Notes
[Sunday 1 November 2009 - Saturday 7 November 2009]
[Notebook: DB 68: Salalah
[page 33]
Sunday 1 November 2009
Khinchin: Mathematical foundations of quantum statistics Khinchin
[page 34]
Continuous mathematics works on the nineteenth century fiction that the points in a set are discrete and addressable and that between any two points one can always find another point, assumptions which seem in the real atomic world to be contradictory.
The search always is for orthogonality, a set of clear distinct steps from beginning to end. In quantum mechanics the vectors in Hilbert space are not an ordered set, although when we talk about a basis we index the orthogonal basis vectors by (say) i.
We cannot have time ordering without space.
When we observe a quantum system we can build up a picture of a source, knowing the structure of our observable, which is a basis of the same cardinality as the system we are observing. We can learn the statistical structure of our source, but learn nothing about the sequence of letters emitted when the source says something meaningful. In quantum mechanics, we might say, the source emits one letter sentences. In quantum computation, we begin to become interested in sequences of operations (observations) on the system of interest, ie sentenced with more than one letter. Our aim is to control the evolution of the quantum system.
Grgin. Thank you for your essay. I am in close agreement with you on the first part. I find the second part a bit obscure but I have sent away for your 2007 book. However, I prefer a different tack,
[page 35]
which I can express succinctly in terms of your methodology.
Veltman: A Hilbert space is a sequence which is represented by a differential equation governing the ordered evolution of the sequence, dogs chasing cats etc. Veltman
In quantum mechanics, probability is measured by harmony. So the probability of a tuned circuit receiving a photon is high at resonance, low to negligible elsewhere, and the sharpness of the tuning depends on the Q of the system. Quantum systems, being frictionless and reversible, have infinite Q. Q factor - Wikipedia
You will hear my words or not hear them depending upon the resonance of my text and your minds.
Quantum mechanics has no memory, but we can make memory (or at least order) by introducing spacetime and see a process moving through spacetime instead of simply through time, or more properly on the time axis, with no particular direction. Space introduces a direction to time.
The birth of quantum mechanics approximately coincided with an explosion in the development of discrete mathematics (logic) beginning with Cantor and developing through Gödel, Turing and Shannon. Meanwhile physicists, faced with a totally discrete world, continue in the hopeless task of describing the discrete world with classically continuous and differentiable functions. Differentiation is equivalent to discerning the logical operation occurring at a point in a process, and integration the understanding of a sequence of logical operations. The writing of each letter in this text is in this context a logical operation.
Khinchin page 1: discrete = denumerable.
The concrete foundation of information theory is the same [as a quantum source]: a source that emits a statistically normalized sequence of orthogonal 'letters' from an 'alphabet' that we represent in quantum mechanics by a basis for a Hilbert space of suitable dimension.
The discrete numerical basis of the world is spin, whose changes are always one unit (measured with the dimensions of Planck's constant) and which can have any integral or half-integral value. Spins do not add but they follow one another.
Khinchin page 1: 'In classical mechanics, the state of a system uniqueluy determines the values of all the physical quantities associated with it . . . In quantum mechanics, the state of a system defines the physical parameteres only as random variables, ie it determines the laws of distribution obeyed by the physical quantities, and not their values.'
The discrete numerical basis of the world is spin, whose changes are always one unit (measured with the dimensions of Planck';s constant) and which can have any integral or half-integral value. Spins do not add but they follow one another [? - what is the spin of n photons?]
Khinchin page 1: 'In classical mechanics, the state of a system uniquely determines the values of all the physical quantities associated with it. . . . In quantum mechanics, the state of a system defines the physical quantities only as random variables, ie it determines the laws of distribution obeyed by the physical quantities, not their values.
ie quantum mechanics goes as far as one can go in an abstract description of a concrete system which is always in one state or another, but which no abstract system can predict, only the concrete system itself knows that. One could not know it without being it, while knowing abstractly allows tone partial knowledge of the other system through partial modification of one's own system,: 'acquiring the knowledge'.
[page 37]
Spin is king, bifurcating into Bose-Einsten (integral) and Fermi (half integral).
. . .
Grgin page 6: '. . . we must first search for the "principle of quantum mechanics", which implies the axioms of the latter.
We cannot observe a continuum, since there is nothing to see, no marks. It is not surprising them that every physical observation without exception is a quantum, a discrete, marked, event.
Dirac's delta describes resonance of infinite Q. Dirac delta function - Wikipedia
Feynman's Thesis: Path Integral integrates an ordered set of (degenerate) points) Feynman
Feynman page xiii (Preface, L M Brown quoting F): 'an atom alone in empty space would, in fact, not radiate . . . and all the apparent quantum properties of light and the existence of photons may be nothing more than the result of matter interacting with matter directly, and according to quantum mechanical laws.'page xv: 'From the overall space-time point of view of the least action principle the field disappears as nothing but the bookkeeping variables insisted on by the Hamiltonian method.'
PATH = COMPUTATION (dynamic), STRING (static).
[page 38]
The principle of least action easily generalizes to the computational view of nature if we see each logical operation as representing one quantum of action and the process of evolution by natural selection selecting those computation processes for which the action is a minimum.
Dirac (1931) 'remarks that the Lagrangian, a function of coordinates and velocities, is more fundamental because the action defined by it is a relativistic invariant and also because it admits a principle of least action. Feynman page xvii Dirac
Our world is a discrete (= denumerable) communication network which we model as a set of memories inhabited by a set of computers able to change the states of the memories according to some algorithm. What we want to do here is to create the minimal cellular automaton that imitates quantum mechanics encoded as the phasing of independent processes. [path integral method phase = action, 'closed' phase (2 pi = completed computation]
Synchronous communication does not require memory (= buffering), asynchronous does, since messages must be stored and forwarded.
Once we have the denumerable network described we apply Cantor's theorem to it to get a hierarchy of non-denumerable networks and we apply the Löwenheim–Skolem theorem to the non-denumerable network to distill out a working denumerable model of our non-denumerable environment. Löwenheim–Skolem_theorem - Wikipedia
ACTION = DEFINITION (moving from infinite to finite, unbounded to bounded, continuous to discrete).
[page 39]
The fundamental reductionist fact is that we can (?) build a Turing machine out of NAND gates. Sheffer stroke - Wikipedia
Is it true that the physics community is s hidebound as the religious community? as any community? How do you measure hideboundness: by the lag in coming to terms with reality, a lag in perception.
Quantum mechanics is analogous to a discrete, memory free (instantaneous) communication network that conserves probability, ie normalization. Let us say that observable process in the Universe is normalized to ℵ0.
Feynman page 2: '. . . we will concern ourselves in this thesis with the theory of action at a distance worked out in 1941 by J A Wheeler and the author.
'The new viewpoint pictures electrodynamic interaction a direct interaction at a distance distance between particles. The field then becomes a mathematical construction to aid in the solution of problems involving these interactions.'
page 4: 'In empty space an atom emits light and there is no potential to perturb the systems and so force it to make the transition. The explanation of modern quantum electrodynamics is that the atom is perturbed by the zero point fluctuations of the quantized radiation fields.
'It is here that the theory of action at distance gives a different viewpoint. it says that an atom alone in space would, in fact, not radiate. Radiation is a consequence of interaction with other atoms (namely those in the matter which absorbs the radiation).'
What we (me and my book) have called full duplex communication.
[page 40]
Feynman page 6: 'One should therefore take the viewpoint that the present paper as concerned with the problem of finding a quantum mechanical description applicable to systems which in their classical analogue are expressible by a principle of least action, and not necessarily by Hamiltonian equations of motion.'
The quantum mechanical method of counting probabilities is the first layer of meaning in the Universe.
Feynman page 9: 'For most mechanical systems it is possible to find a functional, A, called the action, which assigns a number to each possible mechanical path q1(sigma), q2(sigma) . . . qN(sigma) (we suppose N degrees of freedom [memory locations], each with a coordinate qi(sigma) [stored value], a function of a parameter (time) sigma) in such a manner that this number is an extremum for an actual path q bar(sigma) which could arise in accordance with the laws of motion.
Logical laws of motion = propositional calculus.
Quantum mechanic describes a network . . .
My continued life proves I have done what was necessary to survive in the environment in which I have lived.
Half advanced and half retarded = roundtrip time for a message.
Monday 2 November 2009
[page 41]
Feynman RMP 20 367-387 (1948) in Brown (ed) Feynman
'This paper will describe what is essentially a third formulation of non-relativistic quantum theory. . . . A probability amplitude is associated with an entire motion of a particle as a function of time rather than simply with a position of a particle at a particular time.'
Feynman (Brown) page 73: 'The foundation to be presented contains as its essential idea the concept of a probability amplitude associated with a completely specified motion as a function of time.
Continuous physics is a degenerate version of discrete physics, and so we begin always with discrete formulations that are then degenerated by 'passing to the limit'.
Building walls in the woodheap: transforming a random heap of firewood into an organized wall by digging down to a level foundation - the story of my life!
Quantum mechanics is a discrete-continuous hybrid represented by a function space in which a point is defined by a set of discrete basis vectors parametrized by continuous complex amplitudes. Each dimension corresponds to a point in the domain of a function. These dimensions can be indexed by a discrete natural number n, where we are happy to extrapolate n into the transfinite domain. The difference, in the transfinite domain, is that the n's are constructed (complex) numbers each comprising an ordered set of finite natural numbers [like a decimal number].
Cantor introduced an entirely different way of representing discrete numbers of cardinal greater than ℵ0 by considering ordered sets of [finite] natural numbers. [The transfinite numbers are also discrete, not continuous]
[page 41]
Physics is doomed if it does not recognise the new power thus conferred on discrete numbers.
All the we require of a number is that it be discrete and hae an order among its fellow numbers (peers)
Like the transfinite numbers Gödel number grow exponentially with the length of the string they represent, and we find that Hilbert spaces do the same via the tensor product. Gödel numbering is a sort of one dimensional tensor product. Gödel number - Wikipedia
A transfinite number is and represents a complex point.
Cantor's dream failed (shown by Cohen), the reason being that a continuum has no entropy (it is effectively one thing) whereas a set of transfinite numbers has a very large entropy, so there can be at most a degenerate mapping between them.
The statistical descriptions of the outputs of a quantum source and a communication theoretical source are identical: an alphabet of n letters (eigenfunctions) each with a probability pi of emission, normalized by SUMi pi = 1.
Feynmann page 74: '. . . the essential difference between classical and quantum physics lies in equation (2) [Pac = SUMbPabc]. In classical mechanics it is always true. In quantum mechanics it is quite often false.'
[page 43]
Ie there is something going on between a and c that classical mechanics cannot see.
Ant vs grasshopper
Without loss of anything relevant, and a considerable improvement in the interface with quantum mechanics, we may consider the transfinite number aleph(n) to be the cardinal of the set of permutations of the set of transfinite number whose cardinal is aleph(n-1).
Perhaps the greatest discovery in the history of mathematics is position significant notation. Positional notation - Wikipedia Instead of representing 10 as 1111111111, we write 10, the symbol in the 1 positions being given ten times as much power as the symbol in the 0 position. The central insight of this paper is that position significant notation is also constitutive of the Universe, position significant notation here appearing as the disposition of physically embodied information in physical spacetime.
. . .
Post to Emile Grgin Grgin
Dear Emile,
Thank you for your essay. I am in close agreement with you on the first part. I find the second part a bit obscure but I have sent away for your 2007 book. Here I submit a different possible outcome which also seems to fit your guidelines:“(1) There is no return to more naive intuitions.” So let us abandon the notion that the Universe is continuous, since we observe only discrete events at all scales from fundamental particles to the stars and beyond. Like Heisenberg, let us not talk about things we cannot see.
“(2) Every new physical theory requires some new mathematics which is not an evident extension of the previous one.” The birth of quantum mechanics coincided approximately with an explosion in discrete mathematics (logic) beginning around Frege and Cantor and developing through Russel and Whitehead, Gödel, Turing and Shannon to the internet and its denumerable and non-denumerable (theoretical) extensions.
“(3) The goals of physicists are very different from those of mathematicians.” Mathematicians want consistent and interesting symbolic structures; physicists want to explain how the amazingly complex structure of the Universe grew from the initial singularity. Cantor’s construction of the transfinite numbers from the natural numbers could be a starting point here.
“(4) New theories are best developed as answers to pressing questions.” Both physics and theology are currently moribund. It may even be true that the physics community is as hidebound as the theological community. How might we measure hideboundness? Maybe by the lag in coming to terms with reality, a lag in perception. The principal absurdity of physics is the belief that one can encode information in a continuum. We cannot observe a continuum, since there is nothing to see. It is not surprising that every physical observation without exception is a quantum, a discrete, marked, event.
“(5) The unification of theories takes place automatically once the correct mathematical structure has been identified.” See above and below.
“(6) Two theories cannot be unified until they have been finalized and characterized by principles.” As you say, “Special and general relativity are, but quantum mechanics is defined only by technical axioms. It is therefore not ready for unification. Postponing unification, we must first search for the ‘principle of quantum mechanics’ which implies the axioms of the latter. As I see it, the ‘principle of quantum mechanics’ is that it describes a denumerable communication network.
Without going into detail we see that the concrete foundation of information theory is the same as a quantum source: a source that emits a statistically normalized sequence of orthogonal letters from an alphabet. In quantum mechanics we represent this alphabet by a basis of a Hilbert space of suitable dimension.
By seeing the world as a communication network we get an explanation for quantization. As Shannon showed, the strategy for defeating error in communication is to encode messages into long strings or packets whose letters are equiprobable. We can place these packets so far apart in message space that the chances of confusion are negligible. This is quantization. Messages with no overlap (confusion) are orthogonal, like orthogonal quantum mechanical states.
We may see quantum mechanics as a method for computing the traffic in various channels in the universal network. It may be that an abstract denumerable description can go no further. The Löwenheim–Skolem theorem tells us denumerable theories are are unable to control the cardinality of their infinite models. The only way to get into one-to-one correspondence with infinite reality is to be it.
How does all this relate to relativity? An answer might be be that gravitation is the lowest physical level in the Universe and is not quantized. Gravitation sees only energy, so that it is not prone to error. Quantization (packetization) is not necessary. Such complexity as we see in gravitational theory is a result of the subsequent differentiation of the Universe into discrete packets of energy moving in 4D spacetime. In a practical layered network, any communication between peers involves all the lower layers down to the physical layer.
All the best,
Jeffrey
Mathematicians make structures which might be realized insofar as they have no internal inconsistencies. Engineers try to realize these structures given that they are not only internally consistent, but consistent with the rest of the relevant world.
[page 44]
What conditions must be satisfied for two entities to communicate: 1. a common language. 2. the desire to communicate (potential) [each lacking something the other has got = communication = trade].
Why do I want to communicate with you? Because you look sexy. How do we parametrize sex? By the strength of the attraction, the coupling constant.
All the women's magazines are about bonding or accoutrements thereunto.
A NAND gate with phase and threshold, ie two NAND gates communicating. The probability of 0 or 1 depends on the phase of the interaction of the gates.
The minimal path integral. Two paths, one intermediate stop, ie two slit experiment.
The probability of a certain event is 1. The Universe is certain, therefore its probability is 1, and its entropy 0.
Tuesday 3 November 2009
Feynman (Brown) (RMP) page 80:
'The calculation of the Probability Amplitude for a path . . . [Postulate] II. The paths contribute equally in magnitude, but the phase of their contribution is the classical action (in units of h bar)
[page 45]
ie the time integral of the Lagrangian taken along the path.
'That is to say, the contribution PHI[x(t)] from a given path x(t) is proportional to exp (i/h bar) S[x(t)], where the action S[x(t)] = integralL(x dot(t), x(t)) dt is the time integral of the classical Lagrangian L(x dot, x) taken along the path in question.
Put this in computational terms.
[Why the classical action?] Classical action measures the number of computational operations along the path, counted by the function n = S[x(t)] / h bar. The PHI = k (cos(n) = i sin(n))
An event occurs (x at t) when the relevant computation is complete, ie the relevant phase has completed one full cycle. Nothing happens at places and times where this condition is not fulfilled.
I use the dynamic network picture to see how God can be both 'omnino simplex', living and as complex as the Universe of experience.
At the microscopic scale, every perturbation to a system comes as a packet and the overall rate and size of the perturbation depends on the rate of reception of packets and the total number of packets received, is dS/dt and S = integral dS/dt dy. Each perturbation may be seen as a complex message, or degenerately, simply as a push or pull.
Concrete information is discrete, even though the statistics of information may be expressed as continuous distributions [distribution functions].
This paper is too glossy for my ballpens.
[page 46]
What does it mean, in digital computing terms, to say that the Lagrangian is a function of position and velocity. First we must decide what we mean by a quantity, and the principal unit we have in physics is the quantum of action which we say is the physical embodiment of a completed computation. Position = memory address. Velocity = rate of change of memory address [ie rate of access to memory?]
The classical Lagrangian is a number whose time integral is the action, so Lagrangian has the dimensions of energy, the time rate of action. We convert the action to a pure number by dividing it by h bar, S / h bar, and then use this number in a complex exponent to get a complex amplitude whose absolute square is a probability. What does this mean?
Velocity --> kinetic energy
Position --> potential energy
The Universe like me is always on the lookout for clarity which improves definition and increases entropy.
Physics is concerned with computing real cardinals, and constructs ordered sets of cardinals (like Hilbert spaces) which represent reality in detail and then integrates over them in some way to get its answers. The interesting parts of [physics are the ordinals, which are accessible via cardinals that represent the probabilities of various ordinals. The ordinals represent events, so we have probabilities of events.
[page 47]
The different models of quantum mechanics are different ways of setting up the structures to be integrated.
Why does calculus work? Because it deals with continuous (degenerate) functions.
A book is a path to be followed by reading.
VOLTAGE READER / VOLTAGE WRITER
POTENTIAL = MARK - so every mark has energy [as we should expect as all information is represented physically]
The creation of the world is the creation of marks, definite structures that can interact with different structures. [the Universe is writing itself, with our help and the help of every other creative system]
Wednesday 4 November 2009
So the target becomes a fourth logical formulation of quantum mechanics on the trajectory Schrödinger , Heisenberg, Feynman . . .
AMBITION = TO MAKE A MARK ie to leave something in the collective memory, that is, in the collective structure.
Writing: making marks to correspond to marks on one's mind.
CREATION = making marks where there were no marks before, organizing the grains of sand into a line.
[page 48]
A hundred years after its conception quantum mechanics remains a mystery to us. It shows us how to compute distribution functions for events in the Universe, but it does not tell us about these events because it is indifferent to the particular nature of the events, seeing them only as unity, although the mathematical machinery deals in continuous distribution functions. The main clue it gives us are the ways it computes its distribution function, by addition and multiplication of complex numbers, ie 2D 'vectors'.
Khinchin page 6: '. . . in the statistical problems of quantum mechanics it is necessary to develop computational methods for three fundamental statistical schemes, complete, symmetric and antisymmetric.'
The statistical structure of quantum mechanics gives is about the same understanding of quantum events as road accident statistics give us about road accidents. If we develop road accident statistics as a function of road space, we learn the configurations of roads that are associated with a certain frequency of accident. Nevertheless every road accident (as careful investigation shows) is a complex event to which causes can be assigned.
STATE = ADDRESS + CONTENT
Khinchin: 'Fundamental eigenfunctions' correspond to 'fundamental states' which serve as a memory structure (ordered set). 'Occupation numbers', 'each [of which] specifies the number of particles of the system which is found in a particular state . . . the same definite set of
[page 49]
occupation numbers corresponds to each state in any of the three statistical schemes. Conversely, one or several fundamental states correspond to each set of occupation numbers. The number of fundamental states corresponding to a given set of occupation numbers is different for the three basic statistical schemes. This difference is the most important consequence of the statistical dissimilarity of the schemes.'
Different statistical schemes reflect different operators.
Khinchin page 7: 'Many of the most important physical quantities studied in statistical physics have a "sum" character, ie, they are sums of quantities each depending on the state of only one of the particles which compose the system.'
ie they are real cardinal numbers.
Quantum mechanics uses complex numbers to compute its cardinal numbers. Why?
'Any quantity which depends symmetrically on the states of the particles which compose the system can be of interest in statistical physics.'
ie we are abstracting from the particular nature of the states and treating them all simply as units.
Maybe a thorough study of Khinchin will yield the insights necessary to write Computation and the path integral.
All events have definite outcomes: quantum outcomes appear to be unpredictable; classical outcomes are predictable. [my guess is that the unpredictability in quantum events comes from the interaction of independent nodes in the computer network]
[page 50]
Khinchin page 9: '. . . the primary purpose of this book is to show that there is no need for a special analytical apparatus to justify rigorously the methods of calculation of the mean value of physical quantities.'
Quantum mechanics is the interface between the determinate and the indeterminate. It describes the fundamental manifestation of creative intelligence, the process that brings sharply defined structures like these tests out of the mist of probability. So we are taken back to the intellectual presentation of the Hilbert Oscillator in Divinity of Money para 69. Essay on the Divinity of Money
The m effect speeds up writing even before I have got it properly alight [Pavlov?].
How does the indefinite become definite? We cannot explain this from the indefinite side because there is nothing definite to go on so we have to rely on the definite output of quantum mechanics to arrive at any conclusions.
Grgin: I think that I make a little progress by saying stupid things out loud and then repenting them, which was the case when I berated continuous mathematics in my first comment to you. Now I can begin to see more clearly that quantum mechanics lies at the interface between the indefinite (modelled by the continuous) and the definite, modelled by the discrete.
One of my heuristic principles is that the old timers are no less stupid that we are, as we can see by
[page 51]
putting ourselves in their shoes and trying to look at their questions and answers through their perspective. We can extend this principle to the idea that the Universe was no more of less intelligent in the beginning that it is now. Insight and quantum measurement described the same event at different scales of complexity.
Maybe the randomness of quantum mechanics arises from the randomness of its inputs, coming from the undetermined side of the world: random input + deterministic process = random output. The quantum mechanical process nevertheless gives a definite output when observed by a definite observer ('observable'), represented by a matrix operator.
Think of a child learning to speak, taking in a few years of input and then reproducing it to transmit their own state of mind.
Quantum mechanics (= intelligence) turns the indeterminate future into the deterministic past, or is it vice versa.? Lets say every event couples future to past along the chain of time.
Quantum mechanics predicts the statistics of the two slit experiment but it doe not tell us what is actually happening.
Thursday 5 November 2009
All attempts to get behind the statistical facade of quantum mechanics seem to have failed, leaving the rest of physics without guidance. The distribution laws evolve deterministically (we say) while the process leading to the distributions remain unknown [but we know that on the whole they are not stationary].
[page 52]
From a cybernetic point of view a quantum mechanical preparation does not have the variety required to fully constrain the outcome.
Phase - relative completion of a communication. We cannot know the phase unless we know ghow long the computation takes. Non-halting compiutations remain forever at phase 0. In quantum mechanics all that matters is the relative phase.
Feynman III: Bullets: random input, random output; electrons, random input, structured output. Feynman
A foundation for sexiness is to love one's whole self, body and soul. The opposite is to hate one, other or both or at least some parts of one's reality. A system is defective if people must do hateful thing (like denounce their neighbours) to survive within it.
To understand quantum mechanics maybe we have to develop the idea of logical confinement via logical continuity and the notion of a pipeline of processors which take a random input and generate a random output which is nevertheless a (deterministic) transform of the input. So image processing software gives an output which is correlated to the input but can process any image presented in the right format. The wave pattern produced in the two slit experiment is a transformation of the two slits, and we can use it to reconstruct an image of the slits.
Each slit is a source, and they are correlated because
[page 53]
they have a common source.
Feynman: we follow a path by multiplying amplitudes; we unite independent paths by adding amplitudes.
Each step along the path is represented by one step of action, one unit of action. We use the action of the step as the exponent in a complex exponential; to compute the amplitude corresponding to the step which is a factor in the product representing the whole process.
AMPLITUDE = ELEMENT OF PHASE
Past is fixed and cannot be changed. Future is not fixed and present actions can influence it. The present is not an instant, but the length of time it takes to transform a past state into a future one, the time required to simultaneously assimilate and construct a symbol. When I read I construct symbols (signals = physical) in my mind corresponding to the signals I am receiving.
Thinking in terms of phase, computation and clocks each cycle is a datum. When we look at the software processing of a computer we see a lot of different frequencies, each defined by the execution time of a subroutine, and we use Fourier transforms to study the overall computational element and separate out the 'line weights' of different frequencies, ie how often the subroutine with a certain length (inverse frequency) is executed.
INTRINSIC FREQUENCY = 1/execution time (spectral line)
EXTRINSIC FREQUENCY = how often called (line weight).
[page 54]
Let us Fourier analyze a classical universal computer of a certain cardinality performing a) random operations; b) specified operations. The cardinal of a computer. N is the number of states it has available in space and time. The Number of states of the machine taken as a whole is N!.
Computing addresses - Turing machine walking its tape vs random access memory, moving in spacetime.
Computing results: doing it when the data is at hand (in registers).
Statistics: complete (Boltzmann); Fermi-Dirac; Bose-Einstein.Statistical mechanics - Wikipedia Maxwell-Boltzmann statistics - Wikipedia Fermi-Dirac statistics - Wikipedia Bose-Einstein statistics - Wikipedia
Once we have done the lone computer we then connect into a network where asynchronicity comes into play. This leads to 'interference' across the range from constructive (amplitude = 1) to destructive (amplitude = 0)
The statistics are images of operators: so we might have the Fermion-operator and the Boson-operator.
Friday 6 November 2009
Khinchin page 1: '. . . we avoid mentioning spin whenever possible in this book.'
On the contrary, spin being a digital phenomenon closely related to symmetric and antisymmetric statistics,
[page 55]
would seem to be right at the heart of a computational understanding of quantum mechanics.
Can we turn the tables on quantum mechanics by showing that quantum statistics is the outcome of classical computation?
Heisenberg uncertainty principoe: we do not know the outcome of a computation until it is complete. Before that it is not observable (although in real life we can use debugging software to observe the execution of our code, but this does not hange the situation - we have to wait until the debugger has run to see the outcome of each step).
Feynman's axioms:
1. P = probability, phi = probability amplitude, P= |phi]2
2. Alternative routes: phi = phi1 + phi2, P = |phi1 + phi2|2 (unknowable)
3. Alternatives (known): P = P1 + P2
4. Subsequent events: P = |phi1 phi2|2
The question is how do we look at our model to make it look like the world of experience. Quantum mechanics tells us we must take both the observer and the observed into account when answering the question.
The foundation of our model is a not gate observing itself, that is taking its own output as input and consequently oscillating. This, apart from frequency, is the whole story of quantum mechanics. The frequency of the not gate depends on how long it takes to read and understand its input and formulate a response.
To mark a period, we need first some p and then some not-p.
[page 56]
Feynman III page 2-8: Confined waves have definite frequencies because they have nodes (zero points, fixed points, [stationary [points]) at the boundaries of their confinement, therefore definite energies, ie definite rates of action. If we see the Universe as a whole as completely unconfined perhaps we might argue that its energy is zero.
If we are saying that the Universe as always as intelligent as it is now we are saying that intelligence and insight are not emergent properties. What is emergent is their application to more and more complex situations, but we expect to find everything at the level of the single particle talking to itself. But that way lies the infinite self energy of the electron? [only if it is of zero size in a continuous space etc].
N D Cook page 1: 'There is no greater obstacle to progress than a belief that progress is impossible.' N D Cook
Round and round and round we go, but slowly a picture emerges from the mist. As always the heuristic principle that entellignece created definition and definition is scale invariant.
Feynman's book, growth, development, etc are all a process of starting poorly defined ('babies call all men father') and gradually gaining precision. [Aristotle, Physics 184b12 Aristotle] This is also the course of technological development, a mixture of paradigm changes and gradual evolution.
My next target should be Theological Studies 'is the Universe divine?'.
[page 57]
A transition involves uncertainty because the initial state insofar as it changes, does not know (and cannot predict) the outcome.
Saturday 7 November 2009
The intuitive cause of an event is a force - I pushed it and it fell over; and intuitively the transmission of a force requires contact in space and time, ie spatial distance zero at a given interval of time. We say that all information is encoded physically, so the transmission of information is equivalent to the transmission of force, so the road rules, transmitted to every driver, exert apparent forces on the motor vehicles making them drive on the right side of the road, stop at traffic lights etc. This behaviour can ultimately be traced to the state of physical neurons in the driver's heads. We cannot say exactly how this happens, but we can model it formally by fixed points (such as this writing) in the universal dynamics.
We need a structure of logical events that respects the path integral rules and results ion the statistics of quantum mechanics.
Feynman's rules
1. P = |phi|2
2. alternative : phi = phi1 + phi2
3. Consequent: phi = phi1phi2
. . .
Complex phase goes real - imaginary - real - imaginary . . . as the 'phasor' rotates. . . .
We see everytuhing in terms of the flow of [conserved but compressible] fluid which we call probability (?) = count of charges.
[page 58]
Potential (voltage) = cause. Current will flow until voltage = 0.
Current (amperage) = effect.
No current without potential. LOGICAL POTENTIAL
LOGICAL POTENTIAL = QUESTION
LOGICAL CURRENT = PROCESSING toward an ANSWER at which point the logical potential potential drops to zero [the question is closed]
RESISTANCE: logical potential is dissipated by a series of local questions and answers, reducing the rate of progress toward the final answer (ground)
I = V/RT Ohm's law - FRICTION
INDUCTANCE: v = di/dt (see relativity)
The formal mathematics of electricity does not depend on the nature of the charge, only on the fact that it is measurable and conserved. We connect measurements and events by 'constants of nature.'
CAPACITANCE: v = integral i dt.
Computation as a result of an impressed potential is the transient leading to a steady state.
We understand electrical potential by gravitational potential which is immediately available to our senses and is the driving potential for the physical dynamics of balance, locomotion, heavy construction etc.
GRAVITATION = the most fundamental evidence of God, the foundation for the restrictions on behaviour enforces by religion.
MQG: Charge high fees for facilitating profitable deals near the edge of reality.
PRIVATIZATION - a potential (policy) that introduces a transient into the financial system)
Theological Studies: A series parallel to Lonergan's Verbum explaining creation and the divinity of the Universe.Introduction : Explain this.
Human dynamics is isomorphic to physical dynamics up to a factor of complexity.
KIRCHOFF'S LAWS
1. The algebraic sum of the currents directed toward a junction point is zero (conserved flow)
2. The algebraic sum of the voltages taken in a specific direction around a closed circuit is zero (conserved potential, conservation of energy).
The next most complex analogue of religion is ELECTRODYNAMICS.
SOURCE OF POTENTIAL (EMF) = question
LOGIC is all about questions and answers. We can logically transform the correct answer to a question until its 'fits' the question, ie completes the circuit allowing a flow of current which reduces the potential around the circuit to zero.
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MONEY measures POTENTIAL in the economic system and goods flow to bring the money potential to zero: ie I have no money and everything that i want so I am content vs have much money and nothing that i want so I am out shopping.
Fitzgerald and Higginbotham page 9: 'It is because of the possibility of representing physical systems by electrical networks that circuit theory is such a valuable tool in the study of electrical devices.'
DC circuit: GRAVITATION (zero entropy) (no resistance)
AC circuit capable of carrying information as well as energy.
Series and parallel connection (computation) in quantum mechanics give Feynman's rules for computing amplitudes: add parallel, multiply series How do we establish the analogy with electricity, add series, multiply parallel etc.
Req = R1R2 / R1 + R2 ('normalized product')
Parallel computation (scientific community) same question to many processors.
TRIVIAL QUESTIONS lead to simple operations. If the question is its own answer, the required operation in NOP. If the answer and the question are different (complementary) the operation is some form of NOT.
F & H page 15: Circuit theorems.
page 18: 'principle of superposition'. Superposition is
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possible in any system (electrical, mechanical, etc) where f(x1) + f(x2) = f(x1 + x2).
Augustine, Encyclopedia Britannica II page 364 John Burnaby: 'His mind was the crucible in which the religion of the new testament most completely fused with the Platonic traditions of Greek philosophy; and it was also the means by which the product of this fusion was transmitted to the Christendoms of medieval Roman Catholicism and Renaissance Protestantism.' Goetz
Burnably page 365: 'Neoplatonism, in the work of the 3rd century philosopher and mystic Plotinus, its greatest exponent, is a spiritual monism . . . according to which the Universe exists as a series of emanations or degenerations from absolute unity.'*
Dynamic unity, many fixed points, that map onto themselves day after day (with only small changes) like this house, whose half life is circa 50 years, say 1 billion seconds.
*'From the transcendent One arises self-conscious mind or spirit; from the mind comes soul or life; and soul is the intermediary between the spheres of spirit and sense. Matter is the lowest and last product of the supreme unity.'
'[In Augustine], Neoplatonism had reinforces the Manichaean principle that the way of return to God must be through escape from the body; and for Augustine, this meant primarily and immediately escape from the ties of sexuality.
page 366: '. . . for a Neoplatonist the soul's likeness to God is that of a, so to speak, reduced divinity, for the Christian it is that of a temporal and mutable image of the "eternal and changeless". Augustine was reassured that it is the task of a Christian philosophy, guided by
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the scriptural revelation to seek to know God through his image in the soul; and this was the path he followed to his great treatise De Trinitate (On the Trinity).
God is unique and therefore not predictable, even to itself, which leave the way open for humour. Bloke is pulling in a net in an estuary where fishing is illegal. Fisheries inspector comes by and says 'Ho, ho, ho, I've caught you'. Bloke says 'and who might you be?'. Fisheries inspector says 'Fisheries inspector'. 'Thank God', says bloke, 'for a minute I thought you were the owner of the net.'
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