Rob and Jonathon,

P-p-please, sirs. I want some more. (Pip)

Your remarks are very informative for me of the physical nature that quantum theory addresses. Do continue, thanks. jrc

John,

You can find some more info, in my 2012 FQXI essay, about the nature of Fourier superpositions.

The most important point is this; as it is usually measured, a Gaussian function has the minimum time-bandwidth product. Consider two scenarios, both involving multiplying noisy sinusoids, with a Gaussian "window" function:

Case 1) A single noisy sinusoid, is multiplied by the Gaussian window

Case 2) A sum of several, noisy sinusoids, all too close together in frequency, to be "resolved", once they are multiplied by the Gaussian window.

The Shannon Capacity Theorem states that there is a only a small number of bits of information that can be recovered, from any measurements of such signals. If that number is sufficiently small (low signal to noise ratio), then it will be impossible to distinguish the two cases, impossible to deduce, from any measurements, how many sinusoids were summed together, to create either signal.

In numerical analysis, describing such a signal via Fourier Transforms, is referred to as a non-parametric model; there is no fixed number of parameters, used to describe the signal. However, if an observer knows, a priori, that a certain parametric model, such as a single sinusoid, is in fact valid, then it may be possible to "decide" the values of those parameters (frequency, amplitude, phase), even though it cannot be deduced , from observations, that the single sinusoid is, in fact, the correct model.

Almost all modern communications signaling exploits such parametric models, precisely to enable them to decide upon the information content of messages, that would otherwise be undecidable. Thus, in many operational systems, frequency can be estimated to several orders of magnitude better accuracy, than the Fourier Uncertainty Principle specifies; because the uncertainty principle is not really about the accuracy of measurements, but rather, the inability to deduce the number of signal components (sinusoids) that ought to be measured in the first place.

Rob McEachern

Robert,

Thank-you for the time and attention to elaborate, I'll download your 2012 essay. I do have to work on the notion of considering the wavefront of EM as an infinite 2-D plane, but your remarks on 'deciding' from two sets of sinusoids does help to clarify matters. Classically, the common definition of 'photon' lacks a 3-D+t physical model that could be Lorentz invariant. So the same problem exists of deducing 'how many' discrete wavetrains are transmitted in a signal burst, and whether that burst is a single or composite sinusoid. And of course that is strictly about a straight-line free space transmission, not the fuzziness of junction thresholds and refractive indices. (all beyond me)

Thanks again, patience, I'm slow but good with my hands. jrc

John,

Set aside the expanding, traveling EM wavefront, for the moment, and just consider what a single observer, in one location, can directly observe - only that little point on the wavefront, as it happens to move past. Then imagine trying to construct a parametric model, to describe what is observed. Then consider if any a priori knowledge about the model might help to explain the observations.

Better still, think of gravity, and the concept of "action at a distance". How does a planet "know" how to behave towards the sun and other planets? Even if it "knew" Newton's parametric model (stating that the force depends on the distance and direction of the other objects) for how it ought to behave in the presence of gravity, how could one object determine these parameters, from any measurement of its local field, and thus "know" how to behave? How could it know it a priori either? Einstein solved the riddle, by developing a different parametric model, where local (observable) distortions of spacetime, are what cause the objects to behave as they do. Objects like planets (and electrons) do not have to be able to "deduce" how they ought to behave, if they already "know" how to behave, as a consequence of a priori knowledge "built-in" to their very being. It is only entities like human observers, that do not possess such built-in knowledge, that must seek to construct such knowledge, from our observations. Consequently, what is undecidable for human observers, may not be undecidable for either the entities being observed, or for more knowledgeable observers.

Robert,

I'm mulling over your essay, and will no doubt do re-reads. I think I get the general idea, and agree that such things as actual particles possess a priori information. To observers motion is relative, but some physical property associated with matter might well 'know' the absolute velocity of any particle.

In your example of two frequency amplitudes, extracting the natural log, what is you reasoning to import the exponential rate unit argument in the first place? As an aside; a few years ago reading a history of mathematics I gathered it has been in the relatively recent past that Conventions have come down on the side that 'e' can be used as the base, but not the index, in linear algebra.

I think this is arguably weighted by the long use of natural logarithms in physical equations, and perhaps linearly might lack a confident proof that 'e' used as the index might not somehow distribute as its own root. In an earlier time, simply using the 'shotgun approach' I found that treated as the 'e' root of all possible radii which compresses an exponential rate of extention of each radius back to a length of the initial light second, can account for distribution in terms of a continuum of density of energy variation in a spherical volume. That's a rough sketch, but math wise topologically I wonder if it might not be valid to utilize an 'e' root? If Tom is watching I'd like his thoughts too.

Compressing information out of noise, as what I see your essay's major theme to be, not only applies to the 'how many', 'when' conundrum, but might also be relevant to compressing a particle form out of a waveform. :-) jrc

John,

I don't understand your question "what is you reasoning to import the exponential rate unit argument in the first place?" Would you care to elaborate?

The exponential function given in the paper, is just the definition of a Gaussian function. The natural logarithm is used to convert the equation for the Gaussian, into an equation for the exponent of the Gaussian, since the exponent is where the frequency term resides.

Rob McEachern

Robert,

Okay, textbook. Graphically the exponential function describes a parabola, so the usage of 'e' apparently 'smooths' out the transform of probability distributions from Fourier into Gaussian. I had been thinking Gaussian terms, as in electrostatic or electromagnetic units, so had also misunderstood 'Gaussian window' as a range of intensity rather than probability.

Gettin' there. jrc

Hi Folks,

I've been trying to read the above comments, and encountering a weird truncation of the content on the bottom, when I expand the comments. But I noticed I could read a bit more after John left the comment above, so I'm hoping I can get things to expand out further by increasing the lower page boundary - simply by adding more content to the default page representation. I apologize if this rambling comment annoys anyone, but I am trying to use plenty of lengthy words in the hopes that a voluminous block of text can solve the display problem, without having to get the techies involved.

We will have to see, if the addition of a comment or two on the bottom lets people see the entire block of comments above. I imagine it must be my browser version, OS, or something machine-specific, and I hope this page displays correctly for most everyone else. But I wanted to explore the possibility the issue might be self-correcting.

All the Best,

Jonathan

A little better..

In fact; I can read almost all of the above block now.

So I will write a few more lines.

I see that all the extra space I am creating on the bottom is allowing me to see almost all of the comments above. So I am adding just a bit more content.

This additional verbiage, added for no purpose of communication whatsoever, will help me to see Rob's response to John in the block above.

Thanks for your patience folks!

Have Fun!

Jonathan

    FYI..

    It worked!

    I can read the whole block of comments above now, and collapse them when I'm done. So I can get to see all of the content.

    All the Best,

    Jonathan

    Jonathan,

    I started having the same problems on 1/19/15 when Microsoft tricked me into an update of windows 8.1. Now I can get control of my own idiot box. Same old bait and switch. jrc

    Jonathan,

    What worked for me after choosing 'show all replies', was to click the maximize icon upper right corner (Windows) which resulted in a half screen, then click again and I get full screen and no bottom of page truncation. jrc

    Robert,

    from your essay: "However, any mathematical description of the 'state' [sic; spin] of the entity to be observed must take into account that the state depends on the aspect angle from which the entity is to be observed."

    I see better now, your argument that a high degree of surety can be found for the otherwise Fourier unknown frequency window. Fred Gauss really got around, if you'll pardon the pun. If we look at the few Platonic solids from which we could construct symmetric point distribution on a sphere by superscribing a Gaussian sphere on a chosen polygon, so that a trivial rotation would restore the identity of the initial aspect angle of observation, then we could say that a priori information might exist from the initial condition of the vector of a chosen point prior to the rotation.

    IF, pi were existentially a ratio, even as a finite decimal fraction, the surface area of a sphere could be found stable. Physically however, any choice radii from the center as origin of our superscribed polygon that intersects a vertex would have to be a tad bit too long to fit inside the sphere, so it would become 'buckled'. Like the juggler who spins up a plate to balance on the tip of a flexible rod, by whipping the mid-point of the rod in a circular motion, once the plate is spinning in a balanced centrifugal plane, the rod tip's orbital precession dampens. And there is an illustration of your finite Gaussian window described by the time parameter in tracking an orbital of Gaussian co-ordinates on the sphere surface in physical response to the knowable amplitude and phase in the radius rod's initial sinusoidal whipping motion. That a prior information in the original aspect of observation would naturally follow any trivial rotation, and topologically as a continuous function.

    We might go on within Gauss' scheme of things and make that window descriptive of a point charge on a sphere, and the probabilistic vector of the aspect angle being the emitter 'exitor' of the direction in relativistic space of a single sinusoidal wavetrain of electromagnetic pulse, exhibiting the phaseAmplitude of the initial condition of the energy compressed in the buckled radial length.

    Onward! through the fog! :-) jrc

    • [deleted]

    I thought it had been decided that the relationship between was set at the generation of the entangled particles and so one particle did not, in itself, change when the other was read. Is there a difference between determining a state and discovering a state?

    a month later
    • [deleted]

    is this not Entscheidungsproblem?

    or

    I don't know if I can decide

    calculus, Church-Turing thesis, proving the undecidability of the Entscheidungsproblem, Frege-Church ontology, and the Church-Rosser theorem

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