Notes
[Sunday 15 February 2009 - Saturday 21 February 2009]
[Notebook: DB 65 Symmetric U]
[page 134]
Sunday 15 February 2009
Impasse! Gaussian = intrinsic geometry, ie a catalogue of local relationships = local spacetime intervals (in general relativity). Is relativity of itself three dimensional, or does the three dimensionality arise from some other force? And can the simple idea that [the smallest] space that can be fully connected is 3D [be right]? And is space embedded in the Universe of the Universe embedded in space?
1D: flow back and forth around a circle (U(1))
2D: airflow on the surface of the earth with no vertical component,
so a set of 1D flows, ie circles.
3D: [intersecting circles]
Newton's 3 laws are the foundations of gauge theory.
1. No cause, no effect
2. Effect linearly proportional to cause
3. Two things must change, equal and opposite.
Insofar as every point in the Universe is unique there is no 'actual' symmetry. All the apparent symmetries are the result of abstractions (simplification) and coordinate systems with too much variety, particularly the imposition of continuous coordinates on binary systems like the qubit. Qubit - Wikipedia
Coordinate singularities vs real singularities. Hobson page 37. Hobson et al
[page 135]
For Newton, god is the absolute coordinate system. When we get inside god, we need to do away with coordinate systems altogether, which the network model makes possible. What replaces coordinate systems are the fixed points (ie algorithms, communication protocols) in the transmission of messages. Spacetime is such a fixture in everyday life, so that we can express everything in terms of the spacetime algorithm.
Different function of different coordinates gives the same reality. A scalar can exist at point, but strictly speaking differentials and vectors require two points for their definition, although we use limiting procedures to create the fiction that they exist at a point.
Curved space = varying metric = varying density of some conserved quantity - conserved currents.
Probability density and probability flow. Probability theory starts with discrete events and then, through the law of large numbers, yields various continuous functions like the Gaussian distribution, but it must be remembered that these continua are in effect approximations to discrete outcomes. This is analogous to the foundation of analysis on sets of discrete points and using large or even transfinite numbers of points to arrive at concepts like derivatives and integrals and Bolzano-Weierstrass theorems. Bolzano-Weierstrass theorem - Wikipedia All this is very convenient, but in physical applications we must remember that the Universe is grainy at the scale of Planck's constant so attempts to go to the continuum limit may be misguided except in systems that are not quantized in nature.
The conserved current in quantum mechanics is probability and the fact of conservation of probability is formally
[page 136]
represented by the requirement that the evolution of quantum mechanical systems should be unitary, so that al all times the vectors representing physical event have a length of 1 which translates to a probability of one, since the length of a vector a is defined by its dot product with itself <a |a > = 1, and the probability represented by the event is |a | 2 = |<a |a >| 2.
Each event in the world represents a quantum of action, so a probability of 1 represents an action of 1. How does the world come to have so many independent sources of action? The answer is in time division multiplexing through the network hierarchy. Since energy is the frequency of action, we can also see this as energy division multiplexing, the one god who had all the energy at the beginning gradually breaking into countless quasi-independent little gods each a source of action but all cooperating to some extent for their survival using protocols which they share by common descent. In the physical world common descent from the initial singularity operates in real time, every communication in a complex system going through the singularity (the ultimate unifying physical layer) on its way from a to b.
From LAYERING to GAUGE THEORY.
Parmenides is to some extent right but the Hebrews and Christian traditions added considerable refinement to his view by developing the concepts of the Living God and the Trinity.
The Universe is stationary because to every action there is
[page 137]
an equal and opposite reaction. It is easy to see how this is true in cardinal measures, but the transfinite network gives us a way to see it also at work in ordinal measures, ie in structure, eg matter and antimatter, male and female, you and me and so on.
Maybe the first manifestation of this is the equation <a |b > = <b |a >*, although this is a scalar equality which exists because the structure of the vectors a and b is removed by the integration implied in the dot product.
The struggle always is to get from the local detail down through the layers of abstraction to the general statement. The transfinite network is an epistemological as well as a physical network. Zurek Zurek
The archetype of the living god is the harmonic oscillator e it where t = time.
How much lung are these pages worth? There is always a war on and each of us must sacrifice (in a probabilistic sense) a little bit of health for the common good without working ourselves to death. To me it is worth it to see the relationship of gauge invariance and the eternity and life of the classical god because it enables me to make a link between modern physics and classical theology which narrows the rift between them.
The war is between the solipsist view of the world and the divine view, ie between those who think only of themselves and those who acknowledge that they are parts of a whole.
[page 138]
Newton, then Lagrange, who made explicit the roles of potential and kinetic energy.
KE = 1/2 mv 2
PE = F . s
KE + PE = constant. L = KE - PE
In a given frame, KE has a fixed positive definite value, whereas we may redefine the zero of potential arbitrarily. ['gauge freedom'] To give something KE we must use some PE, eg burn some gas.
KE = action in action
PE = stored (static) action
Keeping still requires 'cybernetic work' = feedback = {observe, transform, act}.
Traditional physics might say that the fundamental constants were put in by the creator, and this is the position of the standard model which relies on 20+ observed parameters and relates them to one another in a pretty consistent manner ('t Hooft, 't Hooft)
I have finally worked out where the apostrophe goes in 't Hooft's name, differentiating it from the standard usage in my language. Gerard 't Hooft
A remarkable feature of spacetime is that is is pseudo-Riemannian, having intervals with zero and even negative length. This brings us a bit closer to the old non-spatial god.
[page 139]
The metric is the gauge that couples neighbouring regions of space in the infinitesimal model and we gain a global picture by integrating over the metric. The metric is a message from one point to the next. [?]
General relativity is an application of vector calculus on manifolds.
A scalar field (like potential) becomes a vector field when we start to move from point to point in the potential, since two scalars define a gradient that is in physics a force. So the scalar field of pressure gives a vector field of pressure gradients which in turn gives us forces, accelerations and velocities.
DIFFERENTIALS are LOCAL [relative] (and have in a sense no origin)
No cloning: the conservation of things (like my glasses) which must be somewhere (until destroyed).
contravariant component x covariant basis = covariant component x contravariant basis, connected by Kronecker delta, to represent an invariant vector by varying (but related) coordinate systems.
Hobson et al page 59: The manifold is in effect embedded in the set of all its tangent spaces instead of being embedded in a space of higher dimension. The tangent spaces all have the same dimensions and signature as the underlying manifold, but there is one of them for every point of the manifold and we might think of a manifold that has Hilbert spaces as its tangent spaces. It would need to be complex. We can represent an n dimensional complex space in a on dimensional real space.
For every motion there us an inverse motion that completes the cycle so that m . m-1 = 1. This looks like ea . e a = 1.
Leibniz theorem: Product rule - Wikipedia
. . .
Monday 16 February 2009
't Hooft Under the Spell page 3: 'Space and time are continuous. This is how it has to be in all our theories, because it is the only way known to implement the experimentally established fact that we have exact Lorentz invariance. It is also the reason why we must restrict ourselves to renormalizable quantum field theories for elementary particles. As a consequence we can consider unlimited scale transformations and study the behaviour of our theories at all scales. This behaviour is important and turns out to be highly non trivial. The fundamental physical parameters such as masses and coupling constants undergo an effective change if we study a theory at different length and time scale, even the ones that have been introduced as being dimensionless. The reason for this is that the renormalization procedure that relates these constant to physically observable particle properties depend explicitly on the mass and length scales used!. ' 't Hooft
So space is a player?
[page 141]
A field is something which has a 'value' at each point in spacetime. This value may be very complex, as in a tensor field, but we consider it to be continuous because the manifold to which it corresponds is continuous. When we come to consider the continuity of something like a tensor (which might, like 'Riemann' have 256 components, we do not consider the ordering of the components but just take them one at a time, as scalars, and apply the usual ideas of scalar continuity and smoothness from analysis. This is a matter of cardinal measure.
Logical continuity, however, cannot be demonstrated in this way. Instead, if we want a spatial vision of it we use Venn diagrams. Venn diagram - Wikipedia Thus A implies B iff B is an element of A. How do we understand this in the physical world? In the continuous world of space-time we can relate nearby points by differentials and devise differential equations that when integrated according to the prevailing boundary conditions give us a picture of the field of study.
Electromagnetism, flux, Circulation. Feynman II 1-5. Feynman
Let us guess that field theories are an attempt to map the local behaviour of entities that interact outside space (like gossiping humans) onto space. The big difficulty is in producing continuous equation in which the spacetime manifold can mimic the behaviour of logically driven events that are projected (in some way) into spacetime and whose large scale structure is effectively spacetime. Space is an artifact of the networked interactions of the fundamental particles.
A big hurdle for physicists is the necessity to do away with 3D machines such as the ether and field lines and see that it is pure logic that drives reality and the projection on space
[page 142]
is secondary.
The quantum of action antecedes space, and when it appears in space we find delta p . delta x = h bar = delta E . delta t. We also find that momentum and energy transform [in the Lorentz sense] like space and time.
LOGICAL SPACE <--> PHYSICAL SPACE
This mapping is a task for a computer hardware engineer. Intel 4004 - Wikipedia
So we can draw a particle network, showing all the possible messages between all the possible fundamental particles and then try to project this onto physical space. Every event in this network is measured by a quantum of action. The conserved flow in the network is action. This is the underlying explanation of gauge invariance. Gauge theory - Wikipedia, Gauge fixing - Wikipedia
'Behind' space we have supersymmetry. All the bosons and fermions are channels, that is Turing machines leading from one state to another.
Spin is the base of the number system
spin 0 = 1
spin 1/2 = 2
spin 1 = 3etc
The stuff that is traded in gauge theories is phase which is the wave theoretical version of action.
[page 143]
Newton's gauge was force. Lagrange reconceived Newtonian mechanics. Instead of a set of masses interacting through forces in an absolute space, he devised a system comprising two disjoint sets of abstract coordinates represented by pi and qi and a set of differential and integral equations that described their behaviour. In his hands, mechanics became a branch of [mathematical] analysis, codified in his Mecanique Analytique. He conceived the principle of least action and the calculus of variations which enabled him to represent Newton's equations as relationships of stationary action between the generalized coordinates. Lagrange's gauge was action, the time integral of the Lagrangian L = T - V, T being kinetic energy, V potential energy.
The next step bears the signature of Maxwell. Whereas Lagrange's duality had been the physical coordinates p and q, Maxwell set up a set of differential equations between the duals, the electric and magnetic fields.
Tuesday 17 February 2009
Hobson page 135: '. . . a consistent theory of electromagnetism follows from saying that there exists a pure 4-force that depends linearly on 4-velocity and also on a certain property of a particle, namely its charge, q.
Newtonian superposition = electromagnetic superposition is the consequence of gauge freedom, which is a consequence (?) of the mapping of the logical particle network into physical space (?).
[page 144]
We consider the logical space prior to the physical space and so begin with the logical space and look for transforms that will map it to physical space. We begin with coding delay, which gives us time. We take it that 3D is needed for connections, and then we think about the way ordinary computers transform their output in order to project it onto the screen or printer. 3/4D space is our interface with the processes of the world. So how does the standard model look logically, and how do we project it into spacetime?
't Hooft: Renormalization means fiddling with the short distance behaviour of certain integrals showing that space is not continuous in the analytic sense, but its graininess (like the pixels on a computer) is a function of the processing power available to compute the pixels - more power, more pixels, shorter intervals.
Pais Inward Bound Pais
Pais page 4: 'I intend to stress how progress leads to confusion leads to further progress and on and on without respite.'
Occam's razor: all the problems in modern physics seems to stem from the use of continuous space to describe the world. We dismiss them at a stroke by deriving space from the particle rather than the particles from space.
Initially this means Mecanique Logique, and we can abstract from time, space energy and momentum and just consider the logical kinematics of the identified
[page 145]
particles.
alpha decay strong force
beta decay weak force
gamma electromagnetic force
beta decay: neutron --> proton + electron + antineutrino (Pais page 143)
Space is constructed by the expansion of the Universe.
Pais page 244: Einstein: 'Since Maxwell's time, Physical Reality has been thought of as represented by continuous fields . . . not capable of any mechanical [= continuous?] interpretation. This change in the conception of reality is the most profound and most fruitful that physics has experienced since the time of Newton.'
page 245: Field has dynamical degrees of freedom.
Thinking of photon as a process which changes the state of a particle like an electron.
page 248: 1924 Einstein: 'There are therefore two theories of light, both indispensable and -- as one must admit today despite twenty years of tremendous effort on the part of theoretical physicists -- without any logical connection.'
And it is logical connection that we seek.
page 249 Bohr: 'the epistemological lesson which quantum mechanics has taught us.'
[page 146]
page 251: 'Six theoretical papers appeared [during the era of the old quantum theory 1900 - 1925] which are revolutionary. . . . If these papers have one thing in common it is that they contain at least one theoretical step which (whether the authors knew it or not) could not be justified at the time of writing.'
Heisenberg 1925 'In this paper it will be attempted to secure foundations for a quantum theoretical mechanics which is exclusively based on relations between quantities which in principle are observable.'
Is spacetime observable, or just rods and clocks or more generally messages. Let us say (tautologically) that only messages are observable and all else is theory used to decode messages, ie to put them all together into a unified whole on the assumption that the sender was a unified whole.
1925 - 1927 Knabenphysik = boy physics.
Newtonian physics is the middle ground, v << c, S >> h bar. Is the Universe bounded by velocity and action?
Pais 255: Dirac: 'The problem of getting the interpretation proved to be rather more difficult that just working out the equations.'
The complex formal relationships of the underlying (or overlying) computer network at the heart of the world are lost in the integrations that give us space, time and gravitation, all of which see only mass, energy and momentum.
Born: 'The motion of particles follows probability laws, but the probability itself propagates according to the laws of causality.
[page 147]
Pais page 238.
So we are thinking of a system full of energy that will try anything, but, if it wants to become complex, must fit in with the constraints of communication and computation.
CAUSALITY ==> DETERMINISM ==> QUANTIZATION ie DISCRETENESS (ORTHOGONALITY).
Increasing entropy = increasing orthogonality.
Born again: 'The classical theory introduces the macroscopic coordinates which determine the individual processes only to eliminate them because of ignorance by averaging over their values; whereas the new theory gets the same results without introducing them at all. . . . We free forces of their classical duty of determining directly the motion of particles and allow them instead to determine the probability of states. Whereas before it was our purpose to make these two definitions of forces equivalent, the problem has now, no longer, strictly speaking, any sense.'
We are on our way from classical mechanics to communication theory.
partial d rho / partial dt - div j 0 ==> conservation of probability.
Pauli 1926: '|psi (qi )|2 . d qi is the probability that, in the relevant quantum state of the system, the coordinates simultaneously lie in the relevant volume of configuration space.'
Pais page 259 note **: '. . . the Schrödinger wave are 'guiding fields' in the configuration space of all the particles at once.'
[page 148]
Computer network is the configuration space of the Universe.
Structures are like trees in the wind (of energy) flexing and moving under the stresses induced but maintaining their topological structure until the wind (energy density) becomes so strong that they are torn apart.
From fully connected (convex) topology to 3- (4- ) space.
Pais 260: Carlyle: 'Men understand not what is among their hands'. Thomas Carlyle & G B Tennyson (Editor), Carlyle
Page 261: Einstein to Born 4 December 1926: 'The theory [quantum mechanics] says a lot but does not readily bring us any closer to the secret of the 'old one'. I, at any rate, am convinced that He is not playing at dice.'
Dice are consistent with Turing machines, Gödel's incompleteness and interrupt driven networks, and we see all this projected on continuous space, which in classical analysis is treated as deterministic.
Quantum field theory is a projection of the divinity onto points in 4 space which are not in themselves connected any more than the pixels on a TV screen. The connection is logical and lies behind the scene. Particles in spacetime are the observables in this system, coupling elements together.
PARTICLE = HALTED PROCESS
I go over and over this scenario, and like true love, it seems richer and sexier all the time, and that much harder to explain.
[page 149]
Particles communicate between independent sources correlating them. We can['t] talk about 'the isolated electron' because really it is hooked to the rest of the world.
High energy physics is a bit like pornography, separating out a certain sort of behaviour without seeing its in context. In reality a fuck is a complex 30 year (ish) relationship between three individuals, two parents and an offspring.
Pais page 261: March 1927 Heisenberg published uncertainty relations. They tell us how one logical operation (the quantum of action) projects onto 4-space and its package of {time, space, energy, momentum }
Pais 'unknowability relations' = pixellation which determines the 'resolution' of spacetime events. In very low energy/momentum environments, this resolution can be very poor, say 1 quantum per cubic metre . second.
The transfinite network naturally embraces both gauge invariance ( = gauge freedom) and total absence of symmetry (no cloning) in the same structure through the layering which means a lower layer is indeterminate vis a vis its user layer.
Complementarity - network and 4-space
Pais page 262: Bohr: 'The very nature of quantum theory . . . forces us to regard the space-time coordination and the claim of causality, the union of which characterizes the classical theories, as complementary but exclusive features of the description symbolizing the idealization of observation and definition, respectively.'
[page 150]
Pais 266: Wigner Zeitschrift fur Physik 40 883 1926: 'There exists a well developed mathematical theory which one can use here: the theory of transformation groups which are isomorphic with the symmetric group (the group of permutations).' Chayut
Boltzmann and the permutation group.
Pais page 266: '. . . The state of a system is given classically by a point in phase space. Symmetry operations act on coordinates and velocities in that space. Quantum mechanically, a state is given by a vector in Hilbert space; symmetry operations are implemented by linear (more precisely unitary or antiunitary) operations in that space. This linearity, the real novelty, is directly related to the superposition principle: in quantum mechanics one can add two states. There is nothing like that in classical physics. Linearity is also the basic reason why symmetries in quantum mechanics determine, all by themselves, many properties of systems.
Wigner 1931 Wigner
Wednesday 18 February 2009
Anything that we can truly write down about a changing situation is a 'constant of the motion', so I am a constant of the motion of my own life.
We could do away with all the philosophical palaver of we could just use the network approach to arrive at a physical result that has otherwise proved elusive. The cosmological constant problem, the renormalization business (including
[page 151]
asymptotic freedom and spooky action at a distance) all seem to fit. One would like to see the concept of logical continuity cut a lot of knots that mathematical continuity has tied. Keep at it.
It seems significant that the fundamental particles all have zero radius, yet they all have personalities of their own.
John von Neumann, Zeitschrift fur Physik 48 868 1928 J C Oxtoby et al
Veltman - the mapping between Hilbert space and 3- 4-space is only relevant at the observational ('collapse of the wave function') level and so we may not need to transform Hilbert space to handle the Lorentz transformation in real space. Veltman
REAL SPACE = MESSAGE SPACE
LOGICAL SPACE = THE COMPUTATIONS (TRANSFORMATIONS) behind the
messages.
Experimenters observe cardinals and the theorists try to devise formal structures that lead plausibly to the observed cardinals. The observed cardinals are all in effect possibilities for various events at various points in spacetime.
Both gravitation and spacetime see only cardinals. Gravitation is blind to structure and so is spacetime - the structure comes from the logic behind the scenes, as on a computer screen.
Thursday 19 February 2009
Pais page 300: 'By a general theorem of quantum mechanics, the intensity of a spectral line is proportional to the statistical weight of
[page 151]
the final state.'
We see each fundamental particle as an 'opcode' in the universal process. Some of these opcodes have the relationship opcode - inverse opcode, the second undoing the first, rather like the interactions of matter and antimatter.
Quantum mechanics postulates invisible process behind the action and there is no reason to believe that this process is not quantized. We take the view that all activities underlying a particular user layers are invisible to the user.
USER : TOOL :: aleph( n+1) : aleph(n).
Spacetime enters the world with baryons qqq.
Let us presume that spacetime is not necessary until the number of processes wishing to communicate reaches a certain minimum.
22 = 4, 24 = 16, 216 = 64k, 264k = lots.
SU (3) ==> space.
Asymptotic freedom in Yang-Mills theories shows is the pixellation of space. Asymptotic freedom - Wikipedia, Gross & Wilczek
The kinetic energy of the initial oscillator (which may have any zero or positive value whatsoever) is numerically equal and opposite to the potential energy
[page 153]
arising from the fact that the oscillator has o =< n < Aleph(0) states?
Fundamental particles are the basis set, the basic set of 'opcodes' for the processing of the Universe. To be represented by a set of Turing o-machines, each of which can be written down as an algorithm (in say, Ruby).
Should we say that fermions have the characteristic of memory through the exclusion principle, while bosons are the processors, changing the values of memory by exciting and de-exciting states conjugate to the boson energy.
The importance of energy lies in its determination of 'phase' via rate of processing. The gauge freedom of phase arises from the mistaken view that it is continuous, whereas in fact its relationship to action and Planck's constant (emphasized by Feynman) means that phase is digitized in lumps of 2 pi.
How does QED look if we think of it as mapping a logical network (run by electrons and photons) onto 4-space. It all works because the electrons are not free to all have the same state, and so photons are needed to communicate between the states. For every differentiation (Fermion) there us a unification (boson).
Pais 324: 'As the reader will see, [from 1926 on] confusion and insight will continue to alternate unabated but there ups and downs will from here on mainly occur within a tight theoretical framework, the quantum theory of fields.'
'At a moment which (quantum mechanics tells us) cannot be
[page 154]
predicted, an excited atom makes a transition to its ground state by emitting a photon. Where was the photon before that time? It was not anywhere; it was created in the act of transition.'
Pais page 325: 'Is there a theoretical framework describing how particles are made and how particles are made and how they vanish? There is: quantum field theory.'
'There is still a Schrödinger equation for the hydrogen atom, but it is no longer exactly soluble in quantum field theory. In fact, in a sense to be described, the hydrogen atom can no longer be considered to consist of just one proton and one electron. Rather it contains infinitely many particles.'
Energy division multiplexed, nevertheless.
page 326: 'From [the 50s and 60s] date the beginnings of a new endeavour: axiomatic field theory. . . . Do we know what we mean when we speak of quantum field theory?'
Zee: quantum mechanics is one dimensional. It does not deal with a spatially extended world, so there is no question of relativity. Relativistic quantum mechanics is a different, 4D beast. [Zee page 18] Zee
Pais QED
Photon is outside time (from an observer's point of view) - eternal. Electron is outside space, unextended, but it has mass and therefore 'internal' process.
[page 155]
Pais page 327 '. . . one of the advances of quantum field theory is to have rendered obsolete all arguments assigning finite extent to the electron.'
. . .
page 327: '. . . special relativity tells us that mass is but one among many forms of energy, ... '
page 329: Einstein: '[It is] a weakness of the theory that it leaves time and the direction of elementary processes to chance.'
ie there is no coupling between spacetime and fundamental events, ie maximum entropy, maximum communication of information.
Earliest reference to quantum electrodynamics 1925.
Coulomb gauge. Eric W Weisstein
Pais page 330: '[Dirac's] realization that the scattering of a photon on an electron is not a two-body but an infinitely-many-body problem.'
They both bring their whole networks with them.
'It may fairly be said that the theoretical approach to the structure of matter began its age of maturity with Dirac's two papers published in early 1927.' P A M Dirac
page 333: Born, Heisenberg and Jordan: 'The quantum number [nk ] of an oscillator is equal to the number of quanta with corresponding [nu ]
[page 156]
Pais: 'In the new interpretation the transition from one level to antoher must therefore mean that particles with energy h nu are either made of else disappear.'
Traditionally physicists have placed great weight on obtaining models that give quantitative agreement with observation, since this validates the model and we can little doubt given the precision of its predictions that the model behind quantum electrodynamics is good. By trying to reinterpret the model in communication terms, we do not interfere with the numerical precision but make more sense of the path integral aspect of the model. Dirac plays with equations; here we are trying to match him by playing with words, so many of them. Creating and annihilating words instead of particles, ie sending and receiving messages.
Communication is regarded as perturbation, and higher order perturbations arise from two or more simultaneous messages (bosons).
A transition from initial to final state goes via intermediate (virtual) states j where the uncertainty principle allows the momentary (and unobservable) existence of the j which we would interpret as invisible (transparent) processing in the transformation (coding) of the message from i [initial] to f [final form].
QFT: The communication paradigm.
Sexuality is orthogonal to everyday office behaviour.
Pais page 341: 'Toward the incorporation of invariance principles in quantum field theory.'
[page 157]
Pais page 345: Weyl: gauge invariance --> charge conservation. Weyl
charge conservation --> fixed interaction rate (coupling constant), ie charge is independent of potential or vice versa, ie charge is fixed, potential is fictitious.
We think that the communication approach automatically takes care of the gauge invariance because it is purely relative, like birds singing on a high voltage line.
Pais page 347: 'How could Dirac's relativistic wave equation for the electron be so successful yet so paradoxical?' ie it breached current ideas of logical continuity.
The guiding wave hypothesis: a field is a potential that guides the dynamics, not by its absolute value, but by its gradient. To be electrocuted one must feel the gradient. The gradient is the message (McLuhan). Marshall McLuhan - Wikipedia
Pais page 348: Klein-Nishina
So with relativity. What matters to me at this moment is our relative velocities (me and the ball) not the absolute velocity of the earth. It is the relative velocity that puts demands (measured by the sampling theorem) on my cybernetic processing systems, and so determines the energy of the interaction, ie the data rate, between us. So why does E = mc2 ? Whence the exponent?
Quantum mechanics of communication: 'There is a person here asking about you. Might be a cop.'
[page 158]
What counts is the energy (frequency) difference between the matrix elements of the Hamiltonian, not their absolute values.
My life is an interaction involving ℵ0 (ish) particles all virtual (invisible, transparent) to me.
QFT: Quantum field theory is concerned with the creation and annihilation of particles, that is (in our language) the emission and reception of messages.
In general a Hamiltonian encodes the rate of communication between various nodes (indices i and j in Hij ) in a network.
Feynman's path integral is a means of calculating the energies of these connections = probabilities = event rates,
The back and forth movements of energy carrying signals correspond to up and down movements of the potentials at each end, Energy = rate of action. Potentials store action, which we see as a conserved set of thing - marbles.
Electron self energy / radius problem points to logical approach, as does the zero size of other fundamental particles.
Pais page 374: 'From the first quantum electrodynamical calculation ever performed [Dirac 1926] until the present, [1983 is] all we know about quantum electrodynamics is based on perturbation
[page 159]
theory. Even though the method has served us well we would like to be free of it. We don't know how.' P. A. M. Dirac
Quantum mechanics and quantum field theory are both forms of network traffic analysis.
The new 'marble' is the quantum of action.
Pais page 388: 'The renormalization techniques of the 1940s have served to reduce all the many infinities in quantum electrodynamics to three fundamental ones, those of mass and charge and non-measurable vacuum quantities,'
which therefore may not exist.
page 418: 'Except for electromagnetic interaction all that can be said about the electron holds for the neutrino as well.'
page 436: 'It is Yukawa's great and lasting contribution that he has changed our thinking about forces. To this day his extension to strong interactions of the relation between forces and virtual particles exchange, his analogy with quantum electrodynamics , has remained our guide to thinking about all fundamental forces.'
A particle exchange is virtual to an observer who is not the source of the particle, but infers its existence from an interaction with a real particle to which the observer is a source (= sink).
Friday 20 February 2009
Saturday 21 February 2009
Pais page 462: 'The need for renormalizing mass and charge implies
[page 160]
that the theory cannot provide these values but borrows those from experiment. Mass and charge are what one calls phenomenological parameters.'
From the logical point of view we take the mass to be the rate of internal processing in a particle necessary to maintain its integrity (m = E / c 2, nu = E/h ) and the charge is its rate of communication with other charged particles.
alpha = e 2/hc is a dimensionless number, so dimension wise e2 = hc = ML3T-2 = E . L
'Quantum electrodynamics is nevertheless a theory with enormous predictive power. A finite number of parameters treated phenomenologically makes possible an infinite number of predictions.'
Pais page 464 'Beginning with O (alpha 2 ) one finds in the guts of the radiative corrections contributions from all species of charged particles in the physical world.'
ie they are all on the same network, communicating through the 'electric charge' protocol.
'This shows that the answer to the question of whether the alpha expansion in the photon-electron world converges or not will not say much about the real world, secondly that the radiative corrections of order alpha 2 and higher contain low energy information about the full particle spectrum in the Universe.'
The earliest and best known field is gravitation. Newton produced a pure and clear mathematical model
[page 161]
of gravitation based on Euclid's geometry with a fixed three dimensional background space and universal uniform time. Lagrange recast these ideas in terms of energy, potential and action, all considered continuous quantities with straight forward functional relationships which [were] amenable to treatment by differential and integral equation. Hydrodynamics, electrodynamics and all continuous dynamical systems benefitted from the Lagrangian treatment, which unified the whole of classical dynamics through the principles of virtual work and stationary action.
Newton's gravitational field is a scalar potential that couples to mass. It has a value at every point in spacetime, and we establish the convention that the gradient of a potential is a force and force causes acceleration, so comprehensive knowledge of potential differences in the world enables us to compute the spacetime trajectories of particles moving in the potential providing they do not collide.
Einstein changes the spacetime canvas of the Universe. Whereas Newton had taken the god's eye view of the world from the outside, Einstein was concerned by what an observer inside the Universe sees. By observer here we do not man just scientist, but any system (atom, galaxy) that communicates with the Universe. The abstract description of this is the quantum theory of measurement. The purpose of theory is to explain what we see, not what a hypothetical god (ie a mathematician) sees.
Einstein's fundamental insight is that inside the world communication takes time, and the relationship between communication time and distance is the fixed velocity of light, c.
[page 162]
From a human scale of view, the velocity of light is so great that all light based communications are effectively instantaneous. Not so in the physical world, where the velocity of light plays a role at both the microscopic and the cosmic scales.
Newton had one space. Special relativity deal with an infinity of spaces all in uniform motion relative to one another at velocities in the range 0 < v < c . From this point of view, spaces whose relative velocity is 0 re the same space, and the relative velocity c is reserved for massless particles. All these spaces are called inertial spaces since the relative velocity of any pair is constant. A physicist might think of this as a system of free spaces.
The spaces communicate by the exchange of photons and einstein found that the Lorentz transformation enabled an observer moving with one space to transform observations taken on another space into [its] own rest frame. This gives the well known phenomena of length contraction and time dilation, but it also establishes that momentum and energy transforms like [duals of] space and time, that is the interval between two points in time distance space is represented by a 4-vector which transforms in the same way as two points in the energy momentum space.
The set of all relatively moving space is Minkowski space which is flat, with no potentials of forces.
[page 163]
The general theory of relativity builds force into the scenario, so endowing the space with a potential field. At low relative velocities the potential field looks just like Newtonian gravitation, which is to be expected since in slow moving systems like the solar system, the effects of general relativity are minimal, measured in seconds of arc. Tests of general relativity - Wikipedia
General relativity describes the simplest consistent dynamic space, which is nevertheless quite complex. We may base the description of this space on three principles. 1. The principle of equivalence, which says that gravitation and acceleration are observationally indistinguishable to anyone cut off from light based communication from the rest of the world. It is only our ability to see our environment which enables us to distinguish them and so have different words for them. 2. General covariance: the world goes its own way independently of any coordinate systems or other specified points of view which we use to observe it. Expressions of natural law must therefore remain unchanged under changes of coordinates; and 3. Changes of coordinates are smooth and continuous and take place in smooth and continuous 4-dimensional manifold with a Minkowski metric in local inertial systems. The curvature of this space is represented mathematically by connections between the different points in the manifold and the same information can be carried by a metric tensor residing at every point in spacetime which transforms s the distance measurements between neighbouring points. The metric tensor field described the curvature of the space and the curvature acts as a potential, which is both created by the presence of energy and acts on energy to move it.
[page 164]
The fact is that I am hopeless at finishing things, I have a few vague ideas about how the world works, and these appear to become more refined every day, but never seem to come to a point where they are publishable, except in the present notebook form. This has to happen, and the product, if it is ever going to succeed must go into the market even though it is in a very inchoate form, like motor vehicles a century ago. Then utility will lead to refinement. So do I have to keep looking for a broad enough viewpoint to compress the whole story into a feasible length.
Quantum mechanically gravitation has proven intractable but the field idea has flourished. Faraday and Maxwell imagines electrical and magnetic fields with the help of field lines and the aether. Maxwell's electrodynamics gave the electromagnetic field a life of its own independent of charged particles and so gave a model for light.
Quantum field theory fills the spatial Universe with a set of fields one for each species of [particles]. Each of these fields is a superposition of creation and annihilation operators for its specific particle. The interaction of particles is modelled by the interaction of fields and it is the aim of the theory to produce descriptions of fields and interactions that mimic the real world. Quantum fields are represented by vectors in appropriate Hilbert spaces, each of which is superposed at every point in the spacetime manifold. Quantum field theories are modelled in an infinite dimensional space attached to every point in the spacetime manifold.
[page 165]
This is an enormously complex system all built on the notion of continuous smooth fields whose squared absolute value predicts the probability for the creation or annihilation of a particular species of particles at a particular point in the spacetime manifold. A major role for this theory is describing and predicting events in accelerator physics. A set of initially free particles at t-infinity interacts around t0 and the products of the interaction are detected (again as free particles) at t+infinity. In practical terms t+- infinity may be measured in nanoseconds and distances in metres.
Pais page 488: 'It took quite a few years before physics became comfortable with the idea that there us no real difference between a resonance and a short lived particle.'
And a stable particle is stable resonance, like a soliton. Soliton - Wikipedia
Pais page 493: '. . . meanwhile [1950s] a search had begin for implications of [quantum field theory] that did not depend on perturbative expansion, which everybody, of course, agreed, were useless for strong interactions. '
1950s yielded deeper insight not so much into the structure of matter as the structure of theories,
. . .
[page 166]
Does the extreme complexity of the quantum field theoretical standard model arise from the struggle to use continuous functions to describe the world? In other words, can we find a simple solution encoded in logical functions executable on a Turing machine? Or is it induced by the complex world in which we observe the activities of fundamental particles being used by higher layers when their peer interactions are really quite simple? In other words, very simple protocols (like the nand gate) can be built up into very complex systems (like the internet).
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