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Notes

[Notebook: DB 60 Spotlights]

[Sunday 8 October 2006 - Saturday 14 October 2006]

[page 6]

Sunday 8 October 2006

Physics is simple when studied locally - the differential approach. Logic (and networks) are also simple when analyzed locally.

We want to explain the interpretation of quantum mechanics. Why does the absolute square of a complex quantity predict the probabilities of certain events? Like a classical source of information the probabilities of measuring different quanta are normalized to 1. Every quantum has two addresses, its birth address and its death address [its two contacts with the rest of the world]

We visualize the conserved flow of probability by the flow of a friction less incompressible fluid in a leak free closed system of pipes. The meaning of this flow, its logical interpretation, is something else. Ultimately we select a set of basis states and use them as a foundation for prediction and measurement, retaining all the while the ability to make deterministic transformations between the bases.

What I see = Me (product) It. I see everything relative to myself even which I apply symmetries such as 'human equality', I see myself as the equal one.

Shannon 1949: By moving from continuous to computable we get an equivalent 8 decibel gain in signal to noise ratio: a definite selective advantage. Shannon

[page 7]

Photon is a minimum delay particles - minimal encoding. And the graviton? Couples to itself? Then massive particles with deeper encoding and more delay.

Quadratic form. Communication is a square. One dimension is the signals I send.The other dimension is the signals you read. When we are in perfect harmony all these points lie on a diagonal; or at random; or on an 'antidiagonal'?

Monday 9 October 2006

Is there a 'Turing group', a group of computable functions?

Superposition means that an entity (like an atom) is an assembly of systems which appear in observations with certain probabilities. Higman page 291. Higman We imagine an entity as a server (source) with a certain source alphabet [eigenvalues of the observer's operator] that is emitted (observed) with certain probabilities.

Non-commuting observables = different questions; questions phrased in a certain basis will tend to yield answers in that basis. (Commuting observables have the same natural basis = eigenvectors). Questions asked in a different basis (changing the subject) will yield answers in a new basis that do not relate to the previous basis (or in quantum mechanics, do, by a suitable transformation? [see Feynman on Stern Gerlach, III 5-1 sqq, Feynman])

Tuesday 10 October 2006

Computational delay. It has taken forty years to

[page 8]

formulate whatever I have formulated, but I am still in the position of a half-built arch, reliant on the 'historical' centering until I put the keystone in. For me this remains a consistent and aesthetically pleasing reformulation of general relativity and quantum mechanics in a logical rather than a continuous formalism. Perhaps I should go and do physics for a few years.

As Parmenides noticed, the essence of knowledge is invariance. Parmenides But as [Heracleitus] discovered, everything moves so invariants are hard to find. As Einstein discovered, the root invariant is the relationship of a thing to itself, geometrically, 'parallelism. This, in the form of the inner product, is common to both quantum mechanics and general relativity.

The theological invariant is that all love is based on self love. The root of self love is that normally we do whatever is necessary to survive, which means communicating with out local gods.

A fundamental perversion of this theological invariant is the notion that my self love gives me the right to expect others to die for me and ultimately to take the power to kill people who will not die for me. . . .

Such is the the power of symmetry that it will 'out' whenever conditions (ie the complexity of the available alphabet) allow.

[page 9]

Eg the history of music and the development of scales and the transposition of scales, based on the development of stable and tunable instruments and good ears.

I am coming to the end of a forty year pregnancy brought on by the Holy Ghost (as it was manifested to me complete with contradictory notions about love and death encoded within said Ghost). We are speaking here of gestation in the head rather than the belly, but the parallels are very strong. At least to those among us who have children, they are also very illuminating.

Explain the connection between energy and covariance.

The only thing we know of in the Universe that is static (that is not moving through time) is the photon. Maybe also graviton if it exists and is massless (yet coupled to mass == energy, and since it carries energy, coupled to itself). Everything moves so that ds/d tau (proper time) = 1, usually dt / d tau) = 1.

Wednesday 11 October 2006

Eigenvalues of an observable are orthogonal and therefore a basis. So an observation consists of finding one of the basis vectors of an observable (ie one of the letters of its alphabet) and the corresponding eigenvalue. This is equivalent to characterizing one letter of the alphabet of a source by its probability. An adequate ensemble of observations will enable us to compute the entropy of the source.

[page 10]

Space is a set of degrees of freedom, as are the dimensions of the Hilbert space of the observed system. In a sense each point of 4-space is itself a Hilbert space (ie we have a 'field' of Hilbert spaces mapped onto the elements of 4-space) The relationships of the observations made in different inertial frames in 4-space are described by the metric gik. Does this say anything at all about the underlying (or overlying) Hilbert spaces? Every observation may be considered to have been made in the 'rest frame' of the corresponding Hilbert space, so only the observations have to be transformed, not the Hilbert spaces, although it may be mathematically convenient to transform the Hilbert space to yield the result actually observed in a space in relative motion. [see Veltman page 20 Veltman] In fact, however, the Hilbert spaces are 'blind' to their spacetime locations, so it is more physical to deal with the actual observations, which are 'classical' as special relativity and general relativity are 'classical'.

On this picture, we have two populations of objects mapped onto the spacetime domain, Einstein's tensors and von Neumann's Hilbert spaces. The problem for physics (ie the 'quantization' of gravitation arises from this condition.

We consider spacetime from a 'god's eye view' where all space and al time are laid out before our eyes ('tota simul') so we can have different Hilbert spaces and different tensors at different points in time as well as different points in space, and the general aim of physics and all knowledge is to use knowledge of the past to derive equations of motion that

[page 11]

enable us to predict the future. An equation of motion connects observations with the same local space coordinates made a different times as we observe the trajectory of a ball with the intention of hitting it into a different (hopefully winning) trajectory.

Operators commute if they have the same basis, ie the same eigenvectors.

The stern-Gerlach measurements show that different spatial directions do not commute, different spatial directions are different observables.

A multi choice question must be a complete basis set.

So the four spatial dimensions are observables, but this does not imply extension insofar as they are identical local observables at every spacetime point which transform in the local tangent space observing (following, driven by, caused by, Lorentz invariance. The connection between these local observations (Stern-Gerlach and spectroscopic etc) in local spaces are described by the metric gik which is sensitive to how much is going on in the neighbourhood. Regions of high traffic are attractive. A traffic analysis of something like Google might fit the Einstein equation.

. . .

So we say space is 'caused by' the pairwise commutation of (x, y, z) and t, but the pairwise non-commutation

[page 12]

of x, y and z. By commutation we mean that we can observe one without changing the other, in other words they are interchangeable (symmetric) observables, giving the same result. Non-commutativity) which is measured in units of action) implies TRANSFORMATION = COMPUTATION

Thursday 12 October 2006

The only photon that I see is one that annihilates itself in my eye.

Car engines go because they are a cycle. Each engine is born (made by its technological matrix) and does (is, hopefully, recycled into new necessary products).

The first thing we do to lighten our struggling ship is to throw all unnecessary products overboard. Then we must start to build a system that takes a load off the plane. Ideally it will be a generalized Carnot machine (reversible?) working between the heat of the sun and the cold of space.

Localized thermodynamic disequilibrium is a sign of life. (Lovelock, Lovelock)

Lonergan (and many others) have failed to appreciate the dynamic range of the time constants of the Universe. We learn about past and future by things which are (relatively) invariant in time.

Basic invariants are

[page 13]

first order and second order calculi, known colloquially as propositional and predicate calculus. Mendelson

The children create endless games.

One [first order calculus] is complete and deterministic. The other in incomplete and uncertain. The latter grows from the former by the Cantor mechanism. This, in a nutshell, seems to be the kernel of a new theology, that is a new theory of everything.

Friday 13 October 2006

Quantum mechanics introduced a huge new degree of complexity relative to classical mechanics. In the extreme it replaced real numbers ('the natural line') with an infinite (or transfinite) dimensional Hilbert space, just as Cantor's investigations revealed the transfinite cardinal and ordinal numbers 'hidden' in the line.

Given the logical model of the world we may take relativity and quantum mechanics as read, since all the mathematics can be derived from (say) logic and set theory. This is rather abstract however, and has been with me (at least implicitly) since the sixties (How universal?). But from a practical point of view we want more than this and would like to show how quantum mechanics and general relativity follow from set theory and the logical calculi. This is what seems to be holding me up, but the network paradigm seems to give a good picture of what is going on, and must provide a foundation for

[page 14]

publication one day.

Much revolves around the difficult relationship between the Hilbert spaces and quantum mechanics and the 4-pace of relativity. Our best solution seems to be the graphic user interface - underlying computation model. User operating in 4-space (clicking on a 'screen') acts as an 'oracle' for the underlying computation which polls the screen for input whenever it has come to a decision point.

I have been in a subtle panic all my life about not being able to please god and being rejected, becoming an outcast doomed to die spiritually. Now I begin to feel that I know god well enough through science and imagination to be at hoe in the world, having nevertheless to deal with home economics, strategy, tactics and actions designed to feed me and my line.

Each observation is the execution of a Turing machine. A quantum system is to the frequent observer a superposition of observations with certain spectrum of probabilities. Since the quantum system is countable it does not have sufficient variety of itself to decode exactly which machine will be executed, hence the randomness of observations. On the other hand, once a machine has been selected by an observation (so that both the observer's matrix and he system matrix have the same basis) a repeat observation (if it is done soon enough to avoid the onset of decoherence) will yield the same eigenvalue of the same eigenvector.

[page 15]

Prove: there exists a set of symmetric (reversible) computable functions which has the same cardinality as the observable system and is identical to the set of functions computable by a universal Turing machine. This is equivalent to the universality theorem in quantum information theory.

What does god expect? Stay alive, grow, reproduce. What does society expect? Ditto, but one has to fit in by doing a certain minimum of approved work, ie carrying one's economic weight. I have not done this but have been working nevertheless. Once my work is 'approved' my account should go into credit.

I understand the whole by extrapolating from local conditions, and local conditions by seeking consistency in the whole, general relativity eg.

Shannon 1949 = Chaitin 1973: white noise = random looking sequence, ie densest coding. Shannon, Chaitin

Saturday 14 October 2006

Somme Mud: On certain hypotheses . . . war is necessary in the sense that (given the local constraints) this place is not big enough for both of us. Ideally, peace is not made by the death of any of us (although this will ultimately happen) but by doubling the effective size of the space. The more I can rely on ecosystem services, the bigger my space (parasitism). Or I can create my own ecosystem (cycling between sun and space).

[page 16]

Ultimately reality has more variety than any fairy tale because fairy tales are part of reality.

Terminus a quo 'the Christian hypothesis whose most succinct expressions are in the symbols and creeds developed n the early days, like the fish and the Apostles' Creed. ad quem 'the scientific hypothesis'.

This article is a condensed log of a lifetime journey from Christian hypothesis to scientific hypothesis.

 

We measure locality by bandwidth n the convention that our rate of communication with ourselves (our 'squared length') is one an all other communications die off with distance. This is a definition of distance, different measures of distance at different peer levels.

I'm with Feynman, redundant knowledge brings security.

Quantum mechanics: distance = overlap integral squared
General relativity distance = spacetime interval ds squared.

E P F Lynch Somme Mud Lynch

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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!)

Feynman, Richard P, and Robert B Leighton, Matthew Sands, The Feynman Lectures on Physics (volume 3) : Quantum Mechanics, Addison Wesley 1970 Foreword: 'This set of lectures tries to elucidate from the beginning those features of quantum mechanics which are the most basic and the most general. ... In each instance the ideas are introduced together with a detailed discussion of some specific examples - to try to make the physical ideas as real as possible.' Matthew Sands 
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Lynch, E P F, Somme Mud, the War Experiences of an Australian Infantryman in France 1916-1919, Random House 2008 Amazon customer review: 'The pure horror of trench warfare is brought home strongly in this book. Private Lynch was eighteen years old when he volunteered for the Australian Imperial Forces, for overseas service. A typical Aussie, Lynch was sent to England and then across the channel to France. Messines, Passchendaele, Villiers Bretonneux and Hamel were the stuff of legends, and Lynch survived the machine guns, snipers, the cold, the mud, the artillery and gas attacks. Private Lynch (1898-1980) wrote in the first person, using the alter ego- Nulla. While this style of writing removes the text from the dry history genre, the minutiae were reconstituted when Lynch wrote the notes that formed the book sometime later. The main characters in the book have never been identified. . . . Dr Neill MacNeill 
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Mendelson, Elliott, Introduction to Mathematical Logic, van Nostrand 1987 Preface: '... a compact introduction to some of the principal topics of mathematical logic. . . . In the belief that beginners should be exposed to the most natural and easiest proofs, free swinging set-theoretical methods have been used."  
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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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Papers
Chaitin, Gregory J, "Randomness and Mathematical Proof", Scientific American, 232, 5, May 1975, page 47-52. 'Although randomness can be precisely defined and can even be measured, a given number cannot be proved random. This enigma establishes a limit in what is possible in mathematics'. back
Lovelock, James, "Gaia as seen through the atmosphere", Atmospheric Environment, 6, , 1972, page 579-580. 'The purpose of this letter is to suggest that life at an early stage of its evolution acquired the capacity to control the global environment to suit its needs, and that this capacity has persisted and is still in use. In this view the sum total of species is more than just a Catalogue, "The Biosphere", and like other associations in biology is an entity with properties greater than the simple sum of its parts. Such a large creature, even if only hypothetical, with the powerful capacity to homeostat the planetary environment needs a name: I am indebted to Mr William Golding for suggesting the use of the Greek personification of mother Earth, "Gaia".' . 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
Wilczek, Frank, "QCD Made Simple", Physics Today, 53, 8, August 2000, page 22-28. Quantum chromodynamics is conceptually simple. Its realisation in nature, however, is usually very complex. But not always.. back
Links
John Burnet John Burnet's Early Greek Philosophy: chapter IV, Parmenides of Elea: 85: The Poem back

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