Notes
[Notebook: DB 59 Draughts]
[Sunday 25 June 2006 - Saturday 1 July 2006]
Sunday 25 June 2006
[page 27]
Monday 26 June 2006
There is no mathematically existent (consistent) way that the present can completely determine the future in the Universe we inhabit. This statement itself cannot be made impregnable to objection but it looks as though it is at least as certain as quantum mechanics, which appears to describe the world perfectly, even though we are still only able to apply it in the simplest of situations.
Quantum mechanics embodies the fundamental properties of networks. We then come to apply this theory through Quantum field theory, where we add the simple proviso that communication between points in a network takes time, and attempts to fit this to the traffic actually observed in the micro=Universe.
Perhaps we should say that quantum mechanics describes the computation made within the interactions which derive the output particles
[page 28]
and momenta from the input. What we observe are the input (prepared) and output particles from an interaction. The transformation from input to output is effected by an algorithm which theoretical physicists try to imitate in their calculations.
Creation and annihilation operators.
What we would like to see is that quantum mechanics gives us the most general description of computation in the Universe, and general relativity shows us the overall (statistically averaged) shape of the resulting communication network.
Any force causes acceleration. Any message causes (is associated with) a change of state in both sender and receiver.
Tuesday 27 June 2006
Wednesday 28 June 2006
Thursday 29 June 2006
To be is to communicate. Quid est hoc quod est esse? [what is this which is to be]. There is a strong tradition that the highest state of being is the empty mind, but there seems to be no particular reason for this except the conviction that ultimate reality is absolutely one and simple a la Aquinas. Aquinas himself adds dynamism to God (life) and this implies some sort of complexity, perhaps the complexity of the Universe we see, which is nevertheless dynamically one, while spaced out and complexified by the nature of fermions, and unified by 'network bosons'.
No two fermions can share the same state. This state contains spacetime data as well as the quantum
[page 29]
mechanical states encoded in Hilbert space. The Quantum field theory people devote a lot of energy to transforming states between the 4-space representation and their Hilbert space representation. Are these simply two different channels for the same data, particle (an entity in 4-space) and field (an entity in Hilbert space) ?
We may gain some insight into this by seeing every emission and absorption of a particle (classical 4-space communication) as an example of teleportation. The entanglement which enables states to be teleported exists ab initio, when the whole Universe was one simple state, U(1).
EVENT == MESSAGE dynamic transfer of formal content.
No action at a distance. Always exchange particles.But does gravitation need to exchange particular particles (called gravitons) or might it be a universal feature of any exchange of particles, that is a universal feature of communication?
General relativity is worked out in a continuous 4-dimensional manifold populated with fields of tensors, which may be imagined as 'oracle machines' at every point receiving input from neighbouring machines and transmitting output to them. We thus have an 'infinitesimal network' whose function is relatively independent of the size of the network 'mesh', given a fixed velocity of communication between nodes.
[page 29a]
. . .
General covariance : meaning is independent of language, so that we can say 'food' in an infinity of languages, but to the organism eating, it is all the same.
Paradigm shift: from the harmonic paradigm (Zee page 5) to the communication paradigm, whose most important feature is that it is scale invariant (recursive) and so serves as a paradigm right across the spectrum of complexity.
. . .
Friday 30 June 2006
Saturday 1 July 2006
Many of the problems of classical and quantum physics arise form the continuous paradigm, particularly the 'self-interaction' problems that arise by extrapolating distances to zero and / or infinity. We might avoid this by redefining continuity as 'logical' rather than 'spatial' continuity and seeing the world as a discrete communication network. To make the paradigm stick, we have to be able to use it to calculate the current results of physics in a more plausible manner.
The paradigm begins with the complex numbers, or little arrows. They are the alphabet of quantum mechanics and all the symbolic statements of quantum mechanics use different labels (or names) to represent various distinct complex numbers of the system. In Feynman's picture, we attach a little arrow to every
[page 29b]
point of interest in the space we are exploring.
Then we add an 'algebra of arrows' and an interpretation. What does it all mean? It gets its meaning from the experiments we do to get the data which motivate the development of a theory both to explain these results and point more clearly to gaps in our knowledge where more experience is required.
Interpretation: all our sensations are physically encoded information. We are observers absorbing various particles from our environment and interpreting the result as a soft breeze on a sunny day next to a sparkling river, etc. All done with photons, phonons and molecules. Maybe atomic senses as well - for noble elements.
Insofar as the Universe is observable it is symbolic. And insofar as it is symbolic it is mathematical, taking the broad view that mathematics = symbol manipulation includes logic. The 'inordinate utility'. Eugene Wigner This results from quantization, since independent quanta (fermions) = distinct symbols, addressability.
Does space come from addressability or addressability from space. Four dimensions is the syntax of as certain paradigm of address generation.
NETWORK = {ADDRESSING, PROCESSING} = {MOMENTUM, ENERGY}
[page 30]
Space and time are a 'broken symmetry', ie some features in common, some differences, like time is in some (mathematical) way imaginary, ie t x t = -t2, whereas s x s = s2. It is this break in symmetry that carries us from scalars to little arrows (vectors). Henceforth arithmetic moves from the bottom line to the top line of a complex exponential exp(ix) where x is some operation that can be performed on a function of little arrows. This idea is captured in the 'path integral' formalism, which exploits the 'harmonic paradigm', explaining the world by superpositions of wavefunctions.
More generally, does the world grow like the number system? And so to Cantor.
We then add relativity to this two dimensional Universe to develop the four dimensional Universe of experience via Dirac's equation.
Let us try generating all the permutations of a set first as a lookup table and then as an algorithm.
Like new people on the job, each of us from birth has to learn how the local god(s) work and how to fit in to best advantage (or not, as the case may be).
We want a proof that there are logical strings (theorems) which describe all the transitions between quantum states and which can therefore be executes by a Turing machine or network of same. Given this, we
[page 31]
can provide a network explanation of the forest of particles revealed by particle physicists as well as of everything else. The wave function is said to be deterministic, but is it? There must be catastrophes, poles, etc (at least in the continuum version). What can go wrong with a digital system? Given functioning algorithms, the only things that can go wrong are noise (inputs out of range) or failure of the physical substrate of computation.
The Cantor Universe is no weirder than the path integral method. The Cantor Universe addresses the complex numbers (arrows) that model the process of the world.
Quantum field theory talks about particles in a given 4-space. We wish to see that the structure of 4-space is a natural feature of a transfinite network. We only allow points = states to be different if they differ (are polymorphic) from one another in their vector notation. Every state stores the four parameters of a spacetime address (?)
As tensors are the natural inhabitants of a differentiable manifold, so interactive computing machines are the inhabitants of a transfinite network.
Each peer layer defines and manipulates an alphabet.
Computation time is inversely proportional to energy.
The usual way to reconcile apparently disparate expressions of the same thing is to create a space in which the different expressions can be shown to be equivalent. So we wish
[page 32]
to use the network to connect quantum mechanics and relativity.
In general relativity everything follows its geodesic, unless (like me) it is bonded to a larger body (the earth) and is pushed off its geodesic by non-gravitational forces, ie the demand of every fermion in the planet to have its own bit of space.
After a lifetime of dithering, I must finally force myself to learn enough physics to effectively criticize and amend it. This means slowly creeping along from certainty (sound footing) to certainty. Bonum ex integro What does this tag have to say about path integrals: the addition law of little arrows says a particle will most probably follow the path of minimum action = minimum phase change [with varying path] = shortest (known) algorithm. known = available.
But by wafting around in vagueness I hope to have covered a lot of territory in search of the paydirt that I now feel the need to refine by mapping it back to the old theory, the post-classical limit. As quantum mechanics guided itself by looking to classical limits, so network theorists want to make contact with the quantum field/particle picture.
To quantize a function is to interpret it as an amplitude.
Communication theory does everything with a source and a channel. The receiver listens at the end of the channel, and so is an 'inverse source' or sink.
[page 33]
Sinks want to take in, sources want to put out.
Quantum mechanical phase = degree of consistency, an additive function. So when we superpose waves, we simply add corresponding values. In phase increase amplitude, out of phase decreases it [in each orthogonal dimension]
Global gauge transformation leaves all phases relatively unchanged so that the structure of the superposition is invariant (?).
When we say that |q> = a |0> + b |1>, |a|2 + |b|2 = 1, we say that there are as many different superpositions in |q> as there are values of a (a complex number). Quantum information people think that this may mean that a qubit carries an infinite amount of information (the entropy of its Hilbert space) although we now that a single measurement will yield either |0> or |1> with probability determined by the value of a (or b [which is bound to a by the normalization]) This is a bit sad, but what does it mean? When I download a new program it comes in a serial bit stream usually packaged as a library of functions and a bit of code to use them to do whatever. The complex dynamic structure of such a program during execution has been packaged as a string of symbols. To learn all the details of a quantum state we have to subject it to repeated observation [run it with different inputs] in order to learn its statistical structure and so conjecture at the wavefunctions that constrain this structure. This is more error prone that reading a bit string. But what about wave function, which are supposed to evolve deterministically and unitarily?
Quantum field theory does not explain things very well but can be made to give good numerical results in some circumstances. Elsewhere there are discrepancies between QFT and the observed Universe
[page 34]
of the order of 10120, so we can at least suggest that something is amiss. We want something that gives both a plausible explanation and pretty good numbers as well. QFT worked well in electrodynamics but . . .
We seek to calculate the probability that a certain source will emit a symbol, ie a momentum state.
Yourgrau page 33: '[Lagrange] considered a system to be specified by means of 'generalized coordinates' ie any set of variables sufficient in number to define unambiguously the configuration of the system', eg Cantor Universe. Yourgrau and Mandelstam
Gravitation: We first recognize that since ancient times people have consider the world as a set of appearances controlled by something behind the scenes. In computer terms, the Universe has a graphic user interface and processors in the backgrounds that drive the interface. Plato first proposed this idea scientifically with his theory of forms. Wikipedia: Theory of forms This idea evolved into Aristotle's unmoved mover and Aquinas' God who sees and controls everything. The modern version of this picture is quantum mechanics which proposes a huge complex of mathematical machinery to account for the behaviour of the microscopic particles which drive our macroscopic world. These particles fall into two groups: structure particles (fermions) and communication particles (bosons) that carry structural information from one structure to another. We will return to the mathematical modelling of this system below. Here we first look at the more general philosophy of understanding nature, . . .
[page 35]
in other words an heuristic structure for understanding the Universe, in other words a metaphysics, a general structure to explain all process, ranging from particle phenomena to human interaction. We propose for this role a transfinite network constructed as follows.
We begin with the natural numbers . . .
Does this system have the power to describe the Universe,m ie the cardinality (memory) and processing power?
Change the emphasis on infinity from big to unbounded.
Quantum field theory describes particles in spacetime, using fields in spacetime.
In genetic space, the genomes of living individuals are as rare and far apart as stars in the Universe., Is this a 'consequence' of Shannon's theorems or gravitation: Is this a link between Shannon and gravitation. Fortune favours the prepared state? mind?
Clone, double star. Two stars can only coexist in a small space if they orbit rather than collide.
Introduce the interrupt.
Cold. On interrupt a processor can save its state very succinctly by saving the contents of its registers and program counters. This does not fully determines what happens when the processor resumes its interrupted task. This depends also on the data returned from memory by the processor executing the task.
[page 36]
Turing machine = vertex
Particles: ephemeral (annihilate on interaction) = Turing
Machine
persistent (continue to exist through interaction, ie atom
interacting with photon - persistent = network = universal Turing
machine: we load it with some data and a library and let it run, eg
save, copy, send, connect etc.
There are only ℵ0 Turing machines in a space of ℵ1 possible functions. This situation is reminiscent of McMillan's E=theorem that lies at the heart of the proof of Shannon's theorems. Khinchin
Cantor Universe is a configuration space. Now we need an arithmetic (cardinal) and algebra (ordinal) expression of the constraints on this configuration space. These constraints affect the rate of observation of the states of the configuration space. Some are excluded completely and others appear with varying probability.
Quantum statistical mechanics. How does this relate to communication theory? What is the equilibrium state of a large number of agents interacting (communicating) in a particular way?
Shannon showed the way to jump from harmonic theory to communication network theory by taking entropy from thermodynamics to the statistics of communication. Harmonic quantum mechanics has a foot in both camps, but the area where the legs join
[page 37]
('the collapse of the wavefunction') is shrouded in mystery.
Is quantum processing (at the level of the wave function) real? Will the quantum information be able to get more computing power out of the field than normal people can get out of the particles? Is there more bandwidth behind the scenes than we can see? Or is it a forlorn hope? Landauer's principle would seem to say so. All information is represented physically. What is the physical representation of the unobserved rotations and additions of little arrows in continuous complex infinite space? How many states are represented by a qubit [at a given moment]?
The network way out of this is that when we talk electromagnetically to an atom, we are in fact logging on to a network where the various possible transitions are stored on different servers, so to speak, so any network user has access to all the states in the superposition.
Where are these other servers? First, they do not have to be distinct servers; one server can imitate any number of servers. It is just a matter of addressing and being fast enough to serve the queues without too much chance of upsetting the customers. Nor do they have to be static files: If their complexity is low enough, the server can generate the pages on the fly, using software that must look a bit like quantum mechanics.
Quantum mechanics is non-local [because it does not see space : 'Notes that (0 + 1) dimensional quantum field theory is just quantum mechanics.' Zee page 18] Quantum field theory is local (introducing
[page 38}
a metric of distance (ds2) as well as of probability | psi |2.
The distance measure in computing is card {operations}.
When we have to generate things on the fly and demand is heavy, there is a premium on the most efficient algorithm. In the language of evolutionary theory, the fittest algorithm. But fittest for what? Let us postulate that fitness is a function of network communication. We can guess that one requirement of fit communication is freedom from error. Shannon showed that this can be done, and now we can download gigabytes of information error free.
Shannon showed that to avoid error, we must be able to send symbols that have a very low probability of being confused with one another. The way to do this is to make elements of the alphabet of the message as far apart as possible. Then, if they are knocked out of position by error, there is still an acceptable chance that they will be read correctly. This, we suggest, explains the quantum nature of communication between interacting systems in the Universe.
The power of bosons is their ability to broadcast. Particles having an identical state can propagate through spacetime.
The implementation of Shannon's theorems requires coding which requires computation, which in the real world takes time. The frequency of local system clocks in the Universe is related to their energy by Planck's constant. The time required for computation causes delay in communication, which introduces the spacetime transformations of special relativity.
[page 39]
We think of quantum field theory taking place in 4-space but here we would like to see 4-space as some sort of addressing system which ,maximises the efficiency of the cosmic algorithm/
No correlation without communication, and since we see correlations as the result of laws, we are inclined to see laws as channels of communication and vice versa. We see law as symmetry. Symmetries are also a product of communication, or of initial simplicity, since we expect to find all stages in the evolution of the physical world fixed in the optimal solutions that have risen.
Like Misner, Thorne and Wheeler ('pregeometry') we are trying to go from the theory of communication to an explanation of spacetime. In other words we think the explanation of spacetime lies in quantum field theory rather than the other way around. With the Cantor Universe as our configuration space, we render it dynamic with the addition of Turing machines. We map this onto the world we observe using Shannon's theory. How does this make 4-space? Time seems easy, as it is carried out by sequential, that is ordered events. In my rest frame my life goes forward event by event, each event at my scale being the result of many, sometimes huge numbers of quantum events. The frequency of these events is measured by energy, so my basic processing frequency is mc2 / h, where m = 80 kg, c = 3 x 108, h [= 10-34, giving my processing frequency as approximately 1053 quantum operations per second].
Is it true that every event, no matter what its scale, is associated with one quantum of action, so that the information content of a quantum of action (information represented or carried by) can vary from (say) 0 to ℵ0 bits.
[page 40]
Whatever we manage to cook up in the way of network quantum theory, we need to keep the overlap idea and probably the general method of quantum mechanical computations using the real numbers (as in probability theory) as a tool for dealing with things in the limit of large numbers.
