Notes
[page 191]
Sunday 7 December 2008
The purpose of 'distributions' 'generalized functions' is to make certain that every function is differentiable. Griffel page 14. Griffel This is in a way equivalent to computable, the minimum step
[page 192]
in digital computation being one logical operation. A proof may be viewed as a continuum of logical operations which are nevertheless points in that each may be considered as happening instantaneously.
Point particle = point operation. We can integrate a series of similar operations (steps) to give us the overall distance travelled; we can differentiate a path of equivalent steps (a 'continuous' path) into individual steps.
[This suggests that we define continuity as 'repeated performance of the same operation'.]
In distribution theory the smooth test functions operate as a space in which we find the delta functions, rather like the smooth space in which Democritus' atoms move.
Griffel page 22: 'The advantage of our theory over ordinary calculus is that every generalized function is differentiable.'
Wolfe, Bonfire: 'Then a great sociological truth dawned on him. All religious credentials are arbitrary, self proclaimed. Who originated the articles of faith on which his own boss, the Episcopal Bishop of New York had been ordained? Did Moses bring them down in stone from the mountaintop? No, some Englishman dreamed them up a few centuries ago, and a lot of people with long white faces agreed to call them rigorous and sacred.' page 143. Wolfe
Any model of the world that cannot describe a party, with multiple channels of communication, public and private, joining multiple people, might be considered too weak. Veltman
Its a party in my brain, all of me, a really big party.
[page 193]
Every string of mathematical symbols can be realized as a computation.
Insofar as things are not quantized they carry no information, and so can give us no insight into the working of the world. We observe, and can only make theories about, quantized events. nevertheless the resolution of a quantum [symbol] is not always easy. We must come through the fog to get near it and see it clearly, like a signpost on a foggy highway. The art of designing scientific instruments is the method we use to get nearer our targets, but what we see has a lot to do with what we are looking for, and our search is guided by current theory, which is not always a good guide.
'Competition was as obsolete as democracy. The twenty first century would be the age of the cartel, the unforgettable era of the new monopolists, when men who were strong and far sighted and, yes, ruthless, would control the globe. And Jose Bermudez would be one of them. Beginning tonight, at dinner with the jaded Colombian patriarch.' page 273 Hiaasen & Montalbano
All of physics comes down to solving differential equations. A differential equation defines the local behaviour of a function, and we integrate it to get the global behaviour. The local behaviour of a Turing machine is defined by the point in its program being executed and the overall effect is the integral of all these steps.
PROOF = INTEGRAL
STEP IN PROOF = DIFFERENTIAL
[page 194]
The epsilon - delta argument is tantamount to making the steps n a calculation smaller so as to more precisely define local behaviour.
Always seeking first for understanding and then using the mathematics to express the understanding; but future understanding depends upon past understanding, so past mathematics can help future understanding.
Bosons are cardinal, in that many can go into the same state without any differentiation so that all orderings are the same. ie all permutations are the same from a bosonic point of view. On the other hand, permutations of fermions are pairwise exactly the opposite of one another, as indicated by sign and Pauli exclusion.
Continuity is a property that requires definition of metric. We might say that series of sentences is continuous if they only differ by one letter at a time so adjacent points have a minimum difference.
LINEAR = MEANINGLESS
Power: ability to influence the behaviour of another. Since we all have the same bandwidth, the key to power is abstraction, that is variety reduction. By reducing the variety of others I can control them and so have power.
Q: is the symmetric Universe a vector space? - it is a union of vector spaces, one for each layer.
Q: Has the symmetric space a metric? Yes, hamming distance.
[page 195]
Banach space = complete, normed vector space.
Spinors in symmetric space?
Each step in the execution of a program is the application of an operator to a (possibly trivial) vector to yield a new vector (which may differ at only on point from the input.)
Continuous = separated by an indiscernible distance.
Monday 8 December 2008
Quantum mechanics (as a mathematical theory) is in effect 2 stages removed from reality compared to classical mechanics. In classical mechanics our measures and counts fed directly into the formalism and the outputs of the calculations are directly applicable, to we measure force and acceleration and use F = ma to arrive directly at the mass. These results (apart from measurement errors) are considered deterministic rather than probabilistic.
It is otherwise with quantum mechanics. First, all measurement is achieved by binning and counting the continuum only appearing in the parametrization of the bins. We make a measurement by filtering a system to get a sample of particles in the same bin (state) and then letting them interact and counting the various outputs. The quantum mechanical formalism is continuous and deterministic but predicts only the statistical distribution of the counts between the various bins. What would be nice would be to develop a logical formalism which relates more directly to the outcomes, but ut seems hard to see how this can be deterministic.
[196]
Segments of the universal process, represented by Turing machines, may be deterministic, but all such machines in the real world are 0-machines, connected into a network and so subject to interrupts that redirect them and destroy their determinacy. Oracle machine - Wikipedia
The volume of an infinite dimensional vector space is infinite as long as the lengths of the basis vectors is greater than 1. The volumes of successive transfinite dimensional Hilbert spaces is successively greater.
The axioms of isolated quantum systems describe a continuous system which becomes apparently discrete when it is observed. This is analogous to the apparently continuous processes in my unobserved mind which appear in my observed consciousness as discrete words ready for writing down. All this is easy to express in English, difficult in mathematics. The mathematics of quantum mechanics selects the observed spectrum out of the continuous process. The stable points in the spectrum are the ones that map into themselves under the action of the relevant operators, so that for them the operation is equivalent to no operation.
Statistical distributions can arise from discrete well defined problems that have multiple solutions in line with Gödel's incompleteness result.
All of continuous mathematics, as such, is concerned with measure, that is cardinality, treating all points in the system as units without any personality of their own. The break toward quantum mechanics comes when we introduce discrete dimensions, implying discrete eigenfunctions and eigenvalues, which differentiate the
[page 197]
cardinal values by giving them addresses in vectors, matrices, etc In quantum mechanics each memory location is analogous to Feynman's little arrows and it is capable of storing a complex number which may be a function of the space-time position of the memory location. Feynman The quantum mechanical processor reads in two vectors of memory locations and computes their inner product as a scalar whose absolute square represents the probability of some event.
We imagine this event as a two way 'conversation' between two vectors [states], the transmission of data from b to a being represented by the complex conjugate of transmission from a to b so that the product of the two directions gives the probability of the complete message which in some way corresponds to the halting of a Turing machine.
Griffel page 156: '. . . essentially all of the calculus can be generalized to arbitrary normed spaces.' ie calculus can deal with anything that has a metric and convergence. A discrete logical operation may this be conceived as a fixed point in the space of continuous transformations. Where these transformations are parametrized by time, such fixed points are stationary states, placing logic (= formalism in general) outside time. The transition from one fixed point to another is a achieved by the combination of clock and an ordered set of operations (which can be mapped onto the natural numbers, the program counter) resulting in a computing machine. This development in mathematics might mirror the development of the real world from continuous confusion to clear and discrete insight.
TO DB 65 SYMMETRIC U
[Notebook: DB 65 Symmetric U]
[page 1]
Continued from DB 64 GRAVITATION
In Hilbert space we add angle and inner product to Banach space, so making it the home of quantum mechanics. Distance is measured by the inner product of a vector with itself, ie the measurement of distance requires parallelism. Independence is represented by orthogonality, ie the inner product = 0. The distance between independent vectors is thus 0, rather than infinity as we might expect.
Quantum mechanics decomposes the natural line into an infinite dimensional Hilbert space, each dimension (orthogonal to and independent of all the others) representing a point in the line, reflecting the Cantorian rather than the Aristotelian definition of continuity. Aristotle thought things continuous only if they had a point in common. The ordering of the basis vectors in the Hilbert space corresponds to the ordering of the points in the natural line.
Hilbert space: complete inner product space with basis Griffel p 181
Gram-Schmidt process is analogous to the operation of intelligence in determining the alphabet of a process. Griffel page 183.
Riesz Representation Theorem: For every continuous linear functional f on a Hilbert space, there is a unique u is an H such that f(x) = (x, u) for all x is an H.
The dot product is a variety reducing mechanism capable of reducing finite or infinite vector to a single (complex) number rather like all the internal machinations of a computer giving one
[page 2]
numerical output.
All the cardinal computations in the Universe can be accomplished by the unions and disjunctions of sets and by counting. Ordinal operations, like formulating a reply to a question in English, cannot be achieved this way, but require parallel computation in networks?
An equation is a constraint whose solution(s) satisfy the constraint, eg x2 = 1.
A degree of freedom may be seen as quantized 'on the outside' while being continuous 'on the inside' like a dimension of a Hilbert space.
We can represent all the reals by ordered sets of natural numbers ('decimals') to base 10, or in fact using any base 2 =< n < ℵ0. Using ℵ0 symbols, we can represent numbers up to ℵ0ℵ0 = ℵ1.
Thus ordering carries us from the ℵ0 naturals to the ℵ1 'minimal' reals, and we shall assume that it can go on to do the same job for any aleph(n) --> aleph(n+1). This is how order creates complexity and is the intuitive foundation of the Cantor Universe:
Out of ℵ0 proceeds by a definite law, the next greater cardinal number ℵ1, out of this by the same law the next greater, ℵ2 and so on. But even the unlimited sequence of cardinal numbers
ℵ0, ℵ1, ℵ2, . . . , aleph(n), . . . does not exhaust the conception of transfinite cardinal number. We will prove the existence of a cardinal number which we denote by aleph(omega) and which shows itself to be the next greater of all the numbers aleph(n); out of it proceeds in the same way as ℵ1 out of ℵ0 a next greater aleph(omega + 1), and so on, without end. Cantor, page 109
CONTROL = {KNOWLEDGE, ACTION}
= {CONFINEMENT, ACTION}
= {FORM (MOMENTUM), ACTION (ENERGY)}
Slipped again; No! This approach deals in
[page 3]
principle with some of the problems currently confronting physics.
Lifetime survival is a set of momentary survivals which is the set of all operations (however microscopic) that contribute to an individual existence.
The new model does not have to fit the old model specifically, but the data. Insofar as the new model and the old model describe the same reality, there must be compatibility between them. So we describe the new model from first principles and then make some connections with the data and the old model.
Whenever we see structure we suspect control, and the aim is to lean the control mechanisms of our structured world so that we can fit in with it and survive.
Symmetry with respect to complexity means that in some way my experiences of life give me some insight into the life of an electron: we both share some sort of life at least.
This all looks beautiful to me. Its my baby. What I want to do now, what I am trying to do, is fit the symmetric Universe to the data by listing some of its properties.
its principal property is the ability to create. This property arises from the formal, mathematical and logical properties of the universal process, which gradually increases the resolution of its control through the power of error correction and ordered sets.
Symmetry is built in (to the symmetric network). Every group is there in possibility.
[page 4]
The actual puts constraints on the possible. The actual can be addressed by the countably infinite set of the natural numbers. The possible is represented by the transfinite numbers aleph(> 0), the continuum.
We all have a continuum, that is the region beyond our resolution, the 'transparent' machinery that keeps us alive.
In particular, the model predicts smooth and organized operation at all discernible levels, unlike the violent vacuum proposed by quantum theory. The vacuum insofar as it carries no information and so is continuous (ie beyond any possible resolution, given the uncertainty principle) . . . unresolvable by any knower, and so irrelevant to both physics and human knowledge.
One can only act with the finesse that one can resolve. One must be able to measure the action to the world to get the desired outcome. The ideal action is reversible, yielding as much information as it took to do it, yielding as much value, ie worth it.
Large numbers need precise resolution, so the recursive growth of the transfinite network requires recursive resolution.
The fog of creation. How the initial singularity made itself into the Universe.
We turn Cantor's quest around and define the cardinal of the continuum as 2 raised to the power of the largest resolved
[page 5]
cardinal.
Mathematics assumes that physics can be carried out with mathematical perfection, but that is before we take the limitations of communication into account.
One would expect the world to represent its own eigenfunction s symbolically as computable (and computing, computed) entities.
The resolution of a computation. Even the world computer is resolution limited by the uncertainty principle.
'The eigenvalues of the operator A correspond to the possible frequencies of vibration.' Au = au.
All we have to do is connect spectral theory with network computation and we are done.
. . .
Abuse of language, a useful technique whereby the meaning of terms is stretched beyond their strict definition, in a way which can be understood by using common sense. Griffel.
We'd be fucked without abuse of language, since it may be a force for growth.
Everywhere we need the 0 for completeness. And the infinity too, the continuum.
[page 6]
Heisenberg: if you cannot observe it do not worry about it and you cannot observe the continuum [by definition!].
Griffel page 277: 'The set of all eigenvectors belonging to one eigenvalue of an operator is called the eigenspace of that eigenvalue. Its dimension is called the multiplicity of the eigenvalue. An eigenvalue of multiplicity one is called simple or non-degenerate, an eigenvalue of multiplicity greater than one is called multiple or degenerate.'
A theorem is a process that gets us from an input to an output.
'Commuting operators: if A and B are compact self adjoint operators which commute (that is AB = BA ) then they have a complete orthogonal set of common eigenvectors (that is vectors that are eigenvectors of both A and B).'
Simultaneously observable = spacelike separated?
Dynamics: differential equations involving time derivatives.
Thinking: encoding the continuum of experience into communicable packets.
Observation - Halted Turing Machine - there are countable observations.
Tuesday 9 December 2008
'A symmetry operation . . . of a physical system is a correspondence which yields for each physically realizable state
[page 7]
phi another phi', such that all transition probabilities are preserved
In other words the symmetry operation does not change the relationship (angle) between phi and psi, simply changes their context without affecting them. Such might be the case when we insert a two dimensional entity like spin into a 3D space.
The simplest system we can observe and talk about is one with two states, say p and not-p. There is really nothing to be said about a 1 state system.
Our principle of symmetry with respect to complexity in communication theory and quantum mechanics suggests that the problem the Universe has in creating itself from the continuous energetic fog of the initial singularity is the same as I have trying to create a coherent theory of everything from the energetic fog of information that I have assembled over my lifetime, and that both processes appear the same as one walking through a thick fog toward a complex structure and gradually discerning (creating in herself) more details as she gets closer.
In some ways I feel that I am outside the human race until i succeed in understanding myself and the world in my own way and 'making my fortune' rather like our forebears felt it necessary to make their fortune before marrying and having children. Probably rarely the case, since in general it takes calm deliberation to avoid having children, not something common in young sexual relationships.
Weinberg: 'John Wheeler has predicted that when we eventually know the final laws of physics, it will surprise us that they weren't
[page 8]
obvious from the beginning.' Feynman and Weinberg, page 63. Feynman & Weinberg
Quantum mechanics is a method for computing the weights in the cosmic neural network. We are trained by our past to predict the future.
Symmetry with respect to complexity is another step in dethroning humanity from the centre of the world, since it shows that our intelligence is nothing special but just a part of the global process of cosmic expansion, that is cosmic complexificiation.
Degrees of modulation: carrier, music, speech, digital.
Feynman's Thesis Feynman
Potential = hardware (stationary)
Wednesday 10 December 2008
Thursday 11 December 2008
We have two criteria for the goodness of a model, internal consistency, beauty and so on, and fit. The standard model does very well on fit but is a bit weak on consistency and beauty. We are therefore inclined to retain the gist of the calculations while replacing their justification with the network model. Since Feynman diagrams look like networks, this should not be too hard.
The network model gives the operational detail of universal process and we can use the symmetric network (and the
[page 9]
myth of instantaneous communication, omniscience and omnipotence (ie the god's eye view)) to describe the Universe as a sequence of local states.
Having established the god's eye view we now come to explore this structure from the point of view of an inhabitant.
1. The structure is locally deterministic, ie quantized
2. There is a minimum delay, related to the velocity of light, associated with each action/operation in the network. To us physicists living in the symmetric network, this delay corresponds to the quantum of action, measured by Planck's constant whose dimensions (to a physicists in the system) are energy.time. The more energy we put into an operation, the more quickly it happens , E = h bar omega, where 2 pi omega is a completed action.
The god's eye view is equivalent to the formalist view which is based on the propositions
1. each symbol is a distinct addressable entity, ie a name
2. symbols are eternal
3. operations are instantaneous.
4. Cantor's theorem says that the god's eye view is self contradictory, like the set of all sets. Both god and perhaps physicists must therefore be local, dealing with subsets of the whole.
LOCAL - BOUNDED SET - SUBSET of the Universe.
Rule of law vs rule of a person, symmetry vs authority.
[page 10]
Since it is clear that the standard model is 'onto' something, we wish to present the same calculations justified in a different manner. At the heart of quantum mechanics is the dot product which, in infinite space, involves integration and the effective destruction of detail by subsuming all the discrete dimensions of a vector into one sum of products. But did we have to differentiate the products in the first place? What if we look at the dimensions of Hilbert space as separate channels of communication?
We solve a lot of problems. By replacing cardinal continuity with logical continuity we do away with lots of the infinities arising from zero denominators. Feynman diagrams fit the network model and it should not take long to transform the path integral method into a network too. Digital logic has natural UV and IR cutoffs, so there goes the cosmological constant problem. And we can explain expansion, complexification and creation and all that stuff too, while placing ourselves squarely in the universal swim.
Friday 12 December 2008
An equation puts a constraint on the world and if we have enough equations constraining the world simultaneously we can constrain the world to one point, a static solution such as engineers and other builders seek. The symmetric network as a space contains all groups and all functions. How do we cut it down to the world we see? We take the view that the computations and constraints of physics, chemistry and biology are good since they give good results, even if their foundations may be a bit weak. We may take the most general features of the world, as seen from inside, to be creation, annihilation and the conservation of energy within a general trend toward expansion and complexification.
A weak spot in the quantum theory is the transition from the deterministic (computable) evolution of the wave function to the random output of observations. In a practical network, everything, insofar as it happens, is deterministic, ie guided by definite software, but the instances that the software has to deal with are relatively random, depending for instance on the actual digits dialled by a caller and whether or not the addressee of the call is available, etc. The randomness in a network is a product of the independence of its nodes [when they are not communicating with one another - time division multiplexing]
Identical particles = instances of the same process in different contexts. How do we understand the difference between fermions and bosons in the symmetric network? We begin with a single boson state with all the energy in the world = initial singularity (or vacuum?) and see how it expands.
All the equations of motion in the symmetric Universe are implemented by Turing machines and thus comprise all computable functions culled by natural selection. [All motions are motions of Turing machines?] All Turing machines, in turn, can be reduced to sequences of nand operations.
Symbols = text = forms are stationary in the Shannon sense that they are isolated from their neighbours, and so can be perturbed without changing. How does this link to the stationarity of the Lagrangian? Dirac, The L in QM. Dirac
For any action at a point in the Universe (eg me) the vacuum (= god) is all the unresolved environment around me which, when
[page 12]
it does resolves itself into a message (eg a phone call from someone with a plumbing problem) moves the action (= me) in some direction, ie finish coffee, load up tools and head out. From my point of view, my environment is the superposition of all possible messages that can come to me and I, the measurement operator, only understand and receive those messages when they are in my language, ie my basis, ie one of my eigenvectors.
Spooky action at a distance acts to preserve the integrity of the network so that no two strings are ever perfectly alike, ie degenerate into one string. This is one way of looking at it, but how do we account for creation and annihilation? The symmetric network is fermionic and creates and annihilates features of itself by exchanging bosons. A fermion state is only occupied if it has energy.
Weinberg: '. . . it is widely believed that it is impossible to reconcile quantum mechanics and relativity, except in the context of a quantum field theory. A quantum field theory is a theory in which the fundamental ingredients are fields rather than particles; the particles are little bundles of energy in the field. There is an electric field, there is a photon field, and so on, one for each truly elementary particle. Feynman and Weinberg, page 78-79.
Why? Is this a consequence of continuity?
Weinberg page 79: 'Now the numbers, [which define particular particles] the energy, momentum and so on are simply descriptions of the way that the particles behave when you perform various symmetry transformations . . . From this point of view, at the deepest level, all we find are symmetries and responses to symmetries. Matter itself dissolves, and the Universe itself is revealed as one large reducible representation of the symmetry group of
[page 13]
nature.'
Lagrangian density tells us how fast state vectors revolve (energy density).
Weinberg page 88: 'All terms in the Lagrangian density must have units [mass]4, because length and time have units of mass and the Lagrangian density integral over spacetime must have no units.' (?)
page 89: '[. . . ] that is exactly what we are looking for: a theoretical framework based on quantum mechanics and a few symmetry principles, in which the specific dynamical principle, the Lagrangian, is only mathematically consistent if it takes one particular form. At the end of the day we want to have the feeling that 'it could not have been any other way'.'
page 90: Standard model has 17 free parameters: he is looking for numbers, not forms.
page 91: 'I have shown you that the condition we require in order not to get uncontrollable infinities when we calculate physical quantities, is that the coupling constant that describe the strength of the interaction should be dimensionless , - the units should not be a negative power of mass.'
Why is mass so important. Why do we have the dimensions, M, L, T? We say c = 1, so L = T. The we say h bar = 1, so L = T = 1/M, but why pick M as our unique dimensions, why not T? E = M and E = 1/T.
Newton's constant G (in h = c = 1 units) = [M]-2.
page 93: 'Most theoretical physicists today have come around to the point of view that the standard model of which we're so proud, the quantum
[page 14]
field theory of weak, electromagnetic and strong interactions is nothing more than a low energy approximation to a much deeper and quite different underlying field theory.'
Why do the behaviours of things have anything to do with energy at all? Insofar as energy equals frequency, we would imagine that higher energy means just doing the same things faster, as when we ramp up the clock frequencies of our processors.
But things do change with energy, so what can the network model say about this? When I enter a high energy environment, I am likely to fall to pieces when the energy becomes comparable to the binding energy which holds me together, and once I become unbound, my behaviour changes. So higher energy physics corresponds to more unbound [less complex] physics.
Weinberg page 93: 'We have two indications that nature will reveal simplicities only at an energy that is higher than the energies that we can now explore. One of these indications is the fact that, if you project the coupling constants of the electroweak and strong interactions upwards in energy from the energies at which we are currently measure them, you find that they all come together. . . .
The digital computations of an information processor are dimensionless in the formalism, but the physical process that underlies them has physical dimensions. We are inclined to let the halting of a Turing machine correspond to a quantum of action which has the dimensions of angular momentum, something suggestive if we think of cycles in terms of recursive functions and closure. When frequency of computation becomes energy and particles with invisible internal computations become massive. Mass, length, time,
[page 15]
energy, momentum are all connected by special relativity.
Our fundamental heuristic principle is that we and the fundamental particles we study [are all inhabitants of the same Universe]. The formalism of communication theory, like most mathematical theorems is indifferent to the cardinal numbers involved so long as they remain countable. There is a difficult boundary between countable and uncountable numbers which we attempt to bridge with analysis. Our bridge is, however, unsound. We act as though uncountable infinities in the world are self representing, like the countable numbers. In fact, insofar as they exist in the Universe, they are represented by ordered sets of sets of countable symbols. In other words the Universe represents uncountable numbers in the same way as human mathematicians learned to do 3 thousand or more years ago.
Cantor said that the endless hierarchy of transfinite numbers is created by a unitary law. Cantor, page 109 That unitary law as we read it here is that if you have a set and want to make a larger one, establish order in your set and then generate all possible orderings (permutation) and you have your larger set. This works as long as the cardinal of your original set is 3 or more, since 2! = 2. Let us call this unitary law symmetry with respect to complexity.
We observe in the world that in many cases reality reduces the number of stable or viable permutations by some process of selection. This selection is a consequence of our ordered representation theorem, since there are far more representations possible than can be actually realized with a countable number of letters [symbols] there is competition among the representations for the resources to represent themselves.
[page 16]
The hypothesis that the continuum is physically, rather than logically represented has caused a lot of pain.
Our assumption is that since, at any transfinite level, the possibilities are continuous with respect to the materials for their representation the same competition goes on at various levels leading to similar solutions. So we suspect that our role in the cosmic network at the human level can give us clues about what happens at other level. it is customary to think that the gods lead rather human lives and so we suspect that humans lead rather electron, atomic, molecular and cellular lives.
It may be that all the physical constants evolved to their present values very quickly in the very early Universe when it was very simple. We have only a few fundamental constants like G, e,. c, h, k etc and each must have once been a degree of freedom which very soon found an optimum fixed value in its peer Universe.
On entering the Universe we find that we are subject to the same constraints as every other particle. Physical constraints. It was dawning human awareness of these constraints that led us to ideas of immortal souls, angels and ourselves as angels trapped in vile bodies one day to be freed when the Universe was returned to its true nature in the last days. Loverly, but unrealistic. A picture which has misled and defrauded billions. Catholic theology of the body - Wikipedia
Every physical structure in the Universe is a numeral, an organized ordered struture representing a numerical value.
[page 17]
The flexibility of mathematics is that anything can be mapped to anything provided only that they can both be uniquely identified. But there are symmetries in mappings, so that many can be represented by one symbol.
ORDERING - BONDING - WRITING, Putting the letters in their place. All interactions are local, and the idea of global 'fields' can only be explained by symmetries represented locally.
Saturday 13 December 2008
A Turing machine permutes itself. Formally it need no driver, but when we make it physical it uses energy like anybody else - not so much uses energy as creates entropy. Quantum computation is said to be reversible and this is true within a peer layer. But as soon as we introduce observation we are working in a more complex layer formed formally by a tensor product and although the operators remain unitary, the increased cardinal of the space is tantamount to an increase in entropy, and there is no going back. We see all this done by Turing machines and place the boundary between past and future at the (local) point where a Turing machine stops. In a normal computation the machine is continually stopping (at the completion of each operation) and then using its stopped state (in a formal representation of its history) as the starting point for its next move, just as I do. Some take the fundamental oscillation of the Universe as a step between formal and dynamic, to be represented by the Hilbert Oscillator so called. The stationary period lasts for less than the clock pulse and the rate of computation is governed by the frequency of the clock which physically means its energy.
[page 18]
So the passage from the symmetric network to the real world is analogous (isomorphic) to the transition from Turing machines to real computers.
We might say that there are isomorphisms within peer levels and homeomorphisms [similarities] between levels. Look it up. Isomorphism - Wikipedia, Homeomorphism - Wikipedia [ie this is not correct, we need a looser sort of similarity]
To keep the computations while giving them new meaning we have to map quantum field theory into the symmetric network which can possibly be achieved by mapping it into two layers and then invoking Cantor symmetry to carry it through the whole system
Mason, Wong p 218: '. . . that only goes to support my contention that the act of making love is of no intrinsic importance and that its importance depends entirely on the point of view.' Mason So, in the evolution of human sexuality, fucking has served as the hardware of love and the Christian plan to make love without hardware is a Platonic error that must eventually lead to its downfall.
Mapping to quantum field theory.
1. Quantum mechanics as such fits network well
2. Delay, encoding, relativity and causality
3. Two way communication and Dirac's symbol <a |b > = <b |a >*, ie complex conjugation = unitarity = duplex = reversible.
4. propagators
5. Spin/statistics.
[page 19]
M, L, T are all reals, rather like the dimensions of a vector space whose transformations and equivalences are contained in velocity and action. Physicists make these two dimensionless, so that we write v = 1 = L/T.
So T = L. Similarly we write h bar = 1 = ML2T-1 = ML so T = 1/L. Field theory seems to find mass as the fundamental dimension, but it could be length or time, as I incline toward time because proper time is fundamental in relativity and seems to antedate (belong to a lower level) that space-time. If we give time the dimension 1, we find T = L = M = 1, something like the initial state of the Universe before it diversified from T to L-T to M-L-T. Mass arrives with the development of stationary particles, ie closed structures that map onto themselves unlike photons which in some way map onto the whole Universe, or we might say massless particles are the Universe mapping onto itself.
The essence of a good bank is mass. Like the central black hole stabilizing a galaxy. Let the mass of the Universe be 1. It cannot be measured because there can be no observer outside the Universe.
Orgasm for intellectuals. Orgasm and insight: not for nothing did the old timers coin the phrase carnal knowledge (the hardware of formal knowledge). Each tick of the clock is an act of insight [a quantum (or Quantum) of action].
There are two conditions for interaction 1 = time sequence correct; 2 = protocol correct,
Delay --> time sequence --> ordering and causality.
[locally all sequences are time sequences]
But there us a lever where there is no time sequence, no causality, just simultaneity brought about by zero distance. This is the quantum world.
[page 20]
The fundamental hardware oscillator that drives the world is represented by the sentence possibilities are probable, and we compute probabilities by counting possibilities.
Permutations are nested like the dials on a petrol pump and we can probably write a short recursive bit of code that generates all the elements of a permutation sequentially, abc, acb, bac, bca, cab, cba .
Clock a, b, a, b then ab, ba, ab, ba, the 3 permutation.
Novel: an interesting series of coincidences: survival = manipulating probabilities, finding good bets. Love story.
TV drama: theories of human action and motivation.
Possibility = formality
Probability = Reality
Louder:
POSSIBILITY = FORMALITY
PROBABILITY = REALITY
Wong page 245: 'Nothing can be more romantic than a man looking for a certain girl and refusing to settle for any other', ie going for the big payoff at long odds rather than small and short. So in general rare events are more exciting. One of the skills of show business is the regular and efficient production of 'rare' and 'exciting' events.
[page 21]
Only those things can be represented for which there are sufficient symbols, and the number of elementary symbols in the Universe is, according to the model, countable.
SYMBOL = COMPUTER = TRANSFORMATION (velocity)
The probability oscillator first manifests as c.
Computers both encode and transmit. When we divide things into sources and channels the channel is the computer, the source the input, the output the sink (= source).
CHANNEL = COMPUTER = PROOF. So a photon is proof carrying information from source to sink. There are therefore ℵ0 distinct channels in the Universe, ie distinct encoding each of which may have ℵ0 different inputs.
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