WTF is a wave of probability? This can't be, surely, the model we use to explain fundamental particles?? Are there any experts here who know what this rather peculiar phrase means? Or is it just pseudoscientific mumbo-jumbo that the "experts" would have us believe actually does explain something? Is probability real? How is it connected to numbers, since numbers are imaginary? Translation: how do we assign or measure probabilities, and how do they "wave" or oscillate, and around what?
If there's a clearly oscillatory pattern for a particle, but the individual constituent parts can't be quantified you use probability. Thus it's notated as a wave with having the 'probability' of any given state given any set of circumstances at any given time. Since you're interested in information-theory; it's interesting to note that the probabilities can be defined as packets of information with unknown causation at this time. The inconsistencies would be as unique as perhaps a full DNA strain...which is not very unique at all.
If there are such things, are they real, these "probability waves", they exist and have real values? Is an apple a real value, or a real object? Ben: if you can't see it, what I'm doing here is working backwards from a conclusion to a premise. Maybe you can recall what the premise was, the one you made?
I don't understand anything anymore. I had assumed that you were doing more than cutting and pasting formulae from Wikipedia in your other thread, and now I see that that's all you were doing. So much for me giving people the benefit of the doubt. The wavefunction has the interpretation of a probability wave, i.e. a probability distribution. In general it is a complex number. However, what you learn very quickly when you study quantum mechanics is that the wavefunction is not physical---in order to get a probability, you have to integrate the absolute square of the distribution over all space, in other words \(P = \int \left|\psi\right|^2\). Now, the quantity \( \left|\psi\right|^2\) is manifestly real. If you'd like a demonstration of this, I direct you to the following resource: http://en.wikipedia.org/wiki/Complex_numbers.
You mean "we can't measure a wavefunction"? But it is a 'real' thing, surely? How can it not be real? When all you have to do to make it 'real' is measure a complex amplitude? Could there be a bit of an ontology snafu here? How does something with no physical reality (a wavefunction) "produce" something that has a physical reality (a probability). Is it something to do with measurement? I thought you might have some understanding of the measurement problem, but I see that you would rather stick to a particular method, which looks a lot like "if I just repeat myself, it will start to make sense eventually".
How does one measure something? How do we measure the strength of a magnetic field? We have to interact with the sucka. Use something that moves or blinks when we 'put' it in there. If we have an independent measurement of local field strength, we know that things like electrons' spin will move or blink a certain way - this is our expectation. When a beam of coherent light is plane-polarized, there's a wavefunction describing the geometric polarization as a probability of finding a photon polarized in the plane, or at a perpendicular plane (that is orthogonal) direction. There is also a wavefunction describing the geometry of the polarization as two oppositely circular-polarized photons. Where is the information content? Has any been 'put' in there, by polarizing a beam of light?
The answer is, yes of course there is more information in a polarized photon. You just have to 'measure' the content; this requires a 'measurement basis'. You can't analyze a single polarizing filter, as a kind of sieve, but you can if you have two filters (or more than two). Why would that be?? It's because the first polarizing filter 'computes' a plane-polarized 'wavefunction', which additional filters can 'measure' (by computing the existing polarity of each photon [from the first filter]). Thus, the additional filters can process (i.e. compute) an existing informational entropy, the one 'computed' by the first filter. Although photons have to be analysed "non-classically" when they get processed by filters, beam-splitters and mirrors, as wavefunctions with probability amplitudes, they can be encoded; the 'information signal' is polarity (of the wavefunction), in this case.
Bingo. You asked: "is a probability a real number?" The answer is: no, a real number is a number, that we assign to a real probability. As you say, we 'measure' things which are real and physical (have a real extent in space and time). So strictly speaking a probability is a real possibility of an actual event, the real number corresponding to any possible event is always a number between zero and one. Therefore information is both logical (as in real numbers), and physical.
Part (ii), in electrons, nucleons and massless photons, this probability of being spin up or down, or polarity of spin-[1] photons, which they exhibit as polarization wavefunctions, along a dimensional axis, including parallel to the dimension of propagation (i.e. the positive z-axis), but fermions exhibit as superpositions of spin wavefunctions: It has something to do with Taylor's expansion, with imaginary exponents (by which I mean\(\, i \) rotates a number into the Euclidian plane of imaginary roots). Something to do with algorithms too, and computation. Computations are eigenstates, and eigenvalues or measurements But what do these probabilities do? Who can say? Like I've been rabbiting away about information and communication (i.e. measurement of a signal from somewhere), it depends what you define as the sender/receiver, and which is the channel that commutes the signal - the alphabet. If the sender is an electron that 'spins' into a new state wavefunction-wise (i.e wrt to a nucleus) and 'sends' a photon message, then that's the basis of the measure-space, is one way to say "how to send the signal'. If it's photons as the quantum, then they 'send' matter-waves as electrons, or they encode electrons. Because we decide, fundamentally, which is which when it comes to the content.
What about it? 2 is a number, right? So is 2.141267, right? If I have 2 apples, what do I have? A number, or something I can eat? (This is getting a bit strange, once again - you do understand that numbers are things we count with, etc, you can't "do anything" with a number. Except attach it to something - an abstraction. You understand what abstraction is, or didn't they tell you about that one??)
You have something which you can eat which also has a property which represents the concept of 'Two-ness', which is a property shared by all other pairs of objects/entities No, given a number z in the Complex plane (which is not a metric space so the term 'Euclidean' should not be used), iz can be obtained by rotating z about the origin by \(\frac{\pi}{2}\) radians. Please, please, please can you refrain from trying to put your own 'spin' on things you aren't sure about. If you don't know, don't try to fill in the gaps. Not for myself but so you avoid confusing people who don't know about complex numbers but whom are reading this thread. That's part of the reason I don't let mistakes slide so readily, if this forum exists to help educate some or all of its readers and posters, trying to fill in the gaps is worse than just saying nothing about a topic. Ignoring metaphysics for a moment, probability is a Real (I use capital R to show it's not being used in the same way as Pinocchio says "I'm a real boy" but to refer to the field of real numbers. A probability is a Real number between, and including, 0 and 1. Not all Reals can be probabilities but all probabilities are Reals. Circular definition. You cannot use a word to define itself.
A probability is not a number. It's a real probability. Probability is 'numerical' but it's real, not an abstraction. Real things are probabilities, not abstractions. A plane is Euclidean. Please, please, don't have a tantrum.
Vkothii, these funtions describe how particles will behave in a probabilistic way. They are "real" in the sense that particles really do behave in a wave-like manner, and these functions can accurately describe that behavior. Asking if the functions themselves are "real" is a lot like asking if force=mass x acceleration is "real". Perhaps the best way to put it would be to say "it accurately describes something that really happens".
Really? So a number isn't a probability, then? Real probabilities instead are abstracted logically to real 'numbers', and real physical events and objects are abstracted to logical 'information', then? There are all these 'formulas' made out of mathematics, type of thing?
You could even get all carried away and think something like: "a point can be information, a number, because it can describe a position." If you want to 'send' or 'transmit' this single bit of positional information, you absolutely require another dimension, so information is dimensional, and it isn't 'one-dimensional' or static, either. To get a point from one position to another position, you translate it in another dimension, or 'project' the point, and make a line. This one-dimensional line is the integral of all positions the zero-dimensional point can be in, wrt the start and end positions, so each point can be assigned a position (a number) along this 1-dimensional object. You can call this object "one line", or a line that is "one unit" of information. Actually then, unless a point has a position, it is 'informationally zero', or it's 'dimension' of information is zero. As soon as you give it a number, it has a dimension which is a position in a 1-dimensional space. This encapsulates the idea of having to project something (a physical position), to get it to be information (a number which is logical, not physical). You give something an informational dimension, by translating, or projecting it, generally.
So can you point at a '1 in 5' dog. Or a 2 in 7 table? I can point at 4 dogs or 2 apples. Why are probabilities, which are a measure of the ratio of numbers, more real than the numbers themselves? No, 'Euclidean' is a specific type of space which has a metric or inner product defined upon it. Given a 2 dimensional space, you cannot say if it's Euclidean or not (say Lorentzian) till I tell you how to measure distances between points on that plane. If I tell you that the distance, s, from (0,0) to (x,y) is measured as \(s^{2} = x^{2}-y^{2}\) it is clearly not Euclidean, despite the (x,y) plane being 'flat'. There's a great many other ways of measuring distances, which you would know if you knew about 'metric spaces'. Which you don't. Is this going to be another thing you can't accept you're wrong on or are you going to actually admit you're incorrect on this one? Hopefully the fact I've given an example of a non-Euclidean plane will not fall on deaf ears...
Can you point at a dog and not identify it as "a dog"? Is there a probability you will call it "a horse"? What value is this probability, in numerical and in social terms? Does this probability have a value in psychological terms? I know what a "metric" is though. I know that it can be whatever we say it is, for example, which you don't.
