vol VII: Notes
2016
Notes
Sunday 10 July 2016 - Saturday 16 July 2016
[Notebook: DB 80: Cosmic plumbing]
[page 133]
Sunday 10 July 2016
von Neumann page 87: An operator R is a function defined on a subset of [the Hilbert space] ℋ with values from ℋ, ie a relation which establishes a correspondence between certain elements f of ℋ and elements Rf of ℋ.Such an operator may be conceived as a computer if R is computable.
page 96: An operator A is said to be Hermitian if A* = A It is aso said to be definite of it is always true that (Af, f) ≥ 0. The operator U is said to be unitary if UU* = U*U = 1.
U, U* encode and decode a message, ie a mapping from one point in a Hilbert space to another and back again, U* = U−1.
[page 134]
v N page 102: '6. The Eigenvalue problem'
Existence of a metric makes geometric physics possible. What are the metrics of a text: letter count, word count . . . count . . . count? Also orderings: better / worse, prize winning / not ditto / I like a more than b etc.
Order is the next level of complexity after cardinality. All of physics, the complexity of the machines <—> the complexity of the world, passes through measurements expressed as real numbers binned in a way specified by the isomorphism between the measuring instrument and the system measured, eg LHC [me/you, predator/prey etc]. Large Hadron Collider - Wikipedia
v N page 207: 'Although we believe that after having specified (φ) we know the state completely, nevertheless only statistical statements can be made on the values of the physical quantities involved.
A count is perhaps the most abstract characterization of a set of particles.
Wave equation enables deterministic evaluation of the phase change since some t0 and the whole output of quantum mechanics is determined by the sum of the phases derived from various sources [in time, in the absence of space, phases from all sources add algebraically and instantaneously: shared memory?].
page 209: 'In reality, φ does not determine the state exactly. In order to know this state absolutely, additional numerical data are necessary. That is, the system has other characteristics or coordinates in addition to φ. If we were to know all these then we could give the values of all physical quantities exactly with certainty . . . It is customary to call these hypothetical additional coordinates "hidden parameters" . . . Whether or not an explanation of this type, by means of
[page 135]
hidden parameters, is possible for quantum mechanics is a much discussed question. . . . We shall show later that an introduction of hidden parameters is certainly not possible without a basic change in the present theory.
Probability is a consequence of indeterminacy which is a product of symmetry. So what is the symmetry underlying the 'uncertainty principle'?
Digitization / limited resolution / the continuum between fixed points which we see as the continuum between the eigenvectors of the observable, the regions of mixed phase or one might say non-resonant or non-selected phase. Selection is for harmony = computability.
v N page 210: '. . . we admit as a fact that those natural laws which govern the elementary processes. (ie the laws of quantum mechanics) are of a statistical nature.' from the point of view of those who have not yet decoded the meaning of the apparently random sequences they observe.
Every sequence is local and hidden by the invisibility theorem but we can nevertheless understand is to be a meaningful message between the sources involved. page 6: Invisibility
Two processes of wave function change: a) continuous deterministic evolution; b) probabilistic, non-unitary, non-local discontinuous change brought about by observation, ie interrupt.
page 223: Dirac δ '. . . the introduction of an eigenfunction which is "improper" . . . gives a less good approach to
[page 136]
reality than our treatment here. For such a method pretends the existence of such states in which quantities with continuous spectra take on certain values exactly, although this never occurs.
v N page 247: '5; Projections as propositions'
Monday 11 July 2016
v N page 254: 'The radiation theory advanced by Dirac is one of the most beautiful achievements in the quantum mechanics field.'In my sexy youth I was inclined to see the beatific vision as an endless orgasm / insight / event. Beatific vision - Wikipedia
I have to take myself slowly because I am not very fast, but only want to achieve one thing, and that is the rebuild the foundations of theology, rooting it in physics, the science of God's body, through which it reveals itself just as we reveal ourselves with our bodies, a parallel sequence of observable fixed points [a 4D kinema pixellated at the scale of Planck's constant].
Tuesday 12 July 2016
We want to say that the uncertainty in quantum mechanics is a consequence of Gödel's incompleteness and Turing's incomputability. From a more practical point of view, we might say we are using a tape graduated in units of h and we are trying to measure quantities which are not precisely h just as we have trouble measuring the diagonal of a unit square when we only have units to
[page 137]
measure with. But is this relevant at all? The uncertainty in measuring the output of a quantum source does not lie in the eigenvalue measured but rather in the actual sequence of eigenvectors, which is not ordered by the measurment operator. Given eigenvalues ai. we may measure them in any one of the sequences ai!, using the same measurement operator repeatedly applied.
The only way to deal with moments of depression is to keep the job moving ahead so that all is not lost, een if not enjoyable.
Wednesday 13 July 2016
Reception of a doctrine. Perhaps we can say that quantum mechanics deals in metadata, but the actual messages transmitted in this layer are invisible to us.
La Civilta Catrolica semi-official Vatican publication. La Civiltà Cattolica - Wikipedia
Faggioli Massimo Faggioli: A Non-Synodal Reception for a Post-Synodal Exhortation
Reading P G Wodehouse in the bath reminded me of the headmaster, now deceased, who first turned me on to him. A little later in my life, L.P. Fitzgerald OP became my black beast, the Master of Studies who I failed to convince that my theological ideas were revolutionary and good. He had a hand in having me blackballed and expelled from the Order for which I am forever grateful, but I find myself up against the same problem, now as then. Once an idea becomes established, it is very difficult to dislodge. The idea in my sights now is the very ancient belief that the universe is continuous even though every possible observation from the quantum scale to the galactic suggests that it is a set of discrete objects. The reason for the continuous conviction, I believe, is the apparent continuity of local motion and the reasoning that sugests that space-time is therefore itself continuous. I am not at all happy with the mathematics of continuity, which seems to me to be an absurd attempt, based on the notion of the limit, to construct a continuous space out of a set of discrete point sets by various arguments, the most popular of which we call epsilon-delta. The latest reason I have for discrediting this argument is that it is not computable insofar as the ε-δ process never halts because it can never end by the definition of continuity itself. I would like to couple this idea to Gödel's ideas of incompleteness via Chaitin's algorithmic information theory but the path is not yet clear. Looking at von Neumann, we can then do away with the continuous approach to infinite dimensional Hilbert spaces and replace it with the discrete complexification of function spaces illustrated by Cantor's transfinite numbers. Now that my head is clearing from infectious microbes, I am feeling that this just might be possible. The image I have in mind is the mellifluous flow of Wodehouses's prose even though it is a composition of discrete words. Like the Shannon-Nyquist theorem, as long as we sample frequently enough we can reproduce the experience of continuity in an observer whose response time is longer than the interval between samples.
[page 139]
Thursday 14 July 2016
If the conservatives were right the universe would still be nothing more than a collection of hydrogen atoms or less.
The 64k q: is continuous linear algebra computable, ie, given that only computable can be realized, can continuous quantum mechanics be realized? Maybe not. So let's look at Gödel and Turing.
Slowly humility is being forced on me. In my monastic days I thought the concept that the universe is divine might bring me big prizes in physics and theology, but as I grow old it seems that nobody is really interested in this idea and they all want to stick to the old physics of an omnipotent creator and the old theology of same. For a long time I thought to go slow because my mother was a devout Catholic and I did not want to upset her. Now she is dead and I I am off the leash, but for what? I have become stuck, and have been stuck for quite a while, but apart from feeling a bit helpless, I do not feel like giving up because I have nothing else to do and nowhere els to go and the idea still excites me. The big hope is to get rich enough to go to school again and study all this over again to see of there is a way forward.
Friday 15 July 2016
Mathematics is a deductive science in the same sense as medieval (and modern Catholic) theology. [Such] science [is] something deduced from 'first
[page 140]
principles' with a bit of guidance from reality, but placing greater weight on the principles than the reality. This is why so much of Catholic dogma is counterfactual, virgin birth, immortality etc, and the management reject anyone who dares to question beliefs allegedly drawn from the writings of people 'inspired' by god, or in the case of Jesus, actually a god whose words have been faithfully recorded by the authors of the New testament. It may be that a lot of pure mathematics a là Hardy is also counterfactual, particularly in the area of rational and real numbers. Physicists and mathematicians like von Neumann may be being led astray by the belief that continuous and complete mathematics, although apparently consistent logically, really aplies to the world. This idea takes the program to digitize the universe to a deeper level, made possible by resolutions of the inconsistencies and incomputabilities embedded in point set theory. Hardy: A Course in Pure Mathematics
Hilbert space is a metric space and the metric is the foundation from which we compute observable parameters so that we can compare our calculations to reality. As a builder, my whole day is measure, cut (if necessary), fix, repeat. Behind the metric side of the job, which is necessary to fit the parts together, we have the architectural task, designing the overall assembly of parts. Insofar as parts are standardized an architect helps the builder and reduces costs if he sticks to the standards. The universe, for instance, has given us a set of standardized atoms and everything is made from them. We cannot change them, but must take them as they are, but their remarkable properties have led to the construction of ourselves and our planetary environment.[page 141]
The relationship between metric and architecture is very simple in the building trade. We might ask which comes first, the design or the measurement, In quantum mechanics we see the close relationship between these actors. As builders build by laying bricks, the whole progresses by executing quanta of action. So we are led to explore the relationship between the quantum of action and architecture, and we are led to the idea that all structure must be quantized since if it is not it is a continuum and so not a structure.
Saturday 16 July 2016
von Neumann: Leon van Hove: Leon van Hove: Von Neumann's Contributions to Quantum Theory
matrix mechanics: Heisenberg, Born and Jordan; wave mechanics: Schrödinger.
Equivalence demonstrated by Schrödinger and embedded in Dirac and Jordan transformation theory, marred, in von Neumann's eyes, by the Dirac delta-functions. van Hove, page 96: '. . . the axiomatically defined separable Hilbert space is a suitable framework for the quantum-mechanical formalism as we know it today, and the recognition of this fact we owe to von Neumann.'
page 99: 'there is only one broad subject which we would like to mention here because von Neumonn, obviously giving it considerable thought, returned to it several times in 1934 and 1936,
[page 142]
(in collaboration with Jordan, Wigner and Garrett Birkoff.) It is the question of the algebraic and logical structure of quantum mechanics, where the hope was raised to reach, through abstract analysis, possible generalizations of the accepted theory.
An abstraction is a reference frame and we know that there are an infinity of reference frames all of which can be transformed into one another in terms of a meta-reference frame. This system is grounded in a system which is its own meta-reference frame which we call gravitation. The principle of gravitation is that anything goes as long as it respects some fundamental symmetries which turn out to be space-time and momentum-energy all coupled together by the velocity of light, the velocity of a machine with no overheads, that is no rest mass. Hilbert space is an abstraction, and like any abstraction can be made concrete in a large number of ways. The variety (entropy) of concretion / entropy of abstraction is as real / rational among the numbers. The important point is that the cardinals ae generated by structures, that is ordered sets and the number of sructures may be approximated by the symbol ! (factorial). The symmetry associated with gravitation is that all permutations of a set of symbols are equivalent and therefore equiprobable from the point of view of gravitation. This equiprobability may be reflected in the weakness of gravitation. A Hilbert space can serve as a model of gravitational space, being of sufficient cardinality to map the permutation space.
[page 143]
All that gravitation sees (and determines) is the rate of action ie energy. How do we divide the role of action between kinetic and potential energy, the energy of change and the energy of stability, that is the energy associated with a physical fixed point (orbital)? Change of orbital, change of [potential] energy. The most fundamental economy is the universe is the energy economy that creates its currency by differentiating actual and potential energy, cashflow and capital. Piketty. Piketty: Capital in the Twenty-First Century
General covariance = general blindness = no interest whatever in structure, so the energy in any region of the transfinite network is measured by the processing rate of the computers coupled to the node in question [and their potential energy]. Each computational sequence is an eigenvector, ie a series of values which in the quantum mechanical world transform into probabilities.
