Notes
[Notebook: Transfinite field theory DB 56]
[Sunday 25 April 2004 - Saturday 1 May 2004]
Sunday 25 April 2004
[page 70]
Monday 26 April 2004
The 'perfectly prudent particle' is concerned for its personal welfare at all scales of time. Only the Universe fulfills the definition, since it lives forever.
Functional Analysis Riesz: a function of bounded variation is one that does not map and point of its domain aleph(n) into any point of the range aleph(> n), except at a maximum of aleph(< n) points? Riesz.
All the events of the universal alphabet are of the same size, measured by Planck's constant. All our other physical measures are based on this unit, the 'plank'. The plank maybe decomposed into the orthogonal [non-commutative] dualities energy.time and distance.momentum which handle the processing rate and memory entropy of the Universe.
[page 71]
In mathematics everything is measured by 1's (units) and we construct larger measures (the natural numbers) by aggregating ones together. Peano's axioms specify a machine for making natural numbers.
The passage from natural numbers to real numbers in mathematics in some way models the transition from the planck to specific values of various natural constants (e, g, c, k) which maybe considered as points in a continuous space. Perhaps not. Rational would do from a measurement point of view, but what about formal constants like e, pi? They are limits (1 + 1/n) n --> e, n --> infinity, etc.
What is e? A communication rate __ > force (or at least an element in the calculation (by a physicist or the world)) of the force acting in given circumstances.
In Hilbert space, the 1 condition is achieved by normalization. Let us say that as general relativity is the dynamics of geometry, quantum mechanics is the dynamics of arithmetic.
Quantum information theory is conceived in binary, whereas fully fledged quantum mechanics as 'alephzeronary', so we use ℵ0 square matrices rather than 2 x 2's. But this is only the beginning. The matrices (and vectors) grow like the Cantor Universe. All based on countable communication (classical channels)
Quantum teleportation: here we use a classical channel to select quantum states. A quantum channel can carry the information necessary to select classical states? Can we say the quantum world is invisible, all we see is the classical world which is put together by the machinations of the quantum world, which fill the whole of function space (computable and not). The classical world so conceived in computable, the
[page 72]
full quantum world is not, but it does compute, using the transfinite network machine.
Hypercomputation: transfinite network machine.
Maybe the real numbers are 'artificial' and the real jump is from natural (rational) to complex (real coefficients).
= acts as a communication link, telling us that two items have the same cardinal number.
The invariance of insight with complexity.
Romance: the sexual attractions almost guaranteed, but there are many other transfinite potentials to be satisfied. In a freely mixing community, we can imagine that the goodness of the transfinite fit is measured by the duration of the relationship. of course the arrival of a child converts this into a three body problem and may influence the length of the relationship. We might consider sexual potential and the countable (physical) basis of romance, but living together and reproducing introduce another set of physical potentials, which we might call the facts (ie needs) of life. All this is part of the overall algorithm, stay alive, grow, reproduce.
Tuesday 27 April 2004
Wednesday 28 April 2004
Thursday 29 April 2004
How do we get from the ℵ1 permutations of the natural numbers to the metric function space? We represent each real number by a function which is in turn represented by an ordered list of ℵ0 natural numbers. Since the N [naturals] have an order, we can set the functions representing ℵ1 in dictionary order and then relabel them using decimal numbers. Finally we can normalize the relabelling to a maximum of one without losing any information. Or, in the Hilbert space representation, we can normalize them
[page 73]
all to [an absolute value of] one so that the information encoding the distinction of the functions is not encoded as points in a line but as points on the surface of an n-sphere, which are in turn isomorphic to a set of ℵ1 unit vectors in n-space all with different directions.
The relabelling is entropy (cardinality) preserving, but what does it to to the order?
c number is commutative.
q number is non-commutative [matrix]
If we assign every word in the language to a dimension < ℵ0 then every sentence in the language is represented by a point in this space which represents this sentence as a vector. What about word order?
normalization : from the entropy point of view, words, sentences, novels etc do not have size, only probability of occurrence.
Are we working here toward an explanation of the statistical interpretation of quantum mechanics.
Hilbert space is a vector space which has
a) scalar product (a property of cardinal arithmetic)
b) addition and subtraction of two elements
c) inner product [constructed from product and sum or integral]
All of these processes can be performed term by term in the ordered sets (vectors) We keep coming up against the problem of why quantum mechanics is expressed in 'complex' numbers. There must be a natural explanation.
Complex numbers are necessary to allow all algebraic equations to be soluble.
What we are looking for is the physical/metaphysical
[page 74]
intuition that connects the numerical world of physics with the meaningful world of language. The link is networks and communication. So we want to think of quantum information and computation as the foundation of the whole global process.
Why complex numbers? Because they give us a handy way to write solutions in the wave equation, which are periodic functions, which are parametrized by a phase which is related to time by energy and to space by momentum. The rate of action (ie d phase) is a function of energy and momentum giving is rate per time and rate per space. The symbols of the world are not text in the way we see it, but real time 'events' whose characteristic time rate and spatial resolution are measured by the energy and momentum associated with them. The size of an event is measured by the quantum of action.
Friday 30 April 2004
Schiff: Quantum mechanics. Schiff.
It is easier (and more fun) to get bogged down in detail when one is trying to fly high and get the broad view. But the broad view must be consistent with the detail if it is true: the 1:10n map must be consistent with the 1:1 map.
Spacetime is an exceedingly simple data structure, in which all the discrete countable things of life happen, ie it is an addressing system for all observable entities. In spacetime entities express themselves and observe on another's expressions. What goes on behind the expressions is described by quantum mechanics. The world of expression and observation is countable and classical, the world of quantum mechanics is uncountably infinite and has the quantum mechanical features of non-commutativity, superposition, no cloning, infinite information content etc.
[page 75]
Saturday 1 May 2004
We can understand superposition as a consequence of resolution, ie a sort of blurring. So we can analyze a process into an infinite series of subprocesses which converges on a process. At the level of resolution of the top process, all the other processes are blurred or superposed, that is deprived of order. Aleph (n) can only count aleph(n+1) it cannot see the order in it (which may be a property of aleph(n+2).
aleph(n) = potential -= time
aleph(n+1) = act - space
