Eckard,

I did not "tacitly assumed a segment out of a line of all time or all frequencies from minus infinity to plus infinity." I explicitly stated that I was considering only a finite line SEGMENT, rather than an infinite line, precisely to avoid any infinities of time.

"Perhaps I need not explain why we are using integral transformations in signal processing." Most actual signal processing, involving transformations, is performed digitally, and uses discrete rather than integral transformations. Being discrete, they do not extend over either infinite times or infinite frequencies.

The redundancies you mention are real. But they only exist when the function being transformed is a single, real function. When one considers pairs of real functions, like the real and imaginary components of a complex function, the redundancies no longer exist. Pairs of real functions are just as real as individual real functions. The decision to treat these pairs as a single complex function was merely done as a matter of convenience; in the days before computers existed, multiplying (via pencil and paper computation) complex exponentials was much easier than multiplying trigonometric functions. Hence the popularity of complex notation.

Rob McEachern

Robert,

Why don't I manage explaining to you serious mistakes? I will try it again. Did you understand that IR stands for the entity of all real numbers while IR stands for the entity of all positive real numbers?

You argued that the problems with infinity can be avoided when using "discrete rather than integral transformations". Isn't a discrete CT or FT also an integral transformation? Aren't the (positive or negative) integer numbers and the (only positive) natural numbers IN also infinite?

I wrote IR and IN for blackboard bold letters.

A segment can be part of IR or of IR. You can imagine IR like a symmetrical with respect to zero folded together IR. However, does a symmetrical wrt zero IR make sense? If any negative value is identical except for the sign with its positive counterpart, then this pair is only in mathematical sense a pair without physical meaning.

Did you now understand why CT is tailor-made and why apparently physical symmetries are artifacts of not up to correct interpretation properly performed mathematical tricks?

Eckard

Eckard

Eckard,

You asked:

"Did you understand that IR stands for the entity of all real numbers while IR stands for the entity of all positive real numbers?" Yes.

"Isn't a discrete CT or FT also an integral transformation?" No. Integral Transforms exploit the fact that trigonometric functions are orthogonal with respect to continuous integration. Discrete transforms exploit a different fact; namely that trigonometric functions are also orthogonal with respect to discrete, finite summations.

I fail to see how CT usage of only positive numbers solves any of the problems with physical interpretation. CT still relies on the principle of superposition, just like the FT. Assuming that the MATHEMATICAL principle of superposition is a PHYSICAL principle is the problem. The problem is not the usage of real versus complex functions.

Rob McEachern

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  • Post #142

Robert,

As far as I know, orthogonality is a pretty general property: Two vectors are orthogonal if and only if their dot product is zero. This holds in IR,IR, and IN.

Well, I should better have said "Isn't DFT a FT and DCT a CT?" Of course, the DFT is a transform for Fourier analysis of finite-domain discrete-time functions. This is however irrelevant for my argumentation.

You wrote: "I fail to see how CT usage of only positive numbers solves any of the problems with physical interpretation. CT still relies on the principle of superposition, just like the FT. Assuming that the MATHEMATICAL principle of superposition is a PHYSICAL principle is the problem. The problem is not the usage of real versus complex functions."

I am curious how you will explain why superposition is not a physical principle.

Did you not understand my Fig. 1? Let me tell a joke that ridicules unphysical mathematical reasoning:

There are three people in a room. Then five people are leaving this room. Ergo two people have to come in in order to make the room empty.

The consequence I am trying to make aware of is found in the essay by Ken Wharton.

Eckard

8 days later

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  • Post #143

Hi Robert,

From what I've been told about the original papers on the holographic principle, it is a statement that there is a kind of gauge redundancy that ultimately separates the root states from the alias states brought on by noise and phase distortion. As far as I'm concerned, this is too similar to the dimensional reduction of signal space in Shannon's theory for Shannon's theory to be cast aside in a cavalier fashion.

The major critical difference between this Shannon-esque point of view and the traditional pre-holographic view is that the Shannon POV goes to show that the states "leak" out of the black hole right from the start; there is no information loss paradox when you go this non-traditional route; the traditional view sees the energy leak out due to the noise and phase distortion in a similar fashion, but there is ambiguity as to what occurs to the states (do they stay behind with the black hole proper, do they leak out, do they vanish). If you prefer to say that the holographic principle is bunk because it does not account for Shannon's work, then fine, but you're basically discrediting Shannon too and I'm left unimpressed by the raw butchery.

I'm not really sure if your other comment is for or against black hole complementarity.

- Shawn

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  • Post #144

P.S. Perhaps it's a more appetizing concept if I don't mention the word "state" and instead talk about signals, alias signals, and the Hawking radiation being a manifestation of the alias signals. Or, perhaps not.

Shawn,

I am not sure what you mean by saying "alias states brought on by noise and phase distortion." Aliasing is brought on by sampling. It has nothing to do with input noise or phase distortion. It would be more correct to say that, improper sampling brings about a type of output noise and/or distortion, called aliasing, that degrades the improperly sampled signal.

In your earlier post, you stated "how there is dimensional reduction in the message space when compensating for phase "differences" (same message, different phase) that can arise when sampling of the signal."

I think you might be confusing what is meant by "dimensional reduction" and "different phase". I am not sure how these terms are being used by physicists discussing the holographic principle. But I do know what they mean in signal processing. When a signal is "oversampled", for example when it is sampled at twice the rate required to preserve all its information content, then one can reduce the "dimension" by eliminating half the samples. For the case just described, there are two sampling "phases"; all the even numbered samples, and all the odd numbered samples - "same message, different phases", as you said. But you only need to preserve one of those two sampling phases. This concept and terminology is usually referred to as 'Polyphase" filtering.

With regards to "discrediting Shannon", as I have stated elsewhere, I view Shannon's proof (that it must be possible to achieve error-free information transmission, at rates right up to the Shannon Capacity) as of much greater significance that his expression for that capacity, which is the only thing most people every take note of, in his work. I agree that Shannon's theory cannot be "cast aside", and I am not proposing that anyone do so; rather, I am suggesting that the holographic principle might have to be cast aside, since it seems to be based on a misinterpretation of Shannon's Theory.

Rob McEachern

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  • Post #146

Robert,

You did not yet clarify your position. Do you agree with e.g. Ken Wharton on that there is a symmetry between past and future or may we agree that this symmetry is an artifact of the theory?

Eckard

Eckard,

You asked if I agreed with "Ken Wharton on that there is a symmetry between past and future or may we agree that this symmetry is an artifact of the theory?"

I agree that most of the equations of mathematical physics exhibit time symmetry. But, as stated in my own essay, one needs more than just these equations in order to make any predictions; one also needs auxiliary information, like initial conditions, and these do not have time-symmetry. Hence, reality does not have time symmetry.

In his Essay Abstract, Ken stated that:

"some of these alternate models already have a well-established importance, but are thought to be mathematical tricks without physical significance."

Regarding our early posts concerning superposition, I do indeed think that superposition is one such mathematical trick, without physical significance. Superposition is sufficient, but not necessary, for the mathematical description of quantum probabilities, which are the only outputs from quantum theory that can actually be compared with observations. As I described in earlier posts, a "filter-bank" can produce results that are mathematically identical to a superposition, without ever making use of a superposition; hence, superposition is merely a useful "mathematical trick", not a physically necessary "computational structure".

As a simple example, consider the relation a(b+c) = ab+ac. These two computations produce mathematically identical results, but do not have the same physical manifestation; implemented in hardware, the first requires only a single multiplier, but the second requires two multipliers. By simply re-ordering the computations, a different physical manifestation is produced. My point is this; Fourier Transforms can be re-ordered such that the apparent "superposition" at its heart, is completely eliminated, and replaced by a very different physical manifestation, a "filter-bank", that produces mathematically identical probability predictions.

Rob McEachern

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  • Post #148

Robert,

A hardware filter bank is bound to reality as is Ken Wharton's metaphor computer which stands for what Ken Wharton rejects as anthropocentric. We both agree that the reality is not symmetrical with respect to past and future although the DEQs in physics are invariant under shift or even reversal of time.

You quoted Ken a bit misleadingly. When Ken wrote "are thought", he did not exclude that he does not share this view. He continued: "This essay argues that only by dropping our assumption that the universe is a computer can we fully develop such models, explain quantum phenomena, and understand the workings of our universe."

You might read his essay or better the textbook by Schulman I quoted in an earlier essay of mine. As Hermann Weyl could not explain, the formalisms of quantum mechanics yield an apparent symmetry of past and future. Wharton, Schulman and several others are considering boundary conditions instead of initial conditions. Schulman even argued that there must be a transition between time symmetry in the quantum world and the obviously non-symmetric macro world.

My Fig. 3 relates to an explanation of the putative symmetry in the micro world as a simply wrong interpretation of results of complex calculus. You should be able to check and confirm my compelling reasoning. Please do not hesitate asking for further details if necessary.

In order to understand what you meant with respect to superposition I would like to ask you some details. At first, do you really mean a hardware filter bank, e.g. consisting of lumped elements like R and C or do you refer to a mathematical filter, maybe even an non-causal one? How do you imagine the filter bank related to the Fourier transformation (and possibly also to the cosine transformation)? My perspective is auditory function and MP3, i.e. analysis of a signal, and you will agree that future signals cannot be analyzed because they are not available in advance. You should tell me how you are using the filter bank for predictions without superposition.

Eckard

Dear Rob and Eckard,

I find this Q and A session quite fascinating. As you both know, I am focused on the possible physical consequences of Tajmar's measurement of a gravito-magnetic field that, in coherent circumstances, exceeds the expected field strength by 31 orders of magnitude. The associated G and C fields are 'analogous' to the electro-magnetic E and B fields and this is conceptually quite useful.

For purposes of relating gravito-magnetism to some of your above statements, I'd like to point out that G has dimensions L/T^2 while C has dimension 1/T, where l is length and T is time. Thus G has units of 'acceleration' and C of 'frequency'.

This relates to Rob's comment that "a positive frequency corresponds to an increasing phase angle, [and] if a second hand rotates clockwise, it is said to have a frequency of +1 cycle/minute. But if it rotates counter-clockwise, it is said to have a frequency of -1 cycle/minute. [and] Counting downwards is every bit as "real" as counting upwards."

Yes, one can 'count' downwards. But can the clock run backwards? The mechanical clock can do so, but the C-field cannot. It is a 'left-handed' rotation [accounting for neutrino and other asymmetries] and cannot 'run backward'. In my previous FQXi essay I propose that the C-field established the first cyclical phenomena in the universe, and hence the first instance of the appearance of 'time' in the universe.

Eckard notes that "While time is commonly considered a basic physical quantity," he does not have a problem with "the alternative choice of frequency as a basic physical quantity. Neither the measurable (elapsed) time nor the measurable frequency may change their sign." It is exactly this basic physical quantity that the C-field represents.

Eckard also noted "Dirac was not horribly wrong when he meant that there is no negative frequency in reality."

In "Quantum Mechanics: Myth and facts", Nikolic discusses the fact that relativistic Klein-Gordon equation has solutions with positive and negative frequencies, while the non-relativistic Schrodinger equation has only positive frequency solutions. My current essay relates this QM wave function to the left-handed C-field, and establishes a physical, not a mathematical, reason to throw away the negative frequency solutions of the Klein-Gordon equation.

Thus, in addition to Rob's explanation that, while the mathematical equations exhibit time symmetry, the required initial conditions do not, in the case of the C-field, even the basic equation is asymmetric, as the circulation is clearly left-handed.

While it is impossible to lay out a theory of time symmetry in a comment, I have included more relevant information in my last two FQXi essays, and I suggest that this is the actual physical underpinning for the fact that time does not run backward.

Thanks for exploring these issues.

Edwin Eugene Klingman

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  • Post #150

Dear Robert McEachern,

I think this part of your abstract is deeply mistaken:

"Equations contain very little information. This fact is what makes it possible to symbolically represent them, in a computer memory, by a very small number of bits. As a direct result of this fact, we can conclude, contrary to the fervent belief of most physicists, that equations cannot describe anything other than the most trivial physical phenomenon; those nearly devoid of all information."

This would be exactly true if the terms of an equation did not vary in their numerical values. You have treated a bit of algebra as if it was a single set of numbers. But because the equation is a kind of shorthand, it actually contains a lot of information. When you add this information up, you need to take into account every possible combination of numbers that equation can represent. This shows the limitations of looking at things that way - trying to assess the information content of something is sometimes a futile game, because information is a human idea that is more mental than fundamental.

I hope this information is useful...

    The "terms of an equation" CANNOT "vary in their numerical values", unless the auxiliary conditions vary. Hence all this additional information you make note of, is contained, not in the equations, but in the auxiliary conditions, as I have stated in the essay.

    Rob McEachern

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    • Post #152

    Anonymous,

    "information is a human idea that is more mental than fundamental." ?? I would like to support Robert.

    Moreover there is definitely an objective difference between past and future data. The distinction between them is not a futile game but overdue. See also here and the subsequent correction.

    Eckard

    Eckard,

    I mean a mathematical filter; the equation for a Fourier Transform can, and usually is, interpreted as a superposition. But by simply reordering the computations, it can easily be interpreted as a filter-bank. As I stated in an earlier post, multiplying by the complex exponential, in the argument of the transform, can be viewed as a frequency shift (for a time-frequency transform pair). The integration is a lowpass filter. So the entire transform amounts to a structure that "tunes" to every possible frequency, then lowpass filters these tuned outputs. But the exact same output amplitudes (or powers) can be obtained by a filter-bank of bandpass filters. It is only these output amplitudes that determine observed probabilities.

    In my previous posts, I also noted that future signals can, in fact, be analyzed, if they can be perfectly predicted; for example, the future state of Schrodinger's dead cat can be predicted - "DEAD". It is no accident, that Fourier Transform based QM, can only determine probability estimates, for such highly predictable outcomes.

    Also in previous posts, I described my view regarding "the universe as a computer"

    Rob McEachern

    Edwin,

    Regarding "the actual physical underpinning for the fact that time does not run backward", I have always been amused be descriptions of time-machines traveling back in time. As I wrote, twenty years ago, in the Introduction to my book; "If one could actually build and use a newly invented time-machine to go back in time, the information about how to build and utilize the machine would be the very first information that was lost!"

    How do we know that time is not presently running in reverse? Via cause and effect? By the phenomenon of growth? By the shattering of a dropped glass? But observing such things happen in reverse merely offends our concepts of improbability, not impossibility. However, our concepts for either are based on our experience. If we only experienced time running in reverse, all such experiences would then seem "normal", since our everyday experience is the only measure we have for what is "normal"

    Rob McEachern

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    • Post #155

    Rob McEachern,

    "future signals can, in fact, be analyzed, if they can be perfectly predicted" ?? I see this a typical unnecessarily misleading confusion. In reality there is nothing that can be predicted for sure. You are referring to a model of reality, and in this case the decisive border between the still open future and the finally determined past is shifted. A predicted signal is not really a future signal. Analyzing a predicted signal is equivalent to analyzing the past.

    What about your filter bank, you wrote: "I mean a mathematical filter; the equation for a Fourier Transform can, and usually is, interpreted as a superposition. But by simply reordering the computations, it can easily be interpreted as a filter-bank."

    The FT of a signal yields a complex spectral decomposition, also called a superposition of (by the way positive and negative) frequencies. In order to interpret this complex spectrum as the output of a filter bank, you must omit some parts of it. Perhaps you did not learn the fundamentals from the very beginning. EEs like me are e.g. aware of the non-causality of so called ideal filters. We would also not question the principle of superposition in general. FT and CR are integral transformations, and the symbol for integration is a "s". Integration means limit of an infinite superposition. DFT and DCT even more obviously summations. I do not understand your intention when you are trying to "explain" for instance "the integration as a lowpass filter".

    My intention is to demonstrate foundational shortcomings in the theory of signal processing and in QM. I infer from the restriction in reality to only positive elapsed time and only positive frequency that the shift properties must also be considered restricted. A distance cannot be shifted into negativity. Moreover, representation of a function of positive real values includes after FT in complex domain three redundant copies. They must not be carelessly omitted but correctly interpreted.

    I hope you got this and will support me against those like Feynman, Schulman, and Ken Wharton.

    Eckard

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    • Post #156

    An equation often contains information about a very wide range of auxillary conditions. In one way of looking at it, that's a lot of information. One of the things we find beautiful in physics is that this high information content can be expressed in such an elegant and economical way.

    The amount of information depends on how you look at it, because information is a mental idea, not fundamental to the universe. When we get over the novelty of computing, it will go down in importance. Your statement that "equations cannot describe anything other than the most trivial physical phenomenon" attempts to diminish equations, and physics itself. But you may have stumbled onto a way of showing that information is a thin concept, because of the nonsense that comes out of your approach to it.

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      • Post #157

      Eckard,

      "A predicted signal is not really a future signal. Analyzing a predicted signal is equivalent to analyzing the past." True, but irrelevant. Comparing a predicted signal to a "future signal", is very different from comparing it to a past signal. Making such predictions and then comparing them to future observations is a central part of modern physics.

      "In order to interpret this complex spectrum as the output of a filter bank, you must omit some parts of it. Perhaps you did not learn the fundamentals from the very beginning. EEs like me are e.g. aware of the non-causality of so called ideal filters." Nothing needs to be omitted. It is a mathematical IDENTITY. Perhaps you did not learn that Fourier Transforms are non-causal representations.

      "My intention is to demonstrate foundational shortcomings in the theory of signal processing..." What "shortcomings" are you talking about? How do you propose to algebraically describe the directions of anti-parallel vectors, if not via the use of a negative number for one and a positive number for the other?

      "A distance cannot be shifted into negativity." True, but "distance" is not "location". Distance is the magnitude of a vector, and so it is always positive. But location is the vector itself, not just its magnitude.

      Rob McEachern

      "An equation often contains information about a very wide range of auxillary conditions" Please give an example of such an equation, if you know one, together with the "wide range of auxillary conditions" that you think it contains.

      "showing that information is a thin concept, because of the nonsense that comes out of your approach to it" The approach is not mine, it was invented before I was born. As for it being nonsense, you would not be reading this on your computer, or be able to watch HDTV, or use a cell-phone or any other modern communication device, if it were in fact nonsense.

      Rob McEachern