Notes
[Notebook: Transfinite field theory DB 56]
[Sunday 23 May 2004 - Saturday 28 May 2004]
[page 84]
Sunday 23 May 2004
We take a much broader view than Lonergan of the 'integral heuristic structure of proportionate being'. Lonergan, pp 415 sqq. Lonergan constrains this structure using a theory of knowledge rooted in the experience of insight. We take the view that whatever metaphysics is, if it is to be of any value it must be written down somewhere, on paper, in metastable media or in the minds of the people. Further, since there is no a priori reason to put a size on proportionate being, we start with the biggest writing system imaginable, the Cantor Universe. Instead of using a theory of knowledge to constrain the term proportionate being, we use the theories of writing, known as logic and mathematics. In the formal and natural sciences
[page 85]
we are concerned to separate writings that make sense from those that do not. This occurs at two levels we call them the formal and the natural. So all writings that formally contradict themselves are rejected as possible true accounts of natural systems. [an article of faith is involved here, that the Universe is reasonable] Having passed the formal filter, writings can be passed through the natural filter to see if they correspond to anything actually happening in the world.
We identify the formal filter with metaphysics and the natural filter with the experience based (empirical) sciences.
Next we want to show that quantum mechanics is the local version of the output of the formal filter and quantum field theory the global version of this output which results when we mate quantum mechanics and special relativity.
Pais Inward page 244: 'The rejection of forces acting instantaneously at a distance [the most unrealistic feature of the Newtonian picture, possible because the Universe [was] a puppet of god] in favour of forces transmitted from point to neighbouring point by continuous fields [that determine the probability distribution of discrete events] meant the end of a purely mechanical picture of the physical world.' Pais
Now we have the beginning of a communication network, a field being a continuous communication network, constrained by various conserved flows - probability, action, energy, momentum.
Einstein: 'Since Maxwell's time, Physical Reality has been thought of as being represented by continuous fields [an approximation to discrete networks] . . . not capable of any mechanical interpretation. This change in the conception of Reality is the most profound and most fruitful that physics has experienced since the time of Newton. [Einstein in James Clerk Maxwell, CUP 1931].
[page 86]
Field is a thing with a personality, the dynamical interactions of the elements that it 'drives' (and in the case of gravitation) is driven by. Can we deduce the properties of the continuous fields by taking discrete networks to their infinitesimal limits? This idea underlies Feynman's path integral method. Feynman, Zee.
Monday 24 May 2004
Quantum field theory is the modern version of Aristotle's De Generatione et Corruptione. McKeon pp 470-531, Aristotle.
Why does interfacing special relativity and quantum mechanics introduce so much complexity? Should we kiss simple explanations goodbye, or is there a new simplicity waiting for someone with the right point of view?
What we seek to express in plain terms is 'how the world works' which we take to be equivalent to how god works, so that we are in a position to fit in and modify, to our best advantage, given the constraints of sustainability and justice, which we impose on our unconstrained power probably for reasons of symmetry (beauty, aesthetics). My worst pain is to see people destroying one another and their capital, houses, social structure, languages and so on. What we particularly need is a clear 'sociotype' to handed on from generation to generation through education of our children [maybe better psychotype, the network mind].
These notes mostly talk about what I would like to do without a lot of progress toward doing it. Faith, hope and charity are required here to continue the search so that even if I do not find what I am looking for, I might contribute to a climate in which others will continue the search.
Pais page 333 ''The quantum number [nk] of an oscillator is equal to the number of quanta with corresponding [nu].' Born, Heisenberg and Jordan. Let us have a look at this simple yet quite extraordinary remark which opens the door to quantum field theory.'
page 333: 'To distinct energy levels of the initial oscillator correspond (they propose) distinct numbers of photons. In the new interpretation a transition from one level to another must therefore mean that particles with energy h n are either
[page 90]
made or else disappear.' Photons are massless so E = mc2 does not apply, but the notion of creation and annihilation applies to massive particles as well, provided the energy can be obtained from somewhere or given to something. The creation of a photon involved mass loss in the atom and vice versa. How do we generalize these ideas to the creation and annihilation of people, civilizations, planets etc? Obviously by showing how large events are composed of small events. We see the Universe as a network of events based on countable, planck sized events and working up to the whole.
We conceive of the world as a set [or function] {structure, probability} [f(structure) = probability].
We can guarantee that the structure is somewhere in the TN, ie the 'wave function' of the Universe. The probability we calculate using the methods of quantum mechanics. The rule is that the probability of a certain structure is proportional to the absolute square of the wave function at that point.
The task of the prudent manager is to adjust the probability of events, eg to avoid lost time injury, gain in efficiency, etc. The quantity flowing through the transfinite network is probability, and all the nodes manipulate the probabilities of the links and vice versa [since probability is normalized to one, this flow is conserved, probability moving from one event (form]) to another].
Such a manager will always be testing to find and correct errors in the system to ensure its safe and efficient working. There comes time in the life of every plant, where maintenance and modification are no longer feasible, and the time comes for the plant to be annihilated and a new one created.
We give up the idea that everything has to be made of something (ie we can ignore what it is made of [Landauer] and simply concentrate on what it does (ie things which are in principle observable). This is the same as transfinite symmetry [symmetry with respect to complexity] : when we operate at a certain peer level, the levels below and above us are more or
[page 91]
less invisible. [so we do not have to know nuclear physics or politics to live].
Energy is symmetry with respect to time. Momentum is symmetry with respect to space. Is action symmetry with respect to complexity, ie one act is one act no matter how big it is, god or photon.
Pais page 350: Pais. On the early days of quantum field theory: Heisenberg: 'You know it was not the quantum mechanics. In quantum mechanics everything came out much simpler and much better than what I expected. Somehow, when you touched it and you had a disagreeable difficulty at the end you saw "Well, was it that simple?" Here in electrodynamics it does not become simple. Well, you can do the theory, but it will still never become simple.
Pais: 'So it is to this day and it will never be otherwise.'(?)
'It turned out on several occasions that the context of quantum field theory was much richer than expected - in some ways similar to what happened to the classical theory of general relativity.
page 331: 'There can be no question that a history of our penetration into the mysteries of matter must take cognizance of this inner life of quantum field theory.'
We construct an explanation of something by finding two different points of view that may be equated.
F = ma
Pump = machine for giving potential energy to a fluid.
etc.
So we seek the model that goes into the equation
model = f(world)
[page 92]
Born and Jordan: Born and Jordan.
B & J replace the functions of numbers used in classical mechanics with functions of matrices based on the arithmetic of matrices.
Such matrices must be infinite. May they also be transfinite, so that the outer shell of the Hamiltonian governs the vast complexity of the Cantor Universe. So we can imagine people (represented by suitably complex vectors and operators) are governed by the same form Hamiltonian as electrons and photons, something constructed from kinetic and potential energy.
Quantum mechanics elucidates the connection between structure and probability that controls the world. The constraint is that the sum of the probabilities of different competing structures is 1. This is a profound truth, and lies at the heart of Quantum mechanics as a metric science.
Now we come to life. The wave function determines both the nature and frequency of the outcomes, so if we want to influence the outcomes we must influence the wave functions. The order in my body is maintained by dissipation (Prigogine). I follow the arrow of time, being a heat engine of sorts. How do we build a heat engine on a quantum system, and other sorts of engine can we build, computing engines for instance.
A Carnot engine is reversible, which means that it conserves entropy. The entropy of the hot reservoir is passed conservatively to the cold reservoir and mechanical (zero entropy) energy is made available. In reverse, by absorbing mechanical energy, we have a refrigerator.
[page 93]
Quantum mechanics is also reversible = unitary. The common thread is the conservation of entropy H = sum over i pi log pi. Normally we think of i as a natural number, but what if we make it larger. Normally physics handles this transition from discrete to continuum by the change from sum to integral.
So how do living creatures manipulate the wave functions of wilderness in order to survive? The function of an enzyme is to increase the probability of a certain reaction by lowering its activation energy or by putting in energy in order to make the desired process work, ie active pumping of ions etc.
Tuesday 25 May 2004
Wednesday 26 May 2004
We think of physics as the shell or operating system of the metaphysics, including the hardware. We run on the physical hardware.
We can thus imagine the world as a layered network process analogous to the internet. This seems so clear, but I keep saying it to myself again. We may think of the metaphysics as being spatially inside the physics, while being logically outside it.
Cantor's theorem is complexity invariant, in that it works just as well for any aleph(n) --> aleph(n+1), n >= 0 transition. We would like to think that quantum field theory is also complexity invariant, so that it fits into the transfinite network naturally, and describes it subject to a) finite velocity of communication and countability
[page 94]
of ℵ0, giving us conservation of all countable entities, energy, momentum and action. In each case both the number of different states and the number of quanta occupying the state is countable. This constraint defines the 'ground state' of the transfinite oscillator.
Thursday 27 May 2004
I would like to understand the detail of quantum field theory but I am not well motivated because it seems far too complicated to be the description of something whose existence is unconstrained by historical detail. We see that formally, this condition can apply anywhere as long as there are enough markers (symbols) and weak enough constraints on the rearrangement of markers to enable exponential growth of change as generation follows generation. We might say the transfinite network is constrained by exhaustion, in that there is no way to increase the number of symbols that it generates.
We begin to understand the Cantor Universe by looking at features which are invariant with size (since it is too big to comprehend).
Life is perfect (for me) and yet I worry. Because life is not perfect everywhere, and there is a chance that the disease will spread to me and mine. This is the practical foundation of my worry and the other is aesthetic. I do not like to see suffering.
Kolmogorov: foundations of probability. Kolmogorov
page 1: Every axiomatic (abstract) theory, admits, as is well known, an unlimited number [of concrete interpretations besides those from which it was derived.]
page v: '. . . after Lesbegues publication [of his theories of measure
[page 95]
and integration] the analogies between the measure of a set and probability of an event and between integral of a function and mathematical expectation of a random variable became apparent. These analogies allowed a further extension; thus, for example, various properties of independent random variables were seen to be in complete analogy with the corresponding properties of orthogonal functions.
Kolmogorov page 1-2: '. . . if our aim is to achieve the utmost simplicity both in the system of axioms and the further development of the theory, then the postulational concept of a random event and its probability seem the most suitable.'
Friday 28 May 2004
We have equated metaphysics with both quantum mechanics and quantum field theory, but would a better choice be field theory in general conceived as network theory taken to the continuum limit. The best way to represent the world? Everything we see in the world is an event, a discrete entity with a beginning a middle and an end. Calculus tries to expose the middle by bringing the beginning and the end closer to one another, but this tends to squash the detailed process between points on a curve out of existence. This is natural at the Euclidian/Aristotelian level of geometric complexity. Now from Cantor . . .
The arithmetic shell: despite their great difference in internal complexity 1 (sheep) + 1 (sheep) = 2 (sheep) just as 1 + 1 = 2 holds for atoms, galaxies, in fact any conserved (additive) quantity.
The continuous paradigm could be holding us back. We observe countable events, and we can say that all our information comes in the form of counts taken at different points in spacetime.
[page 96]
We develop the explanation of these counts in terms of an underlying deterministic continuum called (for historical reasons) the wave equations. Waves are are handy because they have both discrete (wavelength, energy) and continuous features, so they are a natural bridge from the continuum. Going the other way, from discrete to continuous, we use the theory of computation. This theory (based on set theory) allows us to detect the boundaries between discrete and continuous and to constrain the functions in the continuum.
Our survival depends on our competence at reading and manipulating the world. We can encode our more predictable manipulations as algorithms for getting from one state to another, the state of independent ingredients, butter, flour, eggs, sugar etc to the state of a cake for instance. These manipulations are often very simple, and often at the high end of research and technology, quite complex and closely controlled in order to get the desired outcome, a state of the art x-ray telescope, say or a system to test the general theory of relativity in the vicinity of earth.
The general algorithm executer is the Turing machine.
Since Kant and perhaps beyond, the ding-an-sich has been considered to be beyond the human mind, but this may not be necessarily so, since of the Universe started from nothing we can expect some parts of it to be quite simple, dual in fact. The unitary simplicity of god is unanalyzable and so not much use to us except as an object of contemplation.
The connection between metaphysics and quantum field theory is via the network model which we map in passing onto function space, Hilbert space and the use [the model] to make a theory of knowledge, so bringing us back to Lonergan's standpoint.
Then we apply Lonergan's axiom as we know, so it is.
