Notes
[Notebook: Transfinite field theory DB 56]
[Sunday 25 July 2004 - Saturday 31 July 2004]
Sunday 25 July 2004
[page 143]
Monday 26 July 2004
It is nice thinking (and writing) about the desirable features of the model. But we have to record the weak spots too (if not in the model, in my current understanding of it). First its principal feature : the space in which we work is the biggest thing imaginable. Every possible ordering of a countable alphabet is represented, and orderings of orderings and so on ad infinitum. Any attempt to construct a bigger formal system encounters degeneracy. Because it is so big, it seems a good place in which to start modelling the whole, whose asymptote is god.
[page 144]
Now we assume that quantum field theory is a really good model of the Universe so we wish to establish a correspondence between quantum field theory and the Cantor Universe. We introduce dynamics into the Cantor Universe by visualizing it as a network of computing machines, each able to transform one symbol of the countable alphabet and (since we consider these machines to be reversible) we need ℵ0 such machines to deal with the ℵ0 letters of the universal alphabet. This is nice, but now here is the question: how do we map this into or onto quantum field theory. The route seems to be through quantum mechanics and Hilbert space.
The stumbling block is that while a Turing machine is not reversible (because it erases things and without a history, it cannot backtrack (?), whereas the fundamental constraint on quantum mechanics is normalization in observable space and unitarity in the complex phase space that is hidden behind the scenes.
A further problem is that the Cantor Universe seems to be concerned with natural and real, not complex numbers. Why does quantum mechanics use complex numbers?
Aside on share market movements - tend to be either equilibria (between buyers and sellers) = constant price or disequilibrium (due to some information) which leads to a rise or fall in price. The 'steepness' of the disequilibrium is related to the steepness of the price change. The psychological perceptions of traders about the prospects of companies are reflected in their share prices, which in turn affect the prospects of the company (particular in financing its dreams).
As Feynman points out, a complex number can be represented by a little arrow (drawn on an 'Argand diagram') QED. Feynman
[page 145]
Such an arrow, rotating at a rate determined by the energy that is associated with it seems to be a food model of the fundamental 'working part' of the Universe.
These working parts we map onto the letters of the alphabet. This is possible because the only thing distinguishing our little arrows (working atomic parts of the Universe) at this point is their energy, ie their rate of rotation. Let us assume for the moment that the rates of rotation (energies) of these atoms are quantized, and so, like the quantized natural numbers, thee are countably many of them. Since we also postulate that energy is conserved in the Universe (being in discrete units, the maximum available is aleph(o)_. So we have ℵ0 distinct entities sharing ℵ0 of energy. A peculiarity of transfinite arithmetic is that ℵ0 x ℵ0 = ℵ0 so that we may have any countable number of atoms all with the same energy, and in this sense degenerate. At least this is a way of imagining things that does not violence to the mapping of atoms ('units') and natural numbers.
This view automatically associates energy with time. We can associate momentum and space in a similar manner.
Why do we need complex numbers to model the Universe. Because of the algebraic fact that only an infinitesimal fraction of polynomials have real roots? We will take this question up after expanding our 1D arrows to ℵ0D and beyond Hilbert space [also, the Argand diagram demonstrates an isomorphism between a little arrow and a complex number.]
In the world of the little arrows, however, there is no distinction between space and time. They live in 'spacetime'. Because of this they do not distinguish either space and time or momentum and energy. Their quantization (and distinction) is called 'action',
[page 146]
the discrete unit here is the quantum of action and there are still ℵ0 units carrying ℵ0 different units of action. [note also that different quantities of energy may be associated with one unit of action. High energy actions simply occur faster - at a greater frequency] These physical units can thus be mapped onto the natural numbers. This is the first step toward building a correspondence between physics and the model.
We can now ask ourselves how the ℵ0 units interact. We built the Cantor Universe by considering permutations of an alphabet of ℵ0 letters, and permutations of permutations and so on. We create permutations by swapping elements , so [1, 2] becomes 2, 1] by swapping the 2 and the one and so on ad infinitum. Units can do this by exchanging quanta of action. So if I am a 10 and you area a twelve we swap by you giving me 2 so that I become a 12 and you a tend. To see how this is modelled in quantum theory, we have to go from little arrows on a two dimensional Argand diagram to little arrows in an infinite dimensional Hilbert space.
Hilbert space : Quantum mechanics : unitarity.
Quantum mechanics shows us in (so far as we know) complete detail how the little arrows of the Universe interact to give us the world we see.
The pattern of transformation between units is seen in its simplest form in the quantum harmonic oscillator where units embodied in the walls of a cavity are transformed from one state to another by the exchange of other units embodied in photons of electromagnetic radiation.
[page 147]
Entropy is an abstract (cardinal) measure of meaning. A Carnot engine moves meaning from a hot reservoir to cold reservoir, extracting meaningless (degenerate) energy, ie 'mechanical' (macroscopic) energy.
Good can only grow. Evil can be established almost instantaneously by violence. The time constant of good is longer than the time constant of evil. Evil may grow too, as the growth of an infection which is good for the bacteria may be evil for the host (who at the very least is contributing to the life of unwelcome strangers). Good helps, evil hinders.
The crux of the matter is whether the permutations of units are as general as the permutations of natural numbers, and we think that the quantum harmonic oscillator says that they are, subject to certain temperature related statistics related to the quantum mechanics of bosons.
The reality principle for quantum mechanics is conservation of probability - every event is by definition normalized to one outcome, as a coin falls on one face. Symmetry is broken, but the sum of the probabilities of the broken symmetries must be one.
One of the most interesting things about quantum teleportation is that it requires a classical channel.
Stock market shows a balance between greed and security.
The little arrows add in a way that is also perfectly modelled by complex numbers, so we can replace a whole series of interacting arrows with a single arrow which names the set of arrows which are 'hidden' within it. Al this is described by the algebra of operators in Hilbert space.
Feynman vol III Quantum Mechanics. Feynman
Selection is powerful because it is exponential, each step building on the one before. Selection takes a random walk (which does cover distance, but in a random direction) and gives it direction, cutting off steps that do not fit the environment to a certain tolerance.
Fitness is always relevant to an environment x, and is equivalent to '
ability to derive an adequate living from x to reproduce'.
Russell page 700: 'To frame a philosophy capable of coping with men intoxicated with the prospect of almost unlimited power and also with the apathy of the powerless is the most pressing task of our time (1940?) Russell
Tuesday 27 July 2004
Each transfinite cardinal represents a discrete entropy level within which reversibility (unitarity) prevails.
Dawkins replicator: communication is replication and the network is a replicator. Dawkins
Intuition : insight : joining the dots. Making aleph(n) into aleph(n+1)
We look forward to the time when natural religion becomes a commodity like coal or computing. Commodities are in the long run a matter of life and death, but for this very reason they become ubiquitous, and therefore in an open and free market, available at reasonable prices.
Reasonable price: everyone makes a living : as the constant
[page 149]
temperature in a thermostatic creature adjusts all the metabolic processes to their optimum rates for the system as designed. So what cam first: the thermostasis or the molecular design? They grew together, cyclically, through the genotype - phenotype - genotype channel.
Language and more generally religion serve as frames of reference through which to communicate and understand our experiences. General covariance does not hold so thoroughly in human space as is does in physical space because our languages and religions modulate our experiences ('a new language is like a new soul') whereas general relativity (as formally expressed by physicists) does not modulate physical experience, although it helps us to understand and communicate it (and do the calculations necessary to guide spacecraft etc).
Our basic scale is time. We are inclined to overlook he intelligence in processes that are too fast or too slow to see easily, and concentrate on our own intelligence at our own scale. This may be the root deign and purpose in anything by the works of Homo sapiens. Human intelligence, we postulate, is an invisible evolutionary process happening in the mind. We see only the results which present themselves as insights, some subjectively instantaneous (like seeing a friend) others long and tedious, like understanding chemistry.
Parts are given meaning by the whole in which they find themselves. So a starter motor is meaningless outside the context of starting motors, and each bit in the execution of a Turing machine is given meaning by its place (address) in the process.
