Notes
Sunday 25 July 2021 - Saturday 31 July 2021
[Notebook: DB 86: Hilbert / Minkowski]
[page 304]
Sunday 25 July 2021
What stops me from writing my magnum opus? Let's call it cognitive dissonance. The way is not yet clear. I have a beginning, the gravitational initial singularity singularity which I identify with the God of Aquinas, the absolutely simple source of the universe. I have an end, the universe I inhabit with my conscious mind. I have a paradigm to get from one to the other, Aristotle's path from physics to the unmoved mover. I have a classical network model built of Turing machines and transfinite numbers. Networks transmit information
[page 305]
that is they copy it through spacetime [ideally using lossless codecs, identical to the unitary flow of information governed by the Schrödinger equation]. All this is summarized in my honours thesis in classical terms. The dissonance arrived at the end of my honours year when I set out to detail the quantum mechanical underpinning of that story and his some problems which, in a nutshell, are the parlous state of quantum field theory documented by Auyang and Kuhlmann. Jeffrey Nicholls (2019): Prolegomenon to Scientific Theology, Auyang (1995): How is Quantum Field Theory Possible?, Meinard Kuhlmann (Stanford Encyclopedia of Philosophy): Quantum Field Theory
An aside: Original sin is an example of guilt by association rejected, according to Martin Powers, by the Chinese Hahn Empire 2000 years ago. Martin Powers: Joe Biden promotes liberal values, but could learn a lesson or two from Imperial China
Back to cognitive dissonance. My plan ran aground on the central problem of quantum mechanics known as the measurement problem associated with the 'collapse of the wave function'. The root of the problem, it seems to me, reflected in Kuhlmann's article, is the idea that Minkowski space is the domain of Hilbert space, so I began to think about the alternative, that Hilbert space describes the true root of the world and Minkowski space, in terms of the layered network idea, is in broad terms the third layer of the universe, the first being god, the second being the bifurcation of action into potential and kinetic energy which is the realm of Hilbert space and quantum mechanics, and the third created by using the structures made available by the second, is the Minkowski space of quantum field theory. The founding insight of this construction is the bifurcation idea that maintains that each new development in the universe is a bifurcation into two new entities that add up to nothing but which are retained in existence by the entropic force that favours complexity [and by the P type computability of the newly formed structure, enabling it to reproduce itself]. So quantum mechanics breaks into Minkowski space with the help of gravitation by the
[page 306]
invention of the null geodesic, and here I am a bit stuck. The way forward is simply to assume this idea as Darwin would suggest, a strategy assumed in st06_creating_new_world section 6.8 [?]. So we just state the case: Minkowski space can emerge from Hilbert space due to the fact that that the speed of light enables the transfer of quantum states unchanged through space. If we can imagine establishing this transition from Hilbert to Minkowski, driven by randomness and selection motivated by the entropic force, the future looks possible. We have three inputs randomness, as a consequence of requisite variety; deterministic selection as a result of P processes establishing the viability [and reproducability] of NP processes, and the entropic force represented by Cantor's theorem, providing the incentive for complexification. Variety (cybernetics) - Wikipedia, P versus NP problem - Wikipedia
The big deal is to introduce the photon as the gauge particle joining Hilbert space to Minkowski space and preserving the masslessness that it enjoys in Hilbert. This requires a discussion of symmetry, continuity and Noether's theorem plus the usual threesome, requisite variety, P vs NP and entropic force all in terms of logical continuity. What I should mention here is Misner, Thorne and Wheeler's statement about the framework for a physical advance requiring a body of established theory as its jumping off point [here I quote:]
Here and elsewhere in science . . . that view is out of date which used to say 'Define your terms before you proceed.' All the laws and theories of physics . . . have this deep and subtle character, that they both define the concepts they use . . . and make statements about these concepts. Contrariwise, the absence of some body of theory, law and principle deprives one of the means properly to define or even to use concepts. Any forward step in human knowledge is truly creative in this sense: that theory, concept, law and method of measurement — forever inseparable — are born into the world in union. Misner, Thorne & Wheeler (1973): Gravitation, page page 71
On this project I would submit the millions of words of notes (46.5 MB) on this site as a diffuse but deliberate record of all the site preparation that went into producing the notion that Hilbert space is the independent source of Minkowski space, a fact reflected in the metric. Minkowski space - Wikipedia
Monday 26 July 2021
Very hard to cut through the complexity of quantum mechanics to get to the points of most importance which are, for my purposes, issues surrounding quantum measurement which have to do with the interfaces between elements of the quantum network to move from "imaginary" to "real" representations which implement the randomness and selection operative in quantum mechanics which leads to creation, that is real increases in entropy [implemented as physically observable and countable particles]. A proper description of this may require a book rather than a chapter so we are constrained to broad statements rather than examination of detail.
To be in motion, speak.
Trying to make sense of the measurement problem may not be central to this project but it is a very interesting problem and the idea of separating Hilbert from Minkowski spaces has a very productive feel to it particularly as an explanation of the Minkowski metric, so in the four years I have given myself to do this job [1.5 already gone] I feel that six months spent on this issue will not be wasted, particularly if it also serves to clarify my understanding of quantum field theory.
According to extant statistics, my page phys04QuantumMec got 109 views in 2008. I read Zurek's paper on the collapse in 2010, so it might be worth looking at the quantum mechanical page on physics in 2011 to see if he was incorporated. Looking back I think this article could be the best starting point for scientific theology chapter 6.
[page 308]
Another go:
0: The bifurcation rule: [to create the universe out of nothing every new act of emergence must be a bifurcation into two elements that add up to nothing, eg potential and kinetic energy for a zero energy universe].
1: Quantum mechanics in Hilbert space with wave equation and unitarity [Feynman, Zurek, von Neumann].
2: Growth of Hilbert space by quanta producing quanta [Like Father and Son in the Trinity], no-cloning, orthogonality and recursion, Hamiltonian and network layers. Jeffrey Nicholls (2008): Natural theology > Development > Physics > Quantum mechanics
3. Superposition and interaction — boson / fermion
4. Observation / interaction on the quantum side — quantum computation. Nielsen (2000): Quantum Computation and Quantum Information
5. Observation on the classical / Minkowski side
6. Zurek on 'collapse of the wave function' Wojciech Hubert Zurek: Quantum origin of quantum jumps: breaking of unitary symmetry induced by information transfer and the transition from quantum to classical
7. Velocity of light / Minkowski metric / unitarity / fixed point [the layered network model suggests that lower layers exist independently of the higher layers built using the resources provided by the lower layer. Here we explore the idea that the quantum layer built in Hilbert space of quantum theory underlies and is independent of Minkowski space, but provides the resources for the emergence of Minkowski space. Maybe the emergence of the Minkowski metric is an instance of the bifurcation rule above, since the existence of null geodesics is made possible by space and time adding up to nothing via quadratic metric ds2 = dx2 - c2 dt2]
8. Particles / entropy / gravitation / QFT [the unitary evolution of the wave function is a perpetual motion. Observation increases entropy and consequently excludes perpetual motion in the domain of real spacetime / particles].
9. Evolution / randomness / selection P ↔ NP [Probabilistic NP processes can go where Turing machines cannot, but some such discoveries can be validated by deterministic P processes, that making it possible to them to be selected for reproduction].
This is the hardest chapter dealing with the hardest problem: how does a universe create itself with unbounded action, no outside control and natural selection. Building a universe out of nothing by bifurcation.
Tuesday 27 July 2021
At the bottom of all this is an effort to boost faith, hope and charity by emphasizing the sublime beauty of our divine universe and identifying the causes of evil in its evolutionary emergence from undifferentiated action.
A peculiarity: One cares the most for those who are the most trouble.
[page 309]
In Minkowski space a system can create and annihilate a photon without breaking unitarity because time does not pass on a photon since it follows a null geodesic. What about Hilbert space? There, where there is no space in the classical sense, everything occurs by action at no distance but the delay is measured by energy, the inverse of frequency. Does this also mean there is no time in Hilbert space , even though we write dψ/dt = Eψ? Here is a puzzle. How do we understand no time and unitary evolution in Hilbert space? Still in the dark. Maybe the Hilbert world like the Platonic world is pure formalism. Perhaps we should then think about a computer, where the interval between clock pulses hides the dynamics so the machine act like a formal logical process independent of time. Einstein solved the problem of relativity by linking the notion of simultaneity to spacetime. So a photon on a null geodesic still takes billions of years to travel across the universe but the wave function 'inside' it does not evolve [which is why we like bosons to be massless and carry their messages of phase faithfully from creation to annihilation].
Parmenides was right. We can only know static things like writing (which is a fixed point in the human dynamics — these words record a thought that flitted through my mind in an eternal form). When we observe a quantum system we get a fixed particle, eg an electron or a photon which is effect eternal between its creation and its annihilation. Parmenides thought the fixed point must be permanently fixed and eternally eternal, enduring forever from s time point of view. Time is marked by events and we can see nothing between events, even though calculus
provides us with a continuous division of measuring every point in a continuum [there is a duality between events and intervals which is realized in spacetime?]. Here we have to think more about the interface between physics, logic and quantum mechanics. Why does it take 4 billion years for uranium to decay.
Schrödinger, Dirac and von Neumann had to solve the equation
wave mechanics = matrix mechanics
Dirac did it with the delta function and von Neumann in Hilbert space taking advantage of the isomorphism of discrete and continuous function spaces. John von Neumann (2014): Mathematical Foundations of Quantum Mechanics
We may think of the space of permutations of the natural numbers as a function space. Given one element mapping to itself the function space has 1 dimension. Given two elements, 0 and 1, we can map 0 to 0 and 1 to 1, the identity function, or 1 to 0 and 0 to 1, the simplest permutation of two objects. With three the function space or permutation group has six possibilities and continuing in this vein there are ℵ1 mappings of the ℵ0 natural numbers to themselves, and we may see this as the set of ordered superpositions in ℵ0 dimensional Hilbert space. An observation then separates out one of these superpositions collapsing the ℵ1 functions to 1. Does this help at all? And given an ℵ0 dimensional Hilbert space, can we establish a one to one correspondence between the ℵ0 states mapped to the dimensions of the Hilbert space. We associate each dimension of the Hilbert space with a Turing machine, an ordered set of operations [since the inner product assumes an order in the elements of the vectors being 'multiplied' (a matrix thing with two 1D matrices = strings)]. We work it out with a qubit q = a|0> + b|1> [where we assume a and b to be real or complex numbers, but is this true, ie realistic?]. At the other extreme of complexity we may associate each dimension of Hilbert space with
[page 311]
a person, a particle or a quantum of action, like myself. Watching soccer. The processes of the relevant Hilbert space are within the players and the team is a tensor product of all the players' internal processes which are expressed on a pseudo-Minkowski space where the speed of light is the speed of the ball.
What would I do if I was the initial singularity? The God of the Trinity can reproduce itself producing the identical Son, which I can only do if I am a dividing cell, but I need matter and energy to do it, ie repeated actions. Note, however, that the quantum universe is in perpetual motion, that is there seems to be no conservation of action, it is a symmetry that can (and does) repeat ad infinitum, too much, in fact, if the quantum field theoretical problems with the cosmological constant are any guide. Cosmological constant problem - Wikipedia
The beauty of abstract discussions as we find a lot of physics is that they can avoid getting down to concrete detail where the action really is and which must actually work if the system is to work, a situation which can be made very clear in digital software.
How de we reconcile perpetual motion with the second law? Quantum mechanics, per se is unitary whereas observation increases entropy, so we need to have observation right at the beginning as soon as there are two states [to observe one another].
So we are thinking that the random variation is in the Hilbert regime where uncertainty reigns and selection is in the Minkowski regime where determinism reigns. The eigenvectors are selected to match precise quanta of action and real numbers for their eigenvalues.
Wednesday 28 July 2021
[page 312]
Have just remembered I wanted to be an astronomer when I was young [but got deflected into theology]. Ben Proudfoot: She Changed Astronomy Forever. He Won the Nobel Prize
From the night comes clarity:
0. Some principles: pure act; cybernetics and randomness; selection; contradiction and logic; quantum theory.
1. Feynman's summary of quantum behaviour.
2. Zurek's six propositions.
3. Hilbert space [as the container for action], normalization, rays, [state vectors], linear operators.
4. Initial singularity, 1D Hilbert space, [structure], probability
5. [Trinity] Father to Son, 2D Hilbert space, tensor product, qubit, [quantum computation,] and observation, thus kicking off the Universe.
6. Quantum perpetual motion, unitarity, constant entropy, determinism.
7. Observation increases entropy [von Neumann]
8. [Cybernetics] requisite variety [Gödel, Turing and Shannon]
9. Selection, eigenfunctions and values, ['collapse of wavefunction' (Zurek again)]
10 Evolution / symmetry with respect to complexity.
< p> 11. Representation, spacetime, Minkowski [metric]We can imagine the Hilbert domain growing toward transfinity by continual addition of new quanta of action, tensor products and superposition. If we imagine this as a pure formal Platonic domain, we have no need of 'conservation of symbols'. As we learn from Peano's axioms, we can go on creating new numbers (symbols) ad infinitum and Cantor's theorem has the effect of putting Peano on steroids since the creation of an actual ℵ1 by permutations of ℵ0 requires ℵ1 copies of each element in ℵ0.
[page 313]
[football] The slo-mo replay turns an instantaneous event into a stationary point which can be observed.
The zero universe is built on duality and the most basic mathematical duality of all appears to be between complex and real numbers paved by the duality of a complex number and its complex conjugate [Dirac: 'conjugate imaginary' ?].
Thursday 29 July 2021
Knowledge of the future makes survival and efficient use of resources much easier, which is why at least some of us study the past [which is the only thing we can study empirically] to seek invariant features which constrain the future. Following Parmenides and Plato and the delusive thinking of many founders of thousand year Reichs, Christianity has endowed itself with an omnipotent and omniscient God who is also benevolent and so will make everything come out right for them like a good father who takes care of every detail. The Catholic Church has logged on to this dream by declaring itself infallible. Experience shows that this position flies somewhat in the face of reality and leads to violent behaviour in those who are powerful enough to control the future of others when they try to make their dreams come true by employing Procrustean measures like police states and autocracy. [W]e see [this] every day, particularly in autocratic societies of which the Roman Catholic Church and the Communist Party of China are the prime global examples. Here we preach a rather blind divinity which starts off with an unbounded capacity for action but finds its powers limited by its inability to implement logical contradictions, as Aquinas notes, alerting us to the fact that the universe and natural selection are governed by logic, and giving us firm guidance, revealed by Darwin, about how things work.
[page 314]
The unit of energy is measured by the distance between acts.
I am slowly getting closer to what I want to say, a clear statement of the hypothesis. Next step, argue for its truth.
Friday 30 July 2021
The Catholic Church and many sports coaches and other urgers favour reason over emotion. The Church in particular claims that the damage done to humanity by the original sin is that God broke the nexus between reason and emotion, so we see authors like Aquinas basing the moral code on the alleged supremacy of reason and the need to force ourselves to do the reasonable thing. This position is not really supported by evolution, where we can imagine that the dichotomy between reason and emotion (or passion) plays no role and that the average creature simply acts in response to the potentials built into it by its evolutionary heritage. I say all this because I am sitting here in the morning pleased with my feelings about the revision of my chapter 6 but feeling the odd niggle about the possible reception of my ideas in the theological world should they ever become prominent enough to elicit a response. In the meantime I am feeling so complacent that I have no inclination to work myself to the extreme of my ability to get it out, as a typical sports coach might recommend, and which would elicit the praise of sports writers who like to see people drop from exhaustion at the end of their races.
[page 315]
One may see the ruling class motivation for claiming the supremacy of reason as a means to get people t sacrifice their personal emotions for the good of the ruler as we are encouraged to die in battle to serve the interests of princes who are almost inevitably far better off than their subjects as a consequence, insofar as they claim a divine right to act outside the norms they impose upon their subjects, often on pain of death. More generally the power of cooperation demands that we should subject personal to collective values but the liberal approach is that this should be achieved by personal collective bargaining rather than imposed by force.
Saturday 31 July 2020
I may claim here to be operating under a theological licence. Since most theological views of the invisible world that they postulate as existing behind the scenes of the material word are quite far out, like eternal post mortem life in Heaven or Hell, I feel that the views expressed here, even if they seem a little far out by many standards. are quite conservative given the constraints on reality that we are aware of.
Looking back I feel that I have often been out of my depth in life trying to achieve dreams with inadequate resources which meant my successes were partial and fragmentary and left a lot of loose ends behind. Now that I am heading home in the final quarter of my life I have the pleasant feeling of finally getting somewhere. I went to university over the last few years but made no impression but now eighteen months after I completed my thesis it begins to seem possible
[page 316]
to make my story stick with help from Aristotle, Aquinas, Feynman, Darwin and von Neumann. I am beginning to see how an active young universe with no possible idea about where it was going would become my home 14 billion years later.
Perhaps quantum mechanics is a model of the universe dreaming. I am often amazed in my sleep by the creativity of my dreams, creating bizarre permutations of my life experiences. I feel that although I am semiconscious of these there is a lot more going on unconsciously and I wonder whether it is possible to create models of neural functioning in complex Hilbert space which give us more insight into our own intelligence than [classical neural network models].
Cordelia: 'I hope I've got a vocation.'
Charles: 'I don't know what that means.'
Cordelia: 'It means you can be a nun. If you haven't a vocation it's no good however much you want to be; and if you have a vocation, you can't get away from it however much you hate it. Bridey thinks he has a vocation and hasn't. I used to think Sebastian had it and hated it — but I don't know now. Everything has changed too much suddenly.' Evelyn Waugh (2000): Brideshead Revisited
Did I have a vocation? Did I think I had? I think I tried but could not stand the falsity of the Catholic message. So here I am fifty years later struggling to invent an alternative and feeling good about it. I am making progress, I think, and will have some sort of win before I go.
[page 317]
There is a long distance between an intuition and the words to express it. How is this link expressed in the mind?
The first bifurcation is action to potential / kinetic,where S = ∫ (PE -KE)dt, when PE = KE, S = 0. Maybe [equals] one up to the Lagrangian integral, so S = (PE -KE)t ie energy-time.
First spit energy-time then [kinetic] - potential and opening the way for a Lagrangian.
