Notes
[Sunday 22 March 2009 - Saturday 28 March 2009]
[Notebook: DB 66 Turing Field]
Sunday 22 March 2009
[page 65]
Monday 23 March 2009
Streater and Wightman Chapter 3 Fields and Vacuum Expectation Values Streater and Wightman
page 96: '[Classical fields] . . . have two basic properties: (1) they are observable, [via test particles?] and (2) they are defined by a set of functions on space-time with a well defined transformation law under the appropriate relativity group.'
We should expect to get the same results whether we apply the relativistic [transformation] to the field and so deduce the motion of the test particles, or directly to the test particles.
'. . . one expects the analogue in relativistic quantum mechanics of a classical observable field to be a set of hermitian operators defined for each point in space-time and having a well defined transformation law under the appropriate group. The first part of the present chapter is devoted to the isolation of mathematical definition of fields in quantum mechanics which accords with these general ideas.'
Our suggestion here is that we apply the relativistic transformation to the particles themselves rather than the operators, thus bypassing a lot of mathematical complexity. Will it work? Read on.
Translation [geometric] makes no sense in Hilbert space.
On the Heisenberg picture, vectors in Hilbert space are invariant, ie memory and they are changed by variable operators (functions, algorithms).
[page 66]
Somewhere near the root of all this is the axiom that the continuum is not observable, only discrete events can be observed.
Streater and Wightman: 'A symmetry operation (sometimes called an invariance principle . . . ) of a physical system is a correspondence which yields for every physically realizable state phi another phi', such that all transition probabilities are preserved.
In other words, in quantum mechanics, a symmetry operation preserves the dot product of vectors, as a rotation does, and simply reflects redundancy in the mathematics.
'Our definition clearly makes every unitary and anti-unitary operator a symmetry operator and Theorem 1-1 shows that these are essentially the only such.' (page 8)
[Theorem 1-1]
Let phi --> phi' be a symmetry of a physical theory satisfying the hypothesis of commutative super-selection rules.
If the symmetry leaves coherent subspaces invariant, then there exists in each coherent subspace a unitary or antiunitary operator V such that for all physically realizable states of that subspace
The operator V is uniquely determined up to a phase.
If the symmetry does not leave coherent subspaces invariant, then restricted to a coherent subspace it is a one-to-one mapping onto another coherent subspace, unitary or anti-unitary, and unique up to a phase.]
When quantum mechanics says a certain hermitean operator is an observable it really means 'a set of observables with normalized probabilities'. In other words, such an operator represents a source in communication theoretical language.
In the Heisenberg picture this means that a unitary transformation of a hermitian operator does nothing because its eigenvectors and eigenvalues will remain the same, and the observable results unchanged, except we are looking at them from another point of view. So we can
[page 67]
see the interaction of two particles as represented by a wrestling match between the two relevant operators as they revolve around one another in infinite dimensional space until they reach a conclusion, pin, submission, or in computational terms halt at the referee's command stop. The wrestlers remain the same throughout the event, but they are continually exploring different relationships to one another, taking into account the rules of the match, both its continuation and its completion. The Hamiltonian models each wrestler as a set of transition probabilities or a set of energies.
As well as learning to talk, we all learn to wrestle and play in childhood and ultimately to mate and kill (in later life).
Although we see the vectors as static, the operators are in continual motion, the probabilities of different relationships rising and falling between zero and one as different current states change the probabilities of future moves.
In my mind I am wrestling with the world trying to come to grips with it enough to see how I and my children fit in.
It is a long step from quantum mechanics, but one any reasonable theory of everything should be prepared to take. We bridge it with the scale invariance of networks. A network is a set of binary relations between processes and memories.
[page 68]
The light cone separates possible messages (defined by their endpoints) from impossible (spacelike) messages. Within the possible, however, there is a broad range of probabilities 0 =< p =< 1, 0 = impossible, 1 = necessary. As we have come to see it, it is up to quantum field theory to tell us what is going on inside the light cone. Let us say that all of these events are to some extent controlled by the relative velocities of the participants. This is the collision point of view.
What we look for is an algorithm converting quantum mechanical probabilities into space-time intervals and vice versa.
Our rate of thought, and so time to complete a given task are governed by physical parameters. Physically computations are pretty simple at the level of adding one, or simply oscillating. Pure local motion may be the simplest visible process. Within any purely locally moving head, more complex interactions are generating this text (and I'm also cooking an apple pie). A rigid local motion preserves the relationships in my brain so life goes on as usual unaware of my inertial motion. The local relationships in my brain are maintained by geometrically choreographed electromagnetic forces. So an atom is a subroutine in a molecule, a molecule in a cell and so on.
'Charge conjugation' c is the inverse of action, the antiaction annihilating the action so that it becomes as if it had never happened, eg as though a particle-
[page 69]
antiparticle pair had never been created. P and T are inverses in space (momentum) and time (energy) respectively.
What is the dimension of space-time? M, L, T or what combination of these. Ie what is the dimension of the interval ds 2 = x mu x mu? Using c to convert between pace and time, we can choose it to be a length or a time or perhaps a dimensionless number, ie a ratio of one interval to another. From a logical point of view, the unit interval would seem to be the quantum of action, the time integral of the Lagrangian.
If ever there is an outcome to this it will be the result of decades staring in disbelief at quantum field theory books wondering if something that began as a singularity could become so complex so soon, down at the fundamental level.
SPACETIME ADDRESSING vs NETWORK ADDRESSING.
In spacetime one is addressed by coordinate, in a network by a name.
In 2D spacetime one can translate but not rotate? One can rotate from space to time, a unitary transformation. [maybe one needs two particle to translate. Can these exist in 2D?]
But in 4D Lorentz transformations not unitary since a series of boosts do not lead back to rest, but approach c. Hyperbolic functions replace trigonometric functions in the rotation matrix. Here we see Lorentz transformation differentiated from unitary transformation. It adds translation to rotation.
[page 70]
Can all the fundamental particles be rotated (transformed) into one another, in other words are they all the same thing from different points of view, the different points of view reflecting their own differentiation, like Wigner's construction of states of a particle by Lorentz transformation.
It is of the nature of different inertial frames that they can only be close to one another or in contact momentarily because of their relative motion.
Another axiom: Invariances are the footprint of an epoch when the Universe was too simple to vary, a the fundamental invariance that the Universe is one (and everything is natural) dates from the initial singularity.
So what does the invariance of ds 2 tell us?
Tuesday 24 March 2009
Space-time (momentum-energy) is pixellated by the uncertainty principle which is in turn the measure of a completed computation, a quantum of action.
I cannot claim that this story is self consistent or consistent with the data, but it does represent a paradigm change, a la Descartes. Descartes He set off the scientific era by insisting that the visible Universe was matter moved by God in a manner consistent with but independent of spirit, also moved independently by God to harmonize (to some degree broken by the Fall) with matter. Here we connect
[page 71]
spirit and matter in principle [by the layered network model] but have yet to work out sufficient detail to make the idea testable.
We can begin the study of the relationship between logical addressing and spacetime addressing by looking at an ordinary computer. Each logical operation is physically located in terms of memory (space) and process (time) and one can, in a way, be swapped for the other so a fast process in a small space may be able to achieve the same results as a slow process in a large space. [large processes are assemblies of small processes] We guess that the stationary action / stationary time principle in physics is in some way connected to optimizing computation in terms of processor speed and memory. The principle of memory may be no double handling. No moving thing around in memory just to make room for something else. Like a building site of limited area, order things so that they are there when they are needed and never 'in the way'.
IN THE WAY / OUT OF THE WAY: 3D enables us to connect every point with a way [not possible in 2D, 4D (space) is overkill?]
WAY = COMMUNICATION ROUTE ie a structure serving to facilitate communication like sealed road vs bush track. The characteristic of a way is continuity, so one can drive from Sydney to Melbourne without having to open any gates or ford any rivers. This continuity is ultimately logical and is equivalent to the repeated performance of some simple operation like the turn of a wheel without complex operations like having to stop to build a bridge over an obstacle in the way.
[page 72]
'I am the Way' = I am the algorithm embodied. John 14:6 John Be like me and all will be cool (except I got crucified and led a fairly stressed life toward the end). Maybe not deliberately, just motivated by the power of an idea like Descartes, Muhammad etc. The task of science is to pierce the personal and political conviction and see if the plan would work in reality.
Jesus was blind to many aspects of reality, like the precision work that goes into being a bird or a flower, both also ways of survival, that is routes from the past to the future.
The dynamic foundation of every personality in the Universe is the passage of its proper time. Lorentz transformations are in effect means to connect the proper times of persons in relative motion.
We define a person as a source and recipient of messages, something with a mouth and ears, although in performed drama the ears are almost irrelevant if everybody has learnt their lines. Some input is needed to cue the speeches, however, although this may be done by giving the actors watches and a time based cue sheet.
Loving thy neighbour in a conservative environment means birth control which means decoupling sex and reproduction, so we can have as much sex as we want without overloading the system. Our ultimate aim must be a stable population in the region of what we've got, while at the same time reducing our footprint say 90%. Tall but feasible with solar energy and closed matter cycles.
[page 73]
'A Lorentz transformation, lambda is a linear transformation mapping space-time onto space-time which preserves the scalar product xmuxmu.' Streater and Wightman page 9.
The most interesting thing about Minkowski space is the change of sign (antisymmetry) between space and time. It points to an epoch where space and time were undifferentiated, ie directly proportional s2 = t2 not s2 - t2 = 0.
Rotation in space-time space = spin?
Streater and Wightman '. . . it makes sense to say that two Lorentz transformations can be connected to one another by a continuous curves of Lorentz transformations.' A curve of increasing relative velocity, ie acceleration, nevertheless asymptotic to zero as the velocity between observer and moving frame approaches c. One cannot communicate with a particle moving away at c with a signal travelling at c any more than one can contact someone walking away at 5 km per hour by walking after them at the same speed.
Streater and Wightman page 10 '[The Lorentz transformation] has four components each of which is connected in the sense that any point can be connected to any other, but no Lorentz transformation in one component can be connected to another in another component.'
Ie we have 'topological differentiation' with no path of smooth deformation from one component to another like doughnuts and cups.
[page 74]
Doing things. 'Doing something' is isomorphic between me and atom insofar as doing something always means sending a message to some other point in the Universe, by voice, gun, jackhammer or whatever. All messages are physically encoded and written in spacetime. Even transparent processing can be visible to someone with the means to examine it (like a software engineer).
But we think this does not go all the way. This is a logical space (represented by quantum mechanics) which can be 'observed' only through its effects and which is prior to spacetime. We imagine this process first as the four components of the Lorentz group and then filling in the mass shells in the light cones.
Unrequited love = unfulfilled possibility at the most fundamental level, in the realm of pure creation and annihilation.
Streater and Wightman page 128: 'The requirement that the fields of a field theory have definite transformation laws under U(Is), U(It) or U(C)uniquely fixes those operators.
'Thus in a field theory the problem of determining whether a given symmetry is a reasonable expression of the operations P, T or C is reduced to the problem of understanding the physical content of the transformation laws of the fields under those operations.'
What we are doing is slowly grafting the classical god onto modern physical rootstock to their mutual transformation and benefit.
[page 75]
Streater & Wightman page 128 'For an observable field a transformation law under P, C or T is a direct statement about observable quantities. and no possibility of ambiguity arises. . . . For unobservable fields, on the other hand, the physical consequences of the assumption of a given transformation are only indirectly ascertainable; an analysis of the super-selection rules of the theory is in general indispensable for deciding whether the assumption of a different transformation law actually leads to a physically distinct theory.'
My feeling in that no field is observable because all fields are represented by continuous functions and continua are not observable by definition. There are no features to observe. This suggests that we can say nothing about applying transformations to fields, they only apply to observables. We let the invisible field (in our case the Turing field) do its thing and then we transform the results into our own rest frame to get a local view of what happened. [this works all right for the 'output particles' from an event, but what about the velocity/momentum relationships of the input particles?]
We want to create some logical Maxwell's equations, an interaction that once started sustains itself (as a photon) until it is terminated by passing its energy to another process.
To love someone can be to want to take possession, to rotate the lover into one's own basis.
The logical Maxwell's equations are Lonergan's schemes of recurrence. Engines that run between one potential and another, like the chemical potential of petrol and the thermodynamic potential of hot air.
[page 76]
Symmetry with respect to complexity is a line of analogy in some way isomorphic to a line of Lorentz transformations imparting increasing boosts to a frame of reference. Such a line is created by a gravitational field, and we can watch the frame accelerating as it falls down the gravitational potential, collecting kinetic energy (four momentum) as it goes.
Streater and Wightman page 117: '3-4 The reconstruction theory: recovery of a theory from its vacuum expectation values.'
The Lorentz transformation is an emergent property of the Universe.
S&W page 97 'There is an invariant state, psi0,
unique up to a constant phase factor (uniqueness of the vacuum)'
VACUUM = ULTIMATE HARDWARE
S&W page 100: 'To be a field theory, a relativistic quantum theory must have enough fields so its states can be uniquely characterized using fields and functions of fields.'
Wednesday 25 March 2009
Given that we cannot observe the continuum we can only observe the Universe insofar as it is not continuous, ie is broken into clearly distinguished elements, observer and observed.
[page 77]
Cardinal numbers count SHIFT operations which merely move something from one memory location to another. The physical size of the location = h = delta E.delta t = 1 unit of spin.
So spinors antedate vectors (ante-date, anti-date)
My life - what to do with it? This is not a question that many people in the history of the world have been able to ask, in that most people . . . have been fixed in societies that give them little personal choice because these societies themselves were fixed by external circumstances relating to food supply, enemies, weather and so on. Diamond
Can we say that all memory locations are isomorphic to the initial singularity: the atoms and other particles seem to have perfect memories of stationary states.
Thursday 26 March 2009
Friday 27 March 2009
One of the most outstanding features of life is its uncertainty, which perhaps motivates the search for certainty and leads many to believe in the existence of a god in absolute predictable (deterministic) control.
