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A new mathematics to describe quantum effects.

Posted by: SuperShuki - Mon Jun 25, 2012 1:33 pm
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A new mathematics to describe quantum effects. 
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Post A new mathematics to describe quantum effects.   Posted on: Mon Jun 25, 2012 1:33 pm
Hello, all.

I thought of a new way to describe quantum effects. I'm not a physics person, so it is probably nonsense, but I'd like to see what you guys think.

The possibility that a specific dimension (spin, energy level, position) has a given value can be described as an angle, out of a circle which describes all possible values of that dimension. For example, an electron is in a given position 10% of the time; so the possibility that the value of the position dimension is that position would be described as having an angle of 10/100 = x/360, or 36 degrees for that position. That means that 300 degrees of the possibility circle describes a different value or values for position.

Entanglement can be described as two particles sharing the same possibility circle for a given dimension.

Matter/energy which is "deterministic", or in the state of being observed, can be described by a state in which all dimensions are describe by possibility circles in which all angles are equal to 360 degrees.

It seems to me that you should be able to merge both classical, deterministic and quantum, undeterministic physical properties using this method.

Anyway, I'd like to see what you guys think.

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Post Re: A new mathematics to describe quantum effects.   Posted on: Mon Jun 25, 2012 8:59 pm
The trouble with this is that 360 degrees in a circle is an arbitrary number created by us and nothing to do with fundamental reality if that is what the quantum foam is at the bottom.

The way i would describe it is my old squaring the circle joke using quantum theory.

If you believe we live in a quantised Universe with Planck's length being the smallest unit of length then pi would depend on the size of the circle as the closest thing to a circle in a quantised Universe would be polygons with sides of Planck's length with the smallest circle being a triangle and the second smallest being a square thus making the square a second rate circle. :wink: :twisted:

So a first rate circle would be a triangle that could only be divided into three degrees as you are at the bottom of the pile where further subdivision is impossible with a second rate circle having four and they would be proportionately different to a three degree circle.

I know the above is only in two dimensions but it is the reason why i think there might be some problems with the math that works out that dark matter should exist or at least some of it and another possible problem with the calculations is that if the universe is still expanding which observation tends to suggest that it is then when looking at big structures many light years across you would also have to take the expansion into account which it is my understanding that it has not been done so far.

I would go for a Nobel with the above but my maths is not up to scratch so if anybody can do the calculations on the above info to prove dark matter is largely a pure maths screw up caused by not taking into account that the universe is quantised and still expanding then i am willing to share a trip to Sweden and split the prize 50/50 :wink: :twisted:

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Post Re: A new mathematics to describe quantum effects.   Posted on: Tue Jun 26, 2012 12:12 pm
Well, even if the universe is quantised that doesn't change the maths of circles, it just means that you can't make a physical system that is perfectly described by that maths. But the Planck length is incredibly tiny, so in by far most real systems any rounding error will be lost in the (thermal) noise anyway.

SuperShuki, about your angles, why not just use [0,1]? Probabilities are usually described by numbers between 0 and 1, not 0 and 360, although the choice is arbitrary (percentages, range [0,100] are also in common use for example). Your probability circle is just a polar plot of a probability density function. I think you're starting to confuse the quantum variable's value and the probability of that value from there on, and I don't think your concept of entanglement is correct.

I'm not a quantum scientist, but here is my take on it.

An ordinary bit is a variable that can take the values 0 and 1. At any point in time, its value is certain, and one of these two, and it can be measured without changing anything. Take two bits, and you can have the combinations 00, 01, 10, and 11, but again, it's always a fixed, certain value and measuring won't change it. If I take two bits and set them to 10, then I will always read 10, until I set them to something else.

A quantum bit, or qubit, can also take the values 0 and 1, but it can also be in a superposition of the two. If you set it to 0, you will subsequently read a 0, and keep reading a 0. The same thing goes for 1. To set it to a superposition of the two, you specify a probability for each possible value, say 0.3 for 0 and 0.7 for 1 (the probabilities have to add up to 1 of course). Now, when you measure the value of your qubit, you will get either a 0 or a 1 at random, with a 0 about 30% of the time and a 1 70% of the time. When you read your qubit, its superposition will collapse, and (rather than 0.7) you will end up with value 0 or 1, whichever you measured. Any subsequent measurement of the same qubit will result in the same value, until you reset it to the other value, or a superposition again. Put it in the same 30/70 superposition again, and you'll again have a 70% chance of getting a 1.

Now, let's say I have two qubits. That gives me four possible exact values as described above for the ordinary bits, and each can be in a superposition. So, I can set one of them to a 30/70 superposition, and the other to a 50/50 superposition, and then I measure the first, and get a 0 with 30% chance and a 1 with 70% chance. Subsequent measurements of that one will keep giving the same result. If I measure the other one, I get a 0 or a 1 with 50% probability, and then subsequently the same value for that one as well.

Two entangled qubits share a single superposition. So, I can put them into the four combined states above, or I can specify a probability for each of the four combinations for a superposition, perhaps 15/15/35/35 for 00,01,10,11. Now when I measure the first qubit, the superposition collapses to one of the four possibilities. If you look carefully, you'll note that the possibilities that have a 0 for the first qubit add up to 30%, and the others to 70%, so it'll look just like the two unentangled qubits above. I can then measure the second one, which will already have been decided, but I can't see that and the decision was 50/50 like above, so again I can't tell the difference. So what's special about entanglement?

Well, let me set my two entangled qubits to the superposition 50/0/0/50 for 00,01,10,11. Next, without measuring anything or breaking the entanglement, I'll give you the first qubit, keeping the second to myself. Then, I'll ask you to measure your qubit, and not tell me the result. When you do so, you collapse the superposition, and if we repeat this trick often, you'll get a 0 50% of the time, and a 1 also 50% of the time, at random. The interesting thing about this entanglement is that I can now predict what you measured, even if you don't tell me the result. I'll just measure my qubit, and note that according to the superposition that the entangled qubit pair was in originally, your measurement must have collapsed it to either 00 or 11 (the others have 0 probability, so can't happen). So, if my measurement is a 0, yours must also have been, and if it's a 1, yours must have been a 1. I tell you my prediction, and I turn out to be right all the time. You wonder how that happened, since you couldn't even predict the outcome yourself, and you didn't tell me, so how could I know? The trick is in the initial superposition.

There's another interesting thing, we can move very far apart, and I can make my measurement a very short time after you make yours, such a short time that any light will not have had a chance to cross the distance, and it will still work. This seems to violate the special theory of relativity, which says that no information can travel faster than light. However, it doesn't, because you cannot influence the outcome of the experiment. You can't choose whether we'll both read a 0 or a 1, so you can't use this trick to communicate anything to me. It's probably better to say that when you measured your qubit and collapsed the superposition, a bit of information came into being in two places at the same time. That's of course still completely crazy, but that's quantum mechanics for you :-).

Incidentally, if you're interested in this stuff you'll like the hammock physicist's blog. He has some good stuff on the holographic principle for example.

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Post Re: A new mathematics to describe quantum effects.   Posted on: Wed Jun 27, 2012 10:32 am
Well, of course degrees are arbitrary, but so is percentages - there is nothing that says that it has to be 1-100, instead of 1-101. It's just a convention.

From 0-1 is of course, radians. When you convert angles to radians, you are converting parts of 360 to parts of 0-1.

The problem with treating systems that you know for certain, and systems that you only know the probability, is that you are dealing with completely different maths. In one system, you never know for sure, and in the other, you always know for sure.

It seems to me that what we are actually dealing with, in quantum physics, is the amount of knowledge we have about the direction that a particle will go, in the future, or that it has gone in the past. And direction is described by angles. If you describe something accurately (i.e., mathematically) then you can formulate laws. But right now, there is no mathematics that describes both systems - deterministic and nondeterministic.

Using probability circles, you could describe both deterministic and nondeterministic systems using exactly the same math. Both are described by a probability circle. You can then make laws, that will apply to both deterministic and nondeterministic systems. In other words, instead of trying to describe nondeterministic systems using deterministic methods, you should try to describe deterministic systems using nondeterministic methods.

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Post Qubits   Posted on: Wed Jun 27, 2012 11:21 am
What you are saying with the qubits is that entangled particles share the same angles - the same possibilities of an entangled dimension being a certain value. If the possibility is 30%/70%, then by the law of entanglement, when one possibility grows to 100%, or 360 degrees (when it is measured/observed) the other entangled particle, which shares the same probability circle, will have the same possibility of that dimension being the given value - i.e. 100%, or 360 degrees. The angles, i.e. the possibilities of a dimension being a certain value, are entangled.

The thing is, this also holds true for all particles, great and small. Deterministic things can be described the same way - with the simple law that if they are deterministic, all angles are always equal to 360. Then you can write laws that tell you when things change from deterministic, to nondeterministic systems - by describing the change in angle.

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Post Re: A new mathematics to describe quantum effects.   Posted on: Wed Jun 27, 2012 2:14 pm
SuperShuki wrote:
Well, of course degrees are arbitrary, but so is percentages - there is nothing that says that it has to be 1-100, instead of 1-101. It's just a convention.


That is my point both systems are arbitrary if you think the the universe is really quantised then you will get different results for a 3 point circle than a four point circle the number of degrees or probabilities will have different margins of error depending upon the scale you are working at.

a 3 point would have 1 in 3 options

a 4 point would have 1 in 4 options

But perfect pure maths using arbitrary numbers 1 in 360 or 1 in a hundred probabilities would have the same spread of results for both but with different margins of error depending upon the scale worked at.

Lourens seems to think these errors would be lost in the thermal noise which could well be true but i think there is the possibility that these errors even tho they are very minuscule by definition could scale up compound interest wise so that when we look at something the size of a galaxy a hundred light years across it might look like it has a slight modification of various inverse square laws making us think that there might be matter that is not actually there. This is only an intuitive guess based upon my above reasoning but as i have said before i don't have enough maths to do a formal proof but i do think it is a sufficiently reasonable hypothesis to be testable if not now then when our measuring capability's have reach sufficient accuracy both up and down.

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Post Re: A new mathematics to describe quantum effects.   Posted on: Wed Jun 27, 2012 5:11 pm
SaneAlex,
I'm not sure that you understand what I'm saying, or maybe I don't understand what you're saying.

A circle does not have 360 discrete points, and a probability doesn't have 100 discrete points. A circle has an infinite amount of points, and the degree simply signifies the value of the instantaneous direction. A value in a probability circle that I am proposing would not have a discrete value; it would be a slice of the pie, signifying the range of possibilities of that value. For example, an experiment that shows that an electron stays in a certain orbit 10% of the time would be described by a possibility circle in which a 10% slice of the circle, or 36 degrees of that circle, would denote that value for which the electron is in that orbit. This also means that for the rest of the circle, the other 227 degrees, the electron is elsewhere (or rather, that the possibility is that the electron is elsewhere). You could write it mathematically as saying that for P(n) (Psubscript n, not P times n), where n is the slice of the possibility circle in degrees, there is a possibility of 1/3.6n that the electron will be in that orbit (n/360=x/100). For entangled particles, for all P(n), n for one particle will equal n of another particle. For particles being measured, n = 360.

The reason you can't simply rely on probabilities, is that it doesn't tell you what the probability is of. It's hard to imagine 60% of the time it does this, and 40% of the time it does that. What you are really talking about is a fraction; not a static fraction, but a fraction that can change, and does change - but a fraction that we can know the value of by observation, or measurement. A measurement is simply measuring the instantaneous value of that fraction, that slice of the possibility circle. Since for entangled particles, the same possibility circle describes both particles, they always have the same value. And since when a certain dimension being measured, all values for all fractions of the circle take up the whole circle, things that are being measured, or are deterministic, can be described by the same system.

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Post Re: A new mathematics to describe quantum effects.   Posted on: Wed Jun 27, 2012 8:02 pm
SuperShuki wrote:
SaneAlex,
I'm not sure that you understand what I'm saying, or maybe I don't understand what you're saying.


It could be a bit of both :wink:

SuperShuki wrote:
A circle does not have 360 discrete points, and a probability doesn't have 100 discrete points. A circle has an infinite amount of points


What i was trying to get across is that if reality is quantised then no real circle has an infinite amount of points the number of possible points always is finite and is dependent upon the scale at which it is at the larger the polygon the smoother looking the circle it is, but real world occurrences are never infinite.

Tho in pure maths you are right that every circle is considered to have an infinite number of points independent of scale. And it is the difference in results of real world compared to the mathematically pure that i am trying to point out as a possible source of weird errors as we may have reached the point where the accuracy of our measurements has got to a level which show up these errors and that they are not thermal noise. :wink: :twisted:

So care should be taken in using pure maths to describe stuff that is passing thru or near a phase change water ice does not become infinitely soft when it melts to become liquid water.

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Post Re: A new mathematics to describe quantum effects.   Posted on: Wed Jun 27, 2012 8:27 pm
Oh, I get it - I think :D

You think that I mean that each possibility would be a discrete angle. Maybe I didn't describe what I meant properly - It's kinda hard without being able to draw. I'll just upload a pic. Forgive my artistry.

Attachment:
quantum math.GIF


n(1) (I'm substituting n for P and 1 for n) is the possibility that the position, or spin, or whatever has a certain value, and its probability is denoted by the fact that it has a 90 degree angle, and therefore takes up a quarter of the pie - exactly the amount that is shown experimentally; for in this case, the particle has that value 25% of the time. The other possibility is described by an angle of 270 degrees - since all the rest of the time, the particle has the other value.

It's both binary - having only two possibilities - and undeterministic - it describes only the probability that a particle will have a given spin, position, etc. Each possibility is given a seperate slice of the pie, but together they make up the total amount of possibilities that can be.

Think of it this way - there are really an infinite amount of posibilities, although there are a finite amount of positions. An electron could be in the first orbit, or the second orbit, but it may be in the first orbit because it was in the 21st possibility, or in the 22.222412451 possibility, or some value. If it is there 25% of the time, it actually does matter where in that 25% it fell; because the cumulative amount of all possibilities for that value make up the total possibility. The 25% possibility is the cumulative of an infinite amount of finite possibilities.


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Post Re: A new mathematics to describe quantum effects.   Posted on: Wed Jun 27, 2012 11:13 pm
SuperShuki wrote:
Think of it this way - there are really an infinite amount of posibilities, although there are a finite amount of positions. An electron could be in the first orbit, or the second orbit, but it may be in the first orbit because it was in the 21st possibility, or in the 22.222412451 possibility, or some value. If it is there 25% of the time, it actually does matter where in that 25% it fell; because the cumulative amount of all possibilities for that value make up the total possibility. The 25% possibility is the cumulative of an infinite amount of finite possibilities.


I think i sort of see where we diverge you think there are real infinity's at the bottom with a major league deity or for the atheists among us just random quantum foam rolling the dice an infinite number of times for each binary decision even if it is at 3 to 1 odds at the bottom. :?:

Where as i suspect that that like in a lot of maths things have gone wrong if not all the fallen eights have cancelled out and that between 10 to the minus 18 m of the size of an electron core and 10 to the minus 35 m of a Planck length there are a lot of positions and possibilities but it is a finite number of both of them even if some of the time weirdly two or more positions are occupied at the same time until the cat is let out for exercise or burial. :wink: :twisted:

Or am i still misunderstanding you :?: *1


*1 And apologies Sigurd if i am leading SuperShuki astray back into the realms of philosophy of some sort.

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Post Re: A new mathematics to describe quantum effects.   Posted on: Wed Jun 27, 2012 11:26 pm
Actually, SaneAlex, you may have it right. You assume that the world is fundamentally deterministic, and I believe that it isn't. The problem is, the experimental data backs up my conclusion. I have no problem using Newton's laws and Einstein's equations for classical mechanics, if the world was fundamentally deterministic according to experimental data. But Quantum mechanics isn't non deterministic for lack of data - it is fundamentally non deterministic. There is no way of knowing what level an electron will be at a given time - the most you can know is the probability that it will be at that level. So instead of trying to force determinism on a non deterministic system, as good scientists, shouldn't we accept the experimental data as it is, instead of trying to force it to conform to our imagination?

I didn't come up with this as a way to demonstrate that point, however - I actually think that I may have something here, a way of describing both quantum effects and classical effects according to the same mathematics. Of course, I am not a physicist, so I am probably wrong - but that doesn't stop me from at least trying. :lol:

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Post Re: A new mathematics to describe quantum effects.   Posted on: Thu Jun 28, 2012 4:45 am
I always thought that since everything is relative to the observer that 3d as angle 1, angle 2, distance and time, that having two or three iterations could describe alot more astronomical about future positions ect. Instead Then using x,y,z,t where everything is referenced from an definitive origin, since everything is moving anyway,

Another thought, if object a is moving at 20 MPH, and object b is moving the opposite direction through TIME and traveling -20 mph in the same orientaion, they still appear to travel the same speed, how could you tell?

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Post Re: A new mathematics to describe quantum effects.   Posted on: Thu Jun 28, 2012 12:54 pm
Sigma wrote:
I always thought that since everything is relative to the observer that 3d as angle 1, angle 2, distance and time, that having two or three iterations could describe alot more astronomical about future positions ect. Instead Then using x,y,z,t where everything is referenced from an definitive origin, since everything is moving anyway,

Another thought, if object a is moving at 20 MPH, and object b is moving the opposite direction through TIME and traveling -20 mph in the same orientaion, they still appear to travel the same speed, how could you tell?



Well, time and space are different for all things, except light. Light is the universal constant, according to Einstein, and it has been proven experimentally. So while time may move differently for something moving very fast and something moving very slowly, the light moves at the same speed for both of them.

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Post Re: A new mathematics to describe quantum effects.   Posted on: Thu Jun 28, 2012 2:57 pm
Okay, I think we now have four or so different topics in one thread.

Probability circles: While the choice of how to represent a probability is arbitrary, it does make things a lot easier if everyone uses the same convention. Your circles don't do anything that a simple probability distribution can't, so I don't see the point in using them. (If you're certain about the value of a random variable, then the probability distribution is a translated Dirac delta function. Or in my qubit example above, you could have a 0%/100% superposition, which is equivalent to a 1.) I don't understand your sum-to-360-if-it's-deterministic rule at all.

I could decide from now on to replace the word "the" with the word "pra" whenever I write English, for after all it's arbitrary which combination of letters is used to represent the definite article. But if I do so, I'm just arbitrarily making life difficult for my readers. Incidentally, converting degrees to radians converts from [0, 360) to [0, 2 pi), a circular arc with radius 1 and covering an angle of 1 radian has a length of 1.

Interpretation of probability: This is a centuries-old debate. The two main interpretations of probability are the frequentist approach, that if you repeat an experiment a large amount of times, you'll get on average p * N successes (with N the number of repeats, and p the probability of success). This fits well with population statistics and things like throwing dice or picking balls out of urns. The other main interpretation is the Bayesian one, which says that a probability is a degree of belief. That is useful if we want to estimate the chances of success of the next launch of Falcon 9. Purely based on frequency, we'd have to go with 100%, since so far it's scored 3 out of 3, but Bayes' rule allows us to combine all sorts of relevant information and get something a bit more realistic.

Thermal noise: to view a molecule directly, you need to cool it down quite a bit, otherwise it moves way too fast and it's just a blur. The Planck scale is many orders of magnitude smaller than that. Measuring the position of anything down to the Planck scale is a lot more difficult than measuring the position of a football down to a single atom's size while it's being kicked around by the Spanish football team. Could this quantisation have a macroscopic effect? Well, this discussion is a macroscopic event that wouldn't have occurred if the universe wasn't quantised, so yes :-).

The interesting thing about a quantised space-time is that any enclosed volume can be described by a finite number of bits. The interesting thing about the holographic principle is that it says that that number of bits grows with the size of the surface area of the volume, not with its volume. Now that is weird! Rumour has it that physicist Erik Verlinde is doing some very interesting things with it. And note that whether the universe is quantised or not is orthogonal to whether it is deterministic or not. The fact that you can't know both the position and the momentum of a particle exactly doesn't mean that it's non-deterministic, just that there's a fundamental limit to the amount of information that describes the particle.

Relativity: an important property of all natural laws is translation and rotation invariance. In other words, if I move my experiment somewhere else, or rotate it, it'll still work the same way. This means that the x, y, t coordinates are completely arbitrary, the result is exactly the same wherever you put your origin. I've written about relative speeds before.

The geometry of the universe is such that nothing can move faster than a certain speed c with respect to anything else, and light travels at exactly that speed in a vacuum.

Richard Feynman has an extensive explanation of this in his famous lectures on physics.

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Post Re: A new mathematics to describe quantum effects.   Posted on: Thu Jun 28, 2012 4:47 pm
Thanks, Laurens. Like I said, I'm not a physics person, but if you don't try, you don't learn. But thanks for taking the time to talk about it. :roll:

I didn't understand radians correctly - I thought it was simply a way of converting angles from parts of 360 to parts of 0-1, instead of a standard relationship between radius and diameter.

With a probability distribution, how can you define the change in the probability? In other words, how do you quantify the change from 50%/50% to 47%/53%? It seems to me that probability distributions only quantify the probability directly, as opposed to the change in the parts of the whole. If you want to quantify parts of the whole, you use fractions, but then you are talking about something static.

I think (tell me if I'm wrong) that an angle is both part of a whole, and something separate by itself. 30 degrees isn't just a part of a circle - it is a quantity. And it isn't just a quantity, it's a part of a whole.

If you have a 30% probability, that doesn't tell you anything else. It could be 30%/1/1/68%, or 30%/2%/68%, or some other combination. With angles, you can name each "section" of the pie, and instead of talking about a section of the pie that is the same as any other section, that section is unique, and will always give a certain result, irrelevant of it's size. You can "name" that result mathematically, which you can't do with probability distributions (or can you?). If you have 30%/30%/40%, there is nothing in that line of math that distinguishes one 30% from the other 30%. You could write, result a=30%/b=30%/c=40%, but then you are separating them from each other by writing that. I guess you could define it by which comes first, and which comes second, but that seems very clumsy to me and inaccurate (again, I'm not an expert)

It seems to me that an angle of 30 degrees is fundamentally different from all other angles, all other slices of the pie; because it is not just any angle, but the specific angle from, say, 300 degrees to 330 degrees. It is a direction, and unique from all other angles by it's nature. There is no other angle from 300 degrees to 330 degrees - it is a unique direction.

There reason this seems important to me is that what is the same in entangled particles isn't just the probabilities themselves, but the specific slices of the pie. The result of one 30 degrees is unique, and that exact same 30 degrees gives the exact same result in the other entangled particle. It isn't just any 30 degrees - it is that 30 degrees from 300 degrees to 330 degrees. You can "name" that section of the pie, in way that you can't (so it seems to me) with probabilities, which by their nature are no different from any other probability. It just seems a more accurate and precise system to describe the nature of quantum particles, exactly what we know, and no more and no less. And the whole purpose of mathematics in physics seems to be to describe what we know about the universe accurately and precisely, and once we do that, once we write a "program" that depicts reality, we can use it to figure out how much we know about the future.

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