This paper is a significant work Lawrence...

From my view; your work is coming to resemble more and more that of Ed Witten. I mean that both as a compliment and a criticism. On the one hand; you are a truly deep thinker with a good grounding in the Maths that let you express your thoughts. On the other hand; your paper is dense with Math and the reader gets immersed pretty quickly, which can overwhelm some who are less Math-agile, and overload some who are - before they get your point. I get the same effect from reading Connes. I know there is something amazing there, and it's exciting to try to follow it, but it's easy to get lost if you don't know some of the technical terms and techniques well.

So I give you high marks but not full credit. This work is patently amazing, and it is obvious that it addresses the organizers' questions in a detailed way, but you make me go back to the books a bit too often - in order to fully grasp your point. I am not 100% sure I understand you completely yet, and I feel that this contest is a forum for people like yourself to prove their ability to reach a broader audience. All in all; I think you did quite well. It almost comes across as a advertisement for Szangolies and Palmer in places, though, and they may not need the boost. But I will help you a bit.

This is a paper I'll probably read multiple times, for what it is worth. There is a lot of meaning to mine from this exploration.

All the Best,

Jonathan

Dear Lawrence Crowell

The only thing I'm sure of in quantum mechanics is your next sentence:

"Quantum mechanics requires an extension of real variables of classical mechanics into complex variables."

After all, I came to that conclusion with my theory, but I can't go further.

Please answer me if you agree with my general position (assumption):

Quantum mechanics occurs, in frame exp (i * pi).

Since I am not educated for the field you are writing about, I do not understand your formulas. So, if you don't understand the formulas in my essay then we are 1: 1.

The text I wrote is nothing special anyway. In any case, your work deserves high marks. By my criteria, and because it contains formulas.

Regards,

Branko

    The equation e^{iПЂ} = -1 is called the Euler equation. It also leads to complex number. Consider the natural logarithm of a negative number ln(-x). In the field of real numbers we are stuck but we can use the rule of logarithms ln(ab) = ln(a) + ln(b). So, ln(-x) = ln(x) + ln(-1) and we can at least peel off the part we don't understand. As a result, with Euler equation write

    ln(-x) = ln(x) + ln(e^{iПЂ}) = ln(x) + iПЂ

    due to the inverse function of the exponent e^{iПЂ}. We can also use e^{iПЂ/2} = I to evaluate the logarithm of an imaginary number ln(ix) = ln(x) + iПЂ/2.

    Quantum mechanics can be thought of as a map of classical mechanics to the quantum according to the Poisson bracket formalism. This bracket {q, p} = (∂q/∂q)(∂p/∂p) - (∂q/∂p)(∂p/∂q) = 1 is replaced with the commutator {q, p} в†' i/Д§[q, p], where these are now operators. The commutator is then equal to i = в€љ(-1). So, in that sense quantum mechanics does have this funny relationship with imaginary numbers.

    The wave function in QM is complex valued. A complex number is z = x + iy. The magnitude of this number is computed with its conjugate z* = x - iy and so zz* = x^2 + y^2. Then with a little trigonometry we have z = zz*(cosОё + isinОё) = zz*e^{iОё}. A polar form of a wave function is of this form.

    Dear Lawrence Crowell

    Thanks for your comprehensive comment, even though you didn't answer my question. Ultimately, it is evident that you are one of the few who speaks about the universe and understands logarithms. My question was related to the boundary case when theta = pi.

    Why is it important. Because in all my works and earlier FQX essays you can see the importance of another borderline case (frame), ie. Exp (2 * pi). You can see the results of such an approach to classical physics through the predictive formulas at the end of my essay this year.

    Regards,

    Branko

    I will try to read your essay in the near future. This issue with the Euler formula hinges around the unit circle in the complex plane with

    e^{iﾎｸ} = cosﾎｸ + isinﾎｸ,

    where this is a way of thinking about trigonometry. the multiplicity of these functions for angles ﾎｸ 竊' ﾎｸ + 2ﾏ\ is a part of the reason ln(-x) = ln(x) + iﾏ\, but also the entire set {..., -3iﾏ\, -iﾏ\, iﾏ\, 3iﾏ\, ...}. This leads to the concept of Riemann sheets, where the complex plane has multiple copies.

    Cheers LC

    Lawrence,

    Hope you have time to check mine out before the deadline: https://fqxi.org/community/forum/topic/3396

    Jim Hoover.

    Dear Peter,

    I believe we are kindred spirits on the topic of spin. I regard it as one of the fundamentals of physics along with charge. As you already know I have derived my theories from scratch, from the basic particles of the Universe to a computational TOE, and a working theory of time. I have given small glimpses of what I have found in some of the many posts I have made but I would certainly value your opinions and insights on a deeper level. It seems that my math and philosophy and even physics are sorely lacking when compared to others, but I have achieved something they haven't.

    Regards

    Lockie

    The quantum numbers such as spin I think are independent. An elementary particle is I think an entanglement of states for each of these quantum numbers. In solid state physics these are called quasi-particles, such as phonons, plasmons, spinons etc. Experiments done recently have shown that an electron can be placed in a state where the spin is in one location and the charge at another. I actually think elementary particles are these entanglements of quantum states corresponding to these quantum numbers.

    LC

    Dear Lawrence,

    Very good and informative essay, with interesting insights and connections, thank you! I liked how you connected different structures and techniques in quantum mechanics to build an argument about the relation with undecidability. Particularly, I am interested in the topological obstructions, but for independent reasons. Thanks again, and I wish you success with the essay!

    Cheers,

    Cristi

      Thanks for the boost. Try to read Szangolies' essay on a related development, and Palmer's on the fractal geometry.

      Your paper works with the connection between Gödel theorem or self-reference and consciousness. I have thought that consciousness is a sort of epiphenomenology that is an illusion having an illusion of itself. I have not read it in its entirty, and I do see you connect with what look like fractals.

      I have been slow. I have had Covid-19. It hit me at the 3rd week of March and lasted about 10 days. It relapsed in April and the fatigue part of this was serious. I still sleep more than I used to, but the most pernicious aspect of this has been dogging me. It is as if my brain has been rewired, or maybe hormone setpoint levels changed. I am not quite the same person I was; I feel as if I am an abruptly changed person. The worst part of this change is that I am more depressed and irritable than I was. It has been hard for me to participate much in this contest.

      Cheers :LC

      Dear Lawrence,

      Here's what I replied to your comment on my page:

      Thanks for the visit. I am very sad that you got Covid-19, I hope it is the easiest form and you'll be well as soon as possible. Don't worry about my essay until you get well, but please get well, because I would love to hear some feedback from you, if possible about the longer version, even if it will be long after the contest ends.

      > Your paper works with the connection between Gödel theorem or self-reference and consciousness. I have thought that consciousness is a sort of epiphenomenology that is an illusion having an illusion of itself. I have not read it in its entirty, and I do see you connect with what look like fractals.

      I didn't appeal to self-reference or fractals, although I'd agree with you that they play a role. But it has strong relation with no-go theorems. As for consciousness, I am interested in the hypothesis that there is something irreducible about it (this irreducible I called "sentience"), and I try to see if this makes testable predictions. My claim is that it does. Indeed, for many who think consciousness is irreducible, the epiphenomenal position seems a good refuge, since it makes the hypothesis unfalsifiable. But I think we should be brave and don't avoid the fact that it does make predictions. So we can test it. We risk, those who deny it risk to see the predictions confirmed, but they can still continue to deny it, since the test of a prediction is not necessarily a proof of what led to the prediction. A rejection of the prediction is a rejection of what led to the prediction, so it is more risky for those who endorse the position that consciousness is not fully reducible. If we want to bring the hard problem into science, we have to take this risk.

      I wish you to get back in shape soon!

      Cristi

      Dear Lawrence,

      Glad to read your work again.

      I greatly appreciated your work and discussion. I am very glad that you are not thinking in abstract patterns.

      While the discussion lasted, I wrote an article: "Practical guidance on calculating resonant frequencies at four levels of diagnosis and inactivation of COVID-19 coronavirus", due to the high relevance of this topic. The work is based on the practical solution of problems in quantum mechanics, presented in the essay FQXi 2019-2020 "Universal quantum laws of the universe to solve the problems of unsolvability, computability and unpredictability".

      I hope that my modest results of work will provide you with information for thought.

      Warm Regards, `

      Vladimir

        Dear Lawrence,

        I beg for the best and a full recovering from COVID19. Here in Germany, everybody try to enforce the goverment to open the restaurants, we have demonstrations agains the lock-down. I hate it. your case show it that it is wrong and we have to be careful to not getting the virus.

        But now back to your very thoughtful essay. I'm a fan of the magic Tits square and now I use very much discrete groups, fractals (or the higher-dimensional generalizations like wild embeddings),...

        I gave your essay the highest vote,

        Good luck for you

        Cheers

        Torsten

          Hi Torsten,

          I somehow missed that you had an essay in lot here. I will queue that up to read. I just looked and today is the deadline! Somehow I thought it was in June. I will surely pull you essay up and make a quick judgment about it. I remember your previous work was interesting.

          The Covid=19 was not that bad at first. Last years beginning 2019 I had influenza that made me a whole lot more sick, threateningly so. I had a mild fever, a dry and not compulsive cough, lost my sense of smell, etc, but the next to worst part is that I have been very fatigued, and that has persisted.. However, the most disturbing part is that I feel as if this has rewired my brain. Seriously, I feel slightly different or that I am a different person now.

          I think Europe is better set than the US, where the hybrid between Godzilla and an orange baboon, Don-the-Con t'Rump, who runs this country is leading us into complete ruin. It can be worse, a friend of mine in Brazil has given me the fill on Bolsanoro, who is a complete moron.

          Gotta get to some essays.

          Cheers LC

          I brought your essay up and I find that today is the deadline. I thought it was a bit later. I have a several of these to read.

          LC