Notes
Sunday 2 August 2020 - Saturday 8 August 2020
[Notebook: DB 85 Science]
[page 69]
Sunday 2 August 2020
The longer one thinks about a new strange idea the more familiar and acceptable it becomes. The notion that quantum behaviour is the first stage in cosmic development after the initial singularity seems to be gaining weight in my mind and I am searching for more evidence (confirmation bias?) the most important line of
[page 70]
which seems to me to be the notion that every quantum event in the universe, which simply means every event in the universe, is accompanied by precisely one quantum of action, as demonstrated by the non-commutative behaviour of quantum operations which I really do not understand given my tendency to skim over the mathematics while trying to create non-mathematical pictures of how the world works.
Stack exchange: is non-commutation a property of 3-space but not of Hilbert space? Is it that before the emergence of space quantum behaviours were all causal not leading to new irreversible situations by permutations of operators, ie AB ≠ BA.
von Neumann page 136: Quantum statistics: '. . . let us emphasize two things: (1) φ has an entirely different appearance and role from the q1, . . . , qk, p1, . . . , pk complex in classical mechanics, and (2) the time dependence of φ is causal, not statistical: φt0 determines all φt uniquely, as we saw above.' von Neumann: Mathematical Foundations of Quantum Mechanics
page 128: φ is normalized, ie it lies on the surface of the unit sphere in Hilbert space.
'[In regard to normalization] it should be pointed out that while φ is dependent on time t as well as on the coordinates q1, . . . , qk of the configuration space of our system, nevertheless the Hilbert space involves only the q1, . . . , qk (because the normalization is related to these alone. Hence the dependence on t is not to be considered in forming the Hilbert space. Instead it is rather to be regarded as a parameter. Consequently, φ—as a point in R∞ depends
[page 71]
on t but is on the other hand independent of the q1, . . . , qk. Indeed, as a point in R∞, it represents the entire functional dependence.'
von Neumann page 137: Compton and Simon experiment with electrons and photons. Compton scattering - Wikipedia
page 144: 'We begin by assuming exact measurability so that [the operators] R and S must have pure discrete spectra.' More generally, we assume that all exact relationships like E = hf in physics rely on discretization as required for error free communication by communication theory. The fact that unitary evolution with respect to time is deterministic suggests that the system at that level must be discrete, working in units of the quantum of action [and 2π radians of phase].
page 146: Expectation value of the sum = sum of the expectation values (Tψ, ψ) = ((R + S)ψ, ψ) - linearity of operators.
page 148: 'We see therefore that the characteristic conditions for the simultaneous measurability of a finite number of quantities R, S, . . . is the commutativity of the operators R, S. In fact this holds for absolutely exact as well as for arbitrarily exact measurements. In the first case, however, it is also required that the operators possess pure discrete spectra, as is characteristic of absolutely exact measurements.
page 150 Uncertainty relation: non commutating operators, derived
[page 72]
from quantities which are canonically conjugate in classical mechanics.
von Neumann page 153: from formal to experimental understanding of uncertainty.
page 159: Projections as propositions
Monday 3 August 2020
Proof required: Every quantum event involving transition between states of a pure discrete spectrum, no matter how small the energy involved requires precisely one quantum of action. This is the discrete foundation from which energy changes are to be calculated. Quantum mechanics is indifferent to any alleged fixed energy zeros, whereas gravitation is strictly proportional to energy [mass], so we have a serious problem with the cosmological constant which is built from many positive quantum energy contributions.
Hilbert space can 'hold' an infinite number of linearly independent (orthogonal) vectors whereas Euclidean space holds a maximum of three. How do we account for this reduction? [Is it related to the "reduction of the wave packet"?] Does the 'crossed wires' theorem have anything to do with this and what about the transition from complex to real numbers?
von Neumann page 164: Radiation theory
page 165: 'Clearly we have questions here which, in accord with the fundamental principles of quantum mechanics, must first be answered classically. The results so obtained can then be
[page 73]
translated into operator form.' This principle is useful and corresponds in some way to Bohr's notion of complementarity, but it may also be the case that the quantum world is so different from the classical world that this idea may serve to hide interesting and meaningful differences, particularly when we want to go the other way, from broad spectrum operators to classical results.
page 166: So we begin with Maxwell's equations in an empty bounded space . . .
These notes are rather boring and long winded, and this sentence is probably a repetition, but there is a certain amount of selection going on insofar as I only write when I feel I have come across something new enough to constitute a discernible step toward my lifelong goal of a scientific theology from which we can derive significant political, social and individual guidance to help us reap the benefits of conforming more closely to reality. As usual I am digging around in the quantum mechanical basement in search of the soul of god when they were just an egg.
An extended piece of music needs memory to hold the sequence of notes and memory requires space, that is a set of [ordered?] orthogonal locations to store the sequential information, like a musical score. The creation of structure is the creation [and population] of memory, that is the creation of fixed points, that is the creation of the different
[page 74]
mappings of the initial singularity onto itself (because it has nowhere else to go).
Ella: Bangarra Dynamics Ella Havelka - Wikipedia
von Neumann and the quantum mechanics in general begin with centuries of Newtonian, Hamiltonian, Lagrangian deterministic physics and then use the translation to quantum operators to convert classical to quantum, but maybe the classical constraint implicit in this approach still limits the quantum possibilities and I am thinking that the freedom of logical consistency, which has real entropy as opposed to information deprived continua. Einstein, I feel, fell down in trying to make particles out of continua. W. Ross Ashby: An Introduction to Cybernetics, page 133
So we look at the fixed points, ie eigenfunctions of linear operators, and the method of growth is through communication in the tensor products of Hilbert space that set the scene for communication /interaction / measurement. Wojciech Hubert Zurek: Quantum origin of quantum jumps: breaking of unitary symmetry induced by information transfer and the transition from quantum to classical
Music drives the dance. Cognition feeding classical space-time, ie spirit driving matter. The creative power of the Hilbert space lies in its linearity as Dirac points out. Paul Dirac: The Principles of Quantum Mechanics (4th ed) Chapter I
So we are trying to make Hilbert space cognitive by linear transformation, as the neural network works by superposition [of synaptic inputs to the body of a neuron]. Synapse - Wikipedia
Tuesday 4 August 2020
[page 75]
The world is inherently dynamic [running on quanta of action] and the messages are fixed points in the dynamics. The archetypical [messenger] is the photon, a fixed point which from the point of view of the massive messengers in space-time is the first fixed point, travelling at the velocity of light when projected into space-time.
Non-linearity arises when systems couple to themselves giving positive or negative feedback as we see in the world of gravitation and of gluons. In linear quantum theory maybe the problem of the infinite self-energy of [zero size] electrons is solved by the fact that electrons do not couple to themselves [although in a two slit experiment a non-localized electron coming through both slits interferes with itself].
Action creates orthogonality, ie not-p is orthogonal to p. Here is the meeting of logic and metric which may be used to measure entropy.
Wednesday 5 August 2020
On constructing cognitive cosmology [e30_cognitive_cosmology.html, in progress]
Where did we go wrong? See von Neumann above page 165 [these notes page 72]. We have been beginning with a lot of old baggage, to wit the logical and mathematical structure that has grown around Newtonian physics, particularly the notion of continuity inherited from Zeno rather than Aristotle. Zeno makes a continuum out of a vast number of discrete points. Aristotle says things are continuous if they share their extremities. Zeno's approach
[page 76]
coupled with calculus led to a lot of mathematical headaches which were cured in effect by Cantor's idea of digitizing the continuum, building it from discrete natural numbers. Aristotle's approach made a logical appearance in the syllogism, where the two premisses are joined by a middle term, an overlapping so to speak [which couples the propositions rather as the overlap integral or inner product in quantum mechanics establishes a statistical bond between two states]. Turing made this idea into a computing machine which proceeds, like a human calculator, in a series of steps, each of which is complete in itself, so that a computer can record their current state, knock off, and the next shift can take up from where the last finished. The two operators overlap. In software engineering a machine processes an interrupt by pushing its current state to the stack, dealing with the interrupt and then popping its previous state from the stack and going on. I call this logical continuity. Continuous processes share states by sharing memory.
Von Neumann, calculating the world, begins with Newton and then converts the variables of classical mechanics like position, momentum and energy into linear operators in Hilbert space which are multidimensional elaborations of Newtonian calculus. Using the inner product metric of Hilbert space, he shows how the infinite spaces of operators can be made continuous by showing that they are complete using the nineteenth techniques devised to demonstrate the continuity of functions by variations of the epsilon delta argument which says that a process is continuous if a small change in its input leads to a small change in its output, and develops this idea to show that there can be deterministic, continuous unitary solutions to wave equations like that devised by Schrödinger. So far so good, mathematically gorgeous in the old continuous paradigm. (ε, δ)-definition of limit - Wikipedia
[page 77]
Now comes the problem of coupling the model to reality, and we are carried back to Planck's discovery that the interaction between radiation and matter only makes sense if radiation comes in discrete units which are products of a discrete physical unit measured by Planck's constant and inverse time, that is frequency, according to the primordial equation of quantum mechanics E = hf. This equation can still be treated as a continuous function since time is considered to be continuous and so frequency must be too.
The next problem to be dealt with was the spectroscopic fact that to the limits of instrumental resolution the spectra of atoms comprised pure discrete frequencies which appeared to be defined in nature with unlimited precision. Bohr and de Broglie took the first steps toward explaining this observation by coupling the quantum of action to stationary states, standing waves, and the existence of such states within an atom whose differences in angular momentum were precisely equal to one quantum of action and these differences could be translated by simple Newtonian electrodynamics into the frequency differences which explained spectral lines. Another step forward which worked quite well for the spectrum of hydrogen [and alkali atoms with one external electron] but did not go much further.
Years went by, and as von Neumann records, 1925 brought a resolution to the difficulties: 'A procedure initiated by Heisenberg was developed by Born, Heisenberg, Jordan and a little later by Dirac into a new system of quantum theory, the first complete system of
[page 78]
quantum theory which physics has possessed. A little later Schrödinger developed "wave mechanics" from an entirely different starting point, . . . On the basis of the born statistical interpretation of the quantum theoretical description of nature, it was possible for Dirac and Jordan to join the two theories into one, the "transformation theory" in which they made possible a grasp of the physical problems which is especially simple mathematically.' (page 6)
In a nutshell the development of quantum mechanics became an eigenvalue problem. In terms of fixed point theory the search for solutions became the search for the fixed points of unitary linear operators represented by the eigenvalues and eigenvectors which solved the unitary operator equation Uψ = λψ. The resulting eigenvalues correspond reasonably well to the observed spectral lines and the Born rule provides an algorithm to compute the probabilities of observing particular lines [corresponding to particular transformations of the states of atomic electrons]. Although this approach provided little insight into what actually happened when a quantum system was observed, raising the perennial problem of the "collapse of the wave function". Eigenvalues and eigenvectors - Wikipedia, Born rule - Wikipedia, John H. Halton: A Very Fast Algorithm for Finding Eigenvalues and Eigenvectors
Using a smaller nutshell, we can say that continuous and differentiable structures are the heart of classical physics. Discrete structures are the heart of quantum physics. Finally computer networks and communication systems represent the best of both worlds. Logically continuous systems are both discrete and continuous, since like Aristotle's syllogisms they establish continuity by shared extremities, that is shared memories, read from
[page 79]
and written to by connected users. So the aim is to build a networked logical system building up from the initial singularity by a process of creating space which is consistent with both classical and quantum physics and automatically produces a unified theory because it is rooted in the initial singularity, a structureless entity whose essence is identical to its existence like the classical God developed by Aristotle and Aquinas.
The minimal nutshell: god is the theory of everything "I am". Every element of the universe represented by a vector in Hilbert space endlessly repeats this sentence "I am ψ". Instead of trying to break down the universe of infinite complexity to find the theory of everything, we begin from the beginning and construct the universe from nearly nothing.
How do we map a creative world onto a creative Hilbert space? The story is coming along but its birth is exceedingly slow.
Thursday 6 August 2020
Kevin Brown: Reflections on Relativity Kevin Brown1.1 From Experience to Spacetime:
'. . . the fact remains that the imperative to reconcile our experience with some model of an objective world has been one of the most important factors guiding the development of physical theories' [not to mention the enormous practical benefit of knowing how the world works].
1.2 Systems of Reference:
We can bring most of the fundamental
[page 80]
dimensions M, L, T, down to ratios of the quantum of action ML2T-1. 2019 redefinition of SI base units - Wikipedia
'Whether or not the principles of quantum mechanics are adequate to justify our conceptions of reference lengths and time intervals, the characteristic spatial and temporal extents of quantum phenomena are used today as the basis for all such references.'
'Arguably we never actually observe fields, we merely observe effects attributed to fields.'
1.3: Inertia and Relativity:
Galileo toward inertia: ' ". . . among things which all share equally in any motion [that motion] does not act, and it is as if it did not exist. . . " '
Newton: ' ". . . the whole burden of philosophy seems to consist in this: from the phenomena of motions to infer the forces of nature, and from these forces deduce the phenomena." ' By considering the transmission of information as a force we radically widen the meaning of 'force'.
The principle of inertia [Newton's first law] 'is the most successful principle ever proposed for organizing our knowledge of the natural world.' Evolution ??
Newton's three laws make no reference to the direction of motion? ['in a right line', 'equal and opposite' ??]
'. . . "the laws of motion" are true by definition. Their significance lies not in their truth, which is trivial, but in their applicability. The empirical fact that there exist systems of inertial coordinates
[page 81]
is what makes the concept significant.'
Brown section 1.4: The Relativity of Light:
Scholastic concepts of lux and lumen: 'The word lux was used to signify our visual sensations, whereas the word lumen referred to an external agent . . . that somehow participated in our sense of vision.'
Finite velocity of light discovered by Roemer and Bradley implies that light has some existence in itself and is not just a relation between entities. More generally, we conceive relations as real entities involving the transmission of physical information between related sources, ie bosons relate fermions to one another. Ole Rømer - Wikipedia, James Bradley - Wikipedia
Wave equation for time dependent scalar field φ(x,t):
∂2φ/∂x2 = 1/v2 ∂2φ/∂t2
. . .
The universal structure effectively starts from nothing (the structureless but nevertheless existent god or initial singularity) and the formal development of structure depends on the emergence of opposites like positive and negative charge, potential and kinetic energy, space and time, particle and antiparticle, up and down, positive and negative, etc which are in effect anti-structures of one or another so that when they are superposed (like fermions) they add up to nothing, just like normalized waves π out of phase.
[page 82]
We can distinguish two cases here. Sometimes we put opposites together (like antiparticles) and get nothing. In other cases we put opposites together (like male and female) and get something new, a baby. In some way we see the second case as 'locking in' the diversity produced by the emergence of the first case from the original nothing.
Brown Section 1.6: A More Practical Arrangement:
' The real content of Einstein's principles is that light is an inertial phenomenon (despite its wavelike attributes). . . . Einstein: "radiation carries inertia between emitting and absorbing bodies." ' ie energy/mass + momentum.
'Einstein: ". . . light propagating as discrete packets of energy . . . cannot be represented as a solution of Maxwell's linear equations".'
'Einstein's contribution was to recognise that "the bearing of the Lorentz transformation transcended the connection with Maxwell's equations and concerned the nature of space and time in general." . . . Lorentz invariance is a key aspect of the modern theory of quantum electrodynamics, which replaced Maxwell's equations.'
1.7: Staircase Wit
1.8: More Symmetry
1.9: Null coordinates.
We might take the existence of the null geodesic in Minkowski spacetime as an indication that space and time are duals of one another and before the emergence of spacetime did not exist except as quantum theoretical possibilities, but how do we derive spacetime from quantum theory?
[page 83]
Brown Section 2.1: The Spacetime Interval:
'[proper time] represents the time that would be measured by an ideal clock comoving with that system.'
' The identification of the spacetime interval with the quantum phase applies to null intervals as well, consistent with the fact that the quantum phase of a photon does not advance at all between its emission and absorption (see section 9.9). Hence the physical significance of a null spacetime interval is that the quantum state of any system is constant along that interval. In a sense the interval represented a single quantum state of a system, so (for example) the emission and absorption of a photon can be regarded as, in some sense, a single quantum act.'
The wave function of a photon is static and therefore exists outside space and time.
'. . . the quantum state of a system gives (arguably) the most complete possible objective description of the system.' Except of course that we cannot objectively observe it!
' . . . relativity rests on both of the assumptions: (1) the zeroth and first derivatives of position [with respect to time] are perfectly relative and undetectable, and (2) the second and higher derivatives of position are perfectly absolute and detectable.'
We may imagine that spacetime is born as null spacetime (as in the initial singularity) and the grows through the advent
[page 84]
of massive particles to spacetime as we know it inside the light cone (and we assume that no communication and therefore no massive structure is possible in the spacetime region outside the light cone.
Brown Section 3.1: Postulates and Principles:
' Einstein: "It should be noted that the laws that govern [the structure of the electron] cannot be derived from electrodynamics alone. After all this structure necessarily results from the introduction of forces which balance the electrodynamic ones." Rather classical. How would explain the electron as a logical software entity [embodying the algorithm that makes electric force so powerful]? A process that emits and absorbs photons which realise the electric field. How does this compare with the gravitational process where the source of the force is mass? We have to imagine very simple software, so simple that it is impervious to error and interacts with energy in all its forms.
' To Einstein the most important quality of his interpretation [of physical theory] was not its consistency with experiment but its inherent philosophical soundness. . . . He may well have realized that any appeal to the Michaelson-Morley experiment in order to justify his theory would diminish rather than enhance its persuasiveness.
Section 3.2: Natural and Violent Motions
' The concept of force is one if the mot peculiar in all of physics' [but very intuitive because of muscular effort].
Force ⇆ communication ⇆ causality
[page 85]
Brown 3.2: 'However, the explanation of phenomena in terms of fields characterized by partial differential equations, as incomplete, because it is not possible to represent stable configurations of matter in these terms.' Linearity / orthogonality /independence.
Human rights : human orthogonality - freedom
' The difference between partial and total differential equations is actually more profound than it appears at first glance. . . . Total derivatives are evaluated over actualized values of variables. In contrast, the partial derivatives over immaterial fields are inherently hypothetical' [because of the multiplicity of variables].
' Einstein: "What appears to me, however, is that in the foundations of any consistent field theory, there should not be, in addition to the concept of the field, any concept concerning particles. The whole theory must be based solely on partial differential equations and their singularity free solutions." Wrong. Messages are particles, like me (and E).
Section 3.3: De Mora Luminis (delay of light):
3.4: Stationary Paths
Fermat's [least time] optical principle can be seen as a remarkable premonition of both special and general relativity [and, recalling that time is of the essence in the macroscopic world, perhaps this essence harks back to the very beginning].
3.5: A quintessence of so Subtle a Nature
Einstein: Gravitational field is identical with space.
[page 86]
' The question of whether electromagnetic phenomena can be accurately modelled as disturbances in an ordinary material medium was quite meaningful and deserved to be explored but the answer is unequivocally that the phenomena of electromagnetism do not conform to the principles governing the behaviour of ordinary material substances.'
' The Minkowskian structure of spacetime is in deed a quintessence of a most subtle nature.'
Brown Section 3.6: The End of My Latin
Section 3.7: Zeno and the Paradox of Motion
Xenophanes: "The all is one and the one is God" [?]
3.8: A Very Beautiful Day
3.9: Constructing the Principles
Einstein, 1949: ' "Gradually I despaired of the possibility of discovering the true laws by means of constructive efforts based on known facts. The longer and more desperately I tried the more I came to the conviction that only the discovery of a universal formal principle could lead us to assured results . . .." '
Friday 7 August 2020
Brown 3.9 (cont) '. . . we might say that Minkowski's spacetime interpretation does for special relativity what Boltzmann's statistical interpretation did for thermodynamics, namely it provided an elementary consuctive interpretation for the theory.'
[page 87]
The idea of making theology a theory of principle is to explain the emergence (within the Pseudo-Dionysian god of absolute simplicity) of the world as we know it by the reflection of this god upon itself, replacing the mysterious notion that a being with no structure can both imagine and construct the enormously complex universe we inhabit, outside itself by pure fiat.
' The modern theory of relativity passed though several phases of development:
1. (1902-1904) Lorentz invariance and Maxwell's equations
2. (1905) Einstein's explicit theory of principle which extended Lorentz invariance to all physical phenomena
3. (1908) Minkowski spacetime: 'merely encodes in a convenient form the physical fact of Lorentz invariance (ie the inertia of energy)'
4. (1928 - ) ' . . . beginning with the Dirac equation (1928) and quantum field theory (discussed in Section 9.4 and 9.10), according to which the advance of the quantum phase of any physical system is proportional to the lapse of proper time, as given by the Minkowski metric. Each stage represented a significant advance in clarity, with the end result being the establishment of a new fundamental principle – Lorentz invariance – which can be constructively understood as a consequence of the inertia of energy.
Section 4.1: Immovable Spacetime
Relativity implies systems of coordinates (artificial?)
Relationism - no need for coordinates: my relationship with people around me is established entirely by the communications we have with one another.
[page 88]
'From a purely relational point of view the concept of absolute inertia on which the principle of relativity is based has no meaning' (?). But, opinion is a cognitive surrogate for inertia, and communication changes opinion. The link is (somehow) Landauer's opinion that information is physical. Rolf Landauer: Information is a Physical Entity
'. . . However, it remains (at least arguably) possible to regard fields as just abstract constructions with no ontological status, and to express all physical phenomena in terms of substantial entities possessing spatio-temporal attributes. In this context the absolute-relational question remains both relevant and unresolved.'
Contra Aristotle and Aquinas: relationship is a substantial entity created by physical communication, as in a family or an atom. You don't know what you have till its gone, so my family's decision to exile me is empirical evidence for the foundation of my theology.
Every relational theory ' has foundered on . . . the apparent absoluteness of acceleration.'
Newton: moon, moving in a circle around the Earth, is falling toward the Earth as required by universal gravitation.
Spinning globes are distinguishable because the one with the most angular momentum (most quanta of action) is more oblate. The situation is incomplete because the angular momentum must have been contributed from somewhere according to the third law.
[page 89]
All symmetries are broken by boundary conditions and the third law applies here so equal and opposite breaks of symmetry are induced by equal and opposite breaks of boundary conditions. Action and reaction are everywhere in a closed universe.
' . . . [general] relativity, no less than Newtonian mechanics, relies on spacetime as an absolute entity in itself, exerting influence on fields and material bodies. . . . spacetime in the theory of relativity cannot simply be regarded as the totality of the extrinsic relations between material objects (and non-gravitational fields), but is a primary physical entity of the theory, with its own absolute properties, most notably the metric with its related invariants, at each point.'
'. . . the lightcone structure of Minkowski spacetime restricts the future of the point P0 to points inside the future null cone, i.e., P0 ± cdt, and as dt goes to zero, this range goes to zero, imposing a well-defined unique connection from each "infinitesimal" instant to the next, which of course is what the unification of space and time into a single continuum accomplishes.'
Brown Section 4.2: Inertial and Gravitational Separations
4.3: Free Fall Equations
4.4: Force, Curvature and Uncertainty
Section 4.5: Conventional Wisdom
'. . . Einstein once hoped that the general theory would not rely on the principle of inertia as a primitive element. However the hope was not filfilled and the underlying physical basis of the spacetime manifold in general relativity remains the set of primitive inertial paths (geodesics) through spacetime.
[page 90]
Brown Section 4.6: The Field of All Fields
The minimum unit of information is the quantum of action, ie one cycle of the minimum unit of process. This can be coupled to a larger event, so one bit of information can initiate Mutual Assured Destruction [Nuclear holocaust].
' The basic point is that although special relativity serves as the local limiting case of the general theory, it is not able to stand alone, because it cannot identify the phenomena to which it is applicable, and this renders it incapable of yielding definite macroscopic conclusions about the physical world.'
From a formal point of view the theory of everything is god, that is pure act, that is formally nothing, to be elaborated by the third law, action and reaction are equal and opposite, time division multiplexed to give energy.
Set of all sets leads to trouble. Gravitation is the field if all fields. So?
' In general relativity the Laplacian is replaced by a more complicated operator [Ricci Tensor], Rg, which, like the Laplacian, is effectively a differential operator whose components are evaluated on the spacetime with the metric g. However, in general relativity the field on which Rg operates is nothing but the spacetime metric g itself. In other words, the vacuum field equations are Rg(g) = 0.'
' The first level of quantum mechanics assigns each classical particle a quantum field. Then, in order to account for the creation and annihilation of particles during their interactions (and also to achieve consistency with special relativity) it was found necessary to introduce
[page 91]
"second quantization" leading to quantum field theory, which is essentially a consideration of (again!) the "field of all fields".' [which seems reminiscent of the layering implicit in the transfinite computer network].
' In his later years it seems Einstein had decided he had made all the progress that could be made on this preliminary basis, and set about the attempt to represent the total field. He wrote the above comments in 1949, after a quarter-century of fruitless efforts to discover the non-linear equations for the "total field", including electromagnetism and matter, so he knew only too well the risks of deteriorating into adventurous arbitrariness.' Where I have spent most of my life from a mythological theological beginning, but hope still burns in my heart as I get little glimpses of a possible future.
Brown Section 4.7: The Inertia of Twins
' Einstein: "No fairer destiny could be allotted to any physical theory, than that it should of itself point out the way to the introduction of a more comprehensive theory, in which it lives on as a limiting case".'
So I would like to produce a theology which sees the material world of STEM as a limiting case, a theory of principle beginning from the Pseudo-Dionysian god. Pseudo-Dionysius the Areopagite - Wikipedia
' Notice that the general theory is operative even in flat spacetime, because . . . all of spacetime (whether flat or curved) is to be regarded as a solution [ie an integration] of the field equations, rather than some a priori structure.
[page 92]
' The spacetime metric field is endowed with its own ontological existence, as is clear from the fact that gravity is itself a source of gravity. In a sense the non-linearity of general relativity is an expression of the ontological existence of spacetime itself. In this context it's not possible to draw the classical distinction between relational and absolute entities, because spatio-temporal relations themselves are active elements of the theory.'
A computer network is an inherently non-linear system often configured to do linear computations.
Brown Section 4.8: The Breakdown of Simultaneity
5.1: Vis Inertiae
5.2: Tensors Contravariant and Covariant: [the magnificent edifice of fully covariant differential geometry is necessary to enable a description of the universe to be crafted using partial differential equations. It may be possible to do the whole job in a very much simpler way by treating the universe as a communication network run by non-linear machines whose local operators are quanta of action.]5.3: Curvature, Intrinsic and Extrinsic: [Gauss made curvature self-documenting, as networks are self documenting].
5.4: Relatively Straight
5.5: Schwarzschild Metric From Kepler's 3rd Law
5.6: The Equivalence Principle
"The important thing is this: to be able at any moment to sacrifice what we are for what we would become." Charles du Bos
[page 93]
' The meaning of the equivalence principle (which Einstein called “the happiest thought of my life”) is that gravitation is not something that exists within spacetime, but is rather an attribute of spacetime.'
' The perfect equivalence between gravitational and inertial mass noted by Galileo implies that kinematic acceleration and the acceleration of gravity are intrinsically identical, and this makes possible a purely geometrical interpretation of gravity.' [which tells us what the spacetime that gives us gravitation looks like, but does not tell us how it works, which must be a quantum mechanical thing.]
"geometrical" → complex numbers → Minkowski space.
'. . . when “action at a distance” theories were replaced by “local action” theories, such as Maxwell’s differential equations for the electromagnetic field, in which only differentials of distance and time appear, we should have, for consistency, replaced the finite distances of Euclidean geometry with the differentials of Riemannian geometry.'
So exposing the hypothesis egregia that a differential is a formal geometrical definition of a quantum of action – so d(p)/d(not-p) is a local function of action, and we can build the whole world out of these units [atoms] of action.
' Thus the only valid form of the Pythagorean theorem is the differential form (ds)2 = (dx)2 + (dy)2. '
' From these considerations it follows rather directly that the influence of both inertia and gravitation on a particle should be expressed by the geodesic equations of motion:
d2xμ / ds2 + Γ μαβ dxα / ds dxβ / ds = 0
Einstein often spoke of the first term as representing the inertial part, and the second term, with the Christoffel symbols Γ μαβ, as representing the gravitational field, . . . '
'. . . it might seem to be a daunting task to attempt to found a viable theory of gravitation on the equivalence principle – just as it had seemed impossible to most 19th-century physicists that classical electrodynamics could proceed without determining the structure and self-action of the electron. But in both cases, almost miraculously, it turned out to be possible. On the other hand, as Einstein himself pointed out, the resulting theories were necessarily incomplete, precisely because they side-stepped the “source” aspect of the interactions.'
The source is the quantum, an actual particle.
' Einstein: "Maxwell's theory of the electric field remained a torso, because it was unable to set up laws for the behaviour of electric density, without which there can, of course, be no such thing as an electro-magnetic field. Analogously the general theory of relativity furnished a field theory of gravitation, but no theory of the field-creating masses".'
Brown Section 5.7: Riemannian Geometry
Section 5.8: The Field Equations
The spacetime metric is the field.
[page 95]
' Spacetime plays a dual role in this theory, because it constitutes both the dynamical object and the context within which the dynamics are defined.
Saturday 8 August 2020
Cognitive cosmology is a broad enough field to bring the classical field equations of relativity into the same field as the social field equations that describe the universe of human interaction. The transfinite computer network seems quite adequate, like the differential manifold, to cover this space but how do we arrive at the field equations that govern this space? Here we turn to quantum mechanics which describes the internal workings of the logical elements ("gates") that drive the network.
Do we see in the competition between the US and China very strong similarities between the GOP which supports Trump [Christians seeking power] and the CPC that supported Xi [Communists seeking power], [a battle totalitarian of ideologies, that seek "purity" and "simplification" running against the creative power of the divine universe].
General covariance = tensor equations = indifference to choice of reference frame (naturally built into network addressing by domain name servers) - how do we map these ideas onto one another using the quantum of action as a differential?
Brown Section 8 (continued): 'It's worth remembering that the generally covariant formalism had been developed only in 1901 by Ricci and Levi-Civita, and the first real use of it in physics was Einstein's formulation of general relativity.' Ricci Curvature - Wikipedia, Levi-Civita connection - Wikipedia
' In this way, Einstein believed he had addressed what he regarded as the strongest of Mach's criticisms of Newtonian spacetime, namely, the fact that Newton's space acted on objects but was never acted upon by objects.' [like Newton's God]
[page 96]
Thus, guided by the belief that the laws of physics should be the simplest possible tensor equations (to ensure general covariance), he proposed that the field equations for the gravitational field in empty space should be
Rμν = 0
'One outcome of the struggle to understand the conservation laws of the relativistic field equations was Emmy Noether’s famous theorem on the relation between symmetries and conservation laws.' ie the consequences of nothing happening, which is a consequence of simplicity, best represented by the Dionysian god. Dwight Neuenschwander; Emmy Noether's Wonderful Theorem
' It's worth noting that Einsteinian gravity is possible only in four dimensions, because in any fewer dimensions the vanishing of the Ricci tensor Rμν implies the vanishing of the full Riemann tensor, which means no curvature and therefore no gravity in empty space.
One breakthrough we need is to couple the above fact (via the notion that a differential = quantum of action is an act of communication) that we can only build computers in 4D space since there it is possible to avoid crossed wires [which would break the orthogonality of the spacetime dimensions]. Another idea maybe comes home to roost. Here's how we make gravitation into a classical network, so we should be able to cook up a network version of the Einstein field equation.
Brown Section 6.1: An Exact Solution: Schwartzschild. Schwarzschild metric - Wikipedia
Section 6.2: Anomalous Precession: Mercury etc.
[page 97]
6.3: Bending Light
6.4: Radial Paths in a Spherically Symmetric Field
6.5: Intersecting Orbits
6.6: Ideal Clocks in Arbitrary Motion
'If no physical phenomena were found to conform to the definition of proper time, then the assertion would indeed be worthless, but experience shows that the advance of the quantum wave function of any physical system moving from the event with coordinates x, y, z, t (in terms of an inertial coordinate system) to the event x+dx, y+dy, z+dz, t+dt is invariably in proportion to dτ wheredτ2 = dt2 − dx2 − dy2 − dz2
This suggests that my idea that quantum phenomena precede the emergence of spacetime may be wrong! [but then how does spacetime work, since according to Einstein it is a dynamic thing and quantum mechanics is the only mechanics we have?]
6.7: Acceleration in Schwartzschild Coordinates
6.8: Moving Sources and Gravitational Waves
Because quanta of action have no meaning (like dollars out of context) they interact linearly, like pure numbers and so can be described by linear operators.
Dynamics equals real time evolution, variation controlled by selection so we can see all the operators in physics being shaped by their interactions with one another as spacetime shapes operators and operators shape spacetime.
[page 98]
In this case very little information need to be carried by the quanta (like molecules of flowing water) but we need a mechanism of population increase to form cooperation by proximity [which requires fermionic rather than bosonic behaviour, so this must be there at the beginning].
My life is a series of dead ends inducing hopelessness which give way in time either to forgetting the problem or creating means to circumvent it to go on to the next problem. While walking each step follows effortlessly but in more complex situations each step must be thought out in detail.
Brown Section 7.1; Is the Universe Closed?
Metric at infinity versus metric at zero [neither make any sense because there is nothing there.
' Of course, in either an open or a closed universe, a theory consisting of differential equations requires boundary and/or initial conditions, but the question is whether the distribution of mass-energy by itself is adequate to define the field, or whether independent boundary conditions on the metrical field are necessary to uniquely determine the field.'
'. . . the metrical field of spacetime is not an observable of the theory.'
7.2: The Formation and Growth of Black holes
7.3: Falling Into and Hovering Near a Black Hole
7.4 : Curled up Dimensions
7.5: Packing Universes in Spacetime
[page 99]
7.6: Cosmological Coherence
7.7: Boundaries and Symmetry
7.8: Global Interpretations of Local Experience
From babyhood the dominant experiences of our lives are our interactions with other people which serve as a reference frame for the interpretation of our interactions with animals, plants and the rest of the world . This is indicated by our attribution of intention to every person and thing that we meet and which gives enormous power to gossip, the cloud of information surrounding us which interprets the intentions of our human, animal, vegetable and mineral environments.
The method of differential equations tries to work out global properties from infinitesimal local assumptions by a combination of integration and initial and boundary conditions.'. . . as Poincare famously summarized it, we can never observe our geometry G in a theory-free sense. Every observation we make relies on some prior conception of physical laws P which specify how physical objects behave with respect to G. Thus the universe we observe is not G, but rather U = G + P, and for any given G we can vary P to give the observed U.' This is how we justify religions, gossip and other mythological beliefs.
'The idea of "bringing the rest of physics into a geometric formulation" refers to attempts to account for the other forces of nature (electromagnetism, strong, and weak) in purely geometrical terms as attributes of the spacetime manifold, as Einstein did for gravity. In other words, to eliminate the concept of "force" entirely, and show that all motion is geodesic in some suitably defined spacetime manifold.' ie the transfinite network.
[page 100]
Brown Section 7.8 (continued): 'Quantum field theory works on a background of spacetime but posits other ingredients on top of that to represent fields.'
8.1: Kepler, Napier and the Third Law:
Kepler: 12 years between the first two laws and the third.
'8 March 1618, something marvellous "appeared in my head". He suddenly realized that:
III: The proportion between the periodic times of any two planets is precisely one and a half times the proportions if their mean distances.
"Proportion = logarithm of the ratio (T1/T2) = (3/2) log (r1/r2). Kepler's laws of planetary motion - Wikipedia
1614: John Napier: Mirifici Logarithmorum Canonis Descriptio - 25 years to produce tables. Logarithm - Wikipedia
Brown Section 8.2: Newton's Cosmological Queries
'Newton: "Is not Infinite Space the Sensorium of a Being incorporeal, living and intelligent, who sees the things themselves intimately, and thoroughly perceives them, and comprehends them wholly by their immediate presence to himself?." '
'[Newton] was interested in the same aspect of science that Einstein said interested him the most, namely, "whether God had any choice in how he created the world".'
Descartes: "I should consider that I know nothing about physics if I were able to explain only how things might be, and were unable to demonstrate that they could not be otherwise."
[page 101]
'This attitude may strike us as naive, but it seems undeniable that it's been an animating factor in the minds of some of the greatest scientists – the desire to comprehend not just what is, but why it must be so.'
Brown Section 8.3: The Helen of Geometers; Cycloid and tautochron (Huygens), brachistochrone (Bernoulli)
8.4: Refractions on Relativity
8.5 Scholium
8.6 On Gauss; Mountains
8.7 Strange Meeting
8.8 Who Invented Relativity>
Dark Waters Dark Waters (2019 film) Wikipedia
8.9: Doubting the Deflection
8.10: Conquering the Perihelion
'The agreement between general relativity and the precession of Mercury’s orbit was, and remains, one of the strongest confirmations of Einstein’s theory because, of all the classical tests, it alone is sensitive to the second-order in m/r.'
8.11: Paths not Taken
nBose(E) = A/(EkT - 1)
nBoltz(E) = A/(EkT)
nFerm(E) = A/(EkT + 1)
[page 102]
'Unfortunately, the assignment of a definite trajectory to a photon is highly problematical because, as noted above, a photon really is nothing but an emission and an associated absorption. To speak about the trajectory of a free photon is to speak about something that cannot, even in principle, ever be observed.' In other words a photon has the same status as a quantum amplitude?
Brown Section 9.1: In the Neighbourhood
Topology - properties preserved under continuous deformation. The world of photons is null, so we may assume that the world of quantum mechanics promotes the emergence of space?
Section 9.2: Up to a Diffeomorphism
Riemann ' "As is well known, physics became a science only after the invention of differential calculus . . .. True basic laws hold in the small and must be formulated as differential equations." ' Diffeomorphism - Wikipedia
'. . . one could argue that such “distant action” was made more feasible by special relativity, especially in the context of Minkowski’s spacetime, in which the null (light-like) intervals have zero absolute magnitude.'
'Apparently unconcerned about the topological implications of Minkowski spacetime, Einstein inferred from the special theory that “physical reality must be described in terms of continuous functions in space”.'
'Einstein: " . . . To be sure, it has been pointed out that the introduction of a space-time continuum may be considered as contrary to nature in view of the molecular structure of everything which happens on a small scale. It is maintained that perhaps the success of the
[page 103]
Heisenberg method points to a purely algebraical method of description of nature, that is to the elimination of continuous functions from physics. Then, however, we must also give up, on principle, the space-time continuum. It is not unimaginable that human ingenuity will some day find methods which will make it possible to proceed along such a path. At the present time, however, such a program looks like an attempt to breathe in empty space." '
'Einstein: "It would be most beautiful if one were to succeed in expanding the group once more in analogy to the step that led from special relativity to general relativity. More specifically, I have attempted to draw upon the group of complex transformations of the coordinates. All such endeavours were unsuccessful. I also gave up an open or concealed increase in the number of dimensions, an endeavor that … even today has its adherents." '
' Einstein to Besso 1954 (see Section 3.8): " One can give good reasons why reality cannot at all be represented by a continuous field. From the quantum phenomena it appears to follow with certainty that a finite system of finite energy can be completely described by a finite set of numbers . . . but this does not seem to be in accordance with a continuum theory, and must lead to an attempt to find a purely algebraic theory for the description of reality. But nobody knows how to obtain the basis of such a theory." '
General relativity - arbitrary transformations up to a diffeomorphism 'which in the absolute sense are not very arbitrary at all.'
Diffeomorphisms have a differentiable inverse so they are only a small set of possible mappings between infinite sets [they are analogous to codecs in communication theory]
'Impressive though [it] is, we should not forget
[page 104]
that general relativity is still restricted to a preferred class of coordinate systems, which comprise only an infinitesimal fraction of all conceivable mappings of physical events, because it still excludes non-diffeomorphic transformations.'
' . . . would it be possible to formulate physical laws in such a way that they remain applicable under completely arbitrary transformations?'
Brown Section 9.3: Higher Order Metrics9.4: Polarization and Spin
9.5: Entangled Events; EPR and Bell Einstein, Podolsky and Rosen: Can the Quantum Mechanical Description of Physical Reality be Considered Complete?, John S Bell: Speakable and Unspeakable in Quantum Mechanics
Section 9.6: Von Neumann's Postulate and Bell's Theorem:
'In his assessment of hidden variable theories in 1932, John von Neumann pointed out a set of five assumptions which, if we accept them, imply that no hidden variable theory can possibly give deterministic results for all measurements. The first four of these assumptions are fairly unobjectionable, but the fifth seems much more arbitrary, and has been the subject of much discussion.' Alexander Wilce (Standord Encyclopedia of Philosophy): Quantum Logic and Probability Theory
9.7: Angels and Archetypes
'The concept of classical determinism relies on each physical variable being a real number (in the mathematical sense) representing and infinite amount of information. One can argue that this premise is implausible, and it certainly can’t be proven. We must also consider the possibility of singularities in classical physics, unless they are simply excluded on principle. Nevertheless, if the premise of infinite information in each real variable is granted, and if we exclude singularities, classical physics exhibits the distinctive feature of determinism.'
9.8: Quaedam Tertia Natura Abscondita
[page 105]
' Thus the "mysterious" and "spooky" correlations of quantum mechanics can be placed in close analogy with the time dilation and length contraction effects of special relativity, which once seemed equally counterintuitive. The spinor representation, which uses complex numbers to naturally combine spatial rotations and "boosts" into a single elegant formalism, was discussed in Section 2.6. In this context we can formulate a generalized "EPR experiment" allowing the two measurement bases to differ not only in spatial orientation but also by a boost factor, i.e., by a state of relative motion. The resulting unified picture shows that the peculiar aspects of quantum mechanics can, to a surprising extent, be regarded as aspects of special relativity.' Spinor - Wikipedia
Imaginary time squared contributes negatively to the line element in the same way that imaginary phase can contribute negatively to probability, yielding interference effects.
Brown Section 9.9: Locality and Temporal Asymmetry
Time asymmetry and neutral kaons. Kaon - Wikipedia
' Subsequent developments (quantum electrodynamics) . . .[lead] us to regard a photon (i.e., an elementary interaction) as an indivisible whole, including the null-separated emission and absorption events on a symmetrical footing. This view is supported by the fact that once a photon is emitted, its quantum phase does not advance while "in flight", because quantum phase is proportional to the absolute spacetime interval, which, as discussed in Section 2.1, is what gives the absolute interval its physical significance.'
'This leads to the view that the concept of a "free photon" is meaningless, and a photon is nothing but the communication of an emitter event's phase to some null-separated absorber event, and vice versa.'
'. . . since the Schrodinger wave function propagates at c, it follows that every fundamental quantum interaction can be regarded as propagating on null surfaces.' [so we speculate that quantum mechanics lives in a null world which is the foundation for our spacetime world].
'Einstein seems to have intuited that quantum mechanics does indeed entail distant correlations that are inconsistent with very fundamental classical notions of causality and independence, but he was unable to formulate those correlations clearly.'
'There are absolute distinctions between the sets of null paths connecting spacelike
[page 106]
separated events and the sets of null paths connecting timelike separated events, and these differences might be exploited to yield a structure that conforms with the results of observation. . . . it's perfectly possible that the objective world might possess a non-transitive locality, commensurate with the non-transitive metrical aspects of Minkowski spacetime.
' . . . many of the seeming paradoxes associated with quantum mechanics and locality are really just manifestations of the non-intuitive fact that the manifold we inhabit does not obey the triangle inequality (which is one of our most basic spatio-intuitions), and that elementary processes are temporally reversible.'Brown Section 9.10: Spacetime and the Mediation of Quantum Interactions
The reduction of spacetime to nothing but the null intervals is one way of reducing the excessive "multifariousness" of the spacetime continuum [ie variety of spacetime is constrained by the variety of the null-space].
Dirac: uncertainty principle suggests that 'all energy must propagate at the speed of light when examined at the microscopic scale'.
'According to Bohr and Heisenberg (in the late 1920s), classical theory had consisted of causal relationships of phenomena described in terms of space and time, whereas the causal relationships of quantum theory cannot be applied to conventional descriptions of phenomena in terms of space-time. They contended that the evolution of a state vector according to the Schrodinger equation describes not a single set of trajectories of the constituent entities in spacetime, but rather a superposition of all possible trajectories.' [in what?]
[page 107]
Brown Section 9.10 (continued): '. . . this reduction to classical terms entails the notorious “jumps” in the state vector, which Bohr and Heisenberg saw as “the limitations placed on all space-time descriptions by the uncertainty principle”. Thus the dichotomy (or complementarity, as Bohr called it) was between the unitary evolution of the wave function on the one hand, and the reduction of observations to classical space-time descriptions on the other.'' Feynman later referred to his work on quantum electrodynamics as “the spacetime approach”, since he conceived of quantum interactions as a sum of all possible paths through spacetime. Hence we might say the totality of spacetime serves to mediate quantum interactions. Hints of this approach could already be seen during the discussions of Bohr and Einstein in 1927, when they considered the one-slit and two-slit experiment, which they might have noticed could be generalized to any number of interposed diaphragms with any number of slits, ultimately leading to Feynman’s view of a particle’s propagation as the superposition of all possible combinations of null segments through spacetime.' Anthony Zee: Quantum Field Theory in a Nutshell
