Notes
[Sunday 22 February 2009 - Saturday 28 February 2009]
[Notebook: DB 65 Symmetric U]
[page 166]
Sunday 22 February 2009
Wigner: 'On the conservation laws of quantum mechanics' Goett. Nachr. 1927, 375.
Pais page 526: 'The physical content of parity is bound to interaction . . . Parity is violated when [the Hamiltonian] cannot be made invariant under reflection by whatever choice of phases,'
page 528 CPT: 'To this day, no violation has been found of the invariance under the application (in any order) of these three symmetries combined. CPT Symmetry - Wikipedia
. . .
[page 167]
Sakurai Invariance Principles PUP 1964 Sakurai
T D Lee: Particle Physics Lee
The next step it to produce actual digital algorithms corresponding to various particles, a tall (impossible?) order
What does a photon do? Carry a quantum of action with a certain polarization and energy from a to b ? What does an electron do? Carry a unit of charge and a unit of action from a to b. In every case, a fundamental particle is carrying some form of action.'
. . .
Pais page 686: Heisenberg: 'The particle spectrum can be understood only if the underlying dynamics of matter is known. Dynamics is the central problem.
We begin to construct the Universe with two distinct states and a particle whose action is to move energy from one state to another. Since for aesthetic and practical reasons we would like the energy of the Universe to be zero we assign negative potential energy to the [unoccupied] state and a positive kinetic energy to the occupied state, so we can write the Lagrangian of this baby Universe L = T - V = 0. Let us call this system the initial oscillator, the first step in the expansion of the initial singularity. This dynamic system effectively breaks the total symmetry of the initial singularity, an entity that has been studied extensively over the last 3000 years labelled in English god, having many other names in other traditions. Western theologians would call the emergence of the
[page 168]
initial oscillator the procession of the Word, the Word being both an element of the one god, but really distinct from the Father. Aquinas 160, Lonergan The boson that joins them might be called the Holy Ghost, the Spirit of love that binds the world together. The initial oscillator has no particular frequency (it is an isolated system) but we can safely assume that each change of state measures one quantum of action and so, insofar as it is relevant, we can say that E = h nu. The initial oscillator is in effect a qubit, whose behaviour is described by the Schrödinger equation. Qubit - Wikipedia, Schrödinger equation - Wikipedia This system need not be spatially extended, nor even in space. It can be described by one constant (Planck) and one parameter (Energy = inverse time). We may think of this little system as energy moving in a circle, a closed flow of energy with a conservation rule defined by Planck's constant and the fact that the system has but one frequency (unmeasurable) which we will dub ℵ0.
We may say that this system is pure energy in the sense that it is the minimal structure to which we can attribute energy, and that, of its nature, energy is conserved. Energy - Wikipedia
As the next step in its expansion, let us imagine that the initial oscillator begins to observe itself and so becomes in effect two entangled qubits. The system is now seen quantum mechanically as a superposition of 4 basis states evolving unitarily with the same total energy as the initial oscillator.
4 basis states communicating with one another gives us
[page 169]
a recognizable network. Each state may communicate with any of the other three at any given moment, but now there are six possible channels, so on the average, the energy (frequency) passing through each channel averages only 1/6 of the total. As the Universe grows more complex, conserving energy all the time the average energy in a channel decreases in inverse proportion.
At this level of complexity we also introduce the possibility of error and therefore the need for quantized encoding a la Shannon. Insofar as we understand this system to be one qubit observing another, we can drive a quantum mechanical picture of their interaction, accompanied (in the Copenhagen interpretation) with the 'collapse f the wave function'. We then interpret the quantum mechanical model in digital network terms.
Back a bit to a 3 state system, conventionally represented in a 3D Hilbert space so that any state of the system is represented by psi = a |1> + b |2> + c |3> |a |2 + |b |2 + |c |2 = 1.
In standard quantum mechanics the qubit has an infinity of states psi = a |1> + b |2>, |a |2 + |b |2 = 1.
Continuous theories attribute far too many degrees of freedom to the world and then need gauge groups and renormalization to cull the excess. We work the other way, matching the degrees of freedom tin the theory to the degrees of freedom in the phenomena: Adequate variety, 64 is fundamental particles = 6 bits.
The initial oscillator has only two states, no superposition?
[page 170]
Or maybe it has the same structure of states as a black body? We take the view that if it is real it can be represented and communicated. The initial oscillator is isolated and so we can say nothing about it. On the other hand, inside it one observes the other through the reception of a real particle = message.
Monday 23 February 2009
We begin with an 'unquantized' qubit, a flow between two states of potential and kinetic energy. [the potential arises simply because they are two?]
It seems that quantum mechanics is logically prior to space and so we may see it as a source of space.
Weyl Groups and Quantum Mechanics page vii; 'I have particularly emphasized the 'reciprocity' between the representations of the symmetric permutation group and those of the complete linear group; . . . '
Weyl page xix: Bohr: '. . . in the general problem of quantum theory one is faced not with a modification of the mechanical and electrodynamical theories describable in terms of the usual physical concepts, but with an essential failure of the pictures in space and time on which the description of natural phenomena has hitherto been based.' Bohr
page xxi: 'All quantum numbers, with the exception of the so called principal quantum number, are indices characterizing representations of groups.'
[page 171]
Weyl page 17: '. . . unitary geometry is the theory of those relationships which are invariant under arbitrary unitary transformations.
page 144: 'One of the most fundamental physical fact, the sharpness of spectral lines.
Although physicists and mathematicians love dealing in abstractions, the Universe is completely concrete, with no unbroken symmetries, every point unique and every process worked out by digital logic against a continuous background of energy and gravitation, ie by somehow utilizing the initial oscillator. The formalism of quantum computation theory is handy here, but we would like to impose the limits of observation onto the observed system so that all deterministic processes are quantized. It has been held that the continuous functions of quantum mechanics and mathematics as a whole are deterministic, but we feel that this is an abstract fiction since continuous (analogue) computations are subject to error unless quantized in some way (eg by dividing the continuum into intervals in which values can be located with certainty).
What can happen is decided by selection rules which are themselves products of various symmetries.
Late editions of Leader and Predazzi and Bjorken and Drell. Leader & Predazzi, Bjorken & Drell
PROBABILITY = EVENT RATE (t) = ENERGY
= EVENT RATE (x) = MOMENTUM
[page 172]
Tuesday 24 February 2009
Weyl page 52 Energy-momentum (4-momentum) space is the dual of space-time (4-space) space. Quantum mechanics operates in the energy domain alone, and we expand this to quantum field theory when we want to deal with quantum mechanics in space-time. From this point of view space is caused by coding delay and so intimately associated with quantization.
Here comes the sun, sending me one dimensional energy.
The fundamental oscillator is error free, so continuous and infinite. When it becomes quantized we assign to it the frequency ℵ0. The n fundamental particles become error prone once there are three of them, so giving us the quantized triad in some way isomorphic to the electrodynamic triad e+ [positron], e- [electron], and gamma [photon]. As the number of particles increases, so the coding must become more robust, which is made possible by the parallel increase in formal computing power (memory), so the Universe bootstraps itself into existence. The energy of the fundamental oscillator is so much greater than our local energy scale that we see this as an explosion.
Initial triad: fermion, fermion, boson. Quantum mechanics sees this as a superposition, but from a network point of view it is a set of quantized time ordered messages. To an outside observer these messages are 'virtual' but to a participant in the triad they are real.
We can characterize the features of the initial triad
[page 173]
by the symmetries C, P, T, and we measure the coupling strength by the frequency of the various events.
The velocity of light (the inverse of coding delay) establishes the time ordering of events in space from the point of view of a given observer at rest in an inertial frame.
Unfulfilled potential is exciting, so we find the bulk of poetry and song concentrates on lost or unattainable love. Love activated is self contained, isolated and does not write to the outside world!
Weyl page 49: 'We unite [the particle and wave] standpoints by retaining the linear wave equation, but by considering the intensity phi*phi as the relative probability that the photon appears at the point (x, y, z ) at time t ; or more precisely, that
is the probability that at time t it will be wound within the small paralleliped with sides of length dx, dy, dz about the point x, y, z ).'
The transfinite numbers apply at the initial level because they are in some sense scale free within each transfinite cardinal as we see by ℵ0 = n . ℵ0, n < ℵ0. So they describe an isolated system whose only scale is itself.
The spatial structure of fundamental particles is irrelevant because it is non existent, and so we can confine ourselves to their logical structure. Out first observation is that despite their lack of spatial structure, they are distinct entities defined by their distinct personalities. By their behaviour we know them.
Wednesday 25 February 2009
[page 174]
Thursday 26 February 2009
Local gauge theories and their attendant gauge bosons are to be mapped to the communications between fermions (in which all communications are to be deemed local), and the processes which account for the changes in fermion states. As with logical processes in an ordinary computer, these processes involve the creation and annihilation of memory states by communication between elements of memory. The probability currents postulated by gauge theories (eg the probabilities of emitting or absorbing a photon in electrodynamics) are to be interpreted as the rate of a given process, each separate current being mapped to a separate process, ie a separately defined subroutine in the universal process whose completion we identify with a particular instance of a particular species of particle.
An important provision, related to the no cloning theorem, is that every process in the Universe is ultimately unique, despite the fact that different processes may fall into different abstract categories, and this unicity is expressed in the Universe by the uniqueness of the points in spacetime where the given process is observed.
The calculations of quantum field theory (specifically in the standard model) are designed to define the communication channels available in the Universe and compute the traffic on each channel. The root channel is simply the power distribution system which sees only rate of action (energy) and nothing else, what we currently called gravitation.
[page 175]
The fundamental thesis here (which I have spent all these millions of words trying to explore and justify) is that transfinite development of the theory that brought us nuclear weapons shows is how to find peace on earth on the principles of peace that guide the quantum mechanical world, even at its most violent [energetic] moment in the infinitesimal initiation of the so called big bang.
Now to a day of constructing human habitation.
LOGICAL CONTINUITY = MAKES SENSE
Does physical (mathematical) continuity make sense. NO. Because for the set theoretical arguments to make, the points of the continuum have to be really distinct, and then it makes no sense to say they are a continuum. This paradox infects the very fundamentals of fundamental physics, the continuity of action and the zero size of the agent particles.
Helen Hanff page 84: . . . We all die (except perhaps protons) when the system can no longer maintain itself error free. The physicists call death decay, the action of annihilation operators, often requiring 'tunnelling'. The appearance of tunelling may be an artifact of the projection of the network onto spacetime, something that needs a lot more thought. Hanff
. . .
Friday 27 February 2009
We may think of a communication channel as one dimension of space and one dimension of time with
[page 178]
a function space at each end to handle the encoding and decoding.
How do we write a description of a computer as a process in a Hilbert space? Ask the quantum computation people. Nielsen & Chuang Maybe quantum computing is analogue but its outputs for purposes of communication are discrete. The Hilbert spaces on each end of a channel must commute ie have a common set of eigenvectors, ie a common codebase or language.
Schwinger QED xiv 'The quantum mechanics of particles had been expressed as a set of operator prescriptions superimposed upon the structure of classical mechanics in Hamiltonian form. When extended to relativistic fields, this approach had the disadvantage of producing an unnecessarily great asymmetry between time and space, and placing the existence of Fermi-Dirac fields on a partly empirical basis. But, the Hamiltonian form is not the natural starting point of classical mechanics. Rather, it is supplied by Hamilton's action principle, and action is a relativistic invariant. Could quantum mechanics be developed independently from an action principle, which, being freed from the limitation of the correspondence principle, might automatically produce two distinct types of dynamical variables?' Schwinger
So we see bosons as characteristic of the error impossible initial oscillator and fermions only enter the picture with the creation of space, the possibility of error, the need for quantization,
[page 179]
computation and memory.
'When the action operator is chosen to produce first order differential equations off motion, or field equations, it indeed predicts the existence of two types of dynamical variables, with operator properties described by commutation and anti-commutation respectively. Furthermore, the connection between the spin and statistics of the particles is inferred from invariance requirements. . . . The practical value of this quantum dynamical system stems from its very nature; it supplies the differential equations for the construction of the transformation, functions that contain all the dynamical properties of the system.
Schwinger page xv ' . . . it took the labours of more than a century to develop the methods that fully express the mechanical principles laid down by Newton.'
'. . . is there a fatal fault in the structure of field theory?'
'. . . the observational basis of quantum electrodynamics is self contradictory.'
page xvi 'The localization of charge with indefinite precision requires for its realization a coupling with the electromagnetic field that can attain arbitrarily large magnitudes. The resulting appearance of divergences and contradictions serves to deny the basic measurement hypothesis. We conclude that a convergent theory cannot be formulated consistently within the framework of present spacetime concepts. To limit the magnitude of the interactions while retaining the customary coordinate description is contradictory, since no mechanism is provided for precisely localized measurements.
[page 180]
Schwinger page xvii '. . . the real significance of the work of the past decade lies in the recognition of the ultimate problems facing electrodynamics, the problems of conceptual consistency and physical completeness.'
Feynman: Space time Approach [in Schwinger] 'This formalism was suggested by some of Dirac's remarks concerning the relation of classical to quantum mechanics. A probability amplitude is associated with an entire motion of a particle as a function of time rather than simply with a position of the particles at a particular time.'
The discrete network view is that the action functional for any process can take only integral multiples of h as its values. The action functional nh corresponds to the halting of n Turing machines to achieve the process considered.
Instead of saying that propagation delay is due to the extent of space we say that the extent of space is due to propagation delay, ie, like travellers, we measure distances by times.
So we arrive at the idea that the accuracy of communication increases with delay (since the message space is made larger by larger packets) and we correlate this with mass.
Increasing mass <--> increasing precision <-->increasing delay (and vice versa), the minimum delay being massless.
[page 181]
Schwinger Theory of Quantized Fields I
page 343: '. . . as a general principle associated with relativistic requirements, we must expect . . . mutually commuting operators to be formed from field quantities at mutually independent spacetime points, that is points which cannot be connected, even by light signals. A continuous set of such points form a spacelike surface; which is a geometric concept independent of the coordinate system.'
SPACELIKE = INDEPENDENT (IN THE PRESENT)
ENTANGLED (BY THE PAST)
On the network principle the past is part of the present since all present processes depend on the real presence of these older and lower layers that provide services to the higher. The leaves of a tree are a spacelike surface not communicating directly but through the branches of the tree. They are nevertheless peers, but only able to influence one another after a lapse of the time necessary to transmit a message.
Spacelike surface = memory, ie a system independent of local events.
Schwinger page 344: 'A description of the temporal development of a system is evidently accomplished by stating the relationship between eigenvectors associated with different spacelike surfaces, or in other words by exhibiting the transformation function 25
The invariance f the lagrange function, and therefore of the action integral guarantees that our fundamental dynamical
[page 1822]
principle . . . is unaltered by a change in the coordinate system.'
Network is its own coordinate system. What we need to see is how the network maps onto the general relativistic spacetime that follows from the energy flow evolved by the developing of fundamental and complex particles.
We generate space by exclusion, brotherhood (entanglement) by inclusion.
In the basic quantum mechanical system we have 'trivial' CPT invariance which, like the layers below it, remains with the Universe forever, just like the energy flow that gives us gravitation, blind to all messages, seeing only power.
Schwinger page 344: 'We shall see that the requirements of invariance under time reflection imposes a general restriction on the commutation properties of fields which is simply the connection between the spin and statistics of elementary particles.'
[ie direction of time is given meaning by spin?]
It all depends on infinitesimal operators.
'. . . the operator principle of stationary action . . . states that the action integral operator is unaltered by infinitesimal variations of the field quantities in the interior of a region bounded by [spacelike surfaces] sigma1 and sigma2, being dependent only on operators attached to the boundary surfaces. The equations of motion for the field are contained in this principle.
[page 183]
1 dimensional spin, gamma at c, e+- at rest, ie 2 states define one dimension, or two dimensions in Hilbert space, one which correlates with time, one with space. So spin is a fully 'spacetime' function.
Schwinger page 352: 'The significant implication of these properties is that time reflection cannot be included within the general framework of unitary transformations.'
Nor can C or P. CPT are throwbacks to the days before unitarity was enforced, ie before energy conservation came in.
