Notes
[Sunday 6 January 2008 - Saturday 13 January 2008]
[Notebook: DB 62 Interpretation]
[page 92]
Sunday 6 January 2008
Energy, like information and entropy, is always embodied.
Quantum mechanics is indifferent to complexity (entropy, information) and is also indifferent to absolute energy, since it is only energy differences that are relevant. This fact opens the way for the net energy of the Universe to be zero, and for energy to be embodied.
So we see momentum as static energy, energy as kinetic energy and 4-momentum (= action in some way) = the combination of the two. So the breaking of action into space-time (momentum-energy) is the first step in the differentiation of the Universe,
Aristotle took change and motion seriously (unlike some of his predecessors, who considered them to be an inferior mode of being). He held quite reasonably that if change is to make sense then there must be two sets of reality which, while constant or static in themselves, combined in different ways to constitute different realities. Change was the equivalent of recombination, as when
[page 93]
a bronze sword was recast as a bronze ploughshare or a marble stature was reproduced in wood. He called the two principles of change matter (analogous to bronze, wood or marble, the material) and form (analogous to sword, ploughshare or a carved image). Aristotle considered that both matter and form were layered. So bronze, wood and marble were constituted from the four elements. The lowest layer he called prime matter, and the higher layers converted lower layers of matter into higher ones. So we have the hierarchy of mineral, vegetable and animal. Although we never allow naked prime matter to exist, Aristotle also postulated a realm of immaterial forms, drawing on he work of his teacher Plato.
Aristotle generalized the matter-form idea under the terms potency [and act]. Potency (like matter) needed an act (a form) to make it actual. So the potential image in a block of wood needed to be actualized by a sculptor who implants the form in the [material].
Aristotle and his successors considered potency to be inferior to act, unable to actualize itself, requiring the services of an agent. This broken symmetry lies at the heart of proofs of god. Aquinas 13
With the discovery of energy two thousand years later, the potency / act dichotomy reappeared renamed as potential and kinetic energy. In this new guise, however, potential and kinetic energy are equivalent, as demonstrated by the pendulum, or in general the harmonic oscillator, which swaps its energy from
[page 94]
potential to kinetic and back again. [terminology: kinetic energy = actual energy (observable energy) potential energy = hidden (unobservable) energy.
We associate potential with spatial structure or spatial frequency, ie momentum and kinetic energy with frequency or energy simpliciter.
SPACE = STATIC = POTENTIAL = MOMENTUM
TIME = KINETIC = ACTUAL = ENERGY
SPACE - FIELD - FORM
TIME - PARTICLE - MATTER
Momentum = amount of motion: it is a potential that can be actualized by stopping the moving object with a force.
A precisely defined momentum defines no position as a precisely defined energy defines no time. The definition of space and timers requires superpositions of different momenta and energies to create a wave packet or ordered set of momenta / energies which may be imagined as an information packet defining a point with a certain resolution determined by the cardinal (bandwidth) of the packet.
So we see different momenta as the letters from which points in space are constructed and different energies as the alphabet of points in time. Between them that define events, things that last for a certain time in a certain places.
[page 95]
How do we relate 4-momentum to angular momentum?
While proper time remains the same, space expands, absorbing energy equal to the mass-energy of the particles to be found in the Universe, so keeping the energy of the Universe to zero. Potential energy (space) and mass-energy (particles) are created simultaneously, meeting in photons which are outside spacetime, experiencing zero spacetime interval between creation and annihilation [although they may spend billions of years traversing billions of light years of space between during their 'lifetime'] Photons in effect have no spacetime extension, following null geodesics.
Is Zurek exactly wrong: the first three hypotheses of quantum mechanics are problematic, the last two intuitive when we look at the world as a corpuscular (packetized) network instead of as a continuous system of interactions. Zurek His proof might then work backwards, along the lines of Cantor's proof, showing how to create continua out of discreta.
Zurek's relationship between continuum and discretum reminds me of the use of Hilbert space by von Neumann to reconcile the wave and matrix versions of quantum mechanics - different representation, same information.
Special relativity shows is that space can appear to substitute for time and vice versa, so that objects in relative motion may appear to e both time dilated and Lorentz contracted, so that the definition of energy and momentum is velocity dependent. Horizons moving away from us at the velocity of light appear to be frozen in time, even though in their own proper frames things
[page 96]
occur as in any other rest frame.
How does this relate to the establishment E = mc2. [the relationship between space and time is reflected in the relationship between momentum p = mv energy, since m = p / v = E / c2 ]
Monday 7 January 2008
In a closed Universe all logic is in some way circular, so we can look at any proof from both ends, in effect. The network view says the Universe is quantized because that is necessary for reliable communication. Continuity follows from this, as Cantor's theorem leads us to transfinity when it is implemented without error.
Science is like sex (and every other event in the world): you keep moving around until you get it right and then it happens - all inhibitions are off. [a definition of love?]
Tuesday 8 January 2008
How does the Universe break the symmetry of pure numbers (counting events) to the distinct dimensions of classical physics - mass, length, time and all the other quantum numbers. These dimensions have meaning which is established by mapping and context [ie the instrumentation used for observation].
Wednesday 9 January 2008
Zee 3: 'It is in the peculiar confluence of special relativity and quantum mechanics that a new set of phenomena rises: particles can be born and particles can die. It is this matter of birth, life and depth that requires the development f a new subject in physics, that of quantum field theory.' Zee
People (including Zee) say this because 'energy can be converted into mass and vice versa'.
Gravitation does not see the conversion, it is all energy.
Nor does the network picture, which sees all logical processing as energy, mass being simple equivalent to closed (non communicating) processes.
Does gravitation see the distinction between potential and kinetic energy> Possibly not. We may equate mass and potential energy as characteristics of a node that us not communicating. So in a pendulum, is the bob more massive at the top of its swing or the bottom?
Lagrangian is the difference between potential energy and kinetic energy. We can set up a pendulum to KE + PE = 0, ie -KE = + PE, L = KE - PE = KE + KE = 2KE.
Thursday 10 January 2008
Zurek: Quantum origin of quantum jumps: breaking of unitary symmetry induced by information transfer from the quantum to the classical.'
[page 98]
Using the assumption that state preparation and measurement transfer information from the classical to the quantum regime, Zurek sets out to connect the 'collapse of the wave function' to the 'symmetric and uncontroversial' postulates of quantum mechanics.
It is but a small step from this picture to see all quantum events [interactions between the quantum and classical worlds] as information transfers, that is communications and to see the universal web of quantum events as a large commnication network. This point of view introduces a complementary explanation for the fact that we observe a quantized Universe.
The stability of a network depends on error free communication and error free communication, as Shannon long ago demonstrated, requires quantization. Communication error occurs when the receiver of a message confuses one of the transmitted symbols with another. [This is quite probable if the transmitted symbols form a continuum]. Shannon saw that by concatenating symbols into blocks, the size of the message space could be increased exponentially while linearly decreasing the number of blocks to be transmitted, so exponentially increasing the distance between blocks in the message space and exponentially reducing the probability of confusion.
This picture is consistent with Zurek's statement of postulate (iv) of quantum theory 'measurement outcomes are restricted to an orthonormal set | sk >| of eigenstates of the measured observable (i.e. measurement does not reveal the state . . . of
[page 99]
the system because it limits possible outcomes to the preassigned outcome states).' Zurek This limitations to be viewed in the light of the first axiom '(i) The quantum state of a system is represented by a vector in its Hilbert space.' Such a vector may be represented in terms of a set of linearly independent basis states |bi > as SUMi ai | bi > where the ai are complex numbers which may be represented as a = x + iy. x and y real (and so continuous). Possible quantum outcomes are a subset of thee vectors selected by the eigenvalue equation M psi = m psi where M is a measurement operator representing the environment into which the quantum state is communicating [and m is and psi an eigenvector of M]. This is consistent with picture proposed by Shannon where the transmitting system, n order to avoid error, must encode its potentially infinite number of internal states into a finite number of packets agreed with the receiver to be legitimate messages. A receiver, n encountering a non agreed block of code knows that there is an error and in technological systems may request retransmission.
We can see this system operating in human commnication where the conventions of language enable is to detect and correct errors or request retransmission 'say again (postulate (iii). [(iii) immediate repetition of measurement yields the same outcome .]
Traditionally the postulates (i) and (ii) 'quantum evolutions are unitary (e.g. generated by the Schrödinger equation) have been considered uncontroversial in relation to the collapse postulate and the 'Born postulate '(v) the probability of finding a given outcome is pk = | < sk || psi >|2, where | psi > is the preexisting state of the system. The communication network picture makes postulate (iv) seem quite natural. The position is reinforced by postulate (v)
[page 100]
which taken in conjunction with the normalization requirement SUMk pk = 1 establishes that a quantum system has a statistical structure identical to that assumed for a source A in communication theory where it is required that the probabilities pi for the emission of letters ai of the source alphabet are normalized so that SUMi pi = 1.
Insofar s the network paradigm renders the controversial postulates of quantum theory natural, we might be moved to reexamne the uncontroversial postulates.
The logical development of the concept of unitary evolution of sate vectors in Hilbert space has a long history whose misty antecedents in the history of ancient mathematics sprang into modern focus with the application of differential and integral calculus to physical problems beginning in Newton's day and continuing until the present.
The study of continuity and its close relations the infinite and the infinitesimal raised many questions that the [physicists generally answered by saying that the methods of calculus work insofar as they produce useful models of reality and that is good enough for us.
The situation is not so easy for mathematicians. Euclid defined a point as an entity with position but no magnitude, and conceived of a line as an entity of measurable magnitude constructed from points. This leads to the naive (but useful) notion that an infinity of infinitesimals add up to a
[page 101]
finite magnitude.
Cantor famously set out to find the cardinal of the continuum and was led to the development of the transfinite numbers. His method, set theory, ultimately led to function spaces (including Hilbert spaces) and many other wonderful developments in mathematics, but it failed in its ultimate purpose. Cohen showed that Cantor's continuum hypothesis is independent of set theory,
Mathematical analysis in its entirety is based on the notion of continuity by closeness. We prove classical results like the Bolzano-Weierstrass theorem by crowding points closer and closer together into ever more confined spaces. We define and prove the continuity of functions by similar processes, showing that if we confine the elements of the domain of a continuous function into epsilon the corresponding elements of its range will be found in another infinitesimal space, delta.
The analytical definition of continuity underlies the notion of unitary evolution of quantum systems and the general notion of an 'argument from continuity' ubiquitous in mathematics and physics. A little reflection will reveal, however, that arguments from analytic continuity have no real force. Instead we must turn to a concept of logical continuity embodied in the idea of mathematical proof and formalized as the propositional calculus.
The space of continuous function or mappings in an infinitesimal fraction of the space of all possible function. Given a
[page 102]
domain of symbol,s the number of possible functions is [at least?] equal to the number of permutations of the domain.
Friday 11 January 2008
Zurek derives the equality < v || w >(1-< Av || Aw >) = 0 and concludes form this that either < v || w > = 0 or < Av || Aw > = 1. In the first case v and w are orthogonal; in the second, no information is transferred, so that if information is to be transferred the outcome states must be orthogonal.
This leads us to question the validity of the first two postulates along the lines of Heisenberg's assumption that we can say nothing about what we cannot observe.
When we observe a classical bit, we get 0 or 1 depending on the [actual] value of the bit. When we observe a qubit, we get 0 or 1 with a certain probability. In the classical case we can say that the observer's phase is locked to the bit phase, whereas in the qubit case the phase relation between observer and observed is random.
In the first case, the observation is deterministic, made within the same Turing machine, so to speak, whereas in the second we have
[page 103]
a communication between two Turing machines beating to different clocks. The quantum uncertainty is a network timing thing, a product of asynchronicity, in other words different energies.
A sale only occurs in a market when buyer and seller agree on a price. A communication only occurs in the world when sender and receiver are in some sense 'in phase' as we find with the energy levels of an atom where the de Broglie waves fit into an orbit (in the Bohr picture) in other words when we have the eigenvalues of an operator. [Distinct orbits within an atom differ by one quantum of action, that is one full turn of phase, so that when a photon is emitted or absorbed it has spin 1, sufficient to ensure conservation of action]
Saturday 12 January 2008
The transfinite symmetric network is a multilayered function space, analogous to a layered computer network, equipped with a metric that is the computing effort required to get from a to b. Since some functions are incomputable, it also incorporates uncertainty, since it is true in the majority of cases that you cannot go (deterministically) from here to there, since the path is incomputable.
How does this fit in with Einstein, Podolsky and Rosen?
How do we fit it with Hilbert space. It does not relate so much to cardinal as to ordinal measures.
Phase is constant within each Turing machine, but different Turing machines may have different energies, frequencies and phases, so that the outcomes of their interactions are not
[page 104]
determined but behave like the observations of qubits phi = a |0> + b |1>. p(|0>) = |a|2 etc.
Each layer of this function space is a set of functions of the functions of the layer beneath it. These functions are symbols or processes that can be executed by Turing machines. [they are related like subroutines of a routine]
How do we deal with unitarity? Do we even need unitarity, since the unitary evolution of a quantum state is invisible and so has no privileged place in a model of observed reality.
We call out metric computational distance, and the actual distance depends upon the algorithm used. The simplest metric is counting which corresponds exactly to the mathematical operation of integration where the function
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