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.
[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
