Notes
[Notebook: DB 59 Draughts]
[Sunday 2 July 2006 - Saturday 8 July 2006]
Sunday 2 July 2006
Monday 3 July 2006
[page 40]
Tuesday 4 July 2006
Pais: Dirac (Goddard ed) page 18: "One was faced with a crisis" how to cope with a theory that works very well approximately but makes no sense rigorously.' Goddard Still true (Veltman)! Veltman
The beauty of messages is that they are naturally born and die so fitting the observation of life in the Universe and (at least to this extent) to quantum field/particle theory.
The entropy (and so the information carried) is a function of the space in which it is carried. Shannon showed that by creating complex particles (in a product space) we can make the particles (like shoals of fish in the ocean) further apart fro the same message and so less likely to confusion ++> error.
Stock market: each stock is a particles (a scalar represented by a single number, the price) existing in a space comprising the actual enterprise (say BHP), physical interactions with the world (industry, environment, etc) and the psychology, resources etc of its shareholders and the general public. From this point of view, even
[page 41]
though I have both won and lost on the market, the experience gained is in a way beyond price since it is the most abstract representation of human industry.
exp(iH) is describing a periodic succession of generations. The process continues as long as the communication between generations is adequate to specify new generations (children)
The simplest processes are such that the parents die to have children.
So we recognize an axiom: particles and spaces are two expressions of the same thing.
Network fits Feynman diagrams.
Class = peer group (we only communicate humanly (personally) with ourselves). The rest are chattels.
A key requirement is a network version of evolutionary theory. We can see it happening in the internet, the foundation of whose growth is the mass production of computers and other information handling hardware. To handle things we need an identification and an algorithm: run those males into the top paddock.
I am trying to get from a hope and a feeling to a mathematical, or more generally, unambiguous formal expression. In the age of Dirac people still expected the Universe to be fundamentally quite simple, and the particle zoo revealed by postwar accelerators was something of a surprise. Now
[page 42]
we suppose that this zoo came or comes about through evolution in a way analogous to but much less (?) complex than biological evolution. (When we look at the bandwidths available in high energy particle interactions, things could be pretty complex.
Things do not couple = do not overlap = do not interact. In this language, the electron and the photon overlap by 1 part in 137 = e2 / hc. Why? I am so far from understanding this but must try.
How much do we have in common = how strongly we bond together (?).
The qubit maps most naturally onto a spin 1/2 particle both because of its duality, but also because the exclusion principle which separates otherwise identical qubits into 'registers'. The quantum origin of space has something to do with spin. Dirac's step (one of many) was to go from two component to 4-component entities. 4-component entities are adequate to describe 4-space, all we need is an adequate cardinality in each component and a transformation expressed as an algorithm.
How much space does a fermion occupy? Momentum maps to space ad spin is angular momentum. . . . Angular momentum involves two lengths which can define an angle by the dot product. Angular momentum is an attribute o a processor, how fast it is going around. It can be measured instantaneously, basically a count
[page 43]
of how many processes with spin h bar are operating in parallel. Energy, on the other hand, is the product of this parallel processing rate with frequency to give total rate of throughput.
What are the software equivalents of the electron and positron wavefunctions. See quantum picture book. Brandt
wavefunction = {amplitudes}. Superposition is the addition of corresponding amplitudes. The correspondences may be addressed by the spacetime coordinates of the corresponding points. Spacetime is the database structure for the cosmic computing system.
SPACE - MEMORY - CONFIGURATION. Any processor using some memory experiences a delay between sending a request for the contents of a particular location and receiving the requested information. Using this delay as a metric, we can determine how accurately a processor can respond to information stored at a certain distance. If information storage capacity varies linearly with the volume of space and the rate of communication is inversely proportional to spatial distance we see that the available information varies as the square of the delay in retrieving it.
Dirac Pais page 28: 'The rules of renormalization give surprisingly, excessively good agreement with experiments. Most physicists say that these working rules are therefore correct. I feel that is not an adequate reason. Just because the results happen to be in agreement with observation does not prove the theory is correct.' Goddard
Strong words.
Lagrange intuitively obeys Chaitin's law, requiring enough generalized coordinates (memory addresses) to describe all the degrees of freedom of the system under study, To make the transfinite network physical, we enliven it with lots of little rotating arrows whose mathematical expression is exp(i/h H).
Dirac Atiyah page 123 Herman Weyl: 'Only with spinors do we strike that level in the theory of representations [of the orthogonal group] on which Euclid himself, flourishing ruler and compass, so deftly moves in the realm of geometric figures. In some way Euclid's geometry must be deeply connected with the existence of the spin representation.
There are ideas (structures) and representation of structures. Every structure and every representation is an ordered sot, and all possible transformations between representations can be be represented by an element of a permutation group of the requisite cardinality.
Wednesday 5 July 2006
The communication model is metaphysical in that it tells us very little about what goes on but gives us a 'heuristic structure of proportionate being' as Lonergan put it. Lonergan, page 416 So if we see two people saying the same thing, we are inclined to loo for a common source or 'cause' of the similarity, so NETWORK <==> CAUSALITY. The theory of probability can tell us from the overlap of two sources the probability that they derive from a common source, and the theory of relativity can tell us whether or not such communication is possible, ie whether the two correlated sources are in the light cone of come potential original source.
[page 45]
From this point of view, the network model has to explain the fundamental constraints on quantum field theory, but the actual details of quantum field theory depend upon the actual messages being transmitted in the network and the algorithms by which these messages are developed from previous messages.
DETERMINIST = DISCRETE
Epsilon-delta arguments proceed in discrete steps by a simple algorithm, and we assume they go to infinity since we can see no reason why not in the formal world. In the real world, things may come to a halt when we come up against a quantum of action which cannot be divided. Nevertheless, the formal argument may be continued in the way mathematics represents real numbers by using a string 3.1515. . .
Legal fiction and the ancient model of god : it is so because I have wanted it and said it. We have the fiction that initially the 'crown' 'owns' all the land (many other meanings may be given to the variable sin inverted commas). Any reasonable and adequately powerful crown that takes this view would injure itself if it tolerated the existence of any other entity that claimed the same ownership of the same land. Some fictions are better than others, and natural selection sorts them. Physicists, looking at their models and data, discern that in all things a quantity called action is associated with every event, large or small, and that those events seem to be selected for which the action is stationary (tradition) or in particular minimized, ie the Universe uses minimal resources to achieve its end.
[page 46]
So we start with two resources, structure and action. We use action to build and destroy structures, structures to guide action. And we hypothesize that action appears where structures are inconsistent, ie it is 'attracted' by or associated with trouble, or as we shall say (?) error. Better maybe difference, since this fits easily into the sentence 'act to settle our differences'.
We associate quanta of action with quanta of structure, and attribute the whole quantization thing to the fitness conferred by accurate communication. Rulers and collectors of statistics are courting failure if they do not seek true information about their environment, even if they use spin or even lies to strengthen their position when communicating the information collected to their subjects or other potential benefactors and malefactors,
Structure and action are most clearly associated in the story of spin and the distinction between fermion (half integral) and boson (integral) spin and statistics.
A Turing machine is a function, or subroutine. We initialize it and await its return. It may call subroutines of its own, and one need not place a priori limits in the depth of calling, although there are obviously machine limits.
We can see the effects of relativity when we model a real time process on a distributed network of processors. DISTANCE is proportional to delay, where the constant of proportionality is speed or velocity.
[page 47]
Tomonaga g = |magnetic moment| / |angular momentum| Tomonaga
Tomonaga page 14: 'When there are two magnetic moments . . . in an atom, there should be a magnetic interaction between them [ie communication by quanta of information in the magnetic channel] and therefore magnetic energy is added to the usual energy of orbital motion . . . '
We can only count discrete objects, so the cardinal of the continuum is a count of discrete objects, not a continuum at all. No wonder the continuum hypothesis is independent of set theory. Cohen
vacuum
The relationship E = h bar omega enables us to measure energy very precisely because we are able to measure temporal and spatial frequencies by counting which is, given discrete countables, perfectly accurate, at least to +- 1 in any count no matter how big.
For astronomers, gravity is the ultimate source of energy. Nature Proga 441:938. Proga
Gravitation - metaphysics - love
Formal systems become real by the attribution of value to their constructs so that e can imagine them competing in the basis of their value in different situations. The fundamental value is fitness, ie the ability to [live and] reproduce. We can imagine that (other things like lifetime being equal) thing that reproduce quickly are more common, ie the probability of encountering one is high.
[page 48]
Communication at a given peer level requires all the lower levels to be working down to the physical layer [henceforth to be known as the material layer].
If we want to write out all the elements of ℵ1 using the elements of ℵ0 as an alphabet, we need ℵ1 copies of each element of ℵ0. Or, time sharing, in which case each element of ℵ0 will have to be used ℵ1 times. This is easy to see in the finite world, where we need 6 As, 6 Bs and 6 Cs to write out the six permutations f A, B, C. Here the ratio ℵ1/ℵ0 = 6/3 = 2.
The simple unitary group U(1) gets us nowhere, just giving a ticking clock until its symmetry is broken by being used as part of a more complex structure that has distinct clocks running at distinct frequencies 2 pi radian = 1 step to nowhere [our material analogue of materia prima Aristotle 191a6]
ACTION = PHASE d(action)/dt = energy, ie d(energy.tie) / d(time) = energy. Oscillators are distinguished (names) by their frequency and phase. Phase obeys modular arithmetic x + 2n pi = x. Congruence mod 2 pi. Trigonometry is pure arithmetic, and its value lies in its fit to the world, to classical Euclidian space.
dot (inner) product encodes angles . . . . Sop much to learn but it will be worth it if it helps me express my ideas with one foot inquantum mechanics and one in metaphysics (theory of communication).
COMMUNICATION = TRANSFORMATION (Dirac) Dirac
Phases add (and subtract)
[page 49]
We can imagine a continuous wave, but it is hard to see a whole lot of numbers interacting.
If all information is encoded physically, the information that defines the physical nature of the actual information carriers must also be defined physically. So a photon defines a photon and that is that? We use discrete arguments to prove that the Universe is discrete because if it is not discrete argument may not hold and we move to the continuum. But here we are prone to error because the elements of the continuum are in some way indistinguishable and liable to confusion when they are close together.
continuum = mystical / discrete / practical.
We measure distance in actions, like the rotations of the wheel on a pedometer (or whatever they're called).
Particle = phenotype
Field = genotype
Thursday 6 July 2006
Sinead O'Connor: I do not want what I have not got
God grant me the serenity to accept the
things I cannot change
The courage to change the things I can
And the wisdom to know the difference. Francis of Assisi (ref ?)
Sad songs = error correcting music. Political, social,
[page 50]
personal errors.
In a moving Universe with delay there is a necessary lag as the solutions to the problem of survival gradually diffuse through the system by conversation between individuals sometimes formalized as education (leading). The basic educational paradigm is follow me (or else) and where teachers were few and students many led to the construction of groups of followers (disciples). Through history we see a strong link between education and discipline. Tradition is not enough, however. On top of it we need an ability to respond to changing conditions, in our case the gradual expansion of physical channels of communication on the globe, leading to the situation that given the software, anyone can talk to anyone.
The minimal public manifestation of human trade is the cashflow(?). From this we can see where something is happening and look. This is too global. Our model should be the immune system ,,,
BIOLOGY - IMMUNITY - CONTROL
We assume that since every peer level comprises a set of discrete entities it has computing power in proportion to its cardinality, and so each peer level is transfinitely more agile (flexible, has more states) that its predecessor. . . .
[page 51]
Our first step toward reality is to map the transfinite network onto the evolutionary paradigm.
A Turing machine is equivalent to an operator and we think of the logical process as a chain of Turing machines and n the mathematical way, we epsilon=delta this into a continuum of Turing machines.
The art of programming is to break the problem down into orthogonal alphabets of subprocesses.
Corresponding to the chain image of continuity is the combinatorial image of infinity which enables us to [construct infinite sets with finite means].
We say the steps in a formal Turing machine are instantaneous, ie 1/aleph(n) in the peer level n.
All that we require for the transition from one peer level to the next is a growth algorithm that yields an exponential gain on its input.
We can understand infinity as not so much big as endless, and so the foundation of some sort of induction based on whatever transformation (like adding one) gets us from one generation to the next.
We can find the theory of evolution in Peano's axioms.
It is a gift from the gods that groups can be realized with matrices, and the group of Turing machines. Is this a group? Turing machines erase and are not therefore
[page 52]
reversibles [I am shivering too much to write and must go for a walk}
Why did we in vent complex numbers? So that equations like ax2 + bx + c could have solutions even when (b2 - 4ac < 0. In this case the two solutions are complex conjugates of one another . . .
This allows a solution to the eigenvalue equation for any physically realizable operator, and so creates an alphabet of eigenvalues and eigenvectors which facilitates error free communication of quantum states ?
What does this say about little arrows? I decomposes the arrow represented by ax2 + bx + c into a product of two arrows . . .
The constraint of multiplication allows three scalars a, b and c to define two arrows (4 scalars) because of redundancy, both sharing -b/2a.
We can represent an electron by a string of symbols sigma(e) e running from 1 to something which gives us enough entropy to establish a correspondence with each possible personal state of the electron. When we put this electron in spacetime we have to add some more symbols to the string, say sigma(s) (for spacetime) so the total string representing an electron is sigma(e) + sigma(s), where the + can mean either concatenation of strings or addition of entropy, since the entropy of spacetime can be added to the
[page 53]
personal entropy of the electron. The resulting string gives us a complete description of the electron in rather an abstract sense. We can imagine this is abstract 'phase' spaces, one representing the spacetime position of the electron, the other its personal state. [The spacetime position allows us to compute the probability that the electron will interact with other particles in its vicinity] And so for all other particles. But there is at least one problem: there is no absolute spacetime to give an address to the electron. Rather we need to se the address string as primary(and perhaps part of the electron personality) and the existence of spacetime as a consequence of the unique personality of every electron (fermion). How does an electron 'know' where it is? it can only do so by communicating with other electrons and storing the answers. This communication takes time and we can use the time as a measure of the distance between electrons.
Probability of an event is determined by a field deployed in spacetime, and this field is a superposition of oscillators giving something like an infinite dimensional sea surface. This is the harmonic version.
What are the chances I will talk to you? This depends on our 'charge', the amplitude for interacting with one another. If love at first sight happens, the amplitude will have an absolute value of 1. Hate at first sight is still a communication, and so it too might lead to a high probability of communication, but repulsive. In the middle we have indifference where the chances of interaction are low. We might do these calculations with vectors ('personality functions') whose dot product measures the probability of interaction.
What is the significance of the different methods of computing probabilities in classical and quantum physics?
[page 54]
Why do we fall in love? As a consequence of some sort of fit: you've got what I want and vice versa. At root, we need eachother to breed after parthenogenesis lost out to sexual reproduction. But there is a lot more to it than that, involving both similarities and differences which combine to give a relationship with a closeness (distance) (ie charge) conformable to both parties. In physics we attribute bonding to a lowering of energy. The analogy of energy in human relationships is tension or insecurity induced by the probability of loss of fitness. Cooperation may be used to increase fitness and reduce stress (stress-energy tensor).
Strain : how far one is deformed from an equilibrium (resting) state y the prevailing level of stress. Two can live as cheaply as one, so they only have to work half as hard.
Does the quantum way of computing probabilities have some meaning that we have not yet fathomed? I hope that it has got something to do with the theory of communication but remains a hunchinsh thing. One point we note here is that Turing machines are not reversible because they erase (?) things, while quantum processes work at constant entropy and so are reversible. The place to try to understand this seems to be quantum information theory which is really just a different way of looking at quantum theory. Here we prepare states delicately rather than with atom smashers and try to isolate them so that we can achieve 'local unitarity', or we can use error correcting algorithms to keep the process unitary = conservation of probability.
[page 55]
In the quantum world, structure arises from interference (superposition) as two superposed sine waves give a beat note, and in general by creating Fourier series to represent various functions.
So what is really going on in the world? For me I was born, have grown and absorbed a lot from the world around me, done may things to assist the survival of myself, my in and friends. Basically, I have always been doing enough to keep going and sometimes even to get ahead. So I have a holiday, buy a car, fix up the house, have a party, etc. I am also mysteriously motivated to work on this project for myself and everybody deriving a model for survival that embraces the advantages and avoids the defects of the Catholicism I was born in. Acting to live [= living]
Do electrons have needs? : I will only exist if I can get 0.5 MeV of energy. I must talk to someone. I cannot go near you while we have the same spin.
Dirac says write beautiful equations and make them simple if you can. This involves a metaphysical assumption about the design of the world. I more concretely, would like to model the world on village gossip and the various perturbations and fluctuations that go with it.
Friday 7 July 2006
The Cantor Universe gives is a large space of symbols to play with and we must select between them to find the world of experience. We would like this selection process to end up with something isomorphic to quantum field theory, though this would be too
[page 56]
[much] to hope for (yet it is a motivation of this work, to reveal in some way the wonders of god). Selection is bases n value, decided (in our world) by a market. So few works (out of billions) have been chosen by the art market to be 'worth' more than $100 million. This certainty has something to do with the intrinsic worth of the objects, but a lot of it may have to do with the financial environment and investment value of art.
What we assume the Universe is selecting for is survival, so (eg) of all possible velocities of light, the one we have is most common (over macroscopic distances) and so the fittest. The fitness may arise simply because it is common to the whole Universe and so provides a common protocol for linking space, time and causality. Causality we see as a foundation for intelligibility : why did you fall over? She pushed me.
'Select natural religion for that natural advantage' : this is generally acknowledged by the correlation between scientific knowledge and national wealth.
Theories (skills) are valued for their ability to show is how to achieve desired outcomes, like getting he gold out of the clay. A metallurgist, faced with a new refinement problem, narrows the field of search for a functional method using past experience, chemistry, quantum theory, etc etc,.
We formalize these ideas by looking for short and effective algorithms, like Feynman's QED : page 85: Feynman
[page 57]
Feynman page 85: Action #1 : A photon goes form place to place
Action #2 : An electron goes from place to place
Action #3 : An electron emits or absorbs a photon.'
Or, more generally: I go from place to place; A book goes from place to place; I read/write a book.
At present I am most interested in what we mean by going from place to place. Given an absolute coordinate system, I might say I erase myself from one place x1 and write myself in place x2. In the relative Universe I say I erase my relationship to the entities at x1 (say my house) and write a new relationship to x2, (say my car). This takes force and energy. In classical and quantum physics, relationships are expressed as potentials.
For physicists, the selection mechanism seems to be 'formulate a Lagrangian for the general coordinate system of interest and then find its stationary points, which will be realized in the world. This introduces some notions of 'cost' and 'value' into the mix. The biggest question here seems to be 'why is the Lagrangian kinetic energy - potential energy' ? If we seek a minimum stationary point this means maximizing potential and minimizing kinetic, (ie maximize capital and minimize spending?).
Siegel page 1: '. . . recent progress {toward finding a satisfactory theory which is consistent with quantum theory and general relativity] has indicated that the restrictions imposed by quantum mechanics on a field theory of gravitation are so stringent as to require that it also be a unified theory of all interactions, and this quantum gravity will lead to predictions for the their interactions which can be subjected to
[page 58]
present day experiment'. Siegel
A 'massive' particle is a 'self-contained' network, whereas a massless particle is a non-network component of a network. Massless particles travel at c, implying no 'process' to slow them down.
Inertial = incomunicando
HALT = MAKE a DECISION = EMIT a PARTICLE: (or ABSORB) Formally, read or write.
Einstein, like Newton, had to simplify to succeed.
The $64 question: will this increase my fitness?
So, at least in the quantum picture, we cannot observe an inertial frame without destroying its inertiality because every observation transfers momentum. Observation in classical theory is thus a repulsive force because it is akin to throwing lumps of mud at one another which stick and the recoil of both throwing and catching is away from one another. As the mass of the inertial systems increases, however, the deviation from inertiality by communication becomes relatively smaller. Nevertheless 'radiation pressure' is a real feature of the Universe.
At the quantum level, however, 'observing an inertial system from an inertial system' is a contradiction in terms which convects to Landauer's proposal that all information (and so all communication) is encoded physically.
[page 59]
In the general theory we observe the effect of gravitation as 'geodesic deviation' between inertial frames. To observe their geodesic deviation, the frames must communicate with particles of a certain momentum, p. The precision with which we can measure this deviation depends on the de Broglie wavelength of the particle . . . . The emission and absorption of these particles will disturb the frame by a certain amount, introducing uncertainty into what was a convergent classical theory.
On the graphic user interface/processor picture, what is the connection between the GUI and the process? Quantum mechanics says that it is not deterministic, even though the hidden 'evolution of the solutions to the wave equation' is considered to be deterministic. This result is particularly puzzling when we consider that there is a lot more information in the quantum process than in the particle 'on the screen' which would seem to be overdetermined if anything. Yet it is asserted that the wave function controls only the probability of events, not the events themselves.
Is an electron with different energy a different particle?
Meanings are scale invariant. Two little words, 'hydrogen' and 'Universe' point to objects whose size differs by 1080 or so.
Gravitation : what does general covariance Additive set theory alone. Quantum mechanics expands to vector algebra.
[page 62]
Saturday 8 July 2006
Quantum mechanics position operator is jut the coordinate x ; momentum operator is 1/h partial d/dx; energy operator is Hamiltonian.
Let us say that space is created by the exclusion principle, space and time in fact. Ie created by fermions, rather than fermions exist in space?
Quantum mechanics works by arithmetic which can be done by computer and works by (count the world in various degrees of freedom); (calculate with the counts); count the world to see if the calculation mimics reality). Quantum mechanics counts events and calculates the probability of events.
Does the Universe, like arithmetic, begin with an 'atom' = indivisible unity? A message is a particle and we can count messages between different personae.
I seem to be going around in circles, or if it is a spiral, it is a very fine pitched one.
So we ask: What is the natural language of the Universe, and what sort of conversation leads to the spacetime we know and love?
What about arithmetic, and the sentence a + b = c ?
Lonergan and his tradition give 'psychological' explanations of the Universe based on introspection and the actions of 'conscious' agents. [an 'anthropic' approach] Intuitively easy.
[page 63]
Using a standard definition of continuity, cardinal(continuum) = aleph(>0), we postulate that aleph(>0) Turing machines are needed to compute he evolution of |q> = a|1>
We must start our model with two sources (transmitters and receivers) . . . which emit packets (particles) of n (random) bits, so that the entropy (log to base 2) of the sources is n and the number of different strings is 2n. This is not a conversation because the output of S1 is not correlated with S2.
Now let us assume that the exchange between the Si is a conversation. What follows? CAUSALITY. My next utterance depends on the whole preceding conversation. If we want a clear conversation we must introduce error correction = Shannon = packet space entropy > entropy per packet.
[page 64]
Feynman arithmetic : adding arrows; multiplying arrows = repeated adding.
DISCOVER = READ
We choose permutations [to develop our versin of the Cantor Universe] because they put us straight into group theory. Also because we believe in no cloning, and it has held consistently since the Universe was just one state as it split into its current tree of states, no two states in the Universe are the same, as is the case with the strings that represent the Cantor Universe.
Splitting of states. The foundation state U(1) is split by combining different elements of U(1) into paths (permutations) that can be designed to lead us to every point in the group by paths whose length ranges from 1 (go directly to jail) to the cardinal of the group (ie for non-intersecting paths) ie no two visits to any element.
The Universe stays still by the coordinate system moving with the moving system.
So we invent the lattice of Turing machines and we allow loops as a process is repeated (but hopefully with different parameters). At the bottom of the hardware we find the little spinning arrows, and we work out how to connect them by logic. Every clock pulse we freeze everything. The way things are at the end of every clock cycle is the starting point for the next step in the process.
Turing machines work instantaneously, so that the ones that are
[page 65]
going to halt do so as soon as they are started, whereas the ones that are not going to halt go on forever.
A non-halting Turing machine = dark matter.
Can we turn dark matter into light matter by interrupting it?
We can introduce communication theory into physics by a lattice type regularization called a transfinite network. The network begins with a set of lattice pints in one to one correspondence with the set of natural numbers N. We say N is infinite, but here that means not so much big as 'size is irrelevant', we can always add 1 to n is a member of N and our proofs, based on induction over the numbers, will still be true. So we can imagine quite small lattices, but must not constrain them when they want to grow. So this lattice has card(N( = ℵ0 points.
We can do arithmetic in this lattice by grouping the points into sets (subsets of the lattice) and combining these subsets in various ways like addition, multiplication, exponentiation and their inverses.
Cantor showed that there were a class of operations which one could perform on sets which would produce new sets of greater cardinal number, even for infinite sets like N. He thus showed the existence, consistency, of a set cardinal ℵ1.
General covariance: to keep saying something in endless different ways.
The simplest 'processor' or process in the Universe. Turing machines, while they are doing their stuff, are incommunicando, (to page 68)
[page 68]
(from page 65)
(ie inertial). We only communicate with them at the beginning and end of their process or function.
Only redundant things are compressible.
One needs all the eigenvalues To reconstruct a transformation, so one needs all the inputs to and outputs from a deterministic black box to figure out what is going on.
To start the Universe, we use U(1) represented by the permutations of the natural numbers. We are not concerned with the cardinal of the group, although not much happens in 0 and 1 and the fun starts at 2. The processor we want is something that can recursively produce permutations.
The simplest linguistic or formal model making more complicated things out of simpler things. Each halting (productive) Turing machine may be considered an eigenfunction in this group, determining which strings are computable (and so useful in the universal process)
INSIGHT = seeing an eigenvector of a problem, in other words an element of the basis for explicating the problem, as electrons and photons are bases for explanation (elements of the alphabet of particles).
Infinite: all our arguments are blind to size, ie 'complexity invariant' cardinality invariant.
Perhaps the heart of knowledge is that we can measure complexity with a scalar, as in the inner product.
[page 69]
Each empty set is the complement of a complete Universe of sets, ie the empty set is outside the Universe : after me the nothing.
Dirac page 21: ',,, any state of our dynamical system at a particular time may be specified by the direction of a bra vector as well as by a ket vector. In fact the whole theory will be symmetrical in its essentials between bras and kets. Dirac
The exclusion principle tells us that two electrons trapped in a small space must be pi out of phase. (see pp 60, 61)
As the Universe grows more and more mathematical structure, ie structure structure, can be realized, moving up through more and more complex groups.
The Universe must be a group by definition, insofar as no operation can lead outside it, although there are non-halting operations that go nowhere.
. . .
Creutz page 6: 'A lattice formalism emphasizes the close connections between field theory and statistical mechanics. Indeed the strong coupling treatment i equivalent to a high temperature expansion. The deep ties between the disciplines ore manifest in the Feynman path integral formulation of quantum mechanics (Feynman, 1948, Dirac 1933, 1945). Feynman In Euclidian space, a path integral is equivalent to a partition function for an analogous statistical system. The square of the field theoretical coupling constant corresponds directly to temperature. Thus the particle physicist has available the full technology of the condensed matter theorist.' Creutz
And from there to network communication theory.
