A philosopher presents the view that physical laws that forbid something are actually very helpful for clarifying the concepts (though searching for loopholes). Even though many find them to be dissatisfying. It's interesting to think about.

PhysicsWeb:

So tell us, Startraveler, what exactly is your position?
Do you abide by the "laws", or do you attempt to defy those "laws"?
Do you consider a mathematical 'proof' as absolute, or do you say 'Well, all it is is mathematics, and mathematics isn't capable of experimentation, so it is possible that it can be done despite the 'proof'.

I do tend to believe the universe follows rules which remain fairly constant over time and can be easily expressed in terms of mathematics. Experiments are worthless unless we use them to learn some lesson about the underlying rules. If we were to have some amnesia after every experiment, no experiment would ever add to our knowledge base. All legitimate physics is derived from experimental results (or, in some cases, logical deductions from thought experiments incorporating certain principles that are then checked via experiment). Mathematical proofs are indeed, in a sense, absolute. Time and again we've seen already-developed mathematical concepts being applied to the physical world: vector calculus applied to the experimental facts of electromagnetism (the mathematical formulation itself revealing new insights and predictions), Riemannian geometry used to describe spacetime, linear algebra utilized to describe quantum phenomena. Do quantum particles have to obey an eigenfunction expansion theorem proved by mathematicians? I don't know if they have to or not, but we've seen that they just do. So do I think we can defy the "laws"? Well, for example, I don't think we can measure two non-commutating properties at once because it doesn't seem that they both have well-defined existences at once. It's difficult to get around something like that. But I imagine there are other rules that can be bent with enough ingenuity. It's difficult to make a blanket statement, it has to depend on which law or principle we're looking at.

To be honest, even though Einstein and Hawking have both said that FTL (Faster Than Light) travel is impossible, it is all based on extrapolated mathematical evidence. I do not believe that mathematics is suffiiently rigorous in terms of external influences to be able to make that assertion. I believe that FTL is possible, regardless of the supposed relativistic consequences of such theories. Can i prove it? No, absolutely not, but I think that Applied Physicist will be able t prove it eventually.

I've always wondered about that - what if it's possible to set up an experiment where one of those properties is forced into being a known value - a clever array of polarising filters, for example - would that mean that it would be impossible to perform any other measurement?

Yes, it would be impossible. Let me try and explain in a slightly mathematical way (and we slip between math and experiment because it works, we can). I'm not sure how much you know about quantum mechanics or linear algebra so if you already know this stuff, my apologies. In quantum mechanics, we often deal with mean values for observables like position or momentum called expectation values (this is actually a concept from statistics and the integral calculation required is borrowed from there). It is possible, however, to get an exact value instead of having to rely on mean values if a certain condition is met.

Suppose we have an operator, A. Operators are recipes for doing something (something like "take a derivative and then multiply by i"). They act on special functions called wavefunctions, ψ (the main characters of QM), which contain all the information about the physical system we're dealing with. Perhaps the ψ we happen to be dealing with obeys the equation Aψ = aψ. This is an eigenvalue equation and it means we took that recipe--the operator--and applied it to our wavefunction ψ and what we got out was that ψ back again (multiplied by a coefficient, a, we call the eigenvalue). A wavefunction that obeys this equation is obviously pretty special and we call it an eigenfunction of A.

Let me take a quick digression to explain what that equation corresponds to physically. Operators represent observables like position or momentum. The wavefunction basically contains all the information abut the system in question. The eigenvalue is the value of the observable that we'll actually measure in an experiment. In the case that ψ is an eigenfunction of A, we don't have to worry about mean values (expectation values) but we can instead determine the value with unlimited precision. That value will be a, the eigenvalue.

Suppose we have another observable of our system in mind, represented by the operator B. In order to determine the expectation value of that observable to unlimited precision (and not just get a mean value), the system has to also obey the equation Bψ = bψ, where b is the eigenvalue associated with that operator. So in order to determine both physical observables with unlimited precision, our wavefunction ψ for the system has to satisfy both equations. That is, it has to be a simultaneous eigenfunction of both the A and the B operator.

Now we can look at the commutator of the operators which is just the quantity [A, B] = AB - BA. Lets hit the wavefunction ψ with that: [A,B]ψ = (AB -BA)ψ = ABψ - BAψ. We know what Bψ and Aψ are from those above eigenvalue equations, since we're working under the assumption we want both to be satisfied. So that last term is ABψ - BAψ = Abψ - Baψ. We can pull those eigenvalues a and b through the operators since they're just numbers and we have bAψ - aBψ which we know is baψ -abψ = 0. In other words the commutator [A,B] = 0--the operators commute.

That is, if we have two observables (represented by operators in quantum mechanics) then they have to commute if we want to measure both observables with unlimited precision. The position and momentum operators, for example, do not commute.

You asked if its physically impossible to perform a second measurement after you've made one measurement with your clever array of polarizing filters. The array of filters is really a set of instructions of things to do to our wavefunction as it passes through, it's a recipe. That is, it's an operator. The measurement process is itself the physical realization of hitting the wavefunction ψ with an operator. So can we do one measurement to the system to get a value with unlimited precision then do another measurement to get a different value also with unlimited precision? Only if the two operators (measurements) in question are commutative can they be done and yield the desired results. What it really amounts to saying is that there's no two experiments you could devise that could yield the desired precision results unless the commutativity condition is met. There are operators that do commute and allow these sorts of double precision results (quantities along different axes, for example) and obviously for those you could set up a set of two measurements to get the results you want.

Apologies to anyone this post confuses, annoys, or exasperates.

Ah. Beautifully explained, Startraveller. Thank you. I think I understand. In terms of simple algebra - the operators for multiplication and addition would be considered commutative, as a*b = b*a and a+b = b+a, whereas division and subtraction are not as a-b != b-a and a/b != b/a.

At least now I can see the problem clearly. Thanks once again.

Well, the question at that point becomes: are you still measuring a property as it exists normally or has your "clever array" made a change to it?

Yes, absolutely. You can get an idea of why AB /= BA in some circumstances if you think of A and B as being matrices. Matrix multiplication involves multiplying the row elements of one matrix against corresponding elements in the column of the other and adding products together to get the corresponding elements of the product matrix. Since this involves a slightly more complicated recipe then just multiplying two numbers, it's not difficult to see why the order of the matrices will usually matter. Indeed, operators can often be represented as matrices.

In the position basis, the position operator is just x and the x-component of the momentum operator is -ih d/dx (that h is actually Plank's constant h divided by 2 pi: h-bar). So you can see that if we were to hit a function ψ with xp = -x ih d/dx we'd, of course, get - x ih dψ/dx. If we were to go the other way and hit ψ with px = -ih d/dx x then we'd instead have -ih d(xψ)/dx which requires using the product rule to get -ih(x dψ/dx + ψ). So if we remember that commutator [x, p]ψ = (xp - px)ψ = -ihx dψ/dx + ih(x dψ/dx + ψ) = ih ψ.

This is a very important result called the canonical commutator: [x,p] = ih. Again, you can see that these don't commute primarily because the order in which you put them determines whether the x sits outside the derivative or if its forced into it, along with the function we operate on.




No problem.



That's one of the primary questions involved in trying to make sense of quantum mechanics (i.e. the famous interpretations of quantum mechanics). Are we measuring pre-existing properties or are we causing something to happen with our apparatuses that wasn't happening before? It's interesting because if it's the former, then something isn't quite right with quantum mechanics because it can't predict for sure the outcome of a measurement, it can only give us probabilities of this or that happening. There must be some extra information hidden somewhere in a way we haven't quite figured out yet. If it's the latter then we have a strange situation where a system seems to exist in multiple states at once before we send it through our apparatus and then it falls into one particular state when we do send it through.

The evidence seems to indicate that the latter possibility is in fact what's happening but the book is far from closed on this one. When I get time, I actually want to write up a thread on my (and others') thoughts on the nature of science/physics itself using this question as an example--a long, musing, quasi-philosophical thread. It'll have to wait for now, though.

It's interesting that you should mention that.

What if the properties of the system are constantly changing all of the time, rather than being in superposition?

As far as I can see, Bell's inequality only holds true if you make the assumption that the properties are static - which for waves, for example, just doesn't feel right...

If we make a measurement (cause the collapse) and we repeat the measurement quickly enough we'll get the same value. If you were of what one might call the realist persuasion, you'd say that of course this should be the case--the measurement returns a state the system was already in before we measured and will continue to be in (assuming we don't change things too much) after we stop looking. In other words, there's no collapse because things are already in some state and we just happen to look sometimes. The more orthodox position is that the system was in the superposition before we measured and the reason a repeated measurement will give the same value is that something indeed forced some kind of collapse (though the system quickly begins to spread out and evolve according to the Schrodinger equation again). In the view you're suggesting, I guess we'd take that time between the initial measurement and the time where we could expect to get a different value in a repeated measurement as a sort of limit on the speed at which the system cycles through the possible states. But suppose we keep making repeated measurements (i.e. keep the wavefunction from spreading out)--the cycling through states will cease and somehow the measurement operation will have halted the natural constant changing of the system. I think that would raise its own questions--within that particular conception--of what's going on.

But let me ask you a question. Suppose we take a particular observable (which we've already said is represented by an operator in quantum mechanics). Maybe this observable is of a variety that's said to have a continuous spectrum, meaning instead of having discrete eigenvalues (eigenvalues being the numbers actually measured in experiments) the eigenvalues actually have a continuous spread. Position is an example of an observable having a continuous spread like that. So if you're suggesting a system is at a particular position and cycles through the different possible positions with time, I'm imagining a system at a particular position and smoothly rolling through the continuous range of possibilities. How does this differ from a "realist" view of a particle having a particular position (or at least a particular range consistent with the uncertainty principle) and just moving with time? It seems like your suggestion is--in some cases, like this one--no different from the idea that a system possesses a particular value for a variable even before we measure it. You're explicitly saying that the system is continuously evolving but I don't know that this really distinguishes your approach from the regular realist one (which, as you've pointed out, runs into trouble with things like Aspect's test of Bell's inequalities).

Another thing to keep in mind is that in QM we can deal with quantum ensembles--large groups of particles prepared in the same state. That way we can do multiple measurements without having to worry about the effects we have on each particle just by measuring it. Suppose we have such an ensemble with two possible eigenstates it can be measured in; let's also suppose that there's a one-third probability we'll measure a particle in this state to be in the first eigenstate and a two-thirds probability it will be in the second eigenstate. So what we're saying is that there's a roughly 33% chance that a measurement of a particle's state will yield eigenvalue 1 and a roughly 67% change we'll get eigenvalue 2 out of the measurements. Of course when we actually measure one particle in this state, we'll get one eigenstate or the other and we won't have shed a whole lot of light on those probabilities we dealt with prior to measurement. But with an ensemble of many particles in this state we can keep measuring different particles and we should find that we measure roughly a third of the particles to yield the first eigenvalue and roughly two-thirds to yield the second eigenvalue. The orthodox bunch will interpret this as reflecting the projections or probabilities as just that--a probability that a particle in that state will collapse this way or that. The behavior of the ensemble is just a reflection of these probabilities playing out on a larger scale--that is, the law of large numbers steps in here (the same way that an individual quarter having a 50-50 probability of landing heads or tails results in a large number of tosses yielding results that are roughly half heads and half tails). The realist would, I think, say that the math just told us that in such an ensemble one-third of those particles would be in the first eigenstate all along and two-thirds would be in the second all along with no superpositions. But in your view of the particles cycling back and forth between the two possible eigenstates, it seems like if I keep measuring particles at random intervals I could fail to get the correct ratios.

Of course, if you're interpreting the 1/3 and 2/3 probabilities to mean that an individual particles spends 1/3 of its time in the first eigenstate and 2/3 in the second, then the law of large numbers should kick in again and help you recover the correct results (in other words, it seems you would have to use that interpretation to make this match up with experiment). But now you've got strange questions about why these particles exist in a definite state but seem to be on a timer that flips them to in a new one (somehow) at the correct time--and this without any (apparent) operation like measurement acting on them to make it happen. I'm not sure this any more appealing than any of the other viewpoints. More than that, remember in the first paragraph we put a sort of limit on the speed at which a system cycles through the possibilities. Imagine we had a system with more than just two possible states coming out of the measurement--how does a system divvy up the time it spends in each possible state if there are now 10, 100, 1000, etc possible states it must cycle through (but divvying up the time in accordance with the numbers usually interpreted to be probabilities)?

Interesting topics, I have 4 years of Solid State Physics. Great topical depth with the mathematics, I always thought of advanced mathematics as a useful scientific language of man, that often is good at describing things in nature, Classical or Modern (Quantum). Wave functions (star-psi-star- Heisenberg) only a probability, according to Heisenberg, not due to ignorance of the system. One of my professors told me, Calculus will not give you the answer to anything (approximations), but often it will get you infinitly close to the answer. Just Food for thought and conversation. Thanks, Robot

That's pretty much the way I saw it working.

For the sake of simplicity, let's talk marbles.

Take 3 marbles, 2 black and one white, packed tight together and rotating in a single plane. If you took a look at them through a pinhole, 2/3 of the time they would appear black, the other 1/3 white.

In terms of combinations - let's say that black marbles have a positive effect to the overall state measured and white marbles have a negative effect to the overall state measured, dependant on their position. For example, if the white marble was at the back, then the overall state would be at it's highest, and if at the front, then it would be at it's lowest.

Spin it and measure the effect and instead of just two states, you get an analog state transition. Mix in enough different marbles, and you could create some fairly wild and varied patterns, yet all cycling.

Obviously, that's a wildly simplified model. For a start, I'd expect it to be spinning in multiple planes and there's no reason why there shouldn't be 3, 4, 5 or more different "marbles", each with their own effects on the state.

I don't however, have any ideas as to why repeated measurement at short enough intervals would produce the same result, other than the act of measurement stops the model from spinning for a short time, and that raises the question of how, and why does it start spinning again?

*Sighs* - It may not be perfect, but it's that or superposition - and for some reason, like Einstein, it just doesn't feel right to me.

The problem is that what you're describing is a local hidden variable. If you pictured the different possible states as being at spokes of a spinning wheel arranged such that we can only see (measure) one at a time then whatever's analogous to the angular momentum of the spinning wheel and the arrangement of the states on the spokes--that is, the stuff that makes the thing deterministic--are the hidden variable. Bell's theorem is interesting in that it doesn't assume anything about the hidden variable (other than that it's local) or the complexity of the bells and whistles underlying the hidden variables, it just assumes there is some such system in pace. The wiki article on Bell's theorem actually presents a setup not all that different from your simplified spinning marbles scenario:



That goes on a little bit to talk about how the inequalities are violated but you get the picture: it's a set of instructions carried by the system that decide what's going to happen we we look at it, similar to your suggestion.

I don't find all of this very appealing either but it seems to be how things are. At least as best we understand now.

This is a little off topic. Has anyone heard of quantum chemistry S1 and also a new theory called S2 chemistry? And if you have, what is your view of this subject? I am a bit of a novice on quantum theory's but it is to me a very interesting subject

Nothing much, they happen to be catalog numbers for university courses.

Thanks for your response, but I am more interested in the aspect of S2 chemistry in which deals more with some of the theory's S1 quantum chemistry does not explain.

exothermic and endothermic reactions can explain alot of this espiecally the faster than light travel... if we can get somthing to move the speed of light using one of these reactions it should speed faster than light.....But in my little known experience I could be wrong

I wonder why this thread was downrated?



I haven't but chemistry's not really my thing. What is it?



Exothermic and endothermic just refer to whether or not something releases or absorbs heat when some reaction occurs. I don't see how chemical reactions could lead to faster than light travel.

Im not quite sure either, nor is chemistry my thing. Are you maybe confused Essene and talking about energy states? As in quantum states?

Maybe the excitation of electrons in orbitals? 1s, 2s, 2p etc?

Can you clarify more essene?

Speaking of physics do any of you by chance have this file or know exactly what Im looking at in the third page of this file?
http://www.wikiupload.com/download_page.php?id=179828

Thanks:
mel

I had been reading in another forum a little joust on occult chemistry and known theory's of quantum chemistry. Pretty interesting and I would love hear any feed back on this theory. By Ron C. <QUOTE>Hello Mike;

Many thanks for inquiring further into this, yes?

As it happens, Occult Chemistry only predicts funnels of ANU. An
Ormusized atom no longer distinguishes between the nucleus and the electron
cloud. In such an atom, the matter distribution will have all sorts of
structures which are made of ANU and are very "delicate".

Any attempts that we know of using ordinary S1 chemistry observation
tools to observe them with electro-optical means destroys the ANU configuration
and the atom reverts to the predictions of quantum chemistry in terms of a
nucleus in the center and some electrons in orbit around that.

Indeed, how does one know that these funnels are "real", right?

At this time, there are only indirect observations via the biological
effects of ORMUS and, of course the clairvoyant observations of Besant as well
as a few pieces of mathematical physics.

In the 1980ies a PhD thesis by Philips at Cambridge University in
England came up with a theory of "quarklets", there being 3 quarklets to make up
one quark and, as usual 3 quarks to make up one proton or 1 electron. Philips
concluded that the hydrogen atom would therefore be made of 9 quarklets for the
proton and 9 quarklets for the electron, for a total of 18 quarlets, which by
then, he assumed to be the ANU that Besant had seen decades earlier.

Philips went on to compute the mass of the atoms of the atoms of the
periodic table and found the same results as Besant, which as you noted, are not
all that good.

More recently, the Tetrahedral Relativity model with its 13-dimensions,
9 more than the regular 4-dimensional space time, has given rise to a 9
dimensional harmonic oscillator model whose quanta would of course be the
quarklets. From mathematical physics, it is known that when one wants to
compute the energy levels, one add a 10th dimension, the energy to the
calculation. Not too surprisingly, this is the SU(10) model that was used by
Philips to compute the mass of the elements, with not so hot results. But then,
one would expect all sorts of perturmations on top of a simple harmonic
oscillator model. So, matters remain inconclusive at this time. Couple all
this with the notion that S2 Chemistry involves two kinds of Gravity, not just
the standard one. This can be taken to mean that Ormusized atoms do not
necessarily have the same mass as S1 atoms. Barry can recount many observed
mass anomalies in ORMUS work, yes?

And of course there is the breath catching feat that some yogi
accomplish which is that of levitation, as if the mass what not really the mass
of the body, oh well!

As you can see, there is not yet a great deal of calculable mathematics
in S2 Chemistry, perhaps because of the difficulties of doing experimental
observations as most of the S2 Chemistry results involve biological phenomena
which are notoriously difficult to control and quantify.

For example, there is a yogi in South East Asia who treats people by
emitting sparks from his fingers. Then there are many documented instances of
Quantum Touch practitioners who can charge dead cell phone batteries by sending
QT energy into them (never physically touching the battery). Also, there is
Danae Harding's sparking water and her personal magnetism that is almost 10
times larger than that of the planet. My Liquefied Barley Grass also sparks
sometimes when I bottle it and my personal magnetism is only twice as large as
that of the planet.

Add to this the troubling known fact that Maxwell's equations fail to
correctly predict the fundamental law of magnetism, called Ampere's law for the
interaction of two currrent elements. These are the very Maxwell equations that
are used in Pauling's quantum chemistry calculation, and you begin to get the
uneasy feeling that maybe, just maybe, there is more than meets the eye that
relies only on quantum chemistry, right?

And, of course, there is the ORMUS phenomenon that has no explanation at
all in terms of Quantum Chemistry and you begin to see that the world is getting
ready for an explosion of new knowledge, yes?<END QUOTE>

Hi Star - apologies for not responding sooner - I've been rather busy as of late.

With regards to Bell's Theorem - I think that what I'm trying to say is easier explained using Mathematics:

To help, I'm using the Bell's Theorem Simple proof from mtnmath.com.



It's this fundamental piece of logic which I believe is flawed.

In a world where properties do not change, then I concur that this holds absolutely perfectly true. However, as I stated in my previous post - I don't see why properties of an object would NEED to be static.

Consider the possibility that the properties are not fixed over time , and you'll soon realise that the above logic would no longer hold true.

Let's skip back into English, just to make sure you understand what I'm saying.

We'll take a contrived example, using a room full of glamour models as our objects and give them the following properties:

Wearing Lingerie
Wearing Dresses
Wearing Boots

which are either true or false.

Translating the above, we can say the following:

The number of Models which are wearing Lingerie and are not wearing Dresses + the Models which are wearing Dresses and are not wearing boots is greater than or equal to the number of Models wearing Lingerie and not wearing Boots.

Let's construct a quick table, just to confirm this:

Name
Susan Wearing Lingerie, Dress & Boots
Mary Wearing No Lingerie, No Dress & No Boots
Shiela Wearing Lingerie, No Dress & No Boots
Beatrix Wearing No Lingerie, Dress & Boots
Paula Wearing Lingerie, Dress & No Boots
Bob Wearing No Lingerie, No Dress & Boots

The number of models which are wearing Lingerie or not wearing dresses = Susan + Shiela + Paula + Mary + Bob = 5
The number of models which are wearing dresses or not wearing boots = Susan + Beatrix + Paula + Mary = 4
The number of models which are wearing Lingerie or not wearing boots = Susan + Shiela + Paula + Mary = 4

As the first two counts (5 + 4) 9 is greater than 4, our equation is satisfied.

However you play around with what the models are wearing (or not), the equation will always be satisfied. This is, quite simply, because the first measurement (Lingerie or no dresses) catches all the models wearing Lingerie, whilst the second measurement (Dresses and no boots) catches all the models wearing no boots.

However...

Models, being busy by nature and having to walk the catwalk several times in an evening have a tendency to change what they're wearing.

If the models get changed between counting the number of models which are wearing lingerie or not wearing dresses & counting the number with dresses or no boots...then our equation may no longer hold true.

Imagine the first catwalk involves Dresses and Boots. Our first measurement catches 0 models. They quickly get changed for their second catwalk session, Lingerie and Boots. Our second measurement catches 0 models. Exhausted for the evening, they come back for the after show party and are counted again - this time all 6 are counted, as they're all still wearing lingerie.

Okay. I know it's a long drawn out example, but on the upside I got to post about models wearing lingerie (or not) in a physics thread. The point is, that as two measurements are made, if the properties of the object change between measurements, then the boolean logic underpinning Bell's theorem collapses spectacularly.

There's much to be said here, I think. You raise an interesting point by bringing time into this but I'm not sure it's exactly in the way you think.

Before we go any further, we should recap what's going on here. It's convenient to think of a version of the EPR thought experiment devised by Bohm. Suppose we've got a neutral pi meson (spin zero) which then decays into an electron and a positron flying off in opposite directions. We set up two detectors, one in the direction the electron is going and the other in the direction the positron is going. When we measure the spin of one we'll either get up (which we'll denote at +1) or down (-1). We know, however, that if the detectors are parallel to each other then when we measure an up (+1) on one, the other will have to yield a down (-1). That is, we don't know which will be which but we know the two detectors have to yield opposites. Now a local hidden variable theory is something that suggests the outcomes are determined in advance. If we assume that this is the case and there is no influence of one on the other after they've separated, we can work out the consequences of this idea.

Bell didn't originally derive his inequality in exactly the same way as the derivation you linked to but he did embrace d'Espagnat's formulation in a famous 1981 talk he gave called "Bertlmann's Socks and the Nature of Reality" (which is fantastic and is the basis for most of what I'm about to say). The Bertlmann's socks analogy is one in which a consumer research organization is worried about whether a sock could survive one thousand washing cycles at 0°C, 45°C, and 90°C. This situation is easily translated into the Wigner-d'Espagnat inequality you quoted in your post. "A" just becomes the number of socks that could pass 0°, "B" just becomes the number that could pass at 45°, and "C" the number that pass at 90°. Bell notes, however, as you undoubtedly did:



If I'm understanding your objection correctly, you're imagining that when we talk about "if one and not the other," etc this implies two tests must be done on one particle and something can happen in the time between those two experiments that renders this all useless. As Bell himself just noted, you couldn't do such a test on particle. First of all, if the "sock" didn't survive the first test, it certainly couldn't be subjected to the second. And even if it did survive the first "wash," it wouldn't be a new sock anymore and the whole thing might be suspect.

The reason the sock analogy is even being used is because socks come in pairs. We make the assumption that both socks in the pair act the same way--that is, if one would survive the conditions of a wash, so would its partner if the test were performed on it. So now instead of talking about a condition A and not B in which a sock could survive a thousand washes at 0°C but not at 45°C, we start using the fact that it exists in a pair and alter what A and B (and C) mean a bit. We now talk about things like "the number of pairs in which one could pass at 0° and the other not at 45°" and so on. If we add in some random sampling we can get probabilities that a sock will pass given a certain condition without ever even thinking of doing more than one test on a single particle. We'll instead do a lot of different tests on a lot of different particles.

It should be pretty clear how the socks analogy related to the real world example. The 0°, 45°, and 90° are of course not temperatures but rather orientations of the detectors we've set up to measure the electron and positron spins. The pair of socks is the entangled electron-positron pair. Here we need to make the adjustment that instead of following the sock rule that if one passes a test so does its mate, we must substitute in the fact that the spins of the particles are going to anticorrelated: if one passes the test, its mate in that situation would surely have failed. We are led to the inequality you posted and see that it is violated. The ultimate moral (not entirely crystal clear here) is that the quantum mechanical calculation for the probabilities of the outcomes of measurements in situations where the electron and positron detectors are oriented toward each other at different angles is not the same as a calculation that assumes the existence of a local hidden variable.

It seems to me your objection was based on a picture of multiple measurements being made on a single particle without proper consideration being given to what happens during the time lag. But there is no such time lag because no one particle is subjected to more than one measurement. But it's still interesting to think about what role time could play in all this. We can change the orientation of the two detectors (and independently of each other, of course) while the electron and positron are in flight, long after they've parted company with each other. But we've still got to get the "right" results (i.e. one spin up and one spin down), regardless of when we measure the particle and what the relative orientations of the detector are (not that we don't have to measure the spins at the same time). So you'd need to figure out a way to make the correlations work right to get the expected results, regardless of what hurdles we throw into the situation.

That said there was an objection concerning time in Bell's theorem brought up a few years ago by a pair of physicists. Let me post the news story on this thrown together by Nature at the time:



Hess and Philipp themselves actually wrote this in their paper on the subject:

The proof of eq. (4) [Bell's inequality] involves a number of definitions and considerations that Bell has discussed in his book [10] and in his last paper [11]. They have been derived involving well-known arguments of relativity by use of light cones. However, there are two additional assumptions that Bell uses and that appear in all proofs of Bell-type inequalities. Bell assumes that ρ(λ) is the same over any run of experiments and that the conditional probabilities (given λ and settings) that A, B assume a certain value are also the same. These assumptions permit then certain factorizations that are vital for all proofs (see, e.g., p. 56 of ref. [10]). We argue below that these assumptions exclude time from the parameter space. The Bell theorem thus describes only time-independent processes. We believe that this greatly restricts the relevance of the Bell inequalities for EPR experiments and like to emphasize the following. The instrument settings need to be changed randomly during a run of measurements and given settings need to occur at random times in order to avoid certain well-known “loopholes” [3, 4] of Bell’s reasoning. The proof of Bell’s inequalities, however, is entirely silent about any role of time and the necessity for given settings to occur at random times. It does not exclude loopholes by putting restrictions on time-like correlations between settings and parameters. This clearly is inconsistent with the claim that Bell’s space of hidden parameters is entirely general. If it were and included time-related parameters, then it would need to restrict these parameters to exclude the well-known loopholes [3]. As we will see momentarily, the proofs ala Bell exclude time and time dependencies altogether.

Here λ refers to the hidden variable (i.e. the unknown scheme by which the results of a measurement are determined) and ρ(λ) represents a probability density of that hidden variable.

However, their arguments seem to have been taken apart by a few different authors. To quote one of the main objections:

Secondly, we did not mention time in our derivation [of Bell's theorem] at all because it was completely irrelevant. Our derivation concerned each run of the experiment. We did not compare actual outcomes under different settings at different times, but potential outcomes under different settings at the same time. Therefore, the argument in (1) Eqs [8] and [9], or in (2), end of the paragraph following Eq. [11], is completely beside the point.

To emphasize this point, consider (as a thought experiment) repeating the measurement procedure just described, not as a sequence of successive repetitions at the same locations, but in a million laboratories all over the galaxy. The prediction of local realism is that when we collect the one million sets of observed quadruples (A, B, X, Y ) together and compute four relative frequencies estimating the four conditional probabilities Pr{X =Y | AB = i j}, they will satisfy (up to statistical error) Bell’s inequality. It is of no importance that the distribution of hidden variables at different locations of the experiment might vary.

Hess and Philipp make a large number of criticisms of the assumptions of Bell’s theorem with the main theme being that variables at both locations can vary in time in a dependent way, leading to dependence between the outcomes, which Bell supposedly did not take account of. Before turning to their model, we confront our formalization of the metaphysical assumptions of local realism with the idea of time variation.

How could time variation invalidate the freedom assumption? One would have to argue that because of systematic long-time periodicities in the various component physical systems concerned, the outcomes of a complex series of events involving a card shuffle, a coin toss and the free will of an experimenter at one location are interdependent and highly correlated with the potential outcome of a certain polarization measurement at a distant location. A good experimental design, with rigorous randomization of the choice of settings, makes this totally implausible.

Once again, Startraveler, my thanks. You've explained that beautifully.

I am, however, still a little confused.

Let me just confirm what I think the position is using the entangled positron/electron pair as an example:

We have two detectors, that have three positions: 0 degrees, 45 degrees up and 45 degrees down.

We produce a very large sample of entangled positron / electron pairs.

When the two detectors are set to 0 degrees [a], we have a perfect correlation (100%) between the spin of the two pairs, one positive and one negative.

When the detectors are 45 degrees apart from each other [b], they correlate 71% of the time.

When the detectors are 90 degrees apart from one another [c], they correlate 50% of the time.

substituting in the values into the equation, we get:

[A or not B] = 100 + [B or not C] 71 >= [A or not C] 100

which doesn't seem to violate the inequality. What am I missing?

I think the problem is that the form of the inequality that you're using deals with numbers of particles (or probabilities), not necessarily correlations. Correlations obviously range between 1 (i.e. knowing the value of one means you know the other is the same) to -1 (knowing the value of one means you know the other has the opposite value). A correlation of zero would mean that knowing the value of one gives you no idea of what you're going to measure for the other--this would be when the detectors are at 90 degrees to one another (when the variable we're looking at has only two possible states and you say that joint measurements "correlate 50% of the time," you're really saying that even if we know one, it's still a crapshoot as to what the other will be). Probabilities, on the other hand, are squares in quantum mechanics and here they're going to be squares of sine functions (which, of course, already range between -1 and 1). As the link you provided mentions, the probabilities we're interested in calculating for this inequality are of the form . So when we're dealing with the right hand side of the inequality, which is concerned with the 0 and 90 degree detector orientations, we're looking at a 90 degree separation between the orientations of the two detectors. Such a situation yields correlations of 0 but the probabilities involved in the situation are 0.25 (the largeness of this number compared to what shows up on the other side of the inequality is, of course, the reason the inequality is violated). The numbers we're dealing with this in this situation are the probabilities, not correlations.

Sorry Essene, I didn't mean to skip over your post. I hate to say it but it sounds like the majority of the post you quoted is bunk. There are little scientific inaccuracies sprinkled in here and there (electrons, for example, aren't made of quarks) and references to things I don't believe are real. For example, I've never heard of "tetrahedral relativity" (the newest forms of special relativity I know of are called doubly special relativity and they certainly say nothing about 13 dimensions) and I'm not sure what it means to say that the quanta of a harmonic oscillator are quarklets. That post says this occult chemistry involves two kinds of gravity but I don't know if that's for chemistry (occult or otherwise) to decide. At any rate, I'm not familiar with the other form of gravity. I'm still not sure exactly what this S2 chemistry is but now I fear there might not be very much science in it. Tread carefully.

Okay. I've re-read the wikipedia entry and I think I've (finally) got my head around the correlation thing.

As there aren't two measurements being made, I'm quite willing to accept that the results preclude the possibility of any hidden variables, as I'm sure that far better Mathematicians than I have scrutinised those results thoroughly.

Why is it then, that my gut is still telling me that God doesn't play dice with the Universe?

I've been doing some deep('ish) thinking about the experiment, and I've come to the following conclusions:

1. The results from the Bell test indicate that additional information, other than the starting conditions are being encoded into the end result.

2. If we preclude the possibility of faster than light transfer of information from one particle to the other, then there is no known mechanism that the particles could use to transfer information between themselves.

3. The only other known source of information within the experiment is the angle of the detectors themselves.

Based on those conclusions, I've come up with the following hypothesis:

Tiggs Hypothesis



In short, the angle of the detectors supply the additional information to the particle, as the particle is only formed by the interaction of the detector and the emanating particle wave.

To me, that's far more palatable than having to believe in Superposition or a mysterious requirement for an observer to cause the wavefunction to collapse.

Instead, the wavefunction becomes the possible combinations in which the particle waves can combine. There is still superposition (of sorts) as technically the particle can take any form defined within the wavefunction, the final form of which is only determined by interaction with the "observing" particle wave.

However - the underlying physics is pure old school - and no dice are required

It's only a hypothesis, however. Feel free to rip it to pieces.

Thank you for your caution but I am way past that and truly something is amiss here. I have seen first hand what this material M3 (a form of ORMUS, ORMUS is a general term) which is supposedly in a monatomic or diatomic state. Its rhodium, iridium and gold which have astounding health effects. My fiance has multiple sclerosis and has ingested M3 since December 2006 (seven months) and has had NO symptoms of MS since December and others have had the same effects for MS and other major health issues (we think she is cured). She has stated to me that she has never felt better in her life since ingesting M3. I am just trying to find scientific credence why this can happen (a cure for MS?). It works, I have no doubt of this. I am not what some may call a crystal waver, I am far from it. I am a layman hoping you may help. This is what some call esoteric chemistry and I am finding this very true as no one can totally explain why these effects occur which are many and very positive. <QUOTE From A Forum Member>

Hi Barry,

Excellent articles on resonance, vibration, and quantum coherence concerning
ormus. It appears that the boson box/cage framework is a good model to explain
properties of ormus. Well done!


It reminds me of Michael Grady's theory of a Liquid Phase Transition model of
our Universe. (this actually fits in a more inclusive Fractal Hologram Model).
Anyway, his idea is that the Universe is a giant crystal growing in a
five-dimensional liquid and quantum fluctuations are simply the random sloshing
back and forth of heat in the bulk of the fluid. "These heat fluctuations
continually buffet the phase boundary in which we live.

Now, the subatomic particles that form the building blocks of our world, these
particles come in two types: "fermions" such as electrons, which obey the Pauli
exclusion principle forbidding two particles from occupying the same quantum
state; and "bosons" such as photons, which observe no such restrictions. Grady
believes bosons are "phonons", or vibrations of the crystal lattice, while
fermions are defects of the lattice known as "screw dislocations". Think of the
planes of a crystal as the stacked floors of a multistory parking lot," says
Grady. "A screw dislocation is like the spiral ramp connecting the floors."

Two identical screw dislocations obey the Pauli exclusion principle because
they repel each other when forced together. And a mirror-image pair of screw
dislocations, differing only in the sense in which they spiral, behave just like
a particle and its antiparticle. When they meet, they cancel each other and are
annihilated in a burst of energy. The opposite of this process is "pair
production", in which two screw dislocations of opposite sense pop into
existence if vibrational energy is supplied to the lattice.

According to Grady, screw dislocations even obey Einstein's special theory of
relativity, with the speed of sound in the solid acting like the speed of light
in our Universe. The link between the speed of sound and the rate at which screw
dislocations can travel was first shown by the Russian physicists J. Frenkel and
T. Kontorowa in 1938. According the their picture, as a screw dislocation
approaches this limiting speed it compresses in the direction of motion by
exactly the amount predicted by Einstein. At the same time, the stress energy of
the screw dislocation rises, again in accord with Einstein. "At the speed of
sound, the energy, and hence the effective mass, of the dislocation becomes
infinite," says Grady. "Fermionic matter is therefore prevented from traveling
faster than light.

Anyway, ORMUS elements exhibit quantum properties and behaviors which are only
associated with bosons.

So, since bosons are force carrying particles...ORMUS platinum group metals
etc., must be a type of boson despite the fact that metallic platinum group
metals are fermions. Further, quantum coherence, quantum non-locality,
superconductivity, tunneling and other generally strange behaviors are
associated with bosons but not fermions.

Further, this means that a particle acts as if it were a spherical oscillator.

(Quantum theory requires particles to meet this requirment, that the frequency
of its waves to be proportional to its mass in accordance with the formula f =
mc2/h.)

And, these particles as an oscillator, exhibit a Standing Wave Center that
resists coupling unless both have the same resonant frequency. Thus the term
Particle and Standing Wave Center are interchangable.

So, these Higgs fields are also scalar fields which tap into dark energy.
This gets into Zero Point Energy, the Aether and so on...

Which leads me too Hans Jenny's Cymatics; where a vibrating surface is covered
with sand and the frequency of the vibration creates patterns -- apply this in
3D/4D
and the over sensitivity to frequency processing in our brains points to the
emergence of 'archeypal' forms.

And, ultimately making the leap to our analysis of fundamental particles.
Which has led to the distinctions of two basic types -- fermions and bosons. Is
it coincidence that the general behaviours of these types reflect properties of
objects (fermion-like, no superpositions, discrete) and relationships? And
(boson-like, superpositions, waves) which seem to be a possible link of brain
oscillation behaviour and the resulting patterns to the beginnings of the
universe based on the oscillations, the dynamics of fermion/boson interactions?




A quote from the American Institute of Physics:

"A superfluid is a liquid that flows without viscosity or inner friction. For
a liquid to become superfluid, the atoms or molecules making up the liquid must
be cooled or "condensed" to the point at which they all occupy the same quantum
state. A liquid of helium-3, an atom whose nucleus is made up of an odd number
of particles, is a type of particle known as a fermion. Groups of fermions are
not allowed to occupy the same quantum state.

By cooling the liquid to a low enough temperature, helium-3 atoms can pair up
(left panel). The number of particles in each nucleus adds up to an even number,
making it a type of particle known as a boson. Groups of bosons can fall into
the same quantum state, and therefore superfluidity can be achieved. Helium-4
(middle panel), a boson, does not need to pair up to form a superfluid; groups
of helium-4 atoms condense into the superfluid state at about 2 degrees above
absolute zero. Superfluidity, especially the kind that exists in helium-3, is
analogous to conventional low-temperature superconductivity, in which electrons
flow through certain metals and alloys without resistance. In a superconductor
(right panel), electrons, which are fermions, pair up in the metal crystal to
form "Cooper pairs," bosons which can then condense into a superconducting
state."




http://www.newscientist.com/article/mg1612...ic-crystal.html




Barry > wrote:
Dear David,

Thanks for a very well written description of this concept. I
generally use one of Hans Jenny's videos in my workshop to illustrate
this concept.

If I might add a bit on to your explanation and relate it to ORMUS:
Imagine that the ORMUS elements can function as quantum coherent
resonators. Now imagine that these resonators hang out inside of the
balloon and that the balloon is a water molecule. If the resonators
changed frequency then the shape of the water molecules would change
in response.

I have written three articles on this theory starting in 2002:

http://www.subtleenergies.com/ormus/tw/box.htm

2004 from my Workshop slide show:

http://www.subtleenergies.com/ormus/tw/resonance.htm

2006:

http://www.subtleenergies.com/ormus/tw/spincoherence.htm

At 09:44 PM 7/13/2007, you wrote:

>Hi Bo,
>
> You are on to something!
>
> You are getting into Sympathetic Vibration & Fractal
> Geometry. From a physics [ PHI - CYCLES ] point of view, the
> fundamental basis of all natural phenomena is vibration. All
> natural forces manifest to us in a vibratory or cyclical manner.
>
> Standing Waves - sound and form - When a vibration or wave
> reflects off of something, such as another wave, it can interfere
> with its own reflection. The interference is alternately
> constructive or destructive as the two waves move past each other.
> This creates a standing wave . Only waves with certain frequencies
> can create standing waves. This is because the distance from one
> node to the next must always be some fraction of the total length
> (one half, one third, etc.).
>
> All objects have a frequency or set of frequencies with which
> they naturally resonate when struck, plucked, strummed or somehow
> disturbed. Each of the natural frequencies at which an object
> vibrates is associated with a standing wave pattern. When an object
> is forced into resonance vibrations at one of its natural
> frequencies, it vibrates in a manner such that a standing wave is
> formed within the object. So the natural frequencies of an object
> are merely the harmonic frequencies at which standing wave patterns
> are established within the object.
>
> These standing wave patterns represent the lowest energy
> vibrational modes of the object or complex system. While there are
> countless way by which an object can vibrate (each associated with
> a specific frequency), objects favor only a few specific modes or
> patterns of vibrating.
>
>The favored modes (patterns) of vibration are those which result in
>the highest amplitude vibrations with the least input of energy.
>
> In 3D standing waves, a structure, with all characteristics of a
> platonic solid, is formed for each standing wave mode. Within an
> atom, which is the building block of matter, the platonic solid is
> not formed by salt or known particles, but by electromagnetic waves in vacuum.
>
>Both the students of Buckminster Fuller and his protege Dr. Hans
>Jenny devised clever experiments that showed how the Platonic Solids
>would form within a vibrating / pulsating 3D sphere.
>
>In the experiment conducted by Fuller's students, a spherical
>balloon was dipped in dye and pulsed with pure sine wave sound
>frequencies. A small number of evenly-distanced nodes would form
>across the surface of the sphere, as well as thin lines that
>connected them to each other.
>
>If you have four evenly spaced nodes, you will see a tetrahedron.
>Six evenly spaced nodes form an octahedron.
>Eight evenly spaced nodes form a cube.
>Twelve evenly spaced nodes form the icosahedron
>and twenty evenly spaced nodes form the dodecahedron.
>
>The straight lines that we see on these geometric objects simply
>represent the stresses that are created by the closest distance
>between two points for each of the nodes as they distribute
>themselves across the entire surface of the sphere.

I'm not sure how this deal with the underlying problem--how we recover the correlations between spatially separated particles. There may be some recipe for what happens when the particle encounters a detector at some angle but we haven't explain why the distributions or the probabilities should change based on the orientation of the other detector. One detector doesn't know when the other is parallel or perpendicular to it and--in this kind of view--neither does either particle. Yet still it seems we have to treat this whole system as a whole, despite the fact that its parts could be separated in principle by great distances. I'm not sure what it all means or what picture exactly this is painting of the universe but it does seem to be the way it is.

Well, we already have a recipe. Quantum theory predicts that the correlation between two entangled particles is cos θ, where θ is the angle between the two detectors.

We know that if we aim both detectors at the particles at an angle of 0^o, we get perfect correlation. If they're both aimed at 45^o, 60^o or 37.2129785^o we also get perfect correlation.

When the detectors are angled at 45^o to each other, we get cos 45 correlation. It doesn't matter which detector is angled up or down, or whether the angles they're measuring at are 0^o and 45^o or 15^o and 60^o.

Similarly, when the detectors are angled at 60^o to each other, we get cos 60 correlation. Again, it doesn't matter which detector is angled up or down, or whether the angles they're measuring at are 0^o and 60^o or 30^o and 90^o.

All that matters is the angle between the two.

In my hypothesis, this is because the angle that the detector combines with the particle wave changes the distribution of the spin of the resultant particle produced.

What I believe that the Bell Tests are telling us is that there is no possible combination of hidden variables which can be present within the particle alone that can give us the results that are produced.

In my hypothesis, there is no need for information exchange between the pairs of detectors and particles. The only information exchange is between the particle wave and the detector, via the combination of the particle waves to form the particle.

That's just the thing; unless you're conceding that your particle is at both detectors at once, I don't see how it could have any idea what the angle between the two detectors can be. All it can know is the angle a between its path and the detector it encounters; the other particle will encounter an angle b between its path and the detector it encounters. But neither angle a nor angle b, the things actually knowable to and encountered by the corresponding particles, are the important part. What's important is b-a = θ. It sounds like you're saying that each particle is carrying a set of instructions that say something like "if the detector is inclined at 45 degrees, do this." But even though the particle knows a = 45, the crucial piece of information is the value of b. If the other detector is also tipped at 45 degrees then b-a = 0--that is, the detectors are parallel to each other.

So maybe at some point after the pi meson decays I'm playing with one of the detectors. One of the particles is coming toward it and I don't decide until the last minute what I'm going to set it to. Meanwhile, at the other detector, the other particle has arrived and found the detector to be inclined at 12 degrees. How, in your view, does it choose what happens? The other detector that I'm playing with (which, maybe is a little further along so it hasn't yet registered its particle) could be set to 12 degrees itself or maybe it's at 57 degrees or maybe something even more random like 39 degrees. If nothing is happening between the two particles, how can their instruction sets tell them to produce an outcome which is based just as much on information that they don't have as it is info they do have?

That is what I was saying, more or less.

Well, not instructions as such, as it's the angle that the two waves merge that would produce the result, but essentially we'd expect the same outcome (on average) each time from the same angle.

What I'm proposing is that a translating function, f, exists such that f[a] !&& f[B] = cos[a-B] (where !&& is a boolean Nand).



Imagine a set of eight particles with +ve and -ve spins which go to each detector, which I'll represent as 0 and 1 respectively, such as

Detector 1 - 01110010
Detector 2 - 10001101

For the sake of simplicity, lets say that at 0 degrees, the function does nothing with the spin of particles, so if both detectors are aligned at o degrees then detector 1 receives +ve(0), -ve(1), -ve(1), -ve(1) +ve(0), +ve(0), -ve(1), +ve(0) and detector 2 receives the exact opposite, giving us an exact correlation between negatives and positives.

At 60 degrees, lets say that the function reverses every second polarity. If we set both detectors to 60 degrees, we'd see the following:

Detector 1 - 00100111
Detector 2 - 11011000

which, again, gives us an exact correlation between negatives and positives.

if Detector 1 is set to 0 degrees and Detector 2 is set to 60 degrees, we'd see the following:

Detector 1 - 01110010
Detector 2 - 11011000

which gives us a .5 correlation between negative and positives.



a and b are interlinked, as there's a boolean operation that occurs between the translating function that occurs at each detector.

f[a] !&& f[B] = cos[a-B] (where !&& is a boolean Nand).

However, it's quite possible that such a function doesn't exist, in which case my hypothesis is, as you suspect, fatally flawed. However, as the same variables exist on both sides, I thought that there's a possibility it might, though, in all fairness, I couldn't tell you what it is.

Ok, I see what you're saying now. There's some function governing the electrons, generating that 0 and 1 list of when to be spin down and when to be spin up. The angle of the detectors alters each rule, each sequence of 0s and 1s in just such a way that it always works out to match the quantum prediction. It's an appealing picture and one a lot of people have wondered about. Obviously the simple toy model you presented wouldn't hold up but how do we know one like it wouldn't? Well, let's think about the implications and walk through it.

Let's think about the products of the electron and positron spins when the electron detector is tilted along some unit vector a (i.e. tilted at that angle) and the positron detector is tilted along some unit vector b. I'm going to change your 0s to -1s to represent spin down but otherwise its the same deal. We send eight particles to the detectors when they're parallel at 0 degrees and find, as you wrote, the spins are going to be

Detector 1 : -1 1 1 1 -1 -1 1 -1
Detector 2 : 1 -1 -1 -1 1 1 -1 1

The products of the spins for each pair is clearly going to be -1 every time, since we know in this situation we should get opposite results each time. So we'll take the average of all these products (eight in this case, but in principle it could be many more) and call it P(a,b). In this case, since the detectors are pointed in the same way, along the same vectors we have P(a,a) = -1. We could flip it so that they point in opposite directions (b = -a) in which case we'd have P(a,-a) = +1. In general, quantum mechanics will predict that this product should be

P(a,b) = -ab
This is just another way of writing what we've been saying all along—the cosine of the angle between the detectors is the important thing (not sure if you're familiar with the dot product but it's just the cosine of the angle between the two dotted vectors). To make our stroll through this math easier, I'm adopting the time-honored tradition of numbering the important equations that we'll be looking back to. Anyway, the question is, can an idea like yours agree completely with this?

So now we think about λ. λ is the secret information, the hidden variable that controls your scenario so that everything works out all right without need for non-locality. At the electron detector, there's a function that depends on λ: A(a, λ). This A function depends only the angle of the electron detector itself and the secret scheme. The function then generates all those -1s and 1s (or 0s and 1s, as you wrote it). That is, A(a, λ) is where it all comes from. Similarly, over at the positron detector there's a different function that depends on λ, B(b, λ). This one generates the sequence of 0s and 1s at the positron detector. Note that this is exactly what you're suggesting in your scheme--there's some rule by which the 0s and 1s are generated at each detector and it depends on the angle of that detector and some secret piece of information.

We know that these A and B functions are only going to give us 0s and 1s (or, as I'm writing it, -1s and +1s). They only have the choice of yielding spin up or spin down. That is, we know

A(a, λ) = ±1; B(b, λ) = ±1.
Now I don't think I've said anything controversial, I'm just rewriting the scheme you suggested in a way we can deal with (and if you have any questions up to this point or what's about to happen, please ask). Now let's see what the implications are.

Let's start with what we know. If we align the detectors, the results (the -1s and 1s) are going to be perfectly anti-correlated. Ups always match downs, in that case. We can write this as

A(a, λ) = -B(a, λ). This has to be true for every λ no matter what it is. We don't need to know the secret information to know this, we're just noting that if the detectors point along the same vector a they better be anti-correlated.

We're interested now in writing down the average of the product of the measurements, that number we called P(a,b). To do so we dip into the long-established rules of statistics/probability theory (and usurped by physics). We write this average as

P(a,b) = ∫ ρ(λ) A(a, λ) B(b, λ) dλ
That ρ(λ) represents the probability density of the hidden variable; as a probability density it's nonnegative and it obeys a normalization condition, i.e. ∫ ρ(λ) dλ = 1. Other than that, we don't know anything (or assume anything) about that probability distribution. Presumably it would depend on the particular hidden variable theory you're using but we're staying general and don't really care what the rule is.

We can use the rule we figured out in equation (3) to rewrite that integral a little differently:

P(a,b) = - ∫ ρ(λ) A(a, λ) A(b, λ) dλ
So now that we've got a recipe for writing integrals for that average of the products of measurements we can imagine subtracting from that integral P(a,b) another that's based on a third vector along which one of the detectors could point, c. That is, (using the rule that when you subtract two integrals you can stick the two integrands under one integral sign):

P(a,b) - P(a,c) = - ∫ ρ(λ) [A(a, λ) A(b, λ) - A(a, λ) A(c, λ)] dλ
We can rewrite this in yet another way if we note that [A(a, λ)]^2 = 1 (since that A function only spits out 1s or -1s anyway, the square will always be +1):

P(a,b) - P(a,c) = - ∫ ρ(λ) [1 - A(b, λ) A(c, λ)] A(a, λ) A(b, λ) dλ
If you multiply that back out and use the fact I just noted, you'll see this all works and I haven't pulled a fast one on you. Now there's just one more change we have to make and we'll see what we set out to see.

If we look back at the equation (2) it's pretty easy to see that -1 A(a, λ)A(b, λ) +1. And if such a product can at most be one (and a probability density must be nonnegative), then we also see that ρ(λ) [1 - A(b, λ) A(c, λ)] 0. With these facts in mind we can rewrite this integral we've been playing with for so long now as an inequality:

|P(a,b) - P(a,c)| ∫ ρ(λ) [1 - A(b, λ) A(c, λ)] dλ
which we can also write as

|P(a,b) - P(a,c)| 1 + P(b,c)
Now I'm sure you had an idea as to where this was going to go from the first sentence but this is Bell's inequality and this is pretty much exactly the way he derived it in his 1964 paper. A little bit of gobblygook but overall pretty simple, eh? You can see that no matter what the functions A(a, λ) and B(b, λ) at the electron and positron detectors, respectively, are they lead you to this inequality. All we've assumed is that you have some local scheme in which the up-or-down outcome at each detector is determined by some algorithm that depends only on the orientation of that particular detector and some machinations hidden to us and represented only by the mysterious λ.

The point isn't that the quantum prediction for every possible combination of three vectors a, b, and c violates the inequality. You can sit down and come up with angle combinations that do satisfy it. The important part is that many of the quantum predictions don't satisfy the inequality. It isn't a general result obeyed by quantum predictions. Similarly, you can devise a local hidden variable scheme that matches the quantum prediction at certain points (and obviously it's easiest for the “easy “ angles like 0, 30, 60, 90). But there's a certain point where a scheme like the one you suggested and the quantum scheme must part company and that point is represented in that inequality.

You could add to your toy model by throwing in a third angle (since you've only 0 and 60 degrees so far). Maybe there's a way you can devise rules that are self-consistent (i.e. with the already established rules for how the detector angles for 0 and 60 impact the series of 0s and 1s) and produce roughly the right correlation—for example, if you try a inclined at 0°, b at 30°, and c at 60° then you'll have to still switch polarities every other spot for the P(a,c) correlation but for the P(a,b) and P(b,c) correlations you'll need a rule that gives the -87% correlation (i.e. about 7 of the 8 pairs are pairs of opposites). Maybe you could do it in a way that replicates the results of quantum mechanics. But I can tell you that the inequality we just derived tells us that in general you can't do this.

The sort of scheme you're dealing with is exactly the kind wrapped up in those A and B functions--that is, they were exactly was Bell was originally dealing with 40 years ago.

There is no such thing as division or multiplication. They are merely shortcuts for addition and subtraction.

And here we are, back at the beginning again.

*Grins*

Many thanks, Startraveller - That was an awesome ride. It's certainly helped me to clarify my thinking on the subject.

...and subtraction is just negative addition.

You can do pretty much anything with enough Nand gates.

I realise that this is a necropost of the highest order- but I've been thinking about this, on and off, for some time now, and to be honest - this was a great thread. The point we reached last time, during this conversation, is that it is impossible to create a correlation with the individual angles of the detectors which would replicate the result of the combined angle of the detectors.

You might want to sit down for this. I've come up with the following:

COS[a+b] = COS[a] * COS[b] - SIN[a] * SIN[b]

In short - I believe that this formula shows that there is a function using the individual angle of both detectors that would correlate results exactly with the combined angle of both detectors.

So. What am I missing?

I'm not sure exactly what you're getting at. You still need two different pieces of information if you use that identity--angle a and angle b. Like I said above:



Can you elaborate?

I hope so.

Let's start by stating what we're in agreement about. There is no combination of 1's and -1's alone arriving at the detectors that can replicate the results given by quantum theory. I agree with that entirely. As illustrated via Bell's inequalities, it's a mathematical impossibility.

From numerous scientific experiments, we know that something strange appears to be happening. Somehow, the particles are giving results in terms of matches that suggests that information is being passed between the particles, in order for them to synchronise their behaviour, at speeds exceeding the speed of light, the famed "spooky interaction at a distance". Moreover, we know that the degree of matches returned differ according to the angle of the detectors. It would seem that not only do we have "spooky interaction at a distance", we also have "a cosmic auditor", keeping count of how many readings we've made at our detectors, in order for our large sample to give us the required results.

My alternative solution is this: The two particles are always synchronised. Always.

Figure 1 illustrates the experimental setup:



In short, I believe that the properties of these particles change over time within the shape of a wave and it is purely the angle of the detectors that determines the amount of matches that are registered, over a large number of samples, due to their angle of interaction with the particle waves.

Now, the difficult bit - trying to prove it.

Let's consider two matched particles, X and Y. In my model, the two matched particle's properties vary over time as sine waves, such that X^Sinproperty = - Y^Sinproperty. When the particles are emitted, their properties may be at any point within that Sine wave.

Let's see what happens when they reach the detectors, A and B.

The angle of the detector alters the angle that the detection wave interacts with the particle wave. Let's say that we get a positive (1) result when we get two matching peaks or troughs, and a negative (-1) result when we get a peak and trough in combination.



Notice that changing the angle from 0^o to 45^o changes the point that the detection wave meets the particle wave from a trough to a peak.

Note that the particle waves starting point effects whether a trough or a peak is detected. In other words, by adjusting the initial angle of the particle wave's properties, we can engineer cases where we have two matching peaks, two matching troughs, peaks and troughs and troughs and peaks.

Now - my Maths isn't that great, and working out the possibility of any two angles generating a match for intersecting waves is currently giving me a serious headache, but I strongly suspect that the number of matches for any two angles ultimately resolves to:

Probability of a match being made = COS[A] * COS[B] - SIN[A] * SIN[B] mostly because I know that this is = COS[A+B]

I can't prove it mathematically, yet - though I think I'm very close - but I believe that under the covers this (or something very similar to this) is what's actually happening.

Tiggs, I'm not seeing how this is substantively different than the ideas you set out on the first page, starting around post #10 or so. It looks like a standard hidden variable model. What am I missing?

In short, I think it differs because there are two sets of hidden variables - one set within the particles and one set within the detect