Let's benefit from special mathematics for elapsed time by Eckard Blumschein

Eckard,

Well said.

I would like to see some of your work on complex signal processing so I can understand your ideas better. Do you have any links or documents?

Brian,

Did you already read my IEEE paper?

http://home.arcor.de/eckard.blumschein/M283.html

Eckard

Eckard,

Give me sometime to review your essay and two attachments again as well as review your IEEE paper. It will take me a little while because I have not examined Fourier analysis in depth for over 5 years.

Brian

Brian,

You are quite right. Careful work requires time. While my time is limited, I will also try and explain more in detail to engineers why restriction to functions of positive argument avoids notorious flaws in signal processing.

Many hoity-toity experts are just not ready to believe that my real-valued frequency-versus-time representation is correct. They cannot deny that is very similar to result of physiological measurement. But they learned and stubbornly believe that cosine transform is only a special case of Fourier transform. Among them is Dick Lyon, who wrote a paper "cochlea demystification". In a discussion group there are many mutually excluding explanations and everyone sticks on his own for decades.

Eckard

Dear Eckard,

I have reviewed Fourier analysis somewhat but reading your essay and papers again has reminded me of something I should be doing in quantum.

You mentioned something in your IEEE paper I absolutely agree with, "measurement that is based on complex analysis should generally be considered with precaution."

If I understand acoustics correctly then any signal can be expressed as a series of terms of the fundamental and harmonics. This is very similar to the ground state and excited states of a quantum system. Eventually the measurement problem and collapse must be explained, how would you tackle these problems?

While I'm just barely competent, my gut feeling for the collapse of the wavefunction would be something similar to noise in a signal. I have read that a signal with high fidelity has a large coefficient for the important terms in the Fourier series. In quantum these coefficients are the probabilities, and when we measure the wavefunction (pass the signal through a network) we collapse it to an eigenvalue. In other words one of these coefficients becomes 1 and the others become zero (filtering?). Considering your lifetime of wisdom with signals and Fourier analysis it would be a crime if I didn't learn as much as I can from you.

Brian

Dear Brian,

When I read the sentence you quoted, I was not cautious enough myself. If one knows how to correctly apply and interpret complex calculus then there is no reason for distrust.

My criticism with respect to quantum mechanics mainly focuses on the following details:

1) I did not found application to quantum mechanics of the complex ansatz exp(i omega t) before the paper by Kramers and Heisenberg in Z. Phys. 31, 681 (1925).

They even wrote:

The object of this work is to show how the careful application od correspondence ideas leads to the surprising result that the ansatz ... is very useful ...

To me, Kramers is known for his theory of dispersion and so called Kramers-Kronig relations.

2)

to be continued

2) Kramers expressed the real part n (phase velocity) as an integral of the imaginary part k (absorption). I did not find any indication that he was aware of the limitation for functions of time to either only positive or only negative values of time. This limitation is not just obvious for what already has been reality but also for the opposite point of view where for a closed ideal system the past does not matter.

3) While above mentioned paper was submitted in January 1925, Heisenberg submitted in July 1925 in Z. Phys. 33, 879 a paper that reveals: He intended to perform an inverse transform. At least he wrote for the last time Re for real part. He also correctly explained that a common phase is arbitrary and without physical significance.

4) to be continued

Dear Eckard,

Thank you for 1-3 and I look forward to [4...n]. Your posts have become a priceless resource for me.

I'm working on something which is for the moment a very rough idea, but I think you will like it when it is completed and polished.

4) Heisenberg wrote in letters to Pauli: On June 21, 1925: "Bohr's theory of hydrogen seems to me formally very much the same as Kramers' dispersions theory", on July 9: "My whole troubled efforts are intended to totally destroy the idea of the orbits.." and on July 9: "... Kramers called me the optimist".

Born received Heisenberg's manuscript on July 11, 1925 and wrote on July 15 to Einstein: "... looks very mysterious but is surely correct and deep". Born appreciated in particular Heisenberg's form of the quantization condition:

pq-qp=h/(i 2 pi).

When Pauli refused to join Born's team, Pauli said:"Yes, I know you are fond of tedious and complicated formalism. You are only going to spoil Heisenberg's physical ideas by your futile mathematics".

The paper of Sept. 27 by Born and Jordan in Z. Phys. 34, 858 (1925) gave credit to Heisenberg's assumption that the square of the magnitude of the element of the matrix representing the electric moment of the atom is a measure of the transition probability. It was motivated by Born's observation in 1924 that Einstein's emission probabilities correspond to squares of classical amplitudes.

This seems to answer my questions why Kramers' representation in terms of real and imaginary part was replaced by a representation in terms of magnitude and phase.

5) Born not only found out that Heisenberg effectively used matrix multiplication, and took the Hamiltonian point of view. He also underlined that pq and pq are not identical:

The matrices must be hermitian: q(nm)=q(mn)* for all real t.

6) to be continued

6) In 5) m and n stand for quantum numbers. With omega(nm) = -omega(mn), cf. Equ. (26) in Born/Jordan Z. Phys. 34, 858 (1925), the matrices p and q must be hermitian, of course.

However, I did not find a compelling physical explanation why they must be hermitian. Why not admitting an expansion into cosine terms instead of complex exponential ones?

Born just borrowed the Hamiltonian from classical mechanics where complex calculus considerably facilitates the work but complex numbers are dispensable in principle.

Engineers know: Functions of time and functions of frequency are related to each other by Hermitian matrices if they are considered complex. Then the eigenvalues are real, and the eigenvectors are mutually orthogonal. They are related to each other via Fourier transform.

At least in classical physics, Fourier transform and cosine transform are equivalent to each other for the following reason: Since it would be absolutely nonsensical to have a result of analysis after all future at t=oo, the assumed end of integration is definitely always followed by something not yet existing.

Nobody at the time of Born already understood that complex representation is twice redundant. The sine and cosine function just differ in the arbitrary choice of the point t=0.

7) I did not yet read Chapter 3 Connection with the Theory of the Eigenvalues of Hermitian Forms in Z. Phys 35, 557 (1926) by Born, Heisenberg, and Jordan. In Z. Phys. 43, 172 (1927) Heisenberg refers to "canonically conjugate dynamical variables" without however explaining the essence of this conjugacy. It is common tenet to attribute both the so called commutation relation pq-qp = h_bar/i and the uncertainty to them: "The more accurately the position is known, the less accurately the momentum is known and conversely."

I revealed that the same uncertainty also affects not only the Fourier transform pair f(t) and F(omega) with the conjugate variables time and frequency but also the cosine transform pair of the same while unilateral variables.

I conclude from this that the imaginary commutation relation must not be considered a quantization relation because imaginary values are precluded with cosine transform.

Plancks constant can be reckoned to the left side.

Heisenberg perhaps already understood that Bohmian guiding waves are fictitious envelopes.

In all I have to apologize for a horrible suspicion: Pauli was possibly not even wrong with his judgment on Born. Later he dealt without convincing result with the question whether and if so why quantum mechanics is the only field of physics that absolutely needs imaginary numbers.

I do not hide that I also tend to suspect: Georg Cantor misused Galileo Galilei's method of bijection, and Oppenheim made the idea of cepstrum by Tukey and others tedious and unnecessarily complicated. If we prefer to accept application as the final judge then it seems to be obvious that neither aleph_2 nor the complex cepstrum found any application so far.

What about quantum mechanics, it is my gut feeling that at least time-symmetry is for plausible reasons an aviodable artifact. Schroedinger repeatedly recommended to take the real part of psi: Naturwiss. 14, 664 (1926); Z. Phys. 4. Mitt. (1926). He still stuck on the traditional point of view where f(t) belongs to reality while F(omega) belongs to the complex domain. This is not compatible with a real and always positive Hamiltonian.

8) Meanwhile I got aware of a paper by Schwarzschild, 1916 "Zur Quantenhypothese" where he used the Hamiltonian and the term "canonically conjugate", and he reported that already "Planck has noted that the elementary domains of phase space must be complete". WRT H Schwarzschild refers to Charlier, Mechanik des Himmels, Bd. I (Leipzig 1902).

Eckard,

I'm not sure if you're still checking your forum but you should know that you've been very helpful and encouraging. I'm still working through this idea and I have had to overcome some major sticking points (more like fixed points). I hope to have something very solid in a month but finding the time to play and think about it has been difficult.

Brian,

Thank you for your encouraging words. I did not check fqxi on a regular basis. One reason for that was my effort to convince those who are still using the traditional theory in so called filterbank models of cochlear function, e.g. by Dick Lyon, Roy D. Patterson, or Frank Baumgarte that my method is not by chance superior. Even the physicist Dr. Strube tried to persuade me that my method is something like a special case of convolution.

The main obstacle seems to be ubiquitous thinking in terms of ergodicity. Not just Einstein imagined closed dynamical systems whose trajectories are given for good quasi by god. Consequently Einstein did not understand what he called the merely illusive separation into past, present and future.

I looked into Aseltine 'Transform Method in Linear System Analysis' NY: McGraw-Hill 1958. Already Fig. 2-1. showed "Three functions (of time) which have the same L-transform".

The late Einstein confessed that the Now worries him seriously, and there is something essential about the Now which is just outside the realm of (his) science.

I consider Claude Shannon refreshing the trivial insight that the past is closed while the future is open to influences.

In his quarrel with Ritz Einstein uttered his belief that one may restrict oneself without essentially restricting the generality of the consideration. He obviously missed the point.

Eckard! Thanks for your feedback, ignore that idea because it needs about five more years of gestation. I hope to see another essay from you I suspect you have a lot of ideas about what is physically possible and impossible ;)

Brian,

While I feel not too old for obeying a piece of advice from someone perhaps considerably younger, I am not sure what "idea" of mine you are referring to.

What about my criticism concerning the belief in the cochlear traveling wave model which was selected by v. Bekesy and cast in hydro-mathematics by Lighthill, Tianying Ren recently provided direct evidence that there is no backwards traveling wave while the proponents still are claiming the opposite. Referees tend to defend opinions that were awarded by a Nobel price no matter whether or not they are tenable. Several experimental findings by Ren challenged the old tenets. Prof. Ren repeatedly managed to publish unwelcome results in PNAS and Nature neuroscience. However he was unable to publish all data and in particular papers in JASA.

What about the not yet existing future, I see it related, and my criticism of sometimes dirty use of complex numbers is well founded. At least, nobody so far could refute my arguments, and I conclude from obviously unresolved incompatibilities between quantum physics and other putative fundamentals of physics that there is still room for a basic correction. The ultimate argument of physicists against my hints was: This is a matter of philosophy rather than physics. I do not think so.

Eckard,

My apologies, I didn't mean ignore your ideas because I know they are legitimate. I meant ignore my idea about space and time with the convergent fixed point of the cosine function. It was a random thought and I'm going to have to spend a lot of time thinking it through.

Physics will benefit from my of your "ideas":

1)

"Tianying Ren recently provided direct evidence that there is no backwards traveling wave while the proponents still are claiming the opposite."

Physics uses a similar idea of the backward travelling wave to explain the quantum tunneling in the finite square well. I'm glad you mentioned it because I always accepted it and now I'm curious if it is the right explanation.

2)

"What about the not yet existing future, I see it related, and my criticism of sometimes dirty use of complex numbers is well founded."

I agree Eckard in my essay I wanted to rid the wavefunction of imaginary numbers because physicists use them to justify the multiverse and other ridiculous interpretations. I also wanted to show how the future does not yet exist by connecting it to entropy, combinatorics and probability theory. The imaginary numbers violate the axiom of additivity in probability and the theory is still accepted as fact!

3)

"The ultimate argument of physicists against my hint was: This is a matter of philosophy rather than physics. I do not think so."

Eckard who are referring these journals and how are they qualified? If a theory is axiomatically unacceptable it is not some sort of philosophical interpretation! If experiments contradict the current theory then we are not debating philosophy, it is a cold hard fact of reality (we shouldn't even call it realism because it will be used to justifity philosophical arguments as valid). Historians have a name for the period of time when philosophies were above science. They call it the dark ages. Experiments are the reality check, so fuck them and their philosophies.

My only young advice that you should consider is that when the theory is mathematically unacceptable, and experiments explicitly prove it, then it is your duty as a scientist to rebel against the dominant paradigm.

Every great scientist has been censored I try to think of it as a rite of passage.

Brian,

You wrote:

1) Physics uses a similar idea of the backward travelling wave to explain the quantum tunneling in the finite square well.

----In case of cochlea, there is compelling evidence against reflection, and therefore the TW must not be considered a genuine wave.

Sommerfield's radiation condition says: Infinity does not reflect waves. I offer a modified argument: Reflection from infinity takes infinitely long time.----

2) I also wanted to show how the future does not yet exist by connecting it to entropy, combinatorics and probability theory. The imaginary numbers violate the axiom of additivity in probability and the theory is still accepted as fact!

-----Probability is always related to guesses but never to facts. What already happened is absolutely real. What we may expect to happen is never absolutely sure. Nonetheless we can speculate: Probably, Shakespeare was not Shakespeare. Negative and imaginary numbers are appropriate on the abstract level of speculation, which likewise includes past and future. It is reasonable to calculate with positive and negative temperature unless the negative exceeds 273°C.----

3) EB wrote: -----"The ultimate argument of physicists against my hint was: This is a matter of philosophy rather than physics. I do not think so."----

You wrote: If a theory is axiomatically unacceptable it is not some sort of philosophical interpretation!

----In mathematics, since Dedekind there are perhaps too many competing and strictly speaking rather questionable axioms, e.g. AC. Hilbert even tried and failed to prescribe axioms to physics.----

You wrote: If experiments contradict the current theory then we are not debating philosophy, it is a cold hard fact of reality (we shouldn't even call it realism because it will be used to justifity philosophical arguments as valid). Historians have a name for the period of time when philosophies were above science. They call it the dark ages. Experiments are the reality check, so fuck them and their philosophies.

----I dislike not just your wording. Given the LHC will fail to find the Higgs boson. Would this provide compelling evidence against the standard model? No. Popper spoke of required falsification.

I am arguing that the future is in general predictable merely to some extent, and one cannot measure in advance future data. "Datum est" can be translated into "it has been given". Future is not yet real except for the eyes of theorists. Therefore we may continue to operate with negative elapsed time and belonging complex data but ultimately we do not fully depend on them.

If a theory denies their origin in reality and if it is unable to re-transform the results into reality, then I see it at risk of unjustified speculations.----

You wrote: My only young advice that you should consider is that when the theory is mathematically unacceptable, and experiments explicitly prove it, then it is your duty as a scientist to rebel against the dominant paradigm.

----Yes. However be cautious with allegedly proven results if the interpretations of these results are based on possibly at least partially questionable theories. Reality is never questionable. ----

Regards, Eckard

Eckard,

I'm not sure I can comment on whether or not we will find the Higgs. My gut feeling is no, but when I did the derivation of the Higgs mechanism my professor was helping me a great deal. I derived it nearly two years ago and now all of the standard model seems like a blur of lagrangians. What I have seen looks like background dependence and that makes me skeptic.