Notes
[Sunday 4 January 2009 - Saturday 10 January 2009]
[Notebook: DB 65 Symmetric U]
[page 41]
Sunday 4 January 2009
The layered network model maps neatly onto Dawkins' Climbing Mount Improbable. Dawkins
Feynman + Hibbs page viii: 'In particular, [the path integral method] provides an expression of quantum electrodynamic laws in a form that relativistic invariance is obvious.' Feynman & Hibbs, Path integral formulation - Wikipedia
A path is an algorithm, a road map, how to get there. The path is the map 1?1 scale. On an unmarked map with but two towns, one may imagine an infinity of routes between the towns. In reality, there will be a mapped road, representing a real road which is by far the easiest way to get from one town to another as a result of the capital invested in building the road.
F&H page 2: '[Quantum mechanics] asserts that there are experiments for which the exact outcome is fundamentally unpredictable and that in these cases one has to be satisfied with computing probabilities of various outcomes. But far more fundamental was the discovery that in nature the laws of combining probabilities were not those of the classical probability theory of Laplace. Classical probability - Wikipedia
'The concept of probability is not altered in quantum mechanics. . . . What
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is changed, and changed radically, is the method of calculating probabilities.'
This is the physicists' eye view. What is its root in reality?
The computation network paradigm envisages the periodic repetition of discrete events like the evaluation of a logical function. It is thus potentially consistent with both the wave (periodic) and particle (discrete) features of quantum mechanics.
The 40 year interval between the birth of the communication paradigm (How Universal?) and its definitive publication (hopefully this year, before I 'retire') is a coding delay, the time I have required to absorb and structure enough information to express myself clearly in the languages of physics and theology. How Universal is the Universe?
The basic change from classical Laplacian probability is the vector addition of amplitudes.
Feynman & Hibbs pare 5: 'III There are complex numbers phi1 and phi2 such that
P = |phi | 2
phi = phi1 + phi2 '
Ie the core of quantum mechanics is expressed in the arithmetic of complex numbers.
A complex number z = x + iy has two degrees of freedom represented by x and y. The fundamental constraint on x and y is implied by the classical constraints on p and the relationship P = |phi | 2 known as normalization. Normalisable wave function - Wikipedia
[page 44]
Feynman & Hibbs page 6. None of the spacetime explanations of electron trajectories through two slits make any sense, only the logical explanation via complex arithmetic.
page 9: Uncertainty principle: 'Any determination of the alternative taken by a process capable of following more than one alternative destroys the interference between the alternatives'.
Heisenberg: 'One cannot simultaneously measure conjugate quantities like position and momentum with unlimited precision. Instead the most precise possible measurement if represented by the relationship delta p.delta q ≈ h bar. The Universe is pixellated on a scale of h bar, which we take to represent the execution of a minimal logical operation.
page 11: We find electrons at spacetime points of 'constructive' interference which occur when | phi1 + phi2 | 2 = 1, ie the vectors representing phi1 and phi2 are pointing in the same direction. In a real time computing environment, we may see this as two Turing machines halting with the same rather than opposite values. We may see the processing in a Turing machine as a rotating vector [clock] in a space with as many dimensions are there are different frequencies of different operations in the machine. Nested loops have a hierarchy of frequencies, the highest being the innermost loop, the lowest the outermost. An observation (ie keystroke) is equivalent to an interruption of this loop structure, perhaps sending it off in a different direction.
page 13: Interfering alternatives
footsoldiersrevolt.net
Footsoldier: someone required to do unpleasant things for the
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profit of another.
We understand computation to some extent by an understanding of timing. Ultimately all data is serial, a sequence of events and non-events whose significance can be modelled by points on a natural line, a timeline. Our hypothesis is that the local timeline is our fundamental tool for communication with the rest of the Universe, including all of you.
Apply quantum mechanics to human relationships via the transfinite network.
I once sold my soul to the God of death and lived to regret it. But this passage through hell has inspired a serious attempt at the mathematical expression of the alternative.
The Christian idea of pain and suffering is exactly wrong, as we can see by considering the cybernetic function of pain and suffering. Christians say we should seek pain because it is a source of grace. So the pain of Jesus is said to have redeemed us from the sin of the first humans, who sought a knowledge of good and evil.
A loverly myth, but with deep political content which we must appreciate if ever we are to be free. Cybernetics says pain is negative feedback, a signal that tells us we are moving into dangerous territory, a place for a survival oriented organism to avoid.
Eureka Street: On pain and suffering.
[page 45]
The fundamental desire of a troubleshooter is to solve all problems quickly and painlessly for a reasonable fee.
Dirac's transformation theory effectively operates at constant entropy, ie it is reversible, the total number of states remains constant and all that changes is the form of the states, represented by a vector in Hilbert space.
FORM (PLATO, ARISTOTLE) = VECTOR (ray) in HILBERT SPACE
When we look closely at a computer we see nothing but a set of states continually transforming into new sets of states at constant entropy, ie constant card(states).
Quantum field theory allows us to deal with the fact that particle are created and annihilated while action, energy and momentum remain constant forever.
The spacetime in which we move is the simplest complete representation of quantum field theory in which all events occur.
The two Nobel prize theory physics & peace (= theology). This has been my (secret) ambition since monastical days and unless I am completely deluded (Oh happy thought) I am on the right track. . . .
We will not fight unless you can prove that there is no alternative.
There is real pain and there is fake pain caused by regulations inconsistent with reality.
[page 46]
Monday 5 January 2009
Electron self interaction: when we are speaking and listening, we must cancel out our own sound to hear the other.
We might call the pixels of the Universe fourxels, since they are units of 4D space-time or momentum-energy or action-number.
Interfering alternatives Feynman Path page 13
Exclusive alternatives vs interfering alternatives
Real numbers vs complex numbers (vectors)
[past vs future]
page 14: 'When alternatives cannot possibly be resolved by any experiment they interfere (ie vector add).
Here we are talking about complete and incomplete processes. When alternatives are indistinguishable [ie no decision has been made] processes are incomplete and there is interference. When processes are complete there is no interference in the quantum mechanical sense and classical statistics applies.
Tuesday 6 October 2009
Feynman, Path page 19 '. . . there is a quantity called a probability amplitude associated with every method whereby an event in nature can take place.'
Ie a probability amplitude is associated with every computation or transformation in nature.
[page 47]
'Furthermore, we can associate an amplitude with the overall event by adding together the amplitudes for each alternative method.'
Since the amplitude is a complex (vector) quantity, some elements of this sum increase its value, some decrease it.
'Next we interpret the absolute square of the overall amplitude as the probability that the event will happen.'
So some elements of the sum of amplitudes increase the probability of an event, some decrease the probability.
'If we interrupt the course of an event before its conclusions with an observation on the state of the particles involved in the event, we disturb the construction of the overall amplitude.'
That is, by communicating with an event we [become aware of it,] complete it and prevent any further evolution. This is the 'collapse of the wavefunction' so called.
'The operation of the measuring equipment is sufficient to disturb the system and its probability amplitude. This latter fact is the basis of the Heisenberg uncertainty principle which states that there is a natural limit to the subtlety of any experiment or the refinement of any measurement.'
This is not so much a disturbance of the system as the completion of a communication involving both the measurer and the measuree, which is quantized in order to avoid error by moving from analogue to digital process. We see the same thing happening in an ordinary computer where the digital switching of the clock masks the
[page 48]
analogue processes of the logic circuitry and reveals it to the beginning of the next computational step when all the physical symbols have settled to their logical values.
Feynman & Hibbs page 21: Infinite screens with infinite holes define the path of a particle with infinite detail and we eventually arrive at the path integral formulation on the assumption that the spacetime background upon which we paint this picture is continuous.
page 22 'We shall find that in quantum mechanics, the amplitudes phi are solutions of a completely deterministic equation (the Schrödinger equation). Knowledge of phi at t = 0 implies its knowledge at all subsequent times.'
[this idea is based on the belief that continuous mathematics can be made deterministic]
Ie the Schrödinger equation is computable, but observations are not computable.
'One [problem] is to show that the probability interpretation of phi is the only consistent interpretation of this quantity. We and our measuring instruments are part of nature and so are, in principle, described by an amplitude function satisfying a deterministic equation. Why can we only predict the probability that given experiment will lead to a definite result?
Feynman says 'Almost without doubt the [uncertainty] arises from the need to amplify the effects of the single atomic events to such a level that they may be readily observed by large systems.'
Is this so? It seems more likely that the uncertainty arises because simple systems lack the variety to fully constrain future events, a consequence of Gödel's incompleteness results. Completeness is only achieved (as is
[page 49]
certainty in human communication) when there is a response from the receiver assuring the sender that the message has been received [intact]. It seems that this is encapsulated in the provision P = |phi | 2 = phi.phi* where phi = <a |b >, phi* = <b |a >, ie a two way message is required to achieve certainty and the boundary between uncertainty and certainty is the boundary between future and past.
Feynman & Hibbs: page 22: 'Other problems which may be analyzed are those dealing with the theory of knowledge. For example there seems to be a lack of symmetry in time in out knowledge. Our knowledge of the past it qualitatively different from our knowledge of the future. In what was is only the probability of a future event accessible to us, whereas certainty of a past event can often apparently be asserted?' See previous paragraph.
Wednesday 7 January 2009
Why is the energy of each degree of freedom kT in classical mechanics [or, since k and T have an arbitrary but fixed relationship to reality, why has each degree of freedom the same energy?]
Feynman & Hibbs Classical trajectory vs quantum trajectories.
Our hypothesis is that quantum trajectories are not per se paths that can be mapped onto spacetime, but rather logical processes onto which spacetime may be mapped.
page 26: Classical action is the time integral of the lagrangian, the difference between kinetic and potential energy [mass is potential energy] Kinetic energy is always positive, and so to minimize L should be kept to a minimum, ie move as slowly as possible, while keeping your potential energy high, ie stay as high as you can.
[page 50]
From these we drive the classical Lagrangian equation of motion.
. . .
Lagrangian mechanics - Wikipedia
Feynman & Hibbs page 28: '[All paths] contribute equal amounts to the total amplitude, but contribute at different phases. The phase of the contribution from a given path is the action S for that path in units of the quantum of action h bar.'
Mathematically, this approach assumes the existence of continuous phase and consequently fractional values of h bar, but this is unrealistic -- all actions must be integral multiples of the quantum of action, ie S = n . h bar, n integral, so that 2 pi phase, that is one complete cycle [corresponds to a quantum of action].
It is not hard to visualize Feynman's path integral picture of Dirac's canonical theory as a network. Each 'hole' in Feynman's screens is a node in a network. We make the passage to the limit for analytic tractability, but there is nothing observed that compels us to go to this limit. We can imagine the world as a network whose digital unit is Planck's constant.
The key to sanity and security is friendship, a trusted zone of communication.
The fundamental fact about the internet as in all other systems, is the interplay between good and evil.
[page 51]
For the purposes of this site, the Roman Catholic Church is the embodiment of evil, the structure we use as a paradigm of how not to do things, which serves as the pathological motivation for the search for a healthy way of doing things. In other words the Roman Catholic Church is the embodiment of all the processes which if eliminated would leave us with peace on earth.
The greatest evil, form the software engineering point of view, us the suppression of variety and bandwidth. From a genetic point of view, the Church represents a population of 1, the allegedly infallible Pope. His word is law from which there is no appeal. Such an insult to the polymorphism of the real world can only be sustained by violence to the human person, violence which is encapsulated in the vows of poverty, chastity and obedience and in the disenfranchisement of women.
Thursday 8 January 2009
'The roots of power . . . .' West, Cassidy page 131. West
The root of corruption is bad law. Meaning? Hypocrisy, legal alcohol, illegal dope,
West page 148: 'Every time we pass a new piece of legislation we create a new class of criminals. . . . My father made a fortune out of legislation that closed the pubs at six. He did a back-door trade with anyone who had a thirst after that time. Prostitution? The same thing. Drugs? We've criminalized the addiction and let the traders have a field day.
[page 52]
Friday 9 January 2009
Lagrange: Lagrangian has dimensions of energy, so the rate of change of energy with position on the classical path equals the time rate of change of the rate of change of energy with velocity.
Quantum mechanically energy = frequency = rate of change of phase = rate of computation [clock frequency]
Feynman & Hibbs The quantum mechanical amplitude page 28
'The probability P (b, a ) to go from xa at time ta to the point xb at time tb is the absolute square P (b, a ) = |K (b, a)| 2 of an amplitude K (b, a) to go from a to b. The amplitude is the sum of contributions phi [x (t )] from each path
The contribution of a path as a phase proportional to the action S :
page 31 Sum over paths
'The number of paths is a high order of infinity, and it is not evident what measure is to be given to the space of paths.'
[page 53]
Feynman & Hibbs page 29: The classical limit: The idea is that because h bar is, so small amplitudes are essentially random and so cancel out. The observed macroscopic reality is to be found where the microscopic realities 'cooperate'.
www.change.gov. Office of the President-Elect
Gell-Mann: 'If it is not prohibited it is compulsory'. Nature 457: 1 January 09 Blair Blair
NASA Fermi Gamma-ray Space Telescope Fermi Gamma-ray Space Telescope - Wikipedia
Aristotle thought the heavens were divine and the Earth mundane Galileo et al established that the heavens too, are mundane. Now I want to reverse the lot and establish that all is divine.
Gingerich Nature 457:28 'It is worth a great prize to understand our place in the intricate web of time and space.' And probably a lot more to realize that our spirit goes far beyond time and space. Gingerich
Gingerich: God's Universe Gingerich
Ohtsuki et al: Indirect reciprocity provides only a narrow margin of efficiency for costly punishment. Nature 457:79 ! January 2009. Ohtsuki
'Direct reciprocity is like an an economy based on the exchange of goods, whereas indirect reciprocity resembles the invention of money. The money that feeds the engines of indirect reciprocity is reputation.'
The classical limit works in the path integral approach
[page 54]
because as h bar gets smaller at constant energy, the rate of change of phases off the classical path becomes so rapid that all off-classical paths cancel.
Walking in steep country I try to minimize my lagrangian by moving slowly and staying high.
The size of the uncertainty is measured to come extent by the size of the action. When there is no control, as in the world of fundamental physics, both action and uncertainty are measured by h bar. In larger actions, measured by n h bar there may be enough control to reduce the uncertainty to n/m h bar, where m <= n, m =- n only when we have physically perfect control.
Control requires computation with memory. At the quantum level, the Universe has no memory, and so all things are possible that do not involve contradiction. So, given a set of symmetrical states, ie states with no distinguishing features, the law of indifference holds and all are effectively equiprobable. [this indifference is the foundation of creation]
Saturday 10 January 2009
Paul Collier The Bottom Billion Collier
McGeough Sydney Morning Herald 3-4 January 2009 page 21. Paul McGeough
Tom Segev 'flawed assumption which has accompanied the Zionist movement since its inception" -- a belief that military strikes against the Palestinians will "teach them
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a lesson."' [from McGeough]
'The thousands of Palestinian rockets . . . have killed 19 Israelis in eight years. In the same period 3000 Palestinians have died under Israelis fire in Gaza.'
Feynman & Hibbs" Developing the concepts with special examples.
page 42: The lagrangian of a free particle is L = 1/2 mv2 (kinetic energy).
page 44: The amplitude to go from the origin 0 (in 2d space - time) to the point x, t is
From this we can conclude that if the time is fixed, the amplitude varies in space with wavelength lambda = h i / i p, since to get to greater distances x, the particle must move faster, so having more momentum and a shorter wavelength.
Feynman & Hibbs speak throughout of 'the amplitude for a particle to do a' that is the process that leads from 0 to a.
F&H page 45. In a quasi classical situation 'the change in phase per unit displacement of the endpoint is
but partial dSclassical / partial dxendpoint is the classical momentum of the particle when it arrives at the point xendpoint . This is p = h bar.k, the phase change per unit distance of a wave is called the wave number and
[page 56]
is very convenient to use.'
From the computational / network point of view we might think of the momentum as the processing rate that defines space rather tan vice versa.A similar analysis (pages 45 - 47) defines time through energy.
F&H page 47 ' In this way the concepts of momentum and energy are extended to quantum mechanics by the following rules:
1. If the amplitude varies as exp (ikx ), we say the particle has momentum h bar.k.
2. If the amplitude has a define frequency varying in time as exp (-i omega t ) we say the energy is h bar.omega.
page 47: Diffraction through a slit
A fundamental flaw in the path integral method (akin to allowing continuous values for phase and action) is to assume that the particle does have a definite position in space and time and then to integrate over these positions to obtain final definite amplitudes. Uncertainty is only then introduced by interpreting these amplitudes as probabilities. In nature (we speculate) the whole process is digitized and pixellated, and the application of continuous mathematical analysis is only justified by the fact that it yields reasonable results in many circumstances, while leading to disaster in others.
The computational approach introduces the natural UV and IR cutoff equivalent to action h bar, and we cannot resolve our computations at any stage beyond this limit.
Although particles (ie electrons) may be said to be identical, the no-cloning theorem says that no two electrons in the Universe can be in exactly the same state. No cloning theorem - Wikipedia We must therefore accept that despite their 'local' identity, the energy-momentum 'address' of an electron is also part of the specification of its state [ie its relationship to the rest of the world]
F&H page 57: 'If we forget everything we know bout a particle except its wave function a a particular time, then we can calculate everything that can happen to that particle after that time. All of history's effect upon the future of the Universe could be obtained from a single gigantic wave function.' Up to a probability. An alphabet cannot determine a sentence. Chaitin, Gödel, Ashby. Gregory Chaitin - Wikipedia, Kurt Gödel - Wikipedia, William Ross Ashby - Wikipedia
DYNAMIC = CONCRETE = Connected to the roots of the Universe
KINEMATIC = ABSTRACT = modelling some aspect of universal behaviour
Representations: we suppose that the fundamental representation of the world is the computation (action) representation which may be transformed into spin, position, time, energy, momentum representation at will as seen by more complex layers of the system.
F&H page 96: 'So far we have used the concept of probability in terms of the position of a particle, but suppose we wish to measure the momentum. Is there an amplitude phi (p ) whose absolute square will give is the probability P (p ) that a measurement of momentum will show that the particle has momentum p ? There is in fact such an amplitude, and we can easily find it.'
delta p.delta x in effect implies the time of flight approach to measuring momentum, since the longer the interval over which we measure the flight
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the more accurately we can constrain velocity and momentum [and the less accurately we can constrain position]. In other words, two position measurements are required to establish momentum. Momentum may remain constant while position is changing and (in a collision) position may remain constant while momentum is changing, so in a sense two momentum (wavelength) measurements are required to establish a position.
Classical momentum is a function of velocity.
Quantum momentum is a function of wavelength, ie 'phase velocity'.
Measurement of velocity depends upon both time resolution (clocking accuracy) and space resolution (distance measurement accuracy). Clocking accuracy depends upon energy and counting (which may be considered perfect). Distance accuracy depends on momentum (fineness of graduation) and counting. In both cases we are counting the number of times a phase passes (say) zero, zero corresponding to a mark on the tape or a tick of the clock.
E = hf, ie h = E/f, frequency being the constant of proportionality between action and energy. So if the halting of a Turing machine is equivalent to a quantum of action, the amount of processing done by the machine is measured by its energy or frequency. Thus the machines that communicate between the energy levels of an atom do different amounts of work depending on the levels transformed, but the action difference is proportional to the difference between the principal quantum numbers of the levels involved.
Schiff page 6: 'Thus classical causality, which requires that the motion of a particle at any time be uniquely determined from its motion at an earlier time, must be abandoned. The new theory that is forced upon us in this way is so successful
[page 59]
in other respects that, at the present state of knowledge, we must regard such classically incomplete descriptions as a fundamental property of nature. Schiff
The two slits may be seen as independent sources which, by interrupting one another, break the determinism of the individual Turing machines to give an indeterminate result. Nature is a network of o-machines, each communicating with its oracular environment to arrive at its ultimate conclusion.
OBSERVATION = HALTING
Schiff: Uncertainty and Complementarity page 6:
Uncertainty arises from the digital pixellation associated with communication. Complementarity arises from the mapping of the pixellation onto space/momentum. energy/time and angular momentum / phase.
Schiff page 13: Particles interfere with themselves.
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