Notes
[Sunday 27 January 2008 - Saturday 12 February 2008]
[Notebook: DB 62 Interpretation]
[page 129]
Sunday 27 January 2008
Schiff Quantum Mechanics Schiff
Wilson and Sommerfeld quantization rule (1915, 1916) '[In] Hamiltonian systems in which the coordinates are cyclic variables . . . the integral of each canonical momentum with respect to the coordinate over a cycle of its motion must be an integral multiple of h.' Schiff page 4.
How do we interpret the two-slit etc experiments in communication / computation terms?
Schiff page 6: 'How can the presence of a slit through which the photon does not go prevent the photon from reaching a part of the screen it would be likely to reach if the slit were closed?'
Possibilities influence outcomes: possibilities = potentials.
[page 130]
Monday 28 January 2008
A source is a potential. So a raised pendulum (with potential energy) is the source of motion of the falling pendulum, and the slits in the two slit experiment are the sources of the messages recorded at the screen.
One complete cycle of a cyclic variable (360 degrees of phase) corresponds to the completion of a computation process and the emission of a message (result, electron etc).
Schiff page 13: 'the diffraction pattern disappears whenever a successful attempt s made to determine through which slit each photon passes.' ie possibility (entropy) is reduced to zero, we have only one source.
The uncertainty principle arises naturally from consideration of a wave packet and its Fourier transform. Localization is achieved by a spread of frequencies, ie by a superposition of processes with different periods.
Romance and the all or nothing principle: if one sees that one's lover is taking an interest in another, one is likely to force a decision, requiring either than the lover completely abandon the other, or, on the other hand, split and go to the new interest. A superposition of lovers thus tends to be unstable.
Schiff page 18: 'We shall therefore find an equation for psi, and having found it, shall regard it as a more fundamental attribute of the wave functions that the harmonic forms [solutions given above].
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Requirements: 1. Linear, to allow for superposition
2. Indifferent to parameters describing particular instances of
motion (momentum, energy, wave number, frequency)
Feynman: Quantum Mechanics: 'The new theory asserts that there are experiments for which the exact outcome is fundamentally unpredictable and that in these cases one has to be satisfied with computing probabilities of various outcomes. But far more fundamental [my emphasis] was the discovery that in nature the laws of combining probabilities were not those of the classical probability theory of Laplace.' Feynman
We must distinguish between complete (observed) processes and incomplete (not observed) processes. The outcomes of a process (like the two-hole experiment) depend on where we observe it, near the holes (to see which hole the particle went through) or t the screen (to see the probability distribution of the particles.
Feynman page 9: 'Any determination of the alternative taken by a process capable of following more that one alternative destroys the interference [interaction, superposition (2D)] between the alternatives,'
PARTICLE = COMPLETE a la TURING, Gödel
WAVE = INCOMPLETE FUTURE
PAST == COMPLETE == SPACE / PRESENT = OPEN = KINETIC
The direction of time is set by the transition from incomplete to complete, ie the halting of a computation.
SPACE = PAST (LIGHT, CAUSAL) CONE of an event
SPACE = CAPITAL
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Feynman page 14: 'The concept of interfering alternatives is fundamental to all of quantum mechanics.
Memory is essential to giving a direction to time.
'When alternatives cannot possibly be resolved by any experiment, they always interfere.' Feynman page 14.
Fermions Bosons
Fermions add with 180 degree phase shift which means the probabilities of finding two in the same place is zero, ie they have different spatial addresses, ie they are the source of space.
Energy
fermion/boson
Momentum
Feynman page 16: 'The concept of identity of particles [protocols] is far more complete and definite in quantum mechanics than in classical mechanics. . . . In quantum mechanics we can give a direct test to determine whether or not particles are completely distinguishable.'
BOSON = KINETIC = PROCESS
FERMION = POTENTIAL = MEMORY
Tomonaga page viii:' The existence of spin and the statistics associated with it is the most subtle and ingenious design of Nature -- without it the whole Universe would
[page 133]
collapse.' (?)
phys07 entanglement
Tuesday 29 January 2008
The mathematical machinery of quantum mechanics is remarkably simple and compact, and stands as a magnificent monument to the mathematical developments arising from Newton's work and the discipline of Newtonian Mechanics to which he made the founding contribution/
The whole system can be understood in its simplest guise as an abstract two-state system called the qubit in the theory of quantum computation.
Although the mathematics is pretty simple, it carries many surprises, not least of which is the feature called entanglement.
We can make quantum mechanics recursive in that any system containing a quantum jump can be embedded in a larger system whose evolution remains unitary. In other words every discrete system can be embedded in a continuous one, which is the content of Cantor's theorem.
To make the transition between two state quantum systems and those with transfinite states, we introduce the 'relativity of transfinity' or 'Cantor's principle of finitism'. Hallett
So, from the point of view of its finite discrete
[page 134]
elementary spaces, their tensor product is continuous. This is a matter of resolution which is strictly related to point of view and the principle of requisite variety. Ashby No receiver can receive messages more complex than its own entropy or variety. Quantum mechanically this means that observer and observed are the same cardinality, observation is symmetrical, and the whole observer / observed system is represented by vectors in the tensor product space of the interagents.
All interaction is knowledge, ie communication. The hard sphere approximation is a very abstract implementation of this axiom in terms of force and momentum.
Product space:
The mind is the space of everything the body can do. So each layer of the transfinite network is the mind of the layer below it, constructing itself by exhausting all the possibilities of it body, including its brain (if any).
Kreyszig page 127: 'In fact inner product spaces are probably the most natural generalization of Euclidian space, and the reader should not the great harmony and beauty of the concepts and proofs in this field. The whole theory was initiated by the work of D Hilbert (1912) on integral equations.' Hilbert Kreyszig
Kreysig page 128: 'an inner product space on [a vector space] X is a mapping
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of X x X into the scalar field K of X.' A computable mapping, ie one that can be represented by an algorithm very much smaller than itself.
'IP1 < x + y, z> = <x, z,> + <y, z>
IP2 <ax, y> = a<x, y>
IP3 <x, y> = <y, x> bar
IP4 <x, x> >= 0; <x, x> = 0 <==> x = 0
We assume that the mathematical rapport between Euclidean and Hilbert space is reflected in reality, Euclidean space being in fact a four-dimensional subspace or elemental space of the infinite dimensional Hilbert spaces of quantum mechanics. This, we propose, explains the ubiquity of four-space and opens a window towards quantum gravity.
Hilbert (1912)
Kreyszig p 173: 'Theorem (Isomorphism and Hilbert dimension) Two Hilbert spaces H and H', both real or both complex are isomorphic, if and only if they have the same Hilbert dimension.
Wednesday 30 January 2008
All communication is communication of value, ie trade, although the value may be at different peer levels, ie trading body for mind or vice versa.
A partially ordered set contains elements a and b for which neither a =< b or b =< a holds.
[page 136]
Totally ordered set or chain has no incomparable elements.
Mathematics expresses ideas of creation in extension and completion theorems of various sots. The outer product space is in a logical sense the completion of the multiplicand spaces. The operators in this space are represented by the set of indices which permute the vectors of the space - here we are trying to see how the tensor product looks in the symmetric Universe (permutation space)
Miss Marple: 'The point is', she said. 'that one must provide an explanation for everything. Everything has got to be explained away satisfactorily. If you have a theory that fits every fact, well it must be the right one. [?] But its extremely difficult. Murder Vicarage page 194 Fontana Collins 1986. Christie
Cardinal measure applies to the probability (frequency) of events which are themselves logically specified.
Quantum mechanics computes the probabilities of various messages by simulating the environment in which the messages are relevant.
Linear operators (Quantum mechanics) '. . . observable physical quantities are represented by self-adjoint operators such as energy or momentum, whereas transformative processes are represented by unitary linear operators such as rotations or the progression of time. Wiki Bra-ket notation. Bra-Ket notation - Wikipedia
How does the logic determine the frequency of
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events? How does quantum mechanics encode this logic (as a set or ordered sets) and compute the frequency of certain outcomes? Basically by the inner product, a form of square integration. So what does all this mathematical palaver mean?
Thursday 31 January 2008
Entanglement: communication introduces correlation as we would expect, and as predicted by the quantum formalism. How this works, however, seems to be something of a mystery, as we see from the EPR (thought) experiment. Bell
Nielsen and Chuang: 'Imagine that we are attempting to send quantum information from Alice to Bob through a noisy quantum channel. If that channel has zero capacity for quantum information, then it is impossible to reliably send any information from Alice to Bob. Imagine instead hat we have two copies of the channel, operating in synchrony. [?] Intuitively, it is clear (and can be rigorously justified) that such a channel also has zero capacity to send quantum information. However if we instead reverse the direction of one of the channels . . . it turn out that sometimes we can obtain a non-zero capacity for transmission of information from Alice to Bob. page 9. Nielsen and Chuang
True to our mercenary foundation, we consider entanglement to be a resource as much as a source of wonder. Nielsen and Chuang page 11.
'. . . a fundamental resource comparable to energy, information, entropy or any other fundamental resource. page 12
[page 138]
Construction = moving things into meaningful relationships to one another. One might say that since the creation of space all processes, (physical, chemical, industrial) are basically a matter of selecting pieces and juxtaposing them in a certain way. The whole o science comes down to learning to predict what will happen when various entities are placed close to one another. Since space is endowed with potential, this juxtaposition process is associated with the transfer of energy.
If we take it as axiomatic that nothing happens without communication (since subjectively, a happening is a set of messages, the experiences of a rock concert, a car accident, childbirth or any of the myriad interactions we have with ourselves and our environment every day).
So when we put things further apart (separate them adiabatically) nothing happens between them because there is no communication. As we bring them closer the role of communication increases, becoming attractive or repulsive as a function of distance and contact.
So we think of spacetime (represented by relativity) as a sort of operating system whose basic metric of distance is rate of communication, normalized in some way. This is all that gravitation sees, concentrations of energy.
Quantum mechanics may be used both to define structures given a certain potential structure and to predict the probability
[page 139]
that a particular structure will be excited, that is executed. Thinking abstractly, we hope that every physical structure can be modelled with a Turing machine, that is an algorithm,. Physical excitement corresponds to the execution of an algorithm in the universal network, which we take to be equivalent to an act of communication, sine the only way we know of the existence of an algorithm is by the transformations it works, eg changing the orbit of an electron and absorbing or emitting a photon.
Physicists try to encode these transformation in mathematical equations which can be executed by classical computers, and predict what will happen when we bring various elements of the observable Universe into proximity with one another. This is the procedure in particle accelerators, where we use energy and momentum to bring particles close together to see what they do. Energy is simply rate of communication and momentum is the consequence of giving energy to a system.
Mathematically an entangled state is one that cannot be resolved into a product of independent states. We can measure the degree of entanglement by the degree to which the entangled states lose their independence and so become correlated.
Can't quote this too often: 'On its own quantum mechanics doesn't tell you what laws a physical system must obey, but it does provide a mathematical and conceptual framework for the development of such laws.' An operating system both for the Universe itself and for those attempting to understand the Universe. Quantum mechanics is the kernel of a recursive
[page 140]
Universe. 'In the next few sections we give a complete description of the basic postulates of quantum mechanics. These postulates provide a connection between the physical world and the mathematical formalism of quantum mechanics. Nielsen and Chuang
P1 The system is completely described by its [complex] state vector, which is a unit vector in the system's state space. Nielsen and Chuang, page 80
'The simplest quantum mechanical system is the qubit'. (?) We imagine 0 and 1 dimensional Hilbert spaces as ingredients for the construction of two dimensional systems.
We can build bigger and bigger Hilbert spaces by adding a dimension to an existing space or by multiplying spaces to get a tensor product.
P2 'Unitary evolution'
(of amplitudes). We see a system evolving with various transitions becoming more or less probable as time goes by. The unitary evolution describes the evolution of the probabilities of communication between various participants in the meeting being described.
A meeting: physical closeness to encourage and control communication.
[Pauli ] X = not = bit flip
Z = phase flip |1> --> -|1>
Nielsen & Chuang page 82: ', , , at least in principle every open system
[page 141]
can be described as part of a larger closed system (the Universe) which is undergoing unitary evolution.
Schrödinger 's equation
H is a fixed Hermitean operator known as the Hamiltonian.
'If we know te Hamiltonian of a system then . . . we understand its dynamics completely, at least in principle. H is an energy matrix = traffic matrix = measure of space. How do we transform part of the global H into spacetime?
Domestic, industrial, security arts.
METRIC = ENERGY FLOW Close = High, far = low.
VELOCITY OF COMMUNICATION
Forces change the velocity of communication, acting between each pair of nodes.
The rate of change of [the wave function psi] is energy .
Page 82: 'In general figuring out the Hamiltonian needed to describe a particular system is a very difficult problem . . . '
How do we write the Hamiltonians of Turing machines?
Stationary state in time = constant energy, the energy
[page 142]
of a state, ie the potential of a state. Stationary state in space = constant momentum (no force)
STATE = POTENTIAL which guides action.
Nielsen page 82: Suppose a single qubit has the Hamiltonian H = h bar omega X
A matrix is a data structure defining a transformation. If the world is to make these transformations we would expect it to have enough memory to store the relevant vectors and matrices. Or are these structures dynamically controlled by consistency so that the follow a majestic dance with no score?
Entanglement arises because the Universe was once a one state system and we are all descended from that system through thirteen billion years of fast and detailed evolution
Force couples to energy through distance. Yet energy = 1/distance, so force is constant, energy x 1/energy.
The force constant is the consequence of one quantum of action = force x distance x time.
Momentum = 1/distance. One day (I hope) it will all suddenly come into focus like those 3D stereoscopic diagrams of proteins in Nature.
Friday 1 February 2008
Saturday 2 February 2008
Quantum dreaming. We have got ourselves an energy operator acting on two states to swap them. Both the existence of the states and of the operator imply some sort of memory, but the best we can say about the states is that one is not the other and their amplitudes are normalized [ie the Universe always exists in one state or another. This holds no matter what the cardinal number of the set of states]. We want to connect this little system to Minkowski space and so grow the quantum Universe in Hilbert space and the spacetime Universe in parallel. We might find a clue to this in the metrics: the Hilbert space and Minkowski space inner products. Insofar as the quantum operator is in contact with the qubits, the interval between them is zero. Insofar as something exists, the probability flow between them is normalized to 1. What does this mean in network terms?
Because space and gravitation is of itself blind to meaning, all we re interested in is the measure of completed process., the quantum of action and the rate of action, measured by energy. We want conserved flows, we acknowledge that spacetime has some structure, 3+ 1 dimensions. The spatial dimensions are, in some way complete, the time dimension is incomplete to the future but lays down space in the past as this moving pen writes and having writ moves on. We have three degrees of freedom and one degree of compulsory motion whose combined metric is invariant.
How does the mathematical use of a complex number for time connect, if at all, to the complexity of Hilbert space? The principle that the Universe started of simple suggests that these problems are soluble only if we could see them from the
[page 144]
right point of view!
In the absence of meaning we may consider networks as transmitters of mass / energy rather than entropy. Given that mass comes in discrete units (particles) entropy is proportional to log(mass). When we introduce meaning, (order) entropy becomes proportional to mass.
Mass flows accompanying meaning flows as we wee with the global Carbon, Nitrogen, Phosphorus etc cycling which occurs as a result of solar energy driving the immensely complex and meaningful processes of the biosphere. Kump, Heimann, Gruber Can we take spacetime as the machine (isomorphic to a Turing machine) which processes the meaning flows that we recognize in Hilbert space, action occurring when the interval between agents is [locally] zero? We have one energy driven program counter and three degrees of motional freedom.
The logical structure of linear algebra is built into quantum mechanics (and there is not a lot more to it) so we imagine a set of linear algebra software would form the basic structure of the Universe of written to be scale invariant, that is recursive.
All the features of linear algebra are connected by proofs, which can be executed by Turing machines.
SPACE = rewritable memory, but serial rather than random access in that to get from q to b we have to pass through all the intermediate points (like a tape rather than a disk). However, if memory is organized in 3D spacetime a and b may be a
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be a long way apart if we follow a sequence, but close if we take shortcut across the space. We se this with a 2D disk where we may either follow a spiral track from a to b or do a head seek across a load of tracks. In general we would like to see a close analogy between the [topological] structure of computing machinery in real 3D space and the functioning of the Universe is 3D space.
Nodes are propositions and edges are proofs, ie Turing machines., Nodes are space, edges are operations. Nodes are characterized by momentum, edges by energy. This book as a whole is a lot of nodes (recorded in writing) joined by the thought processes that have led me from one to the other. The repletion (and banality) of these notes reflects the fact that I am continually moving about within a relatively limited set of nodes, finding new tracks from one to the other. This process is formally best reflected in the literature of mathematics which is a large network of propositions joined by proofs, which we might call linked points in Turing space. So the symmetric Universe is supposed to represent all possible propositions and the Turing machines in the symmetric network all possible paths between propositions. In 3-Space, to walk is to execute the recursive process of getting from a to b. Each step on this walk is a quantum of action (which may be any number of Planck quanta).
A Turing machine is a propagator.
Veltman page 15: 'In a quantum mechanical description, the state of a free spinless particles is completely specified by its
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3 momentum. . . . Location in space and time is completely unknown.' Veltman ie does not enter into the state of the particle, or perhaps is common to the state of all particles, which share the symmetry of existing in spacetime which is a physical layer of the Universe providing services to all particle using it.
'A particle with well defined momentum (and energy) is described by a plane wave .'(ie a tesselation of spacetime). 'This is the true content of the Planck - Einstein relation E = h bar nu = h bar c / lambda sub E (= wavelength in time = wave 'period' ).
[diagram]
Veltman page 21: Translation in space is equivalent to a phase factor in Hilbert space. This change of phase changes the superpositions of which the particle is part, changing in turn the probabilities of various events.
21: 'To every Lorentz transformation, or more generally a Poincare transformation, corresponds a transformation in Hilbert space. '
Because a Lorentz transformation change the momentum of the particle (its complete specification)? But not in the particle's own rest frame, where it has its true personality., How does the joint Hilbert space of two particles in relative motion work? The have different phases.
'If the correspondence is unique, then the product of two Lorentz transformations L1 and L2 must correspond to the product of the two corresponding transformations X1 and X2 in Hilbert space.
One is trying to study a very large elephant one hair at a time.
Pan, d'Espagnat etc quantum mechanics is nonlocal - it does not see differences in location, ie location does not enter into the description of a quantum mechanical state any more than it enters (to a first approximation) to the description of my state. Pan, d'Espagnat Wherever I go, there I am.Pan, d'Espagnat
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