Notes
[Sunday 5 April 2009 - Saturday 12 April 2009]
[Notebook: DB 66 Turing Field]
[page 95]
Sunday 5 April 2009
Why do we observe a quantized Universe? Because a continuum is not observable, having no observable features. This idea is somewhat related to Descartes notion of clear and distinct ideas and the mathematician's struggle to develop clear and distinct ideas about the continuum, culminating in Weierstrass, who assumed in effect that the continuum is made of clear and distinct points, eg the Bolzano-Weierstrass theorem. Bolzano-Weierstrass theorem - Wikipedia
Every now and then a clear and distinct idea seems to pop up and I write it in block capitals.
Creutz: Quarks, gluons and lattices Creutz He works out a theory on a lattice and then shows that it still works on the continuum limit. His lattice occupies a Euclidean (complex time) hypercube in 4-space with N4 sites and examines the differences between each pair of sites. (page 14) We would say that these differences are caused by the exchange of messages and so the lattice looks like a network, each site a node, each difference an edge. What we would like to do is see this geometrically imagined lattice as the product of a logical lattice based on the ideas that coded messages have a particular nature and a delay in time, and that three dimensions of memory are required to allow fully random access non-interfering (ie error free) communication. [insofar as a continuum is not observable, the fact that we can go to the continuum limit says there is no information communicated by this model?] Quantum information theory tells us that
[page 96]
we can vouch for the integrity of a message from a known transmitter because any interception by a third party will make observable changes to the message. Nielsen & Chuang
The whole edifice of calculus is based on a lattice whose spacing a is taken to approach zero.
Einstein showed us how to relate geometry to dynamics. Dynamics is the science of the world out of control, that is the kinematic restraints do not condemn the Universe to total immobility (as some of the ancients thought about god) but leave degrees of freedom and each new dimension of constraint adds a new degree of freedom. The geometry is thus the fixed (observable) points of the dynamics, and the unresolvable continuum of motion between the fixed points cannot be observed. The fixed points are connected by the dynamics and so form a consistent system. Assuming 'universal' (unconstrained) dynamics, we assume that there is no limit to the number of points (messages) in the Universe, but that they obey certain protocols whose only constraint is consistency, ie no dead ends where a decision is impossible because of conflicting information. (There is always insufficient information to completely constrain the future, hence dynamics.)
Veltman points out that conceptually quantum field theory is a mess. VeltmanThis mess would seem to arise because continuous mathematics is too tightly constrained by the assumptions of continuity to represent the full variety of the real world. Feynman shows us how to join
[page 97]
snippets pf continuus formalism into a system of discrete events (vertices) which has sufficient variety to describe the world as it is. Each vertex is a source in a network of exchange of discrete messages (particles, 'virtual' or 'real' = observed).
Transfinity is a relative quantity defined by [a break in continuity and] the signs <, >, greater and lesser than. Although we think of the transfinite numbers as very large, there is no reason why we should not set ℵ0 = 2 as an example. In fact it works (I think) so long as we make ℵ0 greater than 1, which in the digital world is at least two.
A system can be both stabilized and destabilized by its environment, depending on how well it fits in.
Creutz Chapter 5: Fermions
'there are a set of 4 x 4 Dirac matrices satisfying the algebra
[gammamu, gammanu] = gammamugammanu + gammanugammamu = delta mu nu.
gammamudagger = gammamu
The dimensionality of space must have a cost. The benefit / cost ratio for message transmission in space peaks at three. But if this is so, why is Hilbert space infinite dimensional? Because 3-space contains realities, physically encoded messages, whereas quantum mechanics is just formalism = possibility. [?]
FORMALISM = POSSIBILITY
[page 98]
Monday 6 April 2009
Quarks couple to all for forces which suggests a certain complexity, as does their number and variety of bosons [gluons]. We might say tht gravitation has zero particles [one particle, the whole Universe?] not being quantized. Then electromagnetism is 1 the photon, weak 3, W and Z, strong 8. 0, 1, 3, 8.
Getting little glimpses of the relationship between communication network and space-time, ie between theology and physics.
Philosoper's block: starting out enthusiastically along a line of thought only to find it fading into a mirage.
Tuesday 7 April 2009
Space is somehow created by the logical operation of the Universe. This creates objects like atoms which we know to have a fixed size and a Universe which is expanding relative to such fixed sizes.
Since gravitation couples only to energy, the expansion of the Universe, that is the creation of more space-time must be a consequence of the existence of energy which we like to be continuous and indeterminate of itself. [does the fact that energy is conserved mean that it is digital?]
On the other hand the definite size of atoms, nucleaons, etc is a quantum phenomenon which we have to explain by a combination of logic / communication / computation.
[page 99]
The essence of theretical physics is to dream up ordinal sets or algorithms which yield measurable cardinals (counts) with ever improved accuracy and ease. The 'good' models are those which deliver precision results that can be verified.
EXPLANATION = ALGORITHM = THEORY
The network explanation explains everything and nothing insofar as it is a network of all possible Turing machines - all possible algorithms, all possible explanations, so we still have to choose which explanations are relevant at any point in the network. In quantum field theory this choice is equivalent to a Feynman diagram. Feynman diagram - Wikipedia
Space-time is analogous to a dynamic Venn diagram of the local dynamics of the world?
Wednesday 8 April 2009
Statistical mechanics studies the cardinal numbers associated with ordered sets of states represented in Hilbert space. Hilbert spaces used in quantum mechanics range from one dimensional to those whose dimension is the second transfinite cardinal, the cardinal of the continuum from the point of view of geometrical mathematical physics. Cantor opened our eyes to the existence of an infinity of transfinite cardinals beyond the second, and we might look to this as the foundation of transfinite Hilbert spaces of any cardinal. Such a transfinite hierarchy of Hilbert spaces may serve us as a theological phase space, that is a complete set of states. We imagine the dynamics of this space to be
[page 100]
the coupled unitary evolution of Hilbert spaces of ever increasing transfinite cardinals. We would like t think that the symmetric network was large enough to represent this idea in detail.
Physics: dream up a complex ordered formal structure whose evolution is constrained by the cardinal of the Universe (space) in which it moves.
A point is the dual of a space. We are points within an isolated transfinite Hilbert space evolving unitarily. Uncertainty is imposed upon the formalism by the requirement that the sum of the probabilities of possible events must be 1.
Here we come to a very simple model of Dirac's delta, it is a spike aleph(n) units high and 1 / aleph(n) units wide whose area (read probability) is 1. Area = integral. So we introduce a hierarchy of deltas, one corresponding to each transfinite cardinal aleph(n). This same formalism establishes a scale of entropy and information so that I am a point in a sace transfinitely larger than the space of an atom.
This system is vastly complex as a represented by physical calculations, but we also suppose that this complex system was born out of the simplicity of the initial singularity. Since this is an isolated system we model its interior by the unitary evolution of a transfinite hierarchy of Hilbert spaces.
[page 101]
Physics studies the dynamics of this system under the constraint of continuity, that is restricted to a cardinal measure (since all cardinals are unique, and so unmeasurable. A measure implies points [of] similarity, like counting sheep. Counting any things, that is counting events as categorized in some way). Fermions are like this, every one unique and exclusive, making room for itself by the exclusion force. Huang, White Dwarf. Huang
A canonical ensemble is a set of non-interacting possibilities - possible configurations
Quantum mechanics exists independently of space-time. It is a formalism represented in Hilbert space which has the property of completeness defined by convergence or unitarity. Every possibility must be taken into account, but as the possibilities are transfinitely diluted, what actually exists becomes a point of ever increasaing information in a space of ever increasing entropy.
So we begin by equating the interior of the initial singularity to the life of God.
Quantum mechanics may be imagined as wave packets propagating through and interacting in 4D geometric space.
CALCULATION == PROOF 32 x 21 = 672 etc,
Statistical mechanics atempts to derive the macroscopically observable properties of the interaction of microscopic systems.
[page 102]
Invariance with respect to complexity (entopy) means the interface between two transfinite cardinals (= sets of ordered sets all of the same 'length'.)
Huang page 65: 'The assumption of molecular chaos' = all states have the same a priori probability.
When physics comes to a dead end we turn to theology. The physical question is how is the mind of the Universe mapped onto spacetime, and the answer may be in the same way as our internal states are mapped onto our body language.
In statistical mechanics as in communication theory what counts [what is countable and counted] is a state.
Social thermodynamics derived from social statistical mechanics.
Lunatic but realizable, insofar as social statistical mechanics is computable = deterministic = predictable.
If we are to control our destiny we must anchor ourselves to the predictable features of our environment. [eg that the sun is with us for the forseeable future]
The probability of a message is its product with itself |phi|2 = <phi* |phi>, that is the probability of it being received and acknowledged, thus creating a closure of truth. When a particular message (eigenfunction) does not come through, some other member of the complete set of messages will, with a certain probability. We assume that the sum of the probabilities of the possible constituents of any event is 1, which
[page 103]
tells us that something always happens (when it happens).
The probability |phi|2 seems to be evidence for the 'full duplex' view of universal communication. Fo it is only by recursive conversation that two sources can reach agreement.
Thursday 9 April 2009Friday 10 April 2009
Still up against it. Insight is a random event like (we keep saying) the 'collapse of the wavefunction'. There are two dimensions of probability in quantum mechanics relating to what will happen and when it Will happen. What is governed by the pi = |psi|2 and when by the half lives for decay and events in the network for interactions (collisions etc). Whereas the nature of a local system will decide what it does, including decay, the question of when involves the environment, that is the history of the event.
Unitary evolution (rotation) in an infinite dimensional Hilbert space ('the transformation theory') lies at the heart of quantum mechanics. Dirac Each cardinal equivalence of Hilbert spaces and the operators on them are isomorphic and can be mapped to one another one-one, ie given equiprobability of states they all have the same entropy. This process is equivalent to encoding and decoding messages and has nothing to do with meaning, operating as the fundamental hardware layer of the Universe. Meaning arises when higher layers use this hardware layer for their own purposes. Here enters the channel through which isolated quantum systems communicate with one another. The theory of quantum communication and
[page 104]
quantum computation need make no explicit reference to real spacetime as long as we keep it abstract and formal. When we want to build a space-time extended real quantum computer propagation delay and error enter the picture.
From a spacetime point of view, quantum mechanics is one dimensional and this dimension is not one of the other four spacetime dimensions, but something which anteceded them and exists in its own right, but which contains the germ of 4D space, ie it can be used to multiply the number of such independent quantum systems by separating them in a spacelike way (fermion like way) and allowing them to communicate with one another like the Trinity which seems to be the apogee of ancient and medieval theology. They were not ready to see that god and the Universe are one.
