vol 3: Development
chapter 3: Physics
page 2: Why is the universe quantized?
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Why is the universe quantized?
Physicists have long believed that the universe is continuous. The was Aristotle's opinion, and even the Atomists that we know of, Democritus and Leucippus, imagines their 'uncuttables' moving in a continuous space.
Quantum mechanics has nothing to say about why the universe is quantized. It takes this as given and provides a way of discussing the nature and frequency of various quantum events.
If we view the universe as a network, however, we may see an explanation for quantization in the mathematical theory of communication. Shannon, a practical communication engineer, wanted to know how fast we can push information along a communication channel. We are not concerned with meaning here, simply the rate of error free transmission of a sequence of symbols.
Shannon found a measure of information in an adaptation of thermodynamic entropy. A Mathematical Theory of Communication - Wikipedia He modelled a communication system as a source and a channel. The source can emit any letter ai of a alphabet A. Each of the letters is emitted with a certain frequency, pi . The frequencies of the letters are normalized to 1, so we write
Using these letter frequencies, we define the source entropy (that is the entropy per letter emitted) to be
H is at its maximum when the letters are equiprobable. At this point, H = - log p = log n, where n is the number of letters in the alphabet. It is customary in information theory to use logarithms to base 2, so that the quantity of information is measured in bits (bi nary digits ).
In most natural text (like this) the letter frequencies are not equal, so the entropy per letter is less than the maximum. Shannon showed how the output of a source could be encoded in blocks in a way that made the letter frequencies equal, and so used the full potential entropy of the source alphabet.
Error occurs in the transmission of information when one symbol is mistaken for another. The risk of such confusion is reduced if we make the symbols further apart in 'message space'. Shannon's encoding achieves this, but at the expense of some delay in transmission. The computer doing the encoding must first receive a string of output from the source before it can transform it into a string for transmission over the channel.
From the point of view of the network model, error proofing messages requires their quantization. In addition the coding process introduces a delay which could be a good candidate for explaining the finite velocity of communications, from the velocity of light on down.
These ideas have an echo in quantum field theory, which we meet below. Weinberg
Further reading
Books
| Deutsch, David, The Fabric of Reality: The Science of Parallel Universes - and its Implications, Allen Lane Penguin Press 1997 Jacket: 'Quantum physics, evolution, computation and knowledge - these four strands of scientific theory and philosophy have, until now, remained incomplete explanations of the way the universe works. ... Oxford scholar DD shows how they are so closely intertwined that we cannot properly understand any one of them without reference to the other three. ...' Amazon back |
| Khinchin, A I, Mathematical Foundations of Information Theory (translated by P A Silvermann and M D Friedman), Dover 1957 Jacket: 'The first comprehensive introduction to information theory, this book places the work begun by Shannon and continued by McMillan, Feinstein and Khinchin on a rigorous mathematical basis. For the first time, mathematicians, statisticians, physicists, cyberneticists and communications engineers are offered a lucid, comprehensive introduction to this rapidly growing field.' Amazon back |
| Weinberg, Steven, The Quantum Theory of Fields Volume I: Foundations, Cambridge University Press 1995 Jacket: 'After a brief historical outline, the book begins anew with the principles about which we are most certain, relativity and quantum mechanics, and then the properties of particles that follow from these principles. Quantum field theory then emerges from this as a natural consequence. The classic calculations of quantum electrodynamics are presented in a thoroughly modern way, showing the use of path integrals and dimensional regularization. The account of renormalization theory reflects the changes in our view of quantum field theory since the advent of effective field theories. The book's scope extends beyond quantum elelctrodynamics to elementary partricle physics and nuclear physics. It contains much original material, and is peppered with examples and insights drawn from the author's experience as a leader of elementary particle research. Problems are included at the end of each chapter. ' Amazon back |
Papers
| d'Espagnat, Bernard, "Quantum theory and reality", Scientific American, 241, 5, November 1979, page 128. back |
| Landauer, Rolf, "Information is a physical entity", Physica A, 263, 1-4, 1 February 1999, page 63-7. 'This paper, associated with a broader conference talk on the fundamental physical limits of information handling, emphasizes the aspects still least appreciated. Information is not an abstract entity but exists only through a physical representation, thus tying it to all the restrictions and possibilities of our real physical universe. The mathematician's vision of an unlimited sequence of totally reliable operations is unlikely to be implementable in this real universe. Speculative remarks about the possible impact of that, on the ultimate nature of the laws of physics are included.'. back |
| Shannon, Claude E, "The mathematical theory of communication", Bell System Technical Journal, 27, , July and October, 1948, page 379-423, 623-656. 'A Note on the Edition Claude Shannon's ``A mathematical theory of communication'' was first published in two parts in the July and October 1948 editions of the Bell System Technical Journal [1]. The paper has appeared in a number of republications since: o The original 1948 version was reproduced in the collection Key Papers in the Development of Information Theory [2]. The paper also appears in Claude Elwood Shannon: Collected Papers [3]. The text of the latter is a reproduction from the Bell Telephone System Technical Publications, a series of monographs by engineers and scientists of the Bell System published in the BSTJ and elsewhere. This version has correct section numbering (the BSTJ version has two sections numbered 21), and as far as we can tell, this is the only difference from the BSTJ version. o Prefaced by Warren Weaver's introduction, ``Recent contributions to the mathematical theory of communication,'' the paper was included in The Mathematical Theory of Communication, published by the University of Illinois Press in 1949 [4]. The text in this book differs from the original mainly in the following points: o the title is changed to ``The mathematical theory of communication'' and some sections have new headings, o Appendix 4 is rewritten, o the references to unpublished material have been updated to refer to the published material. The text we present here is based on the BSTJ version with a number of corrections.. back |
| Shannon, Claude E, "Communication in the Presence of Noise", Proceedings of the IEEE, 86, 2, February 1998, page 447-457. Reprint of Shannon, Claude E. "Communication in the Presence of Noise." Proceedings of the IEEE, 37 (January 1949) : 10-21. 'A method is developed for representing any communication system geometrically. Messages and the corresponding signals are points in two "function spaces," and the modulation process is a mapping of one space into the other. Using this representation, a number of results in communication theory are deduced concerning expansion and compression of bandwidth and the threshold effect. Formulas are found for the maximum rate of transmission of binary digits over a system when the signal is perturbed by various types of noise. Some of the properties of "ideal" systems which transmit this maximum rate are discussed. The equivalent number of binary digits per second of certain information sources is calculated.' . back |
Links
| A Mathematical Theory of Communication - Wikipedia A Mathematical Theory of Communication - Wikipedia, the free encyclopedia 'The article entitled "A Mathematical Theory of Communication", published in 1948 by mathematician Claude E. Shannon, was one of the founding works of the field of information theory. Shannon's paper laid out the basic elements of any digital communication: o An information source which produces a message o A transmitter which operates on the message to create a signal which can be sent through a channel o A channel, which is the medium over which the signal, carrying the information that composes the message, is sent o A receiver, which transforms the signal back into the message intended for delivery o A destination, which can be a person or a machine, for whom or which the message is intended It also developed the concepts of information entropy and redundancy.' back |
| Albert Einstein On the Electrodynamics of Moving Bodies An english translation of the paper that founded Special relativity. 'Examples of this sort, [in the contemporary application of Maxwell's electrodynamics to moving bodies] together with the unsuccessful attempts to discover any motion of the earth relatively to the ``light medium,'' suggest that the phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest. They suggest rather that, as has already been shown to the first order of small quantities, the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good.' back |
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