. This idea was tested using a photon emitter that could be turned down until it emitted only one photon at a time. Surely, we wouldn't see an interference pattern...

D'oh!

So, other experiments were devised, including one in which a photon detector was placed next to the slit, so that it could be seen which slit the photon travelled through. The interference pattern disappeared!

So what's happening here? Well, the short answer is that we don't know. What we do know is that, if 'which path' information can be extracted from the experiment, there is no interference pattern displayed, thus no wave-like behaviour is apparent. If we can't extract which-path information, the interference returns. This can be, and has been, taken even further, to the extent that a photon is 'tagged' (essentially, it's imparted with a particular 'spin' (see following post), which can be detected), and then there is no interference pattern. Another iteration has this tagging removed before it hits the slit, and the interference pattern returns.

There are many descriptions of what might be happening here, though they are essentially ontologies, and thus not physics. Everett has it that, for every possible path a photon could take, there is a universe in which this path is taken, and our observation determines only which universe of all those we actually inhabit. This is the 'Many Worlds' interpretation of Quantum Mechanics. Richard Feynman's formulation (the one I prefer, though not with any great conviction), is that the photon takes every possible path in this one all at once, each path contributing to the final trajectory. Many of these, through interference and phase variance, cancel each other out (the audio boffins here will get this; in Soviet Russia, phase cancels you!) leaving only a small proportion of them actually contributing any measurable effect to the path. This is the 'path-integral' formulation, or 'sum-over-histories'. De Broglie-Bohm theorem has it that the particle is piloted by a wave, and it is the wave portion that generates the interference. You'll note that neither the Many-worlds (MWI) nor the pilot-wave formulations have any kind of collapse of the wavefunction.

Feynman's interpretation is the one I prefer, if only because it gels well with Quantum Field Theory (QFT), which basically has it that neither wave nor particle have any real existence, but that they are particular manifestations of the behaviour of 'fields'. When we measure the field in a certain way, we see behaviour that looks like 'particle', and when we measure it in another way, we see behaviour that looks like 'wave'.

I should probably note that 'spin' as used in the above doesn't precisely fit with our notion of spin in the classical sense. It isn't quite how fast a particle is rotating about a given axis, quite so much as what it looks like from different angles. For example, a spin 1 particle requires a full revolution so that its appearance is the same from the same vantage point, a spin 2 particle requires only half a revolution, and so on. Hawking presents the following analogy in A Brief History of Time:

We can think of the ace as a spin 1 particle, in that it requires 1 complete revolution before it looks the same from a given perspective. The queen, however, only requires half a revolution and it looks the same, so it's a spin 2 particle. When we get to particles of spin ½, we get the extremely counter-intuitive notion that a particle must go through 2 full revolutions before it 'looks' the same.

Firstly, thank you very much for taking up your Saturday morning to produce all this.

Now then, questions!!

Hack Wrote: Anyhoo, the simplest way to think of what the wavefunction is is to think about a particle, and specifically where it is and how quickly it's moving in a given direction (its velocity; velocity is a vector quantity, which means that it contains a directional component, as opposed to a scalar quantity, such as speed, which does not). Heisenberg's Uncertainty Principle tells us that these quantities are a pair of parameters, known as 'conjugate variables' whose values cannot both be known with arbitrary precision. The more precisely one is known, the more 'smeared out' the other becomes.

Ok I get that both position and velocity can't both be known at the same time, I even get that if one were to determine the precise position of a 'particle' then its velocity becomes indeterminable, - but is the reverse equally true, that in knowing the precise velocity of a 'particle' then its exact position becomes indeterminable, and if so, how? I man how can one measure the velocity of a 'particle' who's position is indeterminable?

Hack Wrote: One of the issues concerns something called 'entanglement'. Basically, when a pair of particles are entangled, measuring a given state for one forces the other into an opposing state. This occurs no matter how far apart they are, even to the extent that the particles could be on opposite sides of the universe. When we measure a particle to have a given 'spin' about a particular axis, the entangled particle is forced into having opposite spin. Einstein, along with Podolsky and Rosen, insisted that there were some 'local' hidden variables that hadn't been accounted for and that, therefore, the Copenhagen interpretation of quantum mechanics was incomplete.They were sure that it was possible to add parameters to QM that would be able to predict the exact outcome of a given experiment.

So with regard to 'entanglement', we're imagining something like the 2 seats on a see-saw, - but without the plank in between them, - here? As one seat goes up, the other instantly goes down? And this happens only at the point of measurement, - implying perhaps that the act of measurement itself somehow 'forces' the entangled particles into a single definitive state as opposed to ALL states at once? Sort of?

Hack Wrote: Max Planck had, a few years before in 1900, been toying with a thought experiment dealing with the energy in an oven. Essentially, he worked out that, under a classical treatment, once all the frequencies of energy were added together, the energy in an oven should actually be infinite. This would be great for re-heating pizza very quickly, but it was clearly bunk, so he started thinking carefully about what was going on. What he worked out was that it could only be explained if most frequencies were excluded, so he set about finding out what frequencies were excluded and by what means. Eventually, he determined that the determining factor was actually the walls of the oven, and that any frequency that did not return exactly to the origin at the walls were excluded. In other words, only waves that completed a half or a full cycle could contribute to the energy of the oven. It's a lot like a guitar string, in which only standing waves contribute to the energy of the string. What this means is that we hear only the fundamental (one standing wave between the ends) and the various harmonics, which are all the frequencies that can complete a full cycle on that length of string. By this, he had realised that the energy in an oven was basically quantised, hence the name.

Interesting. So where does the energy from any frequencies that are not either half, or full cycles, go, do they just cancel each other out? The energy from those frequencies can't just disappear, can it?

DOUBLE SLIT

So the 'wave' part of the double slit experiment is easy to visualise and comprehend, in my thought experiment I can happily use water waves flowing towards the 2 slits, and 'see' how in, or out of phase waves would cause the interference pattern.

Moving on then to the single photon...

Hack Wrote: This idea was tested using a photon emitter that could be turned down until it emitted only one photon at a time. Surely, we wouldn't see an interference pattern...

D'oh!

So, other experiments were devised, including one in which a photon detector was placed next to the slit, so that it could be seen which slit the photon travelled through. The interference pattern disappeared!

So then a single photon happily causes an interference pattern, - UNTIL a photon detector is placed next to the slits? Seriously? So what the hell does our 'detector' do to the photon? It must, like in your entanglement example, - 'force' the photon to make a choice between the slits? Or possibly the detector is simply incapable of detecting more than one state at any given instant, just like our eyes? But wait, because then I'm back to "What is an instant actually an instant of?!?

Let me say as an aside that I dislike the 'Many Worlds' interpretation of Quantum Mechanics as much as you appear to, - although having said that I'm not very keen on Feynman's formulation either. I have to admit though that neither one appeals for no other good reason than that neither one 'feels' right. As for the De Broglie-Bohm theorem, not sure, have never heard of it before and don't know anything about it, maybe you can direct me somewhere I can read up on it, preferably reasonably succinctly?

Back to the double slits...

What other versions of this experiment have been tried? What happens if a mirror replaces the usual backboard? Does varying the distance of the photon firing device from the slits produce any measurable difference to the interference pattern? What happens if the slits are vertically positioned, instead of horizontally? Or one on top of the other instead of adjacent to each other? Are any of these questions even relevant???

"TONGUE-tied and twisted, just an earthbound misfit, I"

ghostgirl wrote:Ok I get that both position and velocity can't both be known at the same time, I even get that if one were to determine the precise position of a 'particle' then its velocity becomes indeterminable, - but is the reverse equally true, that in knowing the precise velocity of a 'particle' then its exact position becomes indeterminable, and if so, how? I man how can one measure the velocity of a 'particle' who's position is indeterminable?

Well, that's entirely the point. We can measure the velocity of the particle but, the more accurately we measure that, the more its position is smeared out. You can think of it in terms of the nature of the measurement smearing out the detail. When we measure the position of a given particle, an electron, say, we do so by means of firing a photon at it. The photon interacts with the electron in such a way that the velocity of the particle is affected by the interaction. Similarly, measuring the velocity of the electron smears out the detail concerning its position. Moreover, regardless of which we measure, we can only capture that information for the duration of the measurement, and we can't predict with any certainty what its position or velocity will be even a fraction of a second later.

So with regard to 'entanglement', we're imagining something like the 2 seats on a see-saw, - but without the plank in between them, - here? As one seat goes up, the other instantly goes down? And this happens only at the point of measurement, - implying perhaps that the act of measurement itself somehow 'forces' the entangled particles into a single definitive state as opposed to ALL states at once? Sort of?

That's actually a really good analogy, and one that we might be able to develop some. I'll give it some thought.

Of course, the one difference is that a see-saw is not a perfectly rigid body. Indeed, there is no such thing as a perfectly rigid body, because such a thing would violate Special Relativity, as it would allow faster-than-light transmission of information.

Nonetheless, your see-saw analogy is a good one. And in fact, you've hit upon one of the strangest things in quantum mechanics, and what seems to us like a paradox, or a violation fo the laws of logic (it isn't, of course, it just seems that way). What you're actually talking about here is the principle of 'superposition', the idea that something can be in more than one state or, indeed, more than one place at once. To our intuition, this violates the law of non-contradiction but, in fact, it doesn't. The LNC says that something can't be both what it is and not what it is at the same time. The electron is always an electron, but it can be an electron here and an electron there simultaneously...

Interesting. So where does the energy from any frequencies that are not either half, or full cycles, go, do they just cancel each other out? The energy from those frequencies can't just disappear, can it?

No, it simply isn't there.

The problem here is in thinking about what energy actually is. In physics terminology, energy is the ability to perform work. If those frequencies cannot contribute, they cannot perform work, thus there is no energy and they aren't there. Does that make sense?

So then a single photon happily causes an interference pattern, - UNTIL a photon detector is placed next to the slits? Seriously?

Yes, seriously.

So what the hell does our 'detector' do to the photon?

Well, that's what the various ontologies bolted onto QM attempt to do, namely to answer that question. In MWI, all the worlds exist in superposition, and the photon potentially goes through both slits at once. The observation determines which world we are in, and removes the potential for all the other worlds, restricting the photon to a single slit, and removing the interference pattern. In the path-integral formulation, the observation reduces the probability of any path that doesn't go through the single slit to zero, meaning that the photon goes through a single slit. In the de-Broglie-Bohm formulation, the observation literally forces the result, by removing the pilot wave.

It must, like in your entanglement example, - 'force' the photon to make a choice between the slits? Or possibly the detector is simply incapable of detecting more than one state at any given instant, just like our eyes? But wait, because then I'm back to "What is an instant actually an instant of?!?

Not really. The photon was always going through the slit, or at least that manifestation of the field's behaviour. Again, it depends on the interpretation.

Let me say as an aside that I dislike the 'Many Worlds' interpretation of Quantum Mechanics as much as you appear to, - although having said that I'm not very keen on Feynman's formulation either.

Well, MWI is horribly unparsimonious, IMO. That's not to say that it isn't true, of course.

As for Feynman's formulation, the core principle is at least a demonstrable one, namely superposition, which has been observed. It sounds absurd to say that the photon travels every possible path on the route between source and detector, a bit like the infinite improbability drive from The Hitch-Hiker's Guide To The Galaxy (and in fact, that's what the Heart of Gold was based on), but superposition has been demonstrated to be real.

I have to admit though that neither one appeals for no other good reason than that neither one 'feels' right.

Well, the one thing I learned early on in studying physics is that the universe isn't required to pander to my puny intuitions.

As for the De Broglie-Bohm theorem, not sure, have never heard of it before and don't know anything about it, maybe you can direct me somewhere I can read up on it, preferably reasonably succinctly?

What other versions of this experiment have been tried?

All kinds of iterations have been carried out. The most impressive and mind-boggling would be the delayed choice quantum eraser, a version of the 'tagged photon' iteration I mentioned earlier. Again, the wiki is pretty illuminating (pun intended) on this.

What happens if a mirror replaces the usual backboard?

Not sure what you mean.

Does varying the distance of the photon firing device from the slits produce any measurable difference to the interference pattern?

No. In fact, in the earliest thought experiments dealing with delayed choice, the version discussed is one in which photons from a single astronomical source, lensed by an intervening galaxy, arrive at the detector. No matter how far apart the source and the screen are, the results are the same.

What happens if the slits are vertically positioned, instead of horizontally? Or one on top of the other instead of adjacent to each other? Are any of these questions even relevant???

Not really, although they would present a slightly different interference pattern. That's not really relevant, though, as it's the interference pattern itself that is of interest. That it exists at all (or not, in the case of the 'which path' iteration) is what matters.

You ask some interesting and challenging questions.

Hack Wrote: That's actually a really good analogy, and one that we might be able to develop some. I'll give it some thought.

Of course, the one difference is that a see-saw is not a perfectly rigid body. Indeed, there is no such thing as a perfectly rigid body, because such a thing would violate Special Relativity, as it would allow faster-than-light transmission of information.

Nonetheless, your see-saw analogy is a good one. And in fact, you've hit upon one of the strangest things in quantum mechanics, and what seems to us like a paradox, or a violation fo the laws of logic (it isn't, of course, it just seems that way). What you're actually talking about here is the principle of 'superposition', the idea that something can be in more than one state or, indeed, more than one place at once. To our intuition, this violates the law of non-contradiction but, in fact, it doesn't. The LNC says that something can't be both what it is and not what it is at the same time. The electron is always an electron, but it can be an electron here and an electron there simultaneously...

Your use of the word "simultaneously" here?... That doesn't sound like a word that really makes sense in quantum mechanics, since according to the "What Time Is It?" video only if 2 objects/people/whatever are inhabiting the exact same place at the same time can it truly be said that they experience anything simultaneously. So what are we talking about here? That 'superposition', is the idea that something can be in more than one state/place as seen by a single observer? Plus!... ...Is the exploitation of quantum superpositions then the way forward in designing a quantum computer that can carry out multiple calculations simultaneously?

So I get that one concrete consequence of the feature of superposition in quantum physics is ‘entanglement’, fine, so if two quantum objects are entangled, a measurement of a property of one object can be used to predict the corresponding property of the other object. Is this a case, though, that these "properties" are not actually determined until they the point at which they are measured? AND, do molecules containing, say, a dozen atoms, also exhibit quantum superposition states, or is this phenomena restricted to elementary particles?

This is all very "Buddhist" isn't it?

Double slit

As far as the experiment goes the electron is fired at the slits prior to the detector detecting which slit it goes through, thereby negating manifestation of the interference pattern, but isn't "before" and "after" purely "relative" - in the Einstein sense of relativity I mean - here?

"TONGUE-tied and twisted, just an earthbound misfit, I"

ghostgirl wrote:Your use of the word "simultaneously" here?... That doesn't sound like a word that really makes sense in quantum mechanics, since according to the "What Time Is It?" video only if 2 objects/people/whatever are inhabiting the exact same place at the same time can it truly be said that they experience anything simultaneously.

Well, it's woolly language, but the video isn't dealing much with quantum mechanics, with a few exceptions. The problem of simultaneity in relativity is really an expression of the fact that there is no preferred frame of reference. To explain that in a little more detail, imagine two street lamps. You are position equidistant between them. They both turn on in what seems, in your reference frame, to be the same time. If you position somebody else between them, halfway between your position and one of the street lamps, one of them will come on before the other form that observer's frame of reference. Neither frame is more correct.

When talking about superposition, we're talking about something occupying disparate states at the same time from every frame of reference. If we're talking about spatial position, this might mean that you and your observing counterpart might register different positions for the particle, but you will both agree that the particle is in two places at once. This is what I mean by simultaneity in the context of quantum superposition, and it is unproblematic in that context.

So what are we talking about here? That 'superposition', is the idea that something can be in more than one state/place as seen by a single observer?

Almost, but it's actually 'as seen by any observer, regardless of inertial frame'.

Plus!... ...Is the exploitation of quantum superpositions then the way forward in designing a quantum computer that can carry out multiple calculations simultaneously?

Exactly. I wrote an article some time ago on an experiment in superposition, which you might find illuminating.

So I get that one concrete consequence of the feature of superposition in quantum physics is ‘entanglement’, fine, so if two quantum objects are entangled, a measurement of a property of one object can be used to predict the corresponding property of the other object. Is this a case, though, that these "properties" are not actually determined until they the point at which they are measured?

It would seem to be the case that these properties don't actually exist until they're measured. That's what I was saying about Einstein's position on quantum theory (despite his being instrumental in its inception, even though not directly involved) and his hatred of 'spooky action at a distance'. He wasted many years searching for the 'hidden variables', but John Bell showed that they weren't there.

AND, do molecules containing, say, a dozen atoms, also exhibit quantum superposition states, or is this phenomena restricted to elementary particles?

The simple fact is that ALL entities are subject to quantum effects. The difference lies in observation. One of the problems in the general understanding is that observation requires a conscious mind but, as I have been at pains to point out, this simply isn't the case. Every particle in the universe is, essentially, an observer, thus responsible for the wavefunction collapse of every other particle in the universe, to a greater or lesser degree. For macroscopic objects, every particle is the observer of every other.

This is, of course, a massive over-simplification, but it should serve to highlight the key points.

This is all very "Buddhist" isn't it?

Double slit

As far as the experiment goes the electron is fired at the slits prior to the detector detecting which slit it goes through, thereby negating manifestation of the interference pattern, but isn't "before" and "after" purely "relative" - in the Einstein sense of relativity I mean - here?

Well, you can have 'marker devices' that put a mark on the electron upstream, and the interference pattern will disappear. However, you can put another 'marker device' that removes the mark, and the interference pattern will return. Put in the simplest terms, it doesn't matter how you recover the 'which path' information, it only matters that, if you can tell which slit it went through, by any means whatsoever, the interference disappears.

Hi Hack, sorry not to have gotten back to this fascinating discussion, been kinda busy with packing, hairdoo, etc, cause I'm off to Stobo Castle Health Spa for lovely break with the girls. Quantum Mechanics is fab, of course, but we'll, 5 days steeped in pure luxury tops it by just a tiny bit.

I'll speak to you once I'm back tho!

"TONGUE-tied and twisted, just an earthbound misfit, I"