Notes
[Sunday 1 March 2009 - Saturday 7 March 2009]
[Notebook: DB 65 Symmetric U]
[page 183]
Sunday 1 March 200
From the bosons in the initial oscillator we imagine the evolution of fermions which correspond to asymmetric wavefunctions and the evolution of the whole zoo of pointlike fundamental particles. Quantization is necessary here because of complexity and so the associated time delay which gives direction to causality. All these particles remain pointlike from our point of view and so we assume that while time ordering and causality exist among them we do not yet need space.
The first spatially extended objects we see are the multiplets of quarks, mesons and baryons which are the foundations of atoms and molecules and the whole extended Universe.
Lorentz transformation is a 6 parameter group whose minimal discrete representation is 6 bits, say. We combine this with Wigner's idea that particles are 'ingredients in the representation of the non-homogeneous Lorentz group in the Hilbert space of
[page 184]
quantum mechanics. . .' Weinberg, I, xxi. Weinberg
QUANTUM MECHANIC (TURING MACHINES) --> SPACE -TIME
WIGNER 1939 Wigner
Although physicists like to talk about free particles, the fundamental particles are at no stage 'free' since they are defined by the relationships to one another mediated by communications, ie forces.
There is just one quantum of action, isomorphic to the initial singularity which appears (in the modern Universe) in an infinity of different situations just as there is but one nand gate that appears in an infinity of different contexts in an infinity of computations.
To exist separately, a particle must be unique, having a unique nature, and (if there are many the same) a unique address. In this sense the no cloning theorem is tautological, since two things which are perfect clones are in fact one thing. So spatial location makes a real contribution to the definition of a particles and all those theories that treat space as a homogeneous background miss some of the point. What we can say is that spatially distinguished particles are different instances of the same algorithm [and ordinary physical space is in a message space].
Electron is a two state 'memory cell' that requires one quantum of action to change its spin [state] from +1/2 to - 1/2.
Guess: The Dirac equation defines space rather than
[page 185]
vice versa. (Weinberg page 7 sqq).
The Universe bootstraps itself. In order to have n particles, we need sufficient 'communication resolution' to address them individually and this resolution needs to be achieved by an adequate coding system to guarantee the resolution. The primordial continuum (symmetry) has to be broken into four really distinct segments by some logical process based on the existence of 4 distinct segments.
Weinberg page 28: 'The wave fields phi, psi etc are not probability amplitudes at all, but operators which create or destroy particles in the various normal modes.'
Weinberg page 49: 'Relativistic Quantum Mechanics' ?
Maybe there is no relativistic quantum mechanics. Quantum mechanics is a 1 D (time/energy) theory that exists outside space. It lies at the root of the Universe, explaining the transfinite superposition of harmonic oscillators out of which the system is built, but like all low layers, it maintains its own integrity and is unmodified by the higher layers that use it for their own purposes. So we leave quantum mechanics alone and simply see it as the producer of an alphabet of processes from which more complex processes are built. Every quantum mechanical Hilbert space is sufficient unto itself, and it is only when we introduce space that they can be differentiated by their different spatial locations and connected by communication which, from a quantum mechanical point of view, takes place in the product space of the communicating spaces.
[page 186]
The world is structured by communication. The continuous unitary spaces of quantum mechanics can only communicate is they share eigenvectors. This is like falling in love, where two persons of infinite and continuous possibility develop a common quantized language for error free computation.
Weinberg page 50: 'A symmetry transformation is a change of point of view that does not change the results of possible experiments.' Ie does not change the content of messages. The existence of symmetry transformations arises from the mathematical model having large numbers of irrelevant degrees of freedom which do not change the reality as we move through them. So the Lonergan view of spacetime that it does not matter where or when we are, when in fact a real person in real life it does because of the other inhabitants of the spacetime that we meet at various places and times. Symmetry is a result of abstraction, ignoring certain details. The network picture shows us how this works, each higher layer giving specific meaning to individual instances of a given process which, abstracting from the higher layer, is a symmetry. So we share the symmetry of sex but every mating is unique.
My relationship to you (insofar as it is blind to the environment) is indifferent to our environment, ie independent f our position in spacetime. This is the physical abstraction.
Quantum mechanics is symmetrical with respect to complexity.
Quantum field theory is so complex because it deals with an infinite set of broken symmetries. It fails to order these symmetries into layers? Or perhaps it
[page 187]
does and there are only 4: gravitation, strong, weak, electromagnetic. From these we get the basic processes from which all other processes are constructed, but we can break these down into the initial singularity.
A gauge invariance defines a communication channel since the channel is invariant no matter what message is sent along it. So although particular messages are defined by a set of particular phase relationships, a phase invariant channel handles them all with indifferent justice.
So blind justice is gauge invariant, exerting no influence over the reasonable determination of the rights and wrongs of the system, ie detecting, pointing out and correcting errors without fear or favour. The opposite situation, as in Zimbabwe, is when the personal preferences of the idiot in charge are impressed on everybody, leading to favouritism and injustice.
Quantum mechanical Hilbert spaces are independent until they begin to speak to one another by the exchange of phases along one of the four universal physical channels, gravitation (boson), SU(3) strong, SU(2) weak and U(1) electromagnetic, the symmetry group defining the world length of the channel from 0 in gravitation to 3D in SU(3)
The inner product of two normalized vectors depends only on their phase, and it tells us, via the Born hypothesis, our probability of seeing the one when we look with the other. Distinct Hilbert spaces can communicate only by exchanging particles.
[page 188]
A ray is a 'complex plane' in an infinite dimensional complex space (?)
Weinberg page 53: '. . . it is widely believed to be impossible to prepare a system in a superposition of two states whose total angular momenta are integers and half integers respectively. In such cases we say that there is a superselection rule between the different classes of states, and the phases of phi (T2, T1) may depend on which of these classes the states of the operators U(T2)U(T1) and U(T2, T1) act upon.
The first symmetry breaking is from real to complex numbers and represents the evolution of the initial singularity into the initial oscillator.
The basic evil in the human world is the attempt to control love, that is for the higher social layer to break the gauge invariance (= free communication) of the human level.
Monday 2 March 2009
Why is the photon massless, or, in other words, what is mass?
Tuesday 3 March 2009
Hilbert space works very well in describing the world through its role in quantum mechanics. We might say that it is the basic [dynamic] data structure of the Universe.
[page 189]
On/Off switch = o-machine. Hopcroft et al page 3. Hopcroft et al
How the initial automaton operates is defined by both the on/off switch and quantum mechanics. All come down to strings of on and off. This is the root of the binary digital paradigm. Once we have worked it out in binary it holds for all time and everywhere, ie logical consistency is a feature (the fundamental feature) of spacetime.
The basic physical feature of spacetime is that we observe no contradictions.
A theorem involves an infinite number of related facts. So the set of all theorems may be the set of all symmetries.
The physical world is built from 4 communication channels to be called 0 (gravitation, energy, time) 1, U(1), electromagnetic,. the birth of fermions and bosons. 2. SU(2) Weak, now our codes words are elements of the SU(2) symmetry, a channel with longer words. And so to SU(3) word length in some sense 3, giving rise to 3D space. So we build the dimensions up as layers in a network.
We transform the observations of quantum mechanics, not the Hilbert spaces that lies behind the observations. Does this give us the same answers as Veltman's discussion (and everybody else's) of infinitesimal Lorentz transformations. The result would seem to be the same but transforming the observation seems much simpler than transforming the whole Hilbert space.
[page 190]
Wednesday 4 March 2009
Since quantum mechanics is one dimensional, it does not include any idea of velocity (or even momentum) both of which require spatial extension for their definition. Concepts like 'at rest' and 'relative velocity' do not apply. All we have in the isolated quantum system is an infinite superposition of energy levels, and it is here that we find gravitation.
Next comes spin, ie energy + angular momentum = action
Veltman and all the rest of them say 'Corresponding to a Lorentz transformation, there is a complicated, big transformation in Hilbert space' (page 20) Maybe not. All that needs Lorentz transformation is the observations, in 3 space, which are something added to Hilbert space by going from the pure energy space of quantum interactions to 2 dimensional energy / angular momentum space of spinning particles and the 4-space of spatially extended systems generated by SU(3) and bigger.
So we might be inclined to deny Veltman's 'extremely important statement: 'To every Lorentz transformation, or more generally a Poincare transformation corresponds a transformation in Hilbert space, page 21.
Thursday 5 March 2009
Friday 6 March 2009
The hunch is that spin is a 2D thing, one space, one time, one since it is clearly quantized our general
[page 191]
program suggests that at the spin level of complexity there is possibility of error avoided by error correcting mechanism. The unit of spin difference is the quantum of action and its range in quanta is 0 =< n =< ℵ0 (?) so that it embraces processes from nop to the maximum computable size.
The difficult problem is to see through the century of continuous mathematics that has grown up around the Lorentz group to the (most probably) very simple digital systems which underlie our observations of spin.
The Universe is of its nature a group since it has a) distinct elements which b) interact without c) coming to a dead end or going outside itself. We see it as the broadest possible group, a group of permutations beginning at order 1 and going to order ℵ0 and beyond.
Saturday 7 March 2009
Veltman: 'We assume the existence of an infinite dimensional space, and to every possible physical state corresponds a vector in Hilbert space. (page 16)
Can this be true? The Hilbert space idea works alright for isolated systems, about which we can say nothing anyway. And it has a very hard time explaining observation: we have to step outside the continuous linear paradigm to introduce a non-linear discrete event. . . .
Veltman page 17: 'This plague, having to abandon Lorentz invariance
[page 192]
in order to define the formalism, seems common to all approaches to quantum field theory. One always seems to need some sort of grid. The final results, the Feynman rules, do not suffer from this breaking of Lorentz invariance.
Communication = probability current. Quantum field theory abstracts from the discrete nature of communication and sees the communication network as a flow of probability currents between states. Although this flow is described as something continuous within he continuous Hilbert space, it is in fact mediated by the exchange of particles accompanied by the 'collapse' of wavefunctions.
So much trouble is generated by thinking that the vectors in Hilbert space are affected by Lorentz transformation. They are not (as far as I can see) because they are outside spacetime. Only the real particles communicating between states are subject to communication delay and so Lorentz transformation.
REALITY == MESSAGE (practical definition WYSIWIG)
Much trouble arises from thinking of reality as a linear superposition of states. This is possibility. Actual reality from my point of view is not a set of possibilities, but the realized possibility that I am now experiencing which describes a 'life line' for me Insofar as I am receiving myriad messages from the relevant elements of my environment, this line can be conceived as an infinite dimensional (ie it describes a path through infinite dimensions of possibility).
[page 193]
The cosmological constant problem arises from mistaking possibility for reality. The attribution of zero point energy (1/2 h bar omega) for every possibility is the heart of the mistake. Energy is only associated with actual particles, and the conservation of energy says that this energy is at any moment associated with the actual exchange of action from one state to another. The unitary evolution of a closed system is actually mediated by the exchange of messages (particles) within that system. Elements of the wave function are collapsing (ie no longer endowed with action) and their action is moving to other possibilities [this movement we see as a flow of action, its flow rate the energy of the interaction]. The process is so fine grained (by the quantum of action) that we can approximate it very closely (at least in some domains) with a continuous formalism. Now although continuous unitary evolution is invisible (from the point of view of quantum mechanics) we in fact live in the interior of such an isolated system, the Universe, and we know in our own lives that only a tiny fraction of the possibilities we can image becomes real [for us], that is communicates with us. So our task becomes redeveloping field theory with the aid of personal experience and the principle of invariance of the local behaviour of a communication network with respect to complexity.
It used to be an element of 'sovereign immunity' that one could take the lives of one's subjects without payment, but the slow realization of humanity as a result of the hardware powers of communication that we developed 200 000 years ago [is rendering this element obsolete]. As we communicate we individualize and so increase the size of our message space, which is made possible by the size of the neural networks within
[page 194]
us which are shaped [programmed] by the environment into which we are born.
All the troubles of quantum field theory arise from trying to map it onto spacetime. The theory does not map onto spacetime. What maps onto spacetime are the messages which are generated by the Turing field that permeates logical space.
True communication has variety necessary to reduce the possibilities to zero. Ashby
The basic error underlying the cosmological constant problem (from the network point of view) is mistaking possibility for reality.
Each point on the path of life is surrounded by a space of possibilities one of which will be realized by the next stop on the path, the transformation of action from the past to the future.
The initial singularity is outside space and time and we are in it.
Veltman page 22: 'Matrices in Hilbert Space . . . The matrices we need will fulfill certain basic requirements, in particular the requirement of locality. This is the requirement that physical processes cannot influence each other if they are outside each other's light cone, ie if speeds larger than that of light are needed to connect the events.'
[page 195]
This is not relevant to the Hilbert space of a point on a lifeline. The local space describes the way the entity in question transfers its action from past to future. Locality applies only to the encoding of messages for error free communication which implies delay which established ordered strings of causality in the system. I live because my past causes my future, and once that ceases to happen I will die.
So we see a light cones not as a geometric object but as a logical object. The geometrization program for physics going back perhaps to the Erlanger program has been very fruitful but the world is not continuous so not geometrical but logical; the transfinite network described the Turing field in logical space. The Turing field can perform any computable transformation but, in the physical world, this takes time, and so the structure becomes wonderfully complex.
I am a particle, a message from one state of the world to another.
Veltman page 22: '. . . we have set up our Hilbert space for free particles. We must rethink our procedure when we introduce interactions.'
page 23: '[Locality] we hope will be achieved by insisting that the operator (matrix) describing a process at the space-time point x will commute with a similar operator for the space-time point y if x and y are outside each others light cone.'
[page 196]
But it is not the matrices that communicate but particles resulting from 'the collapse of the matrix' at x and the 'expansion of the matrix . . . ' at y. The particle itself carries information proportional to the size of the matrix of possibilities that bore it. This space of possibilities is an entity in logical space which becomes free (isolated) when the particles are annihilated. The particle is in spacetime Logical processes are not.
Quantum computation theory has learnt how to write logical operators as matrices. But the actual operation of these matrices on (say) qubit states must be mediated by particles ['all information transfer is physical'] Landauer]
Communication theory gives us a very clear and simple understanding of entropy. The information value of a point in communication space is equal to the entropy of the space which it inhabits. The physical realization of a point is a message ie a (possibly composite like me) particle.
U(1) == isolated Hilbert space, exp (i theta), the aleph(0 ℵ0 the roots of unity, qubit.
Insofar as every point on every world line is an isolated Hilbert space, we can see the Universe as a set of free agents interacting with one another, exchanging forms (ordered sets) of action.
I learnt theology in Latin, now I want to express it in mathematics.
[page 197]
Veltman page 24: 'We now have a set matrices a(p) and a dagger (p) called annihilation and creation operators respectively, defined over the whole Hilbert space and that can be used to build up any matrix.'
He puts them all in one Hilbert space, but since all the (p ) exist in orthogonal subspaces, we can easily see this Hilbert space as a set of independent Hilbert spaces, each with their own (relevant) operators. The creation or annihilation of a particles in one of these spaces sends a message to some other space.
V page 25: 'The Fourier transforms of the matrices a and a dagger are called fields.'
'. . . field has a number of properties that make it very useful to construct physical quantities. The main property is that it is local.'
The logical role of god is to answer all unanswered questions, ie to explain everything that currently remains unexplained.
Special relativity is a constraint on communication space, what we call real space.
V page 29: 'The connection between vectors in Hilbert space and measurements in quantum mechanics is through probability.'
The application of Lorentz transformations to Hilbert spaces is done in order to get the probabilities right which trying to retain normalization. What happens if we leave the Hilbert spaces alone
[page 198]
and apply the Lorentz transformations instead to the probabilities of the outcomes so predicted (as mapped onto points local to the Hilbert space) onto the structure of probabilities noted by an observer as rest in its own frame when it looks at what is going on in other frames. What we re saying is that quantum mechanics is logically prior to relativity.
If we follow Veltman, we see that the connection between Lorentz invariance and spacetime are rather thorny and somewhat inconsistent, justified by the fact that the current view has yielded some quite remarkable numerical coincidences after nearly a century of 'analogue' tweaking.
Veltman page 17: 'What we are doing here is of course quite horrible: we are violating Lorentz invariance. A square box of volume V is not Lorentz invariant.'
The problem here comes from integrating quantum mechanical variables over space when it has really nothing to do with them. By the time we get into space requisite variety has established a fixed value for the quantum mechanical variables and they are encoded in a particle. Then we apply the Lorentz transformation in order to get a rest frame reading on the particle.
The first bifurcation is into number . action = spin
The second bifurcation is into action . frequency = energy
[page 199]
We set out to construct the Universe through a series of bifurcations initial singularity, energy, spin, electromagnetic, weak, strong.
The money system is distorted by the value placed on money (its time value, as represented by yields on physical capital) which renders the system non-linear and prone to catastrophe. A financial crisis is a singularity in the cashflow network, a black hole.
Veltman page 30: '. . . states in Hilbert pace are generally not invariant under Lorentz transformations but they transforms according to some representation of these Lorentz transformations.'
Higman : ' Some [realizations of a group] have been geometrical and expressible in matrix form. But the matrix realizations of a group are of such paramount importance that they are known by a distinct name - representations.'
Veltman: 'The trouble now is that we certainly want probabilities to be invariant under Lorentz transformation [but not probability densities?]. In other words, the above defined dot product [<a |b >] must be invariant . . . .'
This problem arises from the wrong ordering of the operations {Lorentz transformation, observation }. In the network model observation is logically prior to Lorentz transformation and so we transform the observation, not the Hilbert space underlying he observation (and outside space).
[Quantum mechanics makes contact with spacetime through the time dimension]
Veltman page 31: 'For scalar or pseudo-scalar particles there is not yet any problem [with the Lorentz transformation of states in Hilbert
[page 200]
space], the transformations in Hilbert space are rather trivial (as for example the Hilbert space vector |p > transforming to |q > where q is the Lorentz transformation of p [as represented in Hilbert space]. In dealing with particles with spin, such as electrons and photons, the transformation of the states in Hilbert space becomes more complicated, and we will actually run into this problem.'
But not in our ordering. We propose to Lorentz transform the actual spin measured in 4-space to our own frame of reference, rather than transforming the spin wave function.
SEX, VIOLENCE, REPRODUCTION, WAR
They tried to castrate me (spiritually, I'm no singer). Fortunately lust was stronger than bullshit and the need to live carried me beyond their moral and intellectual boundaries (the moral errors being driven by their total intellectual misunderstanding of the world).
Veltman: All this has been about 'free particles' ie particles originating in states in free unitary evolution. Of course, in the network paradigm, as soon as a particle appears, the message is fixed, the isolated unitary symmetry broken.
The beginning of physics is the exchange of particles and to keep the number down to two these particles must be bosons which can exchange with themselves. Then in one space and one time dimension, spin 1/2 particles can appear.
(to be continued)
[TURING FIELD DB 66]
[page 1]
Veltman page 32: 3: Interacting fields
'We will now discuss systems of interacting particles.'
page 33: 'A vector in Hilbert space represents a physical state. What is a physical state? A physical state is simply any possible situation with particles moving here and there . . . Often people make the mistake of identifying the physical state with a situation at a given moment, one picture from a movie. But that is not what we call a physical state. The situation at some moment may be seen as a boundary condition [fixed point]: if one knows the whole situation at some moment, and one knows the laws of nature then in principle we can deduce the rest.'
As long as the system remains undisturbed and evolves deterministically. But every wave function may collapse sometime, and then the system is no longer isolated but disturbed, so we know one eigenvector of its state and the corresponding eigenvalue, but our ignorance remains profound.
INTERACTION = OBSERVATION
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