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Undecidability, Uncomputability, and Unpredictability Essay Contest
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What Is “Fundamental”
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Trick or Truth: The Mysterious Connection Between Physics and Mathematics
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How Should Humanity Steer the Future?
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It From Bit or Bit From It
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Questioning the Foundations
Which of Our Basic Physical Assumptions Are Wrong?
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Is Reality Digital or Analog?
November 2010 - February 2011
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What's Ultimately Possible in Physics?
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RECENT POSTS IN THIS TOPIC

Eckard Blumschein: on 2/20/11 at 21:53pm UTC, wrote Dear Andy, Thank you for pointing me to the preprint Division Algebras and...

Andrey Akhmeteli: on 2/18/11 at 8:53am UTC, wrote I don't know about the specific "clarification" that you mention, but the...

Eckard Blumschein: on 2/17/11 at 8:38am UTC, wrote Dear Andy, I am delighted. Thank you very much for your support. What...

Eckard Blumschein: on 2/17/11 at 8:16am UTC, wrote Dear Andy, Yesterday I was short of time. I intended to ask you for help....

Andrey Akhmeteli: on 2/17/11 at 2:32am UTC, wrote Dear Eckard, Thank you for your comment. Some time ago I was quite...

Eckard Blumschein: on 2/16/11 at 19:38pm UTC, wrote Dear Andrey, I got aware of your utterance that seems to confirm my own...

Andrey Akhmeteli: on 2/15/11 at 10:34am UTC, wrote Essay Abstract While it is obvious that no experiment can tell...


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FQXi FORUM
September 18, 2021

CATEGORY: Is Reality Digital or Analog? Essay Contest (2010-2011) [back]
TOPIC: Discrete and Continuous: Can We Tell One From the Other? by Andrey Akhmeteli [refresh]
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Author Andrey Akhmeteli wrote on Feb. 15, 2011 @ 10:34 GMT
Essay Abstract

While it is obvious that no experiment can tell discrete from continuous due to limited resolution, even within reasonably well defined quantum theories telling discrete from continuous or particles from waves can be less than trivial. For example, due to natural line width, it is not so obvious that atom spectra are discrete, and while it is believed that a measurement of a spin projection of a spin 1/2 particle can only yield values +1/2 or -1/2, the situation is not quite clear-cut due to the well-known measurement problem in quantum mechanics. Point-like particles could be another manifestation of discrete, but using some versions of electrodynamics, one can show that matter particles of quantum theory can be emulated by continuous fields. For example, matter field can be naturally excluded from the equations of scalar electrodynamics (the Klein-Gordon-Maxwell electrodynamics), and the resulting equations describe independent evolution of the electromagnetic field. These equations can also be naturally embedded into a quantum field theory. Some surprising new results for spinor electrodynamics (the Dirac-Maxwell electrodynamics) suggest that similar conclusions may be true for that theory, which is more realistic.

Author Bio

Andrey Akhmeteli obtained his PhD in theoretical and mathematical physics from Moscow University and has worked there, in other research and education institutions, and in industry.

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Eckard Blumschein wrote on Feb. 16, 2011 @ 19:38 GMT
Dear Andrey,

I got aware of your utterance that seems to confirm my own reasoning:

"In principle, we can do without complex numbers in quantum theory."

Regards,

Eckard

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Author Andrey Akhmeteli replied on Feb. 17, 2011 @ 02:32 GMT
Dear Eckard,

Thank you for your comment. Some time ago I was quite surprised by Schroedinger's note that complex numbers are not necessary to describe charged particles. I was also quite surprised to find out last year that the Dirac equation can be rewritten as an equation for just one real function. But I guess you are not happy with real numbers either:-) And I agree with you, the notion of analytical signal, while very convenient, is really messy.

Best regards

Andy

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Eckard Blumschein replied on Feb. 17, 2011 @ 08:38 GMT
Dear Andy,

I am delighted. Thank you very much for your support. What about real numbers, I am very happy with C as well as R, even if I am claiming that, in principle, R+ fits to reality without redundancy and without ambiguity. Training students to use R and C blackboard bold as to describe R and C in the sense of resistor and capacitor was my job for more than forty years.

What about the notion analytical signal, I am an EE, and we EEs like it. For us an analytic signal is a function of time in complex plane with Hermitian symmetry. We do not bother about negative frequencies. Why should we bother about a complex function of simultaneously positive as well as negative time. In contrast to many speculative physicist, engineers have at least in the end to know what they are doing.

Best regards,

Eckard

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Eckard Blumschein wrote on Feb. 17, 2011 @ 08:16 GMT
Dear Andy,

Yesterday I was short of time. I intended to ask you for help. Doug Bundy told me that John Baez somewhere "clarified" the question whether the notion number should be understood as distance from zero or as usual since Dedekind like a point.

I consider this trifle crucial in connection with important implications, and I doubt that John Baez can contribute more than perhaps questionable mainstream excuses. I will read your essay, and I hope you can give me hints. Criticism is always welcome to me.

Best regards,

Eckard

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Author Andrey Akhmeteli replied on Feb. 18, 2011 @ 08:53 GMT
I don't know about the specific "clarification" that you mention, but the following Baez' recent preprint may be relevant:

http://arxiv.org/PS_cache/arxiv/pdf/1101/1101.5690v
2.pdf

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Eckard Blumschein replied on Feb. 20, 2011 @ 21:53 GMT
Dear Andy,

Thank you for pointing me to the preprint Division Algebras and Quantum Theory. It refers to the Question R, C, H, or O and does not even mention R+. Mathematicians like John Baez do obviously not devote attention to the question of avoidable redundancy and ambiguity.

Meanwhile Doug Bundy pointed me to what he recalled as a "clarification" of the question whether the notion number should be understood as distance from zero or as usual since Dedekind like a point. According to my reasoning there is no point in the middle left over, provided the community of mathematicians is ready to jump over its own shadow and admit that real numbers are different from rational ones in that there is no trichotomy with the former. Brouwer already understood: the TND is only valid in case of countability. This means, within real numbers there is no difference between open and closed intervals and no singular points, just limits measured from zero.

Regards,

Eckard

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