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FQXi FORUM
May 25, 2012
CATEGORY:
FQXi Essay Contest - Is Reality Digital or Analog?
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TOPIC: Discrete and Continuous: Can We Tell One From the Other? by Andrey Akhmeteli [refresh]
TOPIC: Discrete and Continuous: Can We Tell One From the Other? by Andrey Akhmeteli [refresh]
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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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.
Download Essay PDF File
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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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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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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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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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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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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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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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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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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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http://arxiv.org/PS_cache/arxiv/pdf/1101/1101.5690v
2.pdf
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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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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
report post as inappropriate
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