Notes
Sunday 1 March 2020 - Saturday 7 March 2020
[Notebook: DB 84 Pam's Book]
[page 172]
Sunday 1 March 2020
Jammer page 26:Planck's original derivation of the black body law was inconsistent insofar as he bridged the gap between continuous electrodynamics and discrete statistics. Einstein introduced discrete photons, easing the pain but the continuous / discrete divide remained, and to my mind continues to bedevil physics which is essentially discrete because it is constrained by fixed point theory within the one 'discrete' god. Nevertheless it seems that quantum theorists continue to insist that the fields behind the particles are continuous. Einstein 1906. Max Jammer: The Conceptual Development of Quantum Mechanics, Einstein (1906): Zur Theorie der Lichterzeugung und Lichtabsorption
page 37: Particle = carrier of energy and momentum. Smallest
[page 173]
particle measured by the quantum of action.
Jammer page 38: Einstein (1909) Zum gegenwärtigen Stand des Strahlungs Problems: ' "light quanta" . . . are subject to the same statistical laws as are particles or molecules in the kinetic theory of the ideal gas' [if they are distinguishable?] List of scientific publications by Albert Einstein - Wikipedia
'Einstein corroborate this result with a discussion of the Brownian motion of a mirror which he assumed to be a perfect reflector for radiation within a frequency interval dv and transparent for all other frequencies. For the mean square of the pressure fluctuation he obtained again a sum of two terms, one term corresponding to the fluctuation of waves and the other to that of particles.'
My next thesis will make the jump from Aquinas to Einstein.
page 42: Modern quantum mechanics . . . views Planck's constant as a kind of connecting link between the concept of waves and that of particles . . . '
page 44; 'Planck's theory was an amalgamation of waves and particles' [embracing the contradiction].
As its name suggests, Christianity is a cult of personality and traditional theologians find it very hard to climb out of this mould. Alister McGrath, for instance, writes [in] his three volume Scientific Theology: "Theology is a principled uncovering of the spiritual structure of reality, and a responsible attempt to represent them in a manner appropriate to their distinctive natures. Just as the natural sciences
[page 174]
' develop a distinctive vocabulary, set of hypothetical entities and constructed to represent the levels of reality which they encounter, so Christian theology has evolved it own characteristic terminology, models and conceptualities to represent the reality which is encountered and disclosed in Jesus Christ." About whom we have no evidence whatsoever that could be called scientific. Just a series of stories placing much ancient wisdom in the mouth of a peon, clearly revolutionary and possibly fictitious whose constructed personality as the Son of God carries very little weight outside the genre of similarly constructed and recorded personalities whose ideas often make very little contact with reality, except in the institutional reality built to justify the cult of their personality. McGrath; A Scientific Theology volume II: Reality
Wave particle duality is a consequence of the fact that the creation and annihilation of particle states is an inherently cyclic computational process whose clock speed is measured by energy.
Jammer page 144: 'Thus it became clear that the principle of exclusion, like that of relativity, is not merely another theorem in physics but rather a general precept regulating the very formulation of physical laws.' Pauli exclusion principle - Wikipedia
page 153: Quantum conditions place sharp restraints on the behaviour of 'continuous' classical systems at the quantum scale, making everything digital or discrete, like the spin of an electron.
Monday 2 March 2020
What is the meaning of the mathematical boundary between Φ and |Φ|2? Closely
[page 175]
related to the fundamental quantum mechanical connection between continuous and discrete modelling of the universe further connected to the (flashed through my mind and then lost it) distinction between deterministic evolution of probability functions contrasted with the probabilistic nature of actual outcomes (see Born Wednesday 26/2 Notes page 166).
The big deal in physics is to get the numbers right. How do numbers couple to reality? Through units. In some cases the unit is simply a scalar, a pure count, as is the case with entropy and statistics in general where we count states, or people, or eggs. In more complex cases the number refers to a "dimension", in physics a count of mass or length or time. Initially these units were arbitrary but now they are linked to fundamental constants like the quantum of action, the velocity of light, Boltzmann's constant, the gravitational constant etc. A serious problem in physics is that there is no natural way of numbering a continuum so we have to pick an arbitrary unit such as the metre for length and define it in terms of some natural length, the wavelength of some natural frequency of light determined by an electronic transition in an atom. The upshot of all this is that the continuous numbers are in reality a secondary source of numerical information defined by the primary source, natural numbers or integers which form the foundation of arithmetic and logic. Leopold Kronecker - Wikipedia
Jammer page 151: 'In view of its decisive importance for basic conceptions of modern quantum mechanics the Compton experiment will be discussed in greater detail. Compton scattering - Wikipedia
page 161: Compton: "A quantum theory of the scattering of x-rays by light elements".
[page 176]
' "The present theory depends essentially on the assumption that each electron that is effective in scattering scattered a complete quantum. It involves also the hypothesis that the quanta of radiation are received from definite directions and are scattered in definite directions. The experimental support of the theory indicates very convincingly that a radiation quantum carries with it directed momentum as well as energy." '
Jammer page 172: 'There can be no doubt that the Danish precursor of modern existentialism and neoorthodox theology, Søren Kierkegaard, through his influence on Bohr, affected the course of modern physics to some extent . . . Kierkegaard's philosophy of life and religion . . . contributed to the creation of a philosophical climate which facilitated the surrender of classical conceptions.'
173: 'Man cannot without falsification conceive of himself as an impartial spectator or impersonal observer; he always necessarily remains a participant.'
page 175: 'Bohr "Every sentence I say must be understood not as an affirmation but as a question." ' (Rosenfeld, Niels Bohr - An Essay p. 3)
page 176; 'Bohr was strongly influenced also by William James.'
page 179: '[James's] vivid description [in The Principles of Psychology, chapter 9] of observing introspectively the "transitive parts" in the stream of thoughts and feeling left a strong impression on Bohr's mind, as he admitted himself.' William James: The Principles of Psychology
page 181: '. . . it was hard to understand how, for example, in a system composed
[page 177]
' of an electromagnetic radiation field susceptible only to continuous changes in energy, and an aggregate of atoms, emitting or absorbing only discrete quantities of energy, the sum total of a continuous and a discrete amount of energy could be a constant.'Jammer page 187: Lande: an atom is "a virtual orchestra" Neue Wege der Quantentheorie, Die Naturwissenschaften 14, 455-458 (1926)
page 188: 'A confrontation with the basic problem of quantum theory, the problem of the interaction of matter and radiation [overlooking the fact that radiation is matter, the distinction is not real] . . . eventually led to the birth of the earliest formulation of quantum mechanics.'
page 205: ' Heisenberg: "Almost every progress in science has been paid for by a sacrifice, for almost every new intellectual achievement previous positions and conceptions have to be given up. Thus, in a way, the increase of knowledge and insight diminishes continually the scientists' claim on 'understanding' nature" [no, errors are removed and truth is increased]. '
page 211; Born: ' "I will never forget the thrill I experienced when I succeeded in condensing Heisenberg's ideas on quantum conditions in the mysterious equation pq - qp = h/2πi.
page 219; 'In fact there is simply no ultimate logical connection between classical and quantum mechanics.'
The basic point of non-commutation in quantum mechanics is that we are not dealing with isolated states but changes of state and the passage from, state a to state b is the inverse of the process from b to a where a andb are orbitals of atomic electrons [or any other different physical states].
[page 178]
Jammer page 222: Born and Wiener: "An operator is a rule in accordance with which we may obtain from a function x(t) another function y(t) . . . It is linear if q(x(t) + y(t)) = qx(t) + qy(t), eg differentiation, integration.
page 224: subject, operation, result.
page 226: Boole: "the mere processes of symbolic reasoning are independent of the conditions of their interpretation." So the transfinite computer network is a vast system of symbolic reasoning which I want to interpret as a theory of everything.
page 227: Heaviside: ' "Mathematics", he declared, "is an experimental science, and definitions do not come first, but later on." The widespread use of the delta function prior to its vindication by the theory of distributions shows that also the development of quantum mechanics was not exempt from Heaviside's antipuristic attitude.'
page 237: "Hamilton's optical-mechanical analogy."
' . . . the surface of constant action of a system of particles is propagated in complete analogy to the surface of constant phase in optics, the wave vector corresponding to the momentum and the frequency to the energy of the particle.
Tuesday 3 March 2020
The universe speaks to us using particles which we take to be exactly analogous to the words of this sentence. The most general definition of a particle is a definite and seperate object that carries
[page 179]
energy and momentum, its energy being its time rate of action and its momentum its space rate of action. Photon, electron and myself meet this definition and an important property of particles, like words, is their ability to bind to one another to form more complex particles. According to the standard model, I, and the universe as a whole, is a system based on about 60 fundamental (ie non-composite) particles analogous to the letters of the universal alphabet. The most fundamental of these articles, repeated ad infinitum is the initial singularity. This seems to suggest that even fundamental particles are in some way composite, although we may see their "composition" as a feature of their relationships to one another mediated by the boson-fermion foundation of the universal network. Every object tells a story in the context of the objects surrounding it. From this point of view we can see everything as a mirror of everything around it, sharing the complexity of its environment. At the root of the transfinite network we have the simple Boolean operators not and and.
Wednesday 4 March 2020
Dante's Hell is a potential well, getting more painful as one goes deeper. Heaven is a similar structure, getting more pleasant as one goes higher. Divine Comedy - Wikipedia
Heisenberg and Schroedinger tell us that the quantum conditions constrain everything to proceed in discrete steps of ℏ.
Where do fundamental constants come from? The only available source in an initially structureless universe is consistency. An atom works because of the nature of protons and electrons. This in turn relies on the relationship
[skip to page 182: pp 180, 181 missed][page 182]
between electric charge and space-time that Maxwell found and expressed in his continuous differential equations which explain the dynamics of electromagnetism. What is the quantum version of the relationship between magnetism, charge and relativity?
They say the basic problem with electrodynamics lies in Maxwell's equations so maybe we should look there for a quantum version of Maxwell and eliminate the problems raised by continuity at the root. Then we may find the root of theology in electrodynamics, the body electric. The basic factor that shapes Maxwell's equation is conservation of the flow of current which since it is not managed by leakproof pipes (Cosmic Plumbing, DB 80 of these Notes: 20 January 2016 - 22 April 2017) can be managed instead by quantum bookkeeping, like discrete units of money.
[Back to page 180 from page 182]
Thursday 5 March 2020
Jammer page 237: '[Hamilton's results] published between 1828 and 1837, found little attention for almost 100 years.
Friday 6 March 2020
Looking for a logical explanation for the birth of the universe that avoids the complexity and contradictions of the standard model. Over the years I have cooked up many heuristic ideas and in the next few years I wish to assemble them into a coherent whole by going over and over the story tweaking and polishing it until it shines [, beginning with my honours thesis]. Prolegomenon to Scientific Theology
Jammer page 239: 'Bragg: "One x-ray provides the energy for one beta-ray, and similarly in the x-ray bulb one beta-ray excites one x-ray . . . the speed of the secondary beta-ray is independent of the distance the x-ray has travelled: so the x-ray cannot diffuse its energy as it goes, that is to say, it is a corpuscle." '
Why does E = hf? Because each cycle uses one quantum of action to execute itself. This shows the way to the quantization of Maxwell's equations. Why does the photon have spin 1? Because it is a boson. But why? Because photons of the same frequency are indistinguishable.
[page 181]
Jammer page 239: Particle wave duality arise because it takes cycles of computation to create [execute] a particle, that is to transform the particle to be annihilated into the particle to be created.
page 243: 'De Broglie's hypothesis which associated with every particle a wave was soon elaborated into a new theory of mechanics. If there are waves, it was argued, there must be a wave equation. What was now needed was the discovery of this wave equation.
page 258: 'de Broglie had shown that the Sommerfeld quantum condition can be interpreted as a statement concerning the number of wavelengths which cover the exact orbit of an electron around the nucleus. For Sommerfeld's ∮ dq = hn and de Broglie's p = h/λ implied that ∮1/λ dq = n
'To Schrödinger this equation immediately suggested an eigenvalue problem.'
page 259: '[Schrödinger's quantum condition]: ψ has to be a real,single valued twice differentiable function for which the integral of the just mentioned quadratic form over the whole of the configuration space (q space) is an extremum. The Euler-Lagrange equation corresponding to this variational integral is the wave equation.'
page 260: 'The discrete eigenvalue spectrum thus turned out to be
[From notes page 181]
E = -mc4 / 2 k2 n2, n = 1, 2, . . . which for k = h/2 π was precisely the Bohr energy spectrum for the hydrogen atom.
Jammer page 260: 'Considering the atom as a system of vibrations, Schrödinger thought in a similar vein to have traced Bohr's formula for the frequency of emission ν = E1/h - E2 /h expresses merely the fact, well known from accoustic and electromagnetic wave theory (heterodyne frequency) that the beat frequency is equal to the difference of two simultaneous characteristic frequencies of the emitter.
[page 183]
Jammer page 271: '[Einstein to Born]: "That one has to solve the quanta by giving up the continuum I do not believe. . . . I am rather convinced now as before that one has to search instead for an overdetermination by differential equations so that their solutions are no longer of a continuous type. But how?" ' Something to do with fixed point theory.
Heisenberg: 'algebraic approach which, proceeding from the observed discreteness of spectral lines, emphasises the element of discontinuity [taking us a step toward logical continuity].
Schrödinger: 'analytic approach which, proceeding from a generalization of the classical laws of motion, stressed the element of continuity and, as its name indicates, was a theory whose basic conception was the wave.'
page 286: 'For Einstein the notion of probability, even as he applied it to reconcile his light-quanta hypothesis with Maxwell's theory of electromagnetic waves, was the traditional conception of classical physics, a mathematical objectivisation of the human deficiency of complete or exact knowledge, but ultimately a creation of the human mind . . . '
'For Born probability, so far as it related to the wave function, was not merely a mathematical fiction, but something endowed with physical reality for it evolved in time and propagated in space in accordance with Schrödinger's equation. It differed, however, from ordinary physical agents in one fundamental aspect: it did not transmit energy or momentum. Since in classical physics, whether Newtonian mechanics or Maxwellian electrodynamics only what transfers energy or momentum (or both) is regarded as "real" the ontological status of ψ has to be considered somewhat intermediate.'
[page 183]
Heisenberg: "[probability waves] are a quantitative formulation of the concept of δυναμις, or in the later Latin version potentia in Aristotle's philosophy. The concept that events are not determined in a peremptory manner, but that the possibility or "tendency" for an event to take place has a kind of reality — a certain intermediate layer of reality, halfway between the massive reality of matter and the intellectual reality of idea or image [formalism] — this concept plays a decisive role in Aristotle's philosophy. In modern quantum theory this concept takes on a new form; it is formulated quantitatively as probability and subjected to mathematically expressible laws of nature."
Saturday 7 March 2020
'A comparison with classical mechanics seems to be instructive. Quantum theory prior to the advent of transformation theory may be compared to Newtonian mechanics prior to, say, Poisson's introduction of generalised moments, and just as the development of the canonical formalism in classical dynamics provides in the work of Jacobi, Poincarè and Appell a profound comprehension of the whole structure of classical mechanics, so the development of the quantum mechanical theory of transformation culminated with the work of Dirac, Jordan and von Neumann in the achievement of new vistas which made it possible to view non-relativistic quantum
[page 185]
mechanics of a finite number of degrees of freedom as a logically consistent compact and unified system of thought.'
Jammer page 294: ' . . . it is possible to define the transformation theory as the study of those transformations in quantum theory that leave the results of empirically significant formulas invariant.
page 295: Poisson bracket [x, y] → xy - yx = ih[x, y]
page 298: '. . . just as the differential and integral calculus was the language of classical dynamics or the tensor calculus that of relativity, the medium of expression for modern quantum mechanics turned out to be the theory of linear spaces, and, in particular Hilbert spaces, or, more generally, the theory of functional analysis.
page 300: 'With the advent of modern point set topology . . . which makes it possible to apply topological notions to algebra and analysis, and with the development of the theory of integral equations . . . functional analysis became one of the most important branches of modern mathematics and . . . the mathematical foundation of quantum mechanics.'
page 301: '[Dirac] gave his theory a new and independent logical foundation by boldly introducing one of the most useful mathematical devices in quantum theory: the famous δ-function . . .. Dirac declared "all electrical engineerss are familiar with the idea of a pulse, and the δ-function is just a way of expressing a pulse mathematically.
Kirchoff defined δ-function in 1882: "As to the function F, we assume that it vanishes for all finite positive and negative values of
[page 186]
its argument, but it is positive for such values when it is infintely small in such a way that ∫F(ζ) dζ = 1.
Jammer page 304: Using δ, Dirac showed that the eigenfunctions of Schrödinger's wave function are just the transformation functions that enable one to transition from the (q) scheme of matrix representation to a scheme in which the Hamiltonian is a diagonal matrix.
page 305: Dirac: "the coefficients that enable one to transform from the one set of matrices to the other are just those that define the transition probabilities."
page 307: 'The transformation theory, both in Dirac's continuous matrix formulation and Jordan's semi-axiomatic formalism . . . was the first indication that it might be possible to dispense with the correspondence principle in the construction of the conceptual edifice of quantum mechanics.'
page 310: '[Hilbert's] conception of a separate analytical apparatus through the interpretation of which empirical relations can be brought into the form of mathematical statements became a basic element of the methodology of modern theoretical physics.'
page 315: 'The decisive innovation on which von Neumann based his new approach was his ingenious discovery, early in 1927, that a new formalism of quantum mechanics can be established
[page 187]
in view of the fact that not Z [discrete] and Ω [continuous] but rather the sequence space FZ over Z and the function space FΩ over Ω are essentially identical. Here FZ is the set of all sequences {um}0∞ satisfying Σ |um|2 = 1 and FΩ the set of all summable and square integrable complex valued functions over Ω satisfying ∫ |ψ(q)|2 dq = 1.
Jammer page 316: 'von Neumann reasoned as follows: since FZ and FΩ — and not F and Z — are the "real analytical substrata" of matrix and wave mechanics, and since they are isomorphic, that is ultimately only different mathematical representations of the same abstract relations, the equivalence between matrix and wave mechanics is a logical consequence of this isomorphism. Moreover, he added, a formulation of quantum mechanics which retains only absolutely essential relations . . . has to be based on the common structure of FZ and FΩ, or as he called it the "abstract Hilbert Space".
page 323; The Copenhagen Interpretation.
'With the establishment of the statistical transformation theory the formalism of nonrelativistic quantum mechanics was completed in all its essential points. But a formalism, even if complete and logically consistent is not yet a physical theory. To reach this status, some of its symbols have to be given operationally meaningful interpretation . . ..'
?page 324: 'In short, there still remained the open questions: how to interpret the new formalism, and in particular, how to put its symbols or mathematical expressions into correspondencr with classical physics, for only in this way could operationally meaningful correlations be established.' Maybe we do not need to put it into correspondence with classical physics, but into correlation with human dialogue!
[page 188]
The same could be said of relativity, relating it to travel [through space] and getting to work on time.
Jammer page 332: Is uncertainty ontological or epistemological? My guess: ontological, opening the way for a digital / discrete / pixellated / logically continuous universe.
'Heisenberg's paper . . . directed the attention of atomic physics to the importance of the concept of measurement which from now on was to assume a central position in all questions relating t0 the foundations of quantum mechanics (ref Jammer page 326: Z. fur Physik 43, 172-198 (1927, English translation in Wheeler & Zurek) Wheeler & Zurek: Quantum Theory and Measurementpage 337: Lucretius (De rerum natura II, 292 "exiguum clinamen principiorum"
page 338: 'Having discussed the development of Heisenberg's principle . . . we are now in a position to discuss its implications with regard to the notions of identity, indistinguishability and identifiability of physical objects . . . ..
page 340: Leibniz . . . principium identitatis indiscernibilium: "no two objects in nature can differ in number" since there is no sufficient reason for identical objects to be in different places, except of course that they are in different places [ie "place" is a real attribute rooted in the nature of space, which also accounts for the velocity of light].
page 348: Bohr "complementarity" . . . ' a logical relation between two descriptions or sets of concepts which, although mutually exclusive, are nevertheless both necessary for an exhaustive description of the situation.'
[page 189]
Jammer page 352: '. . . according to the quantum postulate in atomic physics, any observation of the system implies a disturbance. In other words, a system, if observed, is always an open system. A space-time description, however, presupposes observation. Hence, concluded Bohr, the claim of causality excludes spatiotemporal description and vice versa. The usual causal space-time description, that is the simultaneous use of complementary descriptions is made possible in classical physics, argued Bohr, mostly because of the extremely small value of the quantum of action "as compared to the actions involved in ordinary sense perceptions".'
The discussion of complementarity seeks to give an epistemological account of uncertainty whereas in fact, given the reality of quantization (pixellation) it is actually ontological. It is a simple fact, it seems to me, that a pure continuum carries no more information than a blank page. As Kronecker might say today, God made the quantum and continuity is a meaningless fiction of human creation, as is the belief that continuous functions are ontologically deterministic.
page 358: Born and Heisenberg: ". . . Every additional advance in our understanding of the formulas has shown that a consistent interpretation of the quantum mechanical formalism is possible only on the assumption of a fundamental indeterminism . . . the real meaning of Planck's constant h is this: it constitutes a universal gauge of the indeterminism inherent in the laws of nature owing to the wave particle duality."
I would say that it is a universal building block [universal atomic action] of the ontological structure which can only be further divided in fiction.
[page 190]
Jammer page 363: Ehrenfest: ' "by a short elementary calculation without approximations" showed that the expectation value of the time derivative of the momentum is equal to the expectation value of the negative gradient of the potential energy function [ie Newton's second law]. (Z. für Physik 45 455-457 (1927) Paul Ehrenfest: Bemerkung über die angenaherte der klassischen Mechanik innerhalb der Quantenmechanik
page 365: Barger: 'a system which is in a stationary state will stay in that state during an adiabatic process.'
page 365: 'In fact never has a physical theory given a key to the explanation and calculation of such a heterogeneous set of phenomena and reached such as perfect agreement with experience as quantum mechanics.
page 368: Dirac; von Neumann Dirac: The Principles of Quantum Mechanics (4th ed), von Neumann: Mathematical Foundations of Quantum Mechanics
page 368: von Neumann on 'hidden variables'; "the present system of quantum mechanics would have to be objectively false in order that another description of the elementary processes than the statistical one may be possible."
page 371: 'In the particular case of a pure state ψ = Σ anψn expanded in the eigenfunctions of the observable R to be measured, the measurement of R "reduces" ψ to ψk whose eigenvalue λk is the result of the measurement. According to von Neumann, this "reduction of the state" or more generally "reduction of the wave packet" is an unavoidable element in every measurement and assures that the same result will be obtained for an immediately following measurement of the same observable ("projection postulate").
