vol VII: Notes
2012
Notes
[Sunday 10 June 2012 - Saturday 16 June 2012]
[Notebook: DB 72 Energaia]
[page 12]
Sunday 10 June 2012
Local motion the physical world is one dimensional, that is it can be represented by a parameter on a line. The degrees of freedom of a logical motion are counted by the number of communication channels [messages, particles] entering and leaving the event. [motion has two dimensions, the orbit of motion and the proper time parametrizing this line].
Complexity = 0, the eternal classical God
complexity = 1, one degree of freedom, flow of action
Energy momentum tensor made up of two scalar fields, pressure p and density ρ. Hobson page 179. Too is total energy density, T isotropic pressure in the i direction. Hobson, Efstathiou and Lasenby: General Relativity
Pressure is a vector moving normal to a surface, some relative of curl, ∇ × f.
Monday 11 June 2012
Gravitation is its own source so that it can effectively create itself from 'nothing' by a bootstrap process to give us the 'big bang'. One stumbling block in the way of seeing gravitation in the structureless initial singularity ids its extreme complexity, postulating a tensor with ten degrees of freedom at every point on the space of events. A weak spot in the system may be the fact that it draws all of its structure from a system of events which are believed to have definite positions in the real world which are the real world foundation of the invariance of the metric with respect to coordinate transformation, a
[page 13]
'philosophical' requirement since we assume that at least come of the world exists independently of the language we chose to describe it. Although this is assumed to be true in classical physics, we see that in quantum physics and human communication a state of the observer adds half of the information to be found in the observer's account of what it sees. Mathematically this means that for information to be shared by two particles observing each other they must share a coordinate basis, in effect a language,. So we have here in communication another fled that is its own source., since every communication involved two complementary changes of state in the two communicants, and every change of state lays the foundation for a new communication, the cycle which maintain the bond between two particles.
Do we see here a glimmer of a relationship between quantum mechanics and relativity. We describe both in huge spaces; infinite dimensional Hilbert spaces with basis vectors parametrized with continuous complex numbers; a continuous differentiable manifold each point assigned 4 coordinates xμ according to Gauss. Two points are assumed to be identical of their coordinates are the same in a given coordinate system. Both these mathematical schemes are 'Platonic' in the sense that we take them to exist perfectly ad execute all the operations we expect without going into the question of how all this is to be physically represented. Of course we do physically represent it in our literature and our computers, and these representations are felt to be formally consistent and also consistent with the world we observe but we wonder how does the world come to behave like mathematics if it is not computer, and if it is a computer, how does it work. First, we say, if it is to be deterministic, it must be discrete. (Shannon).
[page 14]
Meaningful messages are a sequence of discrete values.
Quantum mechanics tells us that we will be able to understand the physical when we share its coordinate basis. The applicability of mathematics to nature is a suggestion in this direction.
The metric is the incompressible fluid, the rigid foundation of the universal structure. To what does it correspond? It can only be a deterministic computation to have such stability.
We see the structure in the space-time representation of the metric and the dynamics in the energy-momentum, phase-wavenumber representations. Space is simply the simultaneous existence of two or more state in communication (transformation). The root of the transformation is the local 'generation' of the velocity of light.
If it is both discrete and of low complexity it should be easy to understand, that is demonstrate the logical continuity of all the connections (bonds, messages, forces etc) within the structure. The most basic construction seems to be the harmonic oscillator which we see as the flow of an incompressible fluid (energy)_ from a state called potential to another called kinetic.
Potential: mapped onto space, in fact 'part' of space. <./p>
Ways of understanding space and time.
A standing wave : self sustaining (Lonergan: scheme of recurrence) I have to understand physics much better.
[page 15]
So can we say that the whole machinery of quantum mechanics is devoted to identifying the eigenfunctions and eigenvalues which are features of certain standing wave, that is stable elements of structure, letters of an alphabet, elementary computer.
Although we analyze physical networks in terms of wave mechanics, we know that all the relevant elements in the network, both at the quantum level and the data representation level , are discrete events with periodically varying probabilities (or not, in the case of ideal coding).
Tuesday 12 June 2012
The Lagrangian approach to gravitation, Hobson page 359. Einstein Hilbert action align="center"> SEH = ∫R R√-g d4xwhere d4 and √-g both refer to 4D and we are seeking to extremalize something over a volume, R scalar, (Rμνρσμνρσ, eg)
Hobson: 'To construct an action for general relativity, we must define a Lagrangian L which is scalar under general coordinate transformation and which depends on the components gμν of the metric tensor (these are now the dynamical fields) and their first- and possibly higher-order derivatives.'
'. . . we shall derive the corresponding field equations by considering directly the variation in the action resulting from a variation in the metric tensor.'
Field theories are quite abstract. What really happens locally to be represented by the metric tensor? Something to do with the Lagrangian, the difference between potential and kinetic energy, quantum mechanically an oscillator. We can approach this by scale invariance and consider what really happens locally when a pendulum swings or I do something like writing. In writing there is a potential in my mind that drives the pen whose output feeds the potential in my mind. So, as usual, we try to map all this onto a neural network that works by phase, ie when two phases add something happens, and the phasing at each point depends upon the metric at that point which is the rate of change of phase with respect to the interval between events, dφ / ds
The actual events of the two slit experiment depend on the personalities of the particles used, but the basic mathematical symmetry of superposition (the path integral method) applies in every case, right down to gravitation which applies to quanta of action without any personality at all [ie instances of the initial singularity].On the macro scale everything depends (in the construction industry and everywhere else) on getting the right thing to the right place at the right time, a situation that can in its most abstract form be represented by phase of the halting of a computational process. Now chop some wood. [phase identity: it is the wood chopping phase of the the day again as I transcribe this]
Elastic collision makes no observable change in the colliders (although it may change their quantum state) so we can imagine them as 'empty' units, (empty sets). We might take collision in space-time to be the least complex of all interactions 'completely constrained' by the conservation of energy-momentum, 4-momentum. Because Boltzmann statistics work for a gas, we have to attribute sufficient identity to each molecule to /p>
[page 17]
distinguish it from all the others to give meaning to the permutations that describe the 'complexions'. This identification comes from spatial position and each collision (event) has a space-time address which is a discrete location of a discrete event.
We might think of the Universe as a gas of quanta of action. At very early times of high energy density gravitation is 'stronger' than other forces as we see in black holes, singularities in spacetime.
Maybe the Universe is not convex because of black holes?
Gravitational attraction: shorten the routes of high energy flow until the relvant action is stationary.
S = ∫ (T - V) dt. What does this mean physically? E = hν, the inverse of the above : energy to action, action to energy.
It is one thing to understand numerical relationships between the various parameters we measure in physical systems, but quite another to know what these parameters and their values mean " mass, length, time, charge etc.
Wednesday 13 June 2012
Scale invariance: In the human brain (and nervous systems in general) the action potentials are quantized and the decision of a neuron to fire or not is determined by the phasing of the incoming potentials very much like the path integral algorithm.
[page 19]
At the human scale space appears to be continuous an events discrete.
Every logical process requires a reference frequency or clock [to get the phasing right]. In a system without memory the clock is all there is, and since it is all alone we can assign almost any frequency to it. So the isolated quantum system is a superposition of frequencies in the range 0 < f < ℵ0. A system so conceived may be seen as a model of 'pure energy' not so much with no structure as a superposition of all structures. This system is beautifully modelled by the energy equation and the periodic complex exponential functions that solve it. These equations we take to represent a stationary point in the dynamics of the Universe of pure energy.
Energy is eternal delight. William Blake
The essence of quantum mechanics is the eigenfunction which must be computed to complete an observation, interaction.
We have to delete the idea from our toolbox that an 'observer' is a human physicist and replace it with the notion that all the entities in the Universe observe one another in different ways as they go along the trajectory through spacetime. Gravitation in this context is the universal language, and we might think of it in boson terms, seeing the Universe as a set of states each of which may contain an unlimited number of quanta, eg photons, in the quantum harmonic oscillator.
MEMORY = SUSPENDED EVOLUTION OF PHASE
By connecting the operations of computation to memory we can stop and start at random intervals and[page 19]
computation will still proceed deterministically. Time (phase) as measured by a local clock is no longer of the essence. This approach is common to all literary (written) operations and serves perhaps to obscure the importance of phase emphasized by quantum mechanics. In fact the network model (itself 'outside' time, ie in memory) applies to both quantum mechanics and bureaucratic information processing.Current physics imagines an analogue Universe based on ancient ideas of spatial continuity and the mathematics of the continuum has developed since Descartes unified arithmetic, algebra and geometry. The hypothesis here is that the Universe is in fact digital and logical, so we can best begin our modelling with cardinal numbers, set theory, logic and the theories of computation and communication.
Since my monastic days the most exciting part of my life has been theological ideas and speculations. I feel that I have come a long way but have yet to close my path and so get an estimate of the errors in my navigation.
Arithmetic: logical arguments about quantities, often based on set theory.
What is the difference between quantity, index and truth value as symbolized by {0, 1}? Different meanings for the same symbols determined by context. In the absence of context these symbols are simply defined by the fact that one is not the other [and can be applied to all instances of this case].
A physical law is an algorithm whose inputs and outputs are observable quantities. The nature of the quantity is defined by the method of its observation.
[page 20]
Thursday 14 June 2012
Insofar as navigational errors grow exponentially with space=time, it is best to nip[ them in the bud.
We define a continuum as an object with certain properties and then make logical arguments based on these properties. Although logically consistent, [in part, but what are we to make of a continuum of distinct points?] the continuous model does not necessarily represent reality.
How do we translate the quantum mechanical notion of amplitude into the logical paradigm? We go through phase which measures the progress of a computation, φ = 0 = 2π = beginning or end of a computation, the only 'observable' points.
Still hovering high over the quetions trying to get a 'satellite' picture of what is happening before we descend to detail. It seems that ultimately quantum mechanics assigns amplitudes by detecting 'standing waves' (fixed points) in a multi-dimensional space.
More specifically, we want to establish a correspondence between computation and tensors, which are in effect a way or organizing arithmetic computations [with components].
Feynman Gravitation pager 39: '. . . we see a complete 360° rotation corresponds to two complete cycles of phase [a two act operation] -- the spin is two.'
page 40: '. . . the graviton has an energy content equal to ℏω, ad is therefore itself a source of gravitons. We speak of this as the nonlinearity of the gravitational field.
[page 21]
The Trinity is jut the beginning of a transfinite expansion.
Time goes more slowly in regions of lower gravitational potential. Initial singularity has lowest potential of all so maybe time is stopped there and spatial extension = 0.
Feynman page 71: 'Each volume of space must contain a natural measure of size (or time): something to do with the quantum of action.
Feynman page 75: 'Theories not coming from some kind of variational principle, such as Least Action, may be expected to eventually lead tyo trouble and inconsistencies.page 81: 'It is not certain whether features of non-Lagrangian theories reflect some fundamental truth about nature. It is possible that the fundamental truth may be that processes occur according to a principle of minimum phase, and that the actions of classical physics or of quantum physics are expressions of this phase which are correct to some approximation.
Each cycle of phase is an algorithmic step and nature operates by the most efficient algorithm as a result of selection.
Friday 15 June 2012
EQUIVALENCE
The Universe is inherently dynamic, static features being a conseqeucne of fixed points in the motion. So velocity, (eg c, so stasis = ∫v, motion = ∂v. Acceleration [page 22]= force = communication. The principle of equivalence thus points to gravitation and acceleration, being equally consequences of communication, here the rate of interaction is very low.
Feynman page 89: 'The theory of gravitation is as viewed within the framework of Einstein's ideas is something so beautifully exciting that we shall be sorely tempted to make other fields look like gravity. . . . We shall resist the temptation. Potential.
'He had found a way to reconcile the Lorentz invariance of classical electrodynamics with the apparent Galilean invariance of Newtonian mechanics, and many new physical results followed.'
The logical digital approach to the Universe changes very little current knowledge but eliminates the problems of continuity (renormalizability) and open physics to metaphysical interpretations,
page 90: 'It is quite obvious that mechanical experiments performed inside an accelerating box yield results which are indistinguishable from those which would be obtained if the box were stationary but in the presence of a gravitational field.' Symmetry.
We use logic to talk about mathematical objects (like continua) which we dream up ourselves. So logic is in a sense prior to these things and capable of existing without them.
< The emotional consequences of events, eg death = forces in the human (or other entity) space.
[page 23]
Tensors are an artefact of our need to have a reference frame to talk about things. If we can understand directly the interactions between things (as in the network model) we do not need reference frames. [language is itself a reference frame!].
Most of my communication is within myself, as all is the communication of the Universe is within itself, and the global symmetry of such interactions is captured by the principle of equivalence which relates the overall shape of the Universe to the local communication the the elements which at the fundamental level are 'identical' units. Graviton is a boson, so at the beginning we may imagine all the bosons in one state (equivalent to one energetic boson) so we might think if the Universe itself as a 'graviton'.
Time division multiplexing. My maximum theological effort is equivalent to spending all my time on it, ie making it an obsession.
Feynman page 92: "(gravity)' = (gravity) + (acceleration)" 7.1.3
Gravity is the structure arising from the logical confinement of a number of identical units. The expansion of the Universe is analogous to the decay of the initial graviton at all scales.
'. . . it will not be possible to define a "true" gravity, since we cannot ever define how much of an observed force is given by gravity and how much is due to an acceleration.'
'. . . by considering transformations over infinitesimal regions, we expect to learn how to describe the situation in differential form.
[page 23a]
Differential = local rate of communication = velocity. Acceleration = change of velocity = change of rate of communication, ie losing old friends and getting new ones as we can measure by the distribution of email traffic between individuals.
Fyenman page 93: 'How much like Einstein this [principle of equivalence] sounds, how reminiscent of his postulate of special relativity! We know the principle of equivalence works for springs, (as we knew special relativity worked for electrodynamics), and we extend it by fiat to all experiments whatsoever. We are used to such procedures by now, but how originally brilliant it was in 1911&emdash;what a brilliant, marvellous man Einstein was!'
So we are saying insight is not just a property of the human mind, but of the Universe itself, beginning with the self interaction (gnosti seauton) of the initial singularity which (we might guess) created the first graviton, an instance of 4=space, 4-momentum etc which copied and differentiates itself into the Universe.
page 94: gravitation closely related to clock rates&emdash;zero potential, zero clock, zero time. Time arises from the interaction of a twosome in a potential of their own creation (called love at the human scale).
The transfinite symmetruc network is the richest structure we can represent formally, a maximal mix of cardinality, ordinality and computing power.
POTENTIAL is ENERGY (has energy?)
[page 23b]
Feels like a great morning. The sexy bit is the identification LOVE&emdash;POTENTIAL&emdash;ENERGY, which embraces all the layers in the Universal network.
Trying to stand on the shoulders of Feynman. It is a very long climb.
All this seems obvious to me when I am in love, and I am always in love with my God whatever the details of my human relationships.
Maximise proper time. In love (= free fall) time may 'stand still', photon.
Feynman page 96: 'We see that the p[roper time to be a maximum is equivalent to the principle of least action in the classical limit' which defines the path integral method.
Scalar, photon, graviton. The photons move 'in' the graviton and define it in some way as alphabets define words. Definition = what it is made of (physical) + how it is made (metaphysical).
GOD = TOTAL SYMMETRY = Omnino simplex.
Static (fixed) omnino simplex
Dynamic (living) actus purus
We might read Aquinas' technical terms in the same way as a physicist might read γ, as an index, as a value and so on.
[page 24]
the relationship of kinetic and potential energy. We imagine the 'local frame' of any entity is the one in which its Lagrangian vanishes, so S = 0.
Feynman age 97: '. . . a mechanical principle (equivalent roughly to Newton's second law) which will be relativistic. It is that the variation of ∫ds should be zero
δ ∫ ds = 0.
It was Einstein's guess that this principle will give the motion in the presence of gravitational fields. This solved the problem of finding the equations of motion given the fields. The remaining problem is how to connect the potential &PHI; which appears in this expression with the environment.'
A theory is a network of equalities. A reference frame is a point of view and every entity sees things from its local or home point of view and understands the others through some transformation.
We can understand gravitation by considering time alone (since it is an interval) just as we can understand quantum mechanics on time alone, ie both are aspects of our understanding of energy.
page 97: 'We have seen that gravitational forces are inextricably mixed with inertial forces, so that we cannot make a universally valid separation of the two.'
In a general transformation, each new coordinate is a finction of all the old coordinates. To translate from your point of view to my point of view.
[page 25]
Each differential (like the 10 independent components of gμν may represent a communication channel in 4D space [up from 0 channels in 0D space].
GRAVITATION = ACCELERATION (change of velocity, momentum, energy) = COMMUNICATION.
General covariance requires that the reference system (the Gaussian coordinates) have the same cardinal as the system referenced. In differential geometry this is assumed to be the cardinal of the continuum. Dauben: Georg Cantor
There are 10 cross products in 4D space, 6 in 3D, 3 in 2D and 1 in 1D, and we might imagine each of these being the work of a physically embodied computer. These products are 'modulated' by the relevant components of the metric [ds2 = gμν dxμdxν].
Angular momentum (rotation) is in some sense 'closed' since a spinning object (like an electron) can move without going anywhere. This is also true of the Universe as a whole, which is 'closed' and has nowhere outside itself to go and so may be thought isomorphic to a particle with spin. As systems become more complex, their 'spin' increases, and we may think of a computer as a lot of interlocked spins, or one big spin. We might say that the Universe has grown to be a particle with spin ℵ0. Jammer page 105. Jammer: Concepts of Space
We might think of the Hilbert oscillator in terms of contracting and expanding tensors. The actual work is dome by quantum computers, but we can get an abstract view of it through a differentiable manifold. This works because any process no matter how complex depends upon the frequent
[page 26]
repetition of less complex operations. This picture makes gravitation complementary to quantum mechanics since it places logical and cardinal constraints on the observable products of the computation of eigenfunctions.
Our equivalence primnciple : 'collapse of wave function' ≡ decoding ≡ insight ≡ halted colputation (assuming some state of invariance = a message). In wave mechanics the fixed points correspond to standing waves induced by some potential (= structure, eg violin, movement of play in a football match etc).
Jammer Space Newton General Scholium 111 sqq.
Jammer page 117: '. . . in [Newton's] view space is nothing but a system of relations, devoid of metaphysical or ontological existence.' (?)
Until it communicates it is invisible, like an amplitude. We need ψ and ψ* to create a real observable stationary state, the photon being a stationary state measuring the distance between two ordinal states by a cardinal, frequency / energy.
Relative velocity is a relationship that is only realized when the interval between the two particles is zero, either in physical contact or connected by a photon (same thing?).
Theology has become a tool of established religions, and we see in manuy academic settings where the Papal fiat places firm (and incredible) bounds on theological conjectures. In theology, refutation remains a matter of authority rather than evidence. We are seeing a gradual recession of the authoritarian approach to the regulation of human
[page 27]
affairs and the growth of the evidence based approach that seeks to reconcile scientifically attested reality with the will of the people. Melinda Henneberger
Sometimes I feel that I am gradually making the transition from crackpot to professional. If i can close the circle I will know that I am not deluded. This seems to be how Einstein thought of general relativity.
Weyl in Jammer page 58: 'The principle of gaining knowledge of the external world from the behaviour of its infinitesimal parts is the mainspring of the theory of knowledge in infinitesimal physics as in Riemann's geometry.
Reimann, Jammer page 161: 'The basis of mathematical determination [of a space] must be sought outside the manifold in the binding forces that act on it.' From the manifold to network to gravitation.
page 162: 'For Riemann, matter was the casusa efficiens of spatial structure."
