Notes
[Sunday 16 August 2009 - Saturday 22 August 2009]
[Notebook: DB 67: jciii]
[page 110]
Sunday 16 August 2009
Trying for another leap from guess to reality: ie that the restricted entropy of entanglement is a 'vestigium' of Shannon encoding from non-equiprobable to equiprobable alphabets [sources]. The only clue I have at the moment is Nielsen & Chuang's statement 'It turns out, however, that generalizations of this entanglement transformation problem exhibit deep and unexpected connections to quantum error-correction.' Nielsen and Chuang page 573
One of my heuristic principles seems to be that we must start any explanation of the world with the initial singularity = God. Since all states are born in communication with one another all states are to some extent entangled by their birth and so all operations at constant entropy must be seen as entanglement transformations.
A quantum observation: while I am single I can kiss, cuddle, hug and talk to everybody but as soon as I enter a committed public relationship all these activities are deprecated except with my chosen partner.
[page 111]
Rejected by Sciene (Why_q - 'not the sort of work we publish' but although I was dreaming of acceptance, it seemed natural and I do not seem to be too despondent because it drives me back to the drawing board and I seem to have made enough progress in my own mind and opened up enough new vistas for progress to justify the nine months or so I spent on that paper. Now I want to add a section on entanglement and submit it to Foundations of Physics
Nielsen & Chuang page 2: 'One of the goals of quantum computation and quantum information is to develop tools which sharpen our intuition about quantum mechanics, and make its predictions more transparent to human minds.'
N & C page 8: 'In 1995 Ben Schumaker provided an analogue to Shannon's noiseless coding theorem and in the process defined the quantum bit or 'qubit' as a tangible physical resource. However no analogue to Shannon's noisy channel coding theorem is yet known for quantum information.'
Maybe there is no noise in quantum theory since it has no preconceptions about what is noise and what is not. Noise only exists from the point of view of higher layers that restrict the entropy of lower layers to achieve a more deterministic outcome for their processes.
The weak spot in quantum information theory is that there is no information in a continuum.
Alternatively, insofar as the Universe is meaningful and can be spoken about, it is discrete.
[page 112]
We imagine the Universe at the 1D Hilbert space stage beginning to differentiate into 2D, ie to become a 2 state system from a 1 state system, ie becoming a qubit. We see it as sometimes 1 state, sometimes 2 states and as it differentiates the proportion of time it is 1 state decreases from 1 to 0 as the two state proportion goes from 0 to 1. We can model this with a 'statistical continuum,' using the law of large numbers. Law of large numbers - Wikipedia Then we go through a similar process on the way to 3, 4, 5 . . . n states, each new stage involving a certain amount of entanglement until the new states become completely orthogonal and independent. We see this as parallel to the process of growing a baby, learning to speak, etc, ie differentiation in general. This gives us a sort of conditional bootstrapping Cantor's theorem that becomes 'truer' as the symbols in the proof become better resolved.
All continuous models that appear to work in describing nature are probabilistic approximations to discrete events, as we can assign a real number to the entropy of a stationary source if we observe the source emit ℵ1 symbols (and we can assign a rational number if we observe it for ℵ0 symbols (events). The same goes for quantum mechanics, and this is in effect what we compute when we construct the Hamiltonian, the relative frequency of emission of different letters, ie the relative probability of each letter (eigenvalue [eigenfunction]).
Lonergan's genetic method: embryogenesis : phylogenesis :: psychogenesis : embryogenesis, etc, another form of Cantor symmetry.
[page 113]
A proof brings itself to be by fulfilling its own hypothesis.
Nielsen & Chuang: Although the book is about quantum communication all the proofs in it are classical. Although we might say that the variables in the proofs are superpositions of all the elements of their [ranges] and the proofs are structured so that as long as these variables remain in their relevant ranges the proofs hold.
Music has meaning at low resolution, rather analogous to quantum mechanics. Speech can also be understood as a superposition of frequencies or energies, but it is much more defined and meaningful than music.
How can we speak logically about the emergence of logic from not-logic?
Monday 17 August 2009
Science vs Mysticism: Mysticism seeks the one, the continuum, whereas science is looking for the marks, salient factors, details, explanations. In science the theory of probability defines the continuum and provides a background against which to detect significant differences from the continuum.
Majorization appears to be defined in vector spaces of the same dimension and it appears that a lower dimensional space will always majorize a space of higher dimension since its whole weight (sum = 1) occurs in the first part of the series.
Creation is a motion from the continuum to structure.
Irreversibility is the key to structure, something us builders strive for in the screwing of screws and the setting of adhesives.
[page 114]
Similarly with writing. One wishes to write things that are irreversible, ie not contradicting reality and so carrying the seeds of their own destruction.
The Newtonian theological ansatz has become unfashionable, but it may still have the power to give us a context in which to construct models of the world. Newton
FIRE = VIOLENCE erases memories, turns houses to ash, releasing matter and energy. Agatha Passeger Christie
Probability - fluctuation - irreversibility.
Let us propose a new character for the alphabet of personalities in quantum communication. Alice and Bob are well known. Eve, cast as an eavesdropper, appears occasionally. D for Dodgy is a builder. D owns and operates a set of tools that span the gap between the formal operations of architects, engineers and mathematicians and the physical world. His task is to create physical structures to realize forms. He is, we expect, as universal as a Turing machine, since one of his achievements has been to realize the equivalent of a finite universal Turing machine in physical hardware. While Alice, Bob and Eve are formalists, Dodgy is a realist, and , as his name implies, is always on the lookout for a short cut to a result which will please his customer.
Computational complexity theory: complexity = number of operations x complexity of each operation [= card(action)]
D charges by the atomic operation?
[page 115]
Materials = embodied operations
Labour = embodiment of form in materials.
P can be computed
NP can be checked
Tuesday 18 August 2009
What it now seems to me that I am saying is that a transfinite computer network, bound together by logical continuity, can serve as an explanation for all the theory and data of quantum mechanics, quantum field theory and gravitation. the purpose of the Foundations of Physics article is to represent this guess with a plausible set of words.
What I appear to be saying is that there is in a sense no quantum mechanics, but rather the whole observable world is classical and the network model gives us a classical probabilistic explanation for everything that is attributed to the complex linear vector space attributed to quantum mechanics. This seems a bit extreme, but it seems that I have been pursuing it for a long time and making a little step forward here and there, with occasional attempts to put it all together.
Can this be true? A simple disproof would be good. If it is true, then we can say that the transfinite network model is isomorphic to quantum field theory, or, more generally, isomorphic to the Universe.
Eigenvector = stationary point in an operator.
[page 116]
Why do we want this to be true? Because the network model allows us to explain the Universe as divine and self creative, giving a new empirical foundation to theology [whereas quantum mechanics operates reversibly at constant entropy and so cannot explain creation].
How many times have I said this? Every few days for about fifty years. I go round and round and round slowly spiralling through my life. It is this circular component of my mental motion which accounts for the ubiquity of complex numbers in our model of the periodic Universe.
Traditionally, continuity has been the fundamental explanatory mechanism for the behaviour of the world. From the time of Aristotle we have represented this by geometric continuity, first represented by real numbers and since the advent of wave equations, by complex numbers, that is ordered pairs of reals. In order to explain the creative and intelligent side of the world, I want to introduce logical continuity, which is from an arithmetic point of view, real and digital, but I despair of knowing how to do it, which is why I get up so early (0600) to write this. From the logical point of view, the Universe is geometrically continuous when the logical states are so small and similar that they can be approximated by an unresolved continuum.
Logic deals in form and order and cardinal numbers are of secondary importance. Physics puts heavy emphasis on cardinals and one of its proudest boasts is the close numerical coincidence between measurement and calculation in quantum electrodynamics. Feynman
[page 117
Wednesday 19 August 2009
[page 117]The coupling between the cardinal and ordinal features of the Universe is the unit or quantum of action.
Sex and friendship are orthogonal, but they can becomes maximally entangled. So we can imagine a Bell state (|sex sex> + |friendship friendship>) / sqrt(2).
A wrestling match is a state. Wrestlers (sculpture) - Wikipedia
Process: the arrow moves with apparent continuity, but there are quanta of action underlying the appearance. The relationship between the space and the arrow, ie the arrow's address, is constantly changing and this is a reversible quantum process which underlies the first law: a body at rest remains at rest and a body in motion continues its motion in a straight line unless acted upon by the force [conservation of momentum Momentum - Wikipedia] The relationship of a moving object to its environment is real and physical but the protocol is so simple that we do not see it, logical at this level = physical.
This paper explores the relationship between the logical and physical Universes.FIXED POINT = REVERSIBLE PROCESS
Quantum mechanics cannot explain the increasing entropy of the Universe [except by communication].
[page 118]
A hierarchy of operations, arithmetic, algebraic, anything a Turing machine can do. A proof machine, which may use a lookup table to decide its next operation, like a ribosome which maps RNA --> PROTEIN Ribosome - Wikipedia
Conjecture and refutation. Refutable conjectures must be expressed as some algorithm whose input and output can be mapped to the world.
Feynman's method is an algorithm. Feynman Even though an algorithm may manipulate continuous quantities, it is finite and discrete, and the measure of the complexity of a continuous quantity is the algorithmic complexity of the algorithm to generate the continuum. If the computation of every point in a set of points takes the same time, then we find that resolution = k.computing cost.
Variable = superpositions real.
Are physical superpositions real? Yes, look at the structure of a vibrating string and listen to its sounds. It is a superposition of waves with certain amplitude, phase and frequency. Nyquist / Shannon tells us how much information there is in a waveform, the mediator between continuous and discrete.
A hint of the modern relationship between logic and physics first becomes visible in the literary remains of the first millennium bce. Aristotle devoted a considerable portion of his work to logic. Organon- Wikipedia
[page 119]
Anti-Plato: 1: There are no universals; everything is concrete.
2: nevertheless we can separate out subsets of the full concrete definition of something (eg humanity) and discuss this in isolation from the abstracted detail.
Aristotle was the first formal logician (ie he gives the principles of reasoning using variables to show the underlying logical form of the arguments). Wikipedia, History of Logic. History of logic - Wikipedia
Because the Universe is concrete every physical event has a unique physical address, but we or any other agent can assign it whatever logical address we wish. Sop computer assigns logical addresses to physically addressed hardwre.
MODELLING = LOGICAL ADDRESSING
Unitary matrices change the direction but not the length of vectors.
Inner products of two vectors produce a vector whose length is a functiuon of the angle between them ranging from 0 (orthogonal) [uncorrelated] to 1 (parallel) [fully correlated].
Stationary structure = eigenfunction = algorithm.
'The great and chief end of men uniting in commonwealths and putting themselves under government is the preservation of their property'. Locke, Two Treatises on Government, bk ii chap. ix, para 124 quoted in Tawney Religion and the Rise of Capitalism Tawney Locke
Tawney page 21: It was not till a century after Machiavelli
[page 120]
had emancipated the state from religion that the doctrine of a self-contained department with laws of its own begins generally to be applied to the world of business relations, . . .
'Modern social theory, like modern political theory, developed only when society was given a naturalistic rather than a religious explanation, and the rise of both was largely due to a changed conception of the Church, . . . The social functions matured within the Church, and long identified with it, are transferred to the State, . . . .'
Tawney page 41: 'From the middle of the thirteenth century a continuous wail arises against the iniquity of the Church, and its burden may be summed up in one word, 'avarice'. At Rome, everything is for sale.'
page 45: 'Labour - the common lot of mankind - is necessary and honourable; trade is necessary, but perilous to the soul; finance, if not immoral, is at best sordid and at worst disreputable.'
PRIVATE PROPERTY = PRIVATE MEMORY
Property = memory, data, capital.
Thursday 20 August 2009
We may date the interaction of the logical and physical Universe to the origin of language.Tawney page 277: 'For the condition of effective action in a complex civilization is cooperation. And the condition of cooperation is agreement, both as to the ends to which effort
[page 121]
should be applied, and the criteria by which success is to be judged.'
Tawney page 278: Ruskin: 'There is no wealth but life'.
Constraint on quantum mechanics: all eigenvalues of a unitary matrix have modulus 1 and can be represented as exp(i theta) where theta is real and is the symbolic carrier of information in quantum theory.It is always a worry that the texts that ticker through my head as I lie in the bath reading spy stories will have disappeared by the time I am dried and dressed and sitting by the fire.
We are thinking about mapping the logical and physical Universes. Aristotle did it with matter and form, potency and act. Hylomorphism (Aristotelian) - Wikipedia Actus et potentia - Wikipedia The modern era began with Galilei's manifesto: the book of nature is written in mathematics. Galilei Newton followed this line and it has borne brilliant fruit ever since. Newton Perhaps the most fundamental breakthrough was the invention of calculus which enabled us to talk about a supposedly geometrically continuous world using the discrete symbols of mathematical discourse. Calculus - Wikipedia Here, it seems, is the rub. The world is continuous, but logic is not, so our attempt to map logic to physics has met a bumo.
This bump has come to prominence since the beginning of interest in quantum information and quantum computation. Quantum information - Wikipedia Quantum computer - Wikipedia Los Alamos National Laboratory
As far as we know (ie observe) the world is not continuous.
[page 122]
but a source of discrete symbols which can be mapped to the discrete eigenvalues of certain operators.
The Aristotelian continuum, built like a chain with linked ends, is much more logical than the mathematical continuum, built on the Bolzano-Weierstrass theorem. Aristotle Dedekind cut - Wikipedia It assumes that the points in a continuum are really distinct, something of a contradiction.
So, insofar as it can be modelled by the logical Universe, the physical world is not continuous. Here fits in Cohen's work on Cantor. Cohen
Develop the idea that the symmetric Universe is consistent with local gauge invariance via the connection between phase and data, ie logic is encoded in physics by phase - so Shannon-Nyquist. Nyquist-Shannon sampling theorem - Wikipedia
Motion and stillness / continuity and discreteness. See the candle flame and wonder. The continuum is the flow of identical events. A signal is a flow of distinguished events. So continuity works in physics because the software in the physical layer is fundamentally simple, like the and and not gates that work a computer.
One can imagine that in the human world the interplay between logic and physics began with the evolution of language.
Friday 21 August 2009
[page 123]
Mapping QM to TSN (transfinite symmetric network).
<>p> 1. State space, basically a memory with a relationship between the probabilities of excitement of the different memory cells. We might guess that single memories have countable different states, beginning with binary, the qubit. The answers in quantum mechanics are squared amplitudes, the amplitudes being the coefficients of the basis vectors defining a state vector. [in the full catastrophe we can have a transfinite number of basis vectors, and a transfinite set of different amplitudes available to apply to each one, subject only to normalization]
2. Evolution
The defining feature of quantum mechanics is conservation of probability, that is conservation of squared amplitudes expressed in the normalization of basis vectors and the unitarity of evolution. Here we see reasonableness expressed as the probabilities of actual outcomes of a set of possibilities add up to 1, since only one event occurs. The world is a sequence of fixed values exploring the space of possibility.
We say a function is continuous if changes in the range tend to zero as changes in the domain tend to zero. But what is the ratio of this tendency? It depends on the function, and with something like the tangent, an infinitesimal change near the discontinuity can lead to a very large change in the range (as we pass through pi/2 radians tan theta goes from, +infinity to -infinity.
We see a similar discontinuity in the transfinite transition and in logical (digital) functions, where a change on the range from yes to no can lead to a similar change in the domain. Whereas geometric continuity is concerned with cardinal numbers and measures, logical continuity is concerned with affirmation and negation. [at the binary level, logic and arithmetic become almost (?) identical]
Each peer layer has a 'machine infinity' which defines the
[page 124]
boundary between locally rational and locally real numbers. The Dedekind cut defines new reals between the rationals on each side of the cut.
We know quantum mechanics works, in the sense that the mechanism of Hilbert space and linear algebra map quite precisely to reality, but we do not know how or why it works, which would give us a better examination of the mechanism. [maybe quantum mechanics describes a system without mechanism [constraint], ie totally symmetrical]
Saturday 22 August 2009
Logical explanation = mechanistic explanation: hip bone connected to the thigh bone . . . . Traditional Music The notion of Cantor symmetry suggests that such explanations work at all levels of complexity. In some way the neural network that explains this writing is a paradigm of the communication network that explains my existence.
Permutation explains the constancy of probability, in that all the elements of a given permutation have permanent existence, ie are time invariant. What varies in time is the sequence on which the operations of the permutation are executed.
Perhaps the beauty of these notes lies in their banality: the work of a person of no particular brilliance trying to make sense of a world which was once simply explained by the will or God and must now be explained by its own personality. [which is an amalgam of all the personalities within it]
We see the evoultion of an isolated quantum system as a
[page 125]
'rotation' in permutation space 010101 . . . 123123123 . . . 123231 . . . Formally the operations connect, as in a Turing machine, since the final state of one is the initial state of the next. [and this is concrete]
VARIABLE - a fixed point through which values pass, that is a writable memory, like a square on Turing's tape.
CONSTANT = a variable between changes.
From a practical point of view structures are created by bonding. In my case by glue, screws and nails; in the abstract world of physics by things finding a lower energy state by setting themselves at a certain distance apart explained, in the case of atoms, for instance, by the uncertainty principle and de Broglie waves, 'orbits' or processes being closed so that they become cyclic. Matter wave - Wikipedia BOND = CYCLE, CLOSURE. Couples are bonded when each fulfills the needs of the other and they become stable particles rather than free radicals seeking whom they may couple to.
Physics and formalism: capturing the world in language and using the way language talks about itself to model the process of physics, so the action of unitary operators specified by the axioms of arithmetic mimics the supposed action of isolated quantum systems.
Nielsen & Chuang 'The evolution of a closed quantum system is defined by a unitary transformation. That is the state |psi> of the system at t1 is related to the state |phi' of the system
[page 126]
at t2 by a unitary operatorU which depends only on the times t1 and t2, |psi> = U|psi'>. (page 81)
'In the case of single qubits, it turns out that any unitary operator at all can be realized in realistic systems.' Ie physics exploits the whole potential of mathematics.
Between clear pictures of what to do next there are times of confusion when all the possibilities are on the table and being assessed one by one to find the best course of action. The effectiveness of the Lagrangian method suggests that the Universe works the same way and quantum mechanics models the confusion by unitary evolution, the basis vectors (possibilities) being given more or less emphasis as things evolve.
Einstein (Podolsky & Rosen) argued that quantum mechanics was incomplete, in that its formalism did not completely constrain reality. An unsatisfactory explanation, leaving loose ends which Bell showed implied breaches of the assumptions of realism and locality used to derive the Bell inequalities.
Our assumption is that quantum mechanics describes the world before space came into existence, so that locality has no meaning for it. Only later, when we add space and special relativity to get quantum field theory do we go beyond quantum mechanics. From the Shannon perspective, we would say that quantum mechanics is error free because it deals with a world that has only one symbol, the undifferentiated quantum of action.
[page 127]
QUANTUM - TIME
CLASSICAL - SPACE-TIME
Maybe dumbness will pay. The old view that the world was created by a transcendentally intelligent God who made it totally cool and complex may be replaced by the idea that the world started as a totally simple, thoughtless and structureless point that has evolved by trial and error to its present state, so that if we go back far enough we will come to something so simple and obvious that anyone can understand it. Unitary evolution after all, says nothing much more than the the world goes around like a wheel, moving all the time but keeping a constant radius.
What the quantum computation people hope is that even though we get no more data by observing a qubit that we get by observing a classical bit, we can prepare quantum situations which will achieve vast computations in the unobservable realm whose results we can then observe with sufficient resolution to improve overall on classical computation. This is to give a special role to human observers but the reality may be that the qubits are observing one another during this process just as classical bits are observing one another in a Turing machine, so that the transfinite resolution that we assume to exist in the quantum realm is not really there.
The above paragraph is a lot of words trying to capture an idea that came to me suddenly (a 'flash of insight' [from the 'pure light of reason']) and in its mental form seemed very simple and unified, and it was only when I tried to put it into words that it expanded into a long string of symbols. The idea is a
[page 128]
point, with position and no magnitude, but all the complexity of the idea lies not in the 'no magnitude' but in the position in 'word space' which can only be defined by a 'word vector' in approximately 100 dimensional word space. Each dimension of word space contains the whole dictionary. The task of writing is to look at the dictionary once per word and select a word from it to be the next word in the sentence. Sometimes this is easy and obvious, but sometimes one must stop and think (search) quite a while for the right word.
THINK = SEARCH
We construct the space as we search it, creating new possible places to look.
Diamond, Chimpanzee The power of order: The most significant digit is more powerful than all the others combined, so we conclude that the 1.6% of our DNA that differs from chimps is most significant from the cultural point of view, although only weakly coupled to anatomy. Diamond
Linguistic function space: each dimension takes a particular word as its value and if we read these dimensions we get a sentence which may or may not be meaningful. Let us assume that a speaker like me composing a sentence has a particular order in mind which is represented by the placing of these words on this paper.
My most significant bit is the vector Universe (divine, not divine). I set it to divine establishing a context for all the other vectors in my noetic space. [Majorization]
[page 129]
So the Universe, like an embryo, divides internally. From a logical pint of view, we get something like the Yin Yang diagram, half being p and half not-p with no further specification. Given q = not-p this is identical to not-q, q, on the assumption that not not-p = p. Yin and yang - Wikipedia
The via negativa: an hierarchical array of nots [binary tree] can create any structure, no matter how complex, beginning with the most significant not.
From a quantum mechanical point of view the vectors in a Hilbert space are not ordered? But an inner product is a one to one correspondence. When computing the length of a vector we simply multiply each component by itself (or its conjugate). But what about the inner product of distinct vectors? There are n! different ways to construct an inner product of two distinct vectors. The first component of one may be multiplied by any one of the components of the n other, the second has the choice of n - 1, and so one. Do all these give different answers?
. . .
Can we prove that the permuted dot products are always less than or equal to the originals. In the original order big numbers multiply by themselves, whereas in the permuted order a big number is always multiplying a smaller number, so it becomes less significant.
So for reals? Complex numbers . . .
[page 130] Quantum vector is a string of amplitudes for basis states. These amplitudes squared, ie |psi.psi*| give the probability of observing the system in that basis (determined by the observer, since a quantum system has no inherent basis, just a dimension).
Is the world the output of a pre-existent source? Yes and no; yes insofar as it immanates within the initial singularity. No, insofar as requisite variety tells us that the zero entropy initial singularity cannot fully constrain a world with entropy > 0.
We all have the urge to reproduce ourselves, not just genetically and somatically, but noetically, which is what I am doing here now that my child rearing days are coming to an end.
ISOLATED SYSTEM = INITIAL SINGULARITY, so that insofar as a system is isolated, we can take it as isomorphic to the initial singularity. The process inside it can only be observed by an observer inside it.
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