Notes
[Notebook: Transfinite field theory DB 56]
[Sunday 25 January 2004 - Saturday 31 January 2004]
Sunday 25 January 2004
Monday 26 January 2004
Tuesday 27 January 2004
[page 38]
Wednesday 28 January 2004
We are
all on the same treadmill as the vulture which must get food each day
on order to have the energy the fly high enough to find food on the
morrow. Pennycuick. We find the concept in quantum mechanics in the
ideas of tunnelling and virtual particles where energy is borrowed
from the future in order to get there. So also in the economy, where
the young (people or companies) mortgage their future to get houses
and factories in the present. All this comes under capital with
different time constants and volumes ranging from the quantum of
action on up. In the quantum system energy is conserved, so that the
payoff from tunnelling out of a nucleus is exactly equal to the
potential difference between in and out. In the economy, borrowings
often attract interest, a cost arising from the competition for
capital and [from the point of view of the payer a form of
dissipation] in effect increases the total capital of the system by
increasing the fortunes of the lenders.
DISSIPATION <--> CREATION
(Prigogine) Nicolis.
Playing the penny dreadfuls on the stock market is probably akin
to brownian motion, many fluctuations leading to infinite drift.
In the full complexity of the Universe there are no symmetries
except those imposed by the need for consistency, and the combinatory
nature of process. The Universe is not cyclic but all its
subprocesses are and so we can observe them repeatedly and see and
use
[page 39]
their sameness. Similia similiter intelliguntur/aguntur?
Similar things are similarly understood and done. Productivity
increases as we exploit similarity (mass production, data compression
etc).
Physical theology is founded on the common (similar) elements in
the life of every entity, beginning with the triad birth-life-death.
Only the Universe does not die.
In communication terms, we see birth as the establishment of
communication with the world, life as a dialogue with the world and
death as the closure of a channel. So it is a sort of string (world
link) theory that sees every entity as the transmission of a message.
Thursday 29 January 2004
Mathematics =
the search for equalities (symbol =). Equalities are the channels by
which we move around the world, both in logic and in fact. Eg
matching wavefunctions on each side of a barrier. (Park chapter 4)
Park.
In the Hilbert space of quantum mechanics, the foundation of
statements of equality is the metric (norm) which gives us an
abstract view of which functions (vectors) are equal and which are
not. Every statement of equality has an abstract quality, since we
take concrete equality to be identity. So we can say that some things
are more equal than others, in proportion to the number of points of
agreement.
Bonding and orbit. Things that are bound together orbit around one
another. The archetypal orbit is an ellipse in space
[page 40]
but we also accept more complex concepts of orbit like the
'orbits' of electrons in an atom, or the orbit of lovers around one
another.
The progress of human society depends on the formation of
symmetries that previously did not exist, like the equivalence of
males and females. This symmetry formation can be quite painful for
both elements of the broken symmetry, since it requires an adjustment
of attitude. In many societies, for instance, it may require the
repositioning of women from chattels to partners and men from owners
to partners. One of the most difficult symmetries to establish is a
preferred outcome of divorce where both parents maintain a meaningful
relationship with the children without involving the children in the
bitterness of the breakup.
Friday 30 January 2004
Saturday 31 January 2004
To reconcile gravitation and quantum mechanics, we might assume
that the geometrical structure we experience as gravitation is
induced by the geometric features of the transfinite network, and the
quantum structure arises from the processing of information in the
nodes and edges of the network (fermions and bosons?)
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Further reading
Books
Click on the "Amazon" link below each book entry to see details of a book (and possibly buy it!)
Ashby, W Ross, An Introduction to Cybernetics, Methuen 1964 'This book is intended to provide [an introduction to cybernetics]. It starts from common-place and well understood concepts, and proceeds step by step to show how these concepts can be made exact, and how they can be developed until they lead into such subjects as feedback, stability, regulation, ultrastability, information, coding, noise and other cybernetic topics'
Amazon
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Everett III, Hugh, and Bryce S Dewitt, Neill Graham (editors), The Many Worlds Interpretation of Quantum Mechanics, Princeton University Press 1973 Jacket: 'A novel interpretation of quantum mechanics, first proposed in brief form by Hugh Everett in 1957, forms the nucleus around which this book has developed. The volume contains Dr Everett's short paper from 1957, "'Relativge State' formulation of quantum mechanics" and a far longer exposition of his interpretation entitled "The Theory of the Universal Wave Function" never before published. In addition other papers by Wheeler, DeWitt, Graham, Cooper and van Vechten provide further discussion of the same theme. Together they constitute virtually the entire world output of scholarly commentary on the Everett interpretation.'
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Hughes, I S, Elementary Particles, Cambridge Univerity Press 1991 Jacket: 'This is an extensively revised and updated edition of a text that has established itself as one of the standard undergraduate books on the subject of elementary particle physics.'
Amazon
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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
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Nicolis, G , and Ilya Prigogine, Self Organisation in Nonequilibrium Systems: From Dissipative Structures to Order through Fluctuations, Wiley Interscience 1977 General Introduction: 'The aim of the present monograph can ... be expressed as the studiy of self-organisation in non-equilibrium systems, characterised by the appearance of dissipative structures through the amplification of appropriate fluctuations. ... The natural approach to the problem of the emergence of new patterns is bifurcation theory. The purpose of this theory is to study the possible branching of solutions that may arise under certain conditions. We have tried to present a readable introduction to this rapidly expanding field ... Our main emphasis is in physical examples and simple but representative models, and our aim is to give the reader an idea of the variety of space-time structures that may arise through bifurcation. ... '
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Park, David Allen, Introduction to the Quantum Theory, McGraw-Hill Book Company 1992
Amazon
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Streater, Raymond F, and Arthur S Wightman, PCT, Spin, Statistics and All That, Princeton University Press 2005 Amazon product description: '
PCT, Spin and Statistics, and All That is the classic summary of and introduction to the achievements of Axiomatic Quantum Field Theory. This theory gives precise mathematical responses to questions like: What is a quantized field? What are the physically indispensable attributes of a quantized field? Furthermore, Axiomatic Field Theory shows that a number of physically important predictions of quantum field theory are mathematical consequences of the axioms. Here Raymond Streater and Arthur Wightman treat only results that can be rigorously proved, and these are presented in an elegant style that makes them available to a broad range of physics and theoretical mathematics.'
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Veltman, Martinus, Diagrammatica: The Path to the Feynman Rules, Cambridge University Press 1994 Jacket: 'This book provides an easily accessible introduction to quantum field theory via Feynman rules and calculations in particle physics. The aim is to make clear what the physical foundations of present-day field theory are, to clarify the physical content of Feynman rules, and to outline their domain of applicability. ... The book includes valuable appendices that review some essential mathematics, including complex spaces, matrices, the CBH equation, traces and dimensional regularization. ...'
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von Neumann, John, and Robert T Beyer (translator), Mathematical Foundations of Quantum Mechanics, Princeton University Press 1983 Jacket: '. . . a revolutionary book that caused a sea change in theoretical physics. . . . JvN begins by presenting the theory of Hermitean operators and Hilbert spaces. These provide the framework for transformation theory, which JvN regards as the definitive form of quantum mechanics. . . . Regarded as a tour de force at the time of its publication, this book is still indispensible for those interested in the fundamental issues of quantum mechanics.'
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Wheeler, John Archibald, and Wojciech Hubert Zurek, Quantum Theory and Measurement (Princeton Series on Physics), Princeton University Press 1983 Amazon customer review: 'This is a must-own collection for anyone studying or working in quantum physics. These are the original papers concerning the so-called problem of measurement. Minority views are included; for instance, both parts of Bohm's 1952 paper are here. Not only physicists, but also historians and philosophers of science, will want to read these papers.' Paul E Oppenheimer
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Wigner, Eugene, Symmetries and Reflections: Scientific Essays , MIT Press 1970 Jacket: 'This volume contains some of Professor Wigner's more popular papers which, in their diversity of subject and clarity of style, reflect the author's deep analytical powers and the remarkable scope of his interests. Included are articles on the nature of physical symmetry, invariance and conservation principles, the structure of solid bodies and of the compound nucleus, the theory of nuclear fission, the effects of radiation on solids, and the epistemological problems of quantum mechanics. Other articles deal with the story of the first man-made nuclear chain reaction, the long term prospects of nuclear energy, the problems of Big Science, and the role of mathematics in the natural sciences. In addition, the book contains statements of Wigner's convictions and beliefs. as we as memoirs of his friends Enrico Fermi and John von Neumann.
Eugene P. Wigner is one of the architects of the atomic age. He worked with Enrco Fermi at the Metallurgical Laboratory of the University of Chicago at the beginning of the Manhattan Project, and he has gone on to receive the highest honours that science and his country can bestow, including the Nobel Prize for physics, the Max Planck Medal, the Enrico Fermi Award and the Atoms for Peace Award. '.
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Papers
| Pennycuick, C J, "The soaring flight of vultures", Scientific American, 229, 6, December 1973, page 102-109. 'The six common vultures of East Africa can make a round trip of as much as 200 kilometres by skilfully riding updrafts. How they do so is examined with the aid of a powered glider.' . back |
| Salart, Daniel, et al, "Testing the speed of 'spooky action at a distance'", Nature, 454, , 14 August 2008, page 861-864. 'Correlations are generally described by one of two mechanisms: either a first event influences a second one by sending information encoded in bosons or other physical carriers, or the correlated events have some common causes in their shared history. Quantum physics predicts an entirely different kind of cause for some correlations, named entanglement. This reveals itself in correlations that violate Bell inequalities (implying that they cannot be described by common causes) between space-like separated events (implying that they cannot be described by classical communication). Many Bell tests have been performed, and loopholes related to locality and detection have been closed in several independent experiments. It is still possible that a first event could influence a second, but the speed of this hypothetical influence (Einstein's 'spooky action at a distance') would need to be defined in some universal privileged reference frame and be greater than the speed of light. Here we put stringent experimental bounds on the speed of all such hypothetical influences. We performed a Bell test over more than 24 hours between two villages separated by 18 km and approximately east–west oriented, with the source located precisely in the middle. We continuously observed two-photon interferences well above the Bell inequality threshold. Taking advantage of the Earth's rotation, the configuration of our experiment allowed us to determine, for any hypothetically privileged frame, a lower bound for the speed of the influence. For example, if such a privileged reference frame exists and is such that the Earth's speed in this frame is less than 10-3 times that of the speed of light, then the speed of the influence would have to exceed that of light by at least four orders of magnitude.. 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 |
| Turing, Alan, "On Computable Numbers, with an application to the Entscheidungsproblem", Proceedings of the London Mathematical Society, 2, 42, 12 November 1937, page 230-265. 'The "computable" numbers maybe described briefly as the real numbers whose expressions as a decimal are calculable by finite means. Although the subject of this paper is ostensibly the computable numbers, it is almost as easy to define and investigate computable functions of an integrable variable or a real or computable variable, computable predicates and so forth. The fundamental problems involved are, however, the same in each case, and I have chosen the computable numbers for explicit treatment as involving the least cumbrous technique. I hope shortly to give an account of the rewlations of the computable numbers, functions and so forth to one another. This will include a development of the theory of functions of a real variable expressed in terms of computable numbers. According to my definition, a number is computable if its decimal can be written down by a machine'. back |
| Zurek, Wojciech Hubert, "Decoherence, einselection, and the quantum origins of the classical", Review of Modern Physics, 75, , 2003, page 715-775. The manner in which states of some quantum systems become effectively classical is of great significance for the foundations of quantum physics, as well as for problems of practical interest such as quantum engineering. In the past two decades it has become increasingly clear that many (perhaps all) of the symptoms of classicality can be induced in quantum systems by their environments. Thus decoherence is caused by the interaction in which the environment in effect monitors certain observables of the system, destroying coherence between the pointer states corresponding to their eigenvalues. This leads to environment-induced superselection or einselection, a quantum process associated with selective loss of information. Einselected pointer states are stable. They can retain correlations with the rest of the Universe in spite of the environment. Einselection enforces classicality by imposing an effective ban on the vast majority of the Hilbert space, eliminating especially the flagrantly nonlocal “Schrödinger-cat states.” The classical structure of phase space emerges from the quantum Hilbert space in the appropriate macroscopic limit. Combination of einselection with dynamics leads to the idealizations of a point and of a classical trajectory. In measurements, einselection replaces quantum entanglement between the apparatus and the measured system with the classical correlation. Only the preferred pointer observable of the apparatus can store information that has predictive power. When the measured quantum system is microscopic and isolated, this restriction on the predictive utility of its correlations with the macroscopic apparatus results in the effective “collapse of the wave packet.” The existential interpretation implied by einselection regards observers as open quantum systems, distinguished only by their ability to acquire, store, and process information. Spreading of the correlations with the effectively classical pointer states throughout the environment allows one to understand “classical reality” as a property based on the relatively objective existence of the einselected states. Effectively classical pointer states can be “found out” without being re-prepared, e.g, by intercepting the information already present in the environment. The redundancy of the records of pointer states in the environment (which can be thought of as their “fitness” in the Darwinian sense) is a measure of their classicality. A new symmetry appears in this setting. Environment-assisted invariance or envariance sheds new light on the nature of ignorance of the state of the system due to quantum correlations with the environment and leads to Born’s rules and to reduced density matrices, ultimately justifying basic principles of the program of decoherence and einselection.. back |
Links
| Emmy Noether English translation of 'Invariant variation problems' "Invariante Variationsprobleme," Nachr. v. d. Ges. d. Wiss. zu Göttingen 1918, pp 235-257 English translation by M. A. Tavel. Reprinted from "Transport Theory and Statistical Mechanics" 1(3), 183-207 (1971). Provided to this site by M.A. Tavel and Henry M. Paynter.
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| Emmy Noether - Wikipedia Emmy Noether - Wikipedia, the free encyclopedia 'Amalie Emmy Noether, . . . (23 March 1882 – 14 April 1935) was a German mathematician known for her groundbreaking contributions to abstract algebra and theoretical physics. Described by Albert Einstein and others as the most important woman in the history of mathematics, she revolutionized the theories of rings, fields, and algebras. In physics, Noether's theorem explains the fundamental connection between symmetry and conservation laws.' back |
| Oracle machine - Wikipedia Oracle machine - Wikipedia, the free encyclopedia 'In complexity theory and computability theory, an oracle machine is an abstract machine used to study decision problems. It can be visualized as a Turing machine with a black box, called an oracle, which is able to decide certain decision problems in a single operation. The problem can be of any complexity class. Even undecidable problems, like the halting problem, can be used.' back |
| Quantum electrodynamics - Wikipedia Quantum electrodynamics - Wikipedia, the free encyclopedia 'Quantum electrodynamics (QED) is a relativistic quantum field theory of electrodynamics. QED was developed by a number of physicists, beginning in the late 1920s. It basically describes how light and matter interact. More specifically it deals with the interactions between electrons, positrons and photons. QED mathematically describes all phenomena involving electrically charged particles interacting by means of exchange of photons. It has been called "the jewel of physics" for its extremely accurate predictions of quantities like the anomalous magnetic moment of the electron, and the Lamb shift of the energy levels of hydrogen.
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| Richard P. Feynman Nobel Lecture Nobel Lecture, December 11, 1965: 'We have a habit in writing articles published in scientific journals to make the work as finished as possible, to cover all the tracks, to not worry about the blind alleys or to describe how you had the wrong idea first, and so on. So there isn't any place to publish, in a dignified manner, what you actually did in order to get to do the work, although, there has been in these days, some interest in this kind of thing. Since winning the prize is a personal thing, I thought I could be excused in this particular situation, if I were to talk personally about my relationship to quantum electrodynamics, rather than to discuss the subject itself in a refined and finished fashion. Furthermore, since there are three people who have won the prize in physics, if they are all going to be talking about quantum electrodynamics itself, one might become bored with the subject. So, what I would like to tell you about today are the sequence of events, really the sequence of ideas, which occurred, and by which I finally came out the other end with an unsolved problem for which I ultimately received a prize.' back |
| Wojciech Hubert Zurek Decoherence, einselection and the quantum origins of the classical 'Decoherence is caused by the interaction with the environment
which in effect monitors certain observables
of the system, destroying coherence between the pointer
states corresponding to their eigenvalues. This leads
to environment-induced superselection or einselection, a
quantum process associated with selective loss of information.
Einselected pointer states are stable. They can
retain correlations with the rest of the Universe in spite
of the environment. Einselection enforces classicality by
imposing an effective ban on the vast majority of the
Hilbert space, eliminating especially the flagrantly nonlocal
“Schr¨odinger cat” states. Classical structure of
phase space emerges from the quantum Hilbert space in
the appropriate macroscopic limit: Combination of einselection
with dynamics leads to the idealizations of a point
and of a classical trajectory. In measurements, einselection
replaces quantum entanglement between the apparatus
and the measured system with the classical correlation.
Only the preferred pointer observable of the apparatus
can store information that has predictive power.
When the measured quantum system is microscopic and
isolated, this restriction on the predictive utility of its
correlations with the macroscopic apparatus results in
the effective “collapse of the wavepacket”. Existential interpretation
implied by einselection regards observers as
open quantum systems, distinguished only by their ability
to acquire, store, and process information. Spreading
of the correlations with the effectively classical pointer
states throughout the environment allows one to understand
‘classical reality’ as a property based on the relatively
objective existence of the einselected states: They
can be “found out” without being re-prepared, e.g, by
intercepting the information already present in the environment.
The redundancy of the records of pointer states
in the environment (which can be thought of as their ‘fitness’
in the Darwinian sense) is a measure of their classicality.
A new symmetry appears in this setting: Environment
- assisted invariance or envariance sheds a new
light on the nature of ignorance of the state of the system
due to quantum correlations with the environment, and
leads to Born’s rules and to the reduced density matrices,
ultimately justifying basic principles of the program
of decoherence and einselection. back |
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