Notes
[Notebook: DB 62 Interpretation]
[Sunday 2 December 2007 - Saturday 8 December 2007]
[page 54]
Sunday 2 December 2007
We may say that quantum mechanics, being indifferent to complexity, operates at all scales of complexity, from complex numbers on up. Gravitation goes one step deeper, being compatible with no coding and so describes phenomena in a channel without meaning at the very basis of the Universe. The differentiation into space and time requires coding to keep messages in these separate channels distinct.
Simple things like gravity look complex when they are applied to complex systems just as arithmetic may look complex when applied to the accounts of a large firm because it is applied at many different points defined by the structure of the firm's business. So gravitational interaction is spread over the whole complex
[page 55]
Universe created by Cantor's theorem acting in the context of quantum mechanics. Shannon's theory in a sense confines the Universe to exist in discrete entities (like 4-D spacetime). This is a sort of 'bootstrapping' - error free communication enables structure; structure enables error free communication.
Quantum mechanics is normalized but is not confined to any particular number of states or entropy. Reversibility requires both normalization and constant entropy, and so we can expect reversibility only within layers of the network where the cardinal number is fixed (ℵ0, aleph(n) etc)
The net energy of the Universe may be zero. Feynman Gravitation 9-10. Feynman
Electromagnetism is based on a conserved current of charge. Quantum mechanics on a conserved current of entropy ?[at each peer level]. Gravitation a conserved current of action? All these currents being conserved with respect to time, binding future to past, ie constraining what can happen (potential) by what has happened (actual)
This is the general import of equations in physics. To say B = A means that the actual value of A fixes all the potential values of B.
Can we say that action space is the root of a tree that branches into energy space and momentum space which are related to one another by the Fourier transform. Fourier analysis - Wikipedia
The fact that we can find a consistent explanation of the world in a 4-D representation
[page 56]
suggests that general and special relativity antedate the bifurcation of the Universe into space and time.
How do we match the field picture to the communication picture?
A roller-coaster ride is a segment of life confined by the tracks of the roller coaster and gravitational potential. My intellectual life (and more general life) is rather like this as I seek constraints to direct my mind to a proper understanding of the structure of the world that constrains my life to the roller coaster ride along the transfinite geodesic that I have experienced.
We write the fundamental equation of quantum mechanics as d psi / d a = B psi, where psi is a vector and B is a matrix. If we differentiate psi with respect to time (a = t), B becomes the energy matrix H; if a = x, space, B becomes the momentum matrix and is a = action, B becomes the action matrix S. We represent action by spin, which counts the number of events per 2 pi change of phase, 0, 1.2, 1, 1 1/2, 2 etc.
Gravitation works around entropy which quantum mechanically corresponds to frequency. Clock (frequencies) are slower in regions of lover gravitational potential. If we assume that the gravitational potential in the
[page 57]
initial singularity is zero, we might say that the frequency of its clock is zero and so that it is eternal, fitting the correspondence initial singularity = god. Feynman, Gravitation page 68.
The relativistic version of Schrödinger 's equation is only necessary if we want to compute how systems in relative motion look to one another. In our rest frame we can write d psi ? d tau = H psi, where tau is our proper time and H is our rest frame energy operator.
Einstein's etaab is four functions describing transformations from one reference frame to another. gab is ten functions describing ds2 in the presence of matter.
FUNCTION = CHANNEL as in the Hamiltonian in which each element of the matrix describes the interaction between some base state and another.
Four separate Turing machines describe flat space, whereas curved space requires 10, since there are ten bilinear products dxa dxb ( 4 + 3 + 2 + 1)
phys05Distance
Timewise the distance between two points is a the number of steps it takes to cover the distance, each step being a unit of time. More generally, the steps are units of action.
[page 58]
If the total energy of the Universe is zero, its potential energy is the negative of its total observable (mass + kinetic) energy.
The quantum harmonic oscillator requires 'material' oscillators (atoms etc) to enable the states of the electromagnetic field (photons) to be created and annihilated. How do we put this in? The material oscillators correspond to Turing machines or algorithms (potential structures( which can move between states of different potential in an inverse manner to the photons which they emit and absorb, so the absorption of an energetic photon moves the oscillator (memory) to a state of higher energy.
So we model kinetic and potential energy by stasis and change. Potential = memory; energy = rate of change of the contents of memory.
Monday 3 December 2007
Our null network (a Universe since it has no outside contact) has 1 node and no edges, and we can say no more about it than the classical god.
Our next model is a duality, with two nodes and two edges joining them, and a superposition of messages flowing in both directions. We represent these messages by z and z bar and measure the probability of a complete circuit by z z bar = |z |2. Normally when we writ z = x + iy we assume that x and y are real
[page 59]
with ℵ1 possible values, but here we have no way of defining real numbers and presume that we have at most two numbers (local machine infinity) at our command. With these the only complex numbers that we can make are 1 + i, 0 + 0, 0 + i, 1 + 0 say.
[diagrams]
Energy of the Universe may be zero if we have two cancelling motions.
Tuesday 4 December 2007
An isolated system is described by an abstract Hamiltonian which tells us how fast each pair of elements in the superposition are communicating with one another.
How complex does a system have to be to have an observer? As soon as it is dual, one pole observes the other. When one annihilates, the other creates.
phys06spin
phys07 quantum information theory. We have been thinking quantum mechanics drives information theory, or is it the other way round? Who can tell, so the two views interfere in some way.
[page 60]
Cybernetics tells us the limits of control which puts limits on error rates and the stability of systems. Biological reproductive systems sustain huge error rates but use high levels of multiplicity to stay fit.
Systems interact positively when the are 'in phase' and negatively if they are out of phase, as we discover in the theory of driven harmonic oscillators. This is the geometric fit of orbits to deBroglie wavelengths that 'explains' the energy levels of atoms.
Each little device that we construct becomes a layer in the network building on previous layers and a component of subsequent ones.
Time's arrow says structural complexity grows monotonically with time, at least for systems not in equilibrium. The distance of a system from equilibrium is a function of the ratio of its internal energy scale to the energy of its environment. So chemical processes are almost unseen by nuclei.
Every now and then this stuff takes on a bit of a mystical glint, looking rather sexy. Not a bad thing, but let not lust cloud judgment.
Since the not operation is not observable, we take them to be represented not by events but by amplitudes. So we see the not situation as a two state event which proceeds cyclically with no memory, a plane wave. From the
[page 61]
digital point of view we see not operating between 0 and 1, not -0 = 1, not-1 = 0, a binary system.
How do we give such a system memory? Lower frequency.
Music is one dimensional since we can represent the music at a point in space by a string of values sampled at the Nyquist frequency. Quantum mechanics in effect describes music by energy operators which are in effect 'informed' by the [musical] notation.
SPACE = MEMORY Dynamics proceed by changing the contents of memory = changing the contents of space.
Wednesday 5 December 2007
Thursday 6 December 2007
Friday 7 December 2007
Saturday 8 December 2007
Mutually exclusive states are represented in Hilbert space as orthogonal vectors.
ORTHOGONAL = XOR
SUPERPOSITION = AND - no, superposition = arithmetic +
We might imagine that the first four dimensions of Hilbert space describe spacetime that then we add ore dimensions to describe the rest of the Universe that exists in spacetime. Would seem to work in 2D, or is it one D, where we see time as
[page 62]
the conjugate imaginary of space. But how do we go from there to 4_d?
4-space = {electron, positron, photon}
We say that energy preserves the unity of a Universe whose symmetry is broken by Cantor's theorem.
A state is said to exist if it has energy which, in a stationary state is called potential energy. This energy has mass (an atom gets heavier when we excite it) but what about the potential energy of the expanding Universe? The Universe is the initial singularity in an excited state? From a quantum mechanical point of view it does not matter where we set our 0 of energy. How does this relate to the scale invariance of quantum mechanics?
ENERGY = WORK (energy on the move) (kinetic)
ENERGY = POTENTIAL (energy stored in a stationary state)
The Universe starts in the bottom of a potential well whose potential energy is -1 and kinetic energy +1 giving zero in total?
A stationary state (like a stable orbit) is nevertheless a state of motion so that its kinetic energy is in effect [potential]
e i theta = cos theta + i sin theta
|e i theta | 2 = (cos theta + i sin theta) (cos theta - i sin theta) = c2 + s2 = 1.
In the unobserved not operator, theta can have two values, 0 and pi/2, so e i theta = {1, i}.
The key to creation is that talking about myself creates a 'conscious copy'.
not operator is the basic recursive function. We can think of the Peano successor as the new number that is not all the previous numbers. not is scale invariant, not-electron = positron, not-Universe = vacuum.
Dirac and Klein-Gordon equations operate with four degree of freedom. We create new degrees of freedom by taking derivatives; we annihilate degrees of freedom by integration?
We maintain the unity of the world by time division multiplexing ie energy, so that the initial singularity splits into two, one not the other, but stays one by swapping them round all the time, since they are equiprobable but it can only be in one state at a time until space is invented.
From a time point of view a space is an interval which can accommodate two intervals half as long (and so double the energy). Pure space has no energy because it is eternal. The weakness of gravitation is related to the slowness (stability) of space. So the timescale of gravitational interaction is 1040+ times slower
[page 64]
than an electrical interaction.
Making a world out of not s. Not, like yes, is determined by what is not.
p union not-p = everything
p and not-p = nothing (impossible)
Past = {completed computations
future = {ongoing computations}
CONTRADICTION requires CONTACT -1 + 1 = 0.
COMMUNICATION = CONTACT interval = 0.
The static structures in the world are equivalent (in wave mechanics) to standing waves, ie interfering waves moving in opposite directions as in a vibrating string or an air column.
Mass/energy is gravitational charge, like + and - are electrical charge; both determine a rate of interaction.
