Notes
[Notebook: DB 62 Interpretation]
[Sunday 30 December 2007 - Saturday 4 January 2008]
[page 88]
Sunday 30 December 2007
If the operator is passed a time value, the frequency of its response is measured in energy. If a space value, the return is in momentum, if a number, the return is in action.
When I first became acquainted with computers (6809 chip, 1970's) the most difficult abstraction for me to comprehend was the distinction between logical and physical addressing. Motorola 6809 - Wikipedia Quantum information theory brings these two ideas together in that the physics is the logic, ie the logic is embodied in the physics. The conceptual problem now (closely related to the 'collapse of the wave function') is to map the continuous models of physics onto the discrete models of logic.
Although the not operator is a discrete function, it can be made to look continuous in a more complex system and represented by a wavelike Hamiltonian because it may be called at any moment and the random superposition of these discrete calls may look like a continuous wave. This is analogous to the idea that although each individual Turing machine is a deterministic entity a network of Turing machines becomes indeterministic because they may interrupt one another at random moments in their individual processes. This ideas is expressed more satisfactorily by saying Turing machines in a network may be concatenated in random order, since then we do not have to interrupt a
[page 89]
deterministic process, a rather self-contradictory idea. The superposition of a set of not operators called at different moments is what looks like a continuous wave.
We represent reversible Turing machines by matrices and the deterministic processes of quantum theory by 'matrix mechanics'.
Another analogy by which the appearance of continuity may arise is the decoupling between meaning and symbol. So any program may call any subroutine and give it any meaning it likes. So it is with natural languages, naming and mapping in general, so that the same object may have an infinity of different names in different languages and the same linguistic symbol may be mapped to an infinity of different objects.
The binary world is modelled (at its most abstract) by connection and disconnection.
Quantum mechanics explains the structure of the world by showing what can and cannot happen. All events are digitized and things can only happen in a digital way, eg de Broglie's hypotheses explains the stationary states ('packets') of the atom by the requirement that integral numbers of waves fit into certain spaces. So confinement digitizes.
Monday 31 December 2007
Tuesday 1 January 2007
Ehrenfest's adiabatic hypothesis: If a system is affected in a reversible adiabatic way, allowed motions are transformed into allowed motions.' Bohr: 'Principle of mechanical transform ability'.
[page 90]
Van der Waerden p 4-5. van der Waerden
Wednesday 2 January 2007
Thursday 3 January 2007
Friday 4 January 2007
Hopefully, the point of view developed by Zurek (2007) is the final plank necessary to develop the logical (digital ) view of the Universe proposed in these pages. Zurek
The trouble for physics lies in the first postulate of quantum theory, that the quantum state of a system is represented by a vector in its Hilbert space. The trouble arises from the continuity assumption embodied in this postulate, since continuity is part of the definition of Hilbert space (von Neumann). von Neumann
Further postulates: second unitary evolution
Third immediate repetition of a measurement using the same
measurement operator yield the same outcome.
Zurek page 1: 'These first three postulates indicate no bias - they treat every state [every vector] in the Hilbert space of the system on an equal footing.'
That is, they do not partition the message space into 'legitimate' and 'illegitimate' (erroneous) messages as required by Shannon's theory if we are to have error free transmission of messages. Shannon, Khinchin
Zurek: 'By contrast, the last two postulates, (iv) measurement outcomes are restricted to an orthonormal set { |sk|>} of eigenstates of the measured observable
[page 91]
(i.e. measurement does not reveal the state of the system because it limits possible outcomes to the preassigned outcome states) [as we do in error resistant coding] and (v) the probability of finding a given outcome is pk = |<sk || psi >| 2 where | psi > is the [assumed by postulate (i)] preexisting state of the system are at the heart of long standing disputes on the interpretation of quantum theory.'
'The aim of this paper is to point out that already the (symmetric and uncontroversial) postulates (i) - (iii) necessarily imply selection of some preferred set of orthogonal states, that they impose the broken symmetry that is at the heart of postulate (iv) - although they stop short of specifying what this set of outcome states is and obviously cannot result in anything non-unitary (eg the actual collapse).
Here we see logical continuity at work 'defeating' the classical 'argument from continuity' that is inherent in the postulate of unitarity.
Saturday 5 January 2007
How much control is a legitimate and much debated question. Since experience suggests that too much can be as bad as too little, the problem presumably has an optimum.
The Universe is a real time system whose elements must be 'tuned' to certain frequencies for processes to work properly. The fundamental structure of the Universe is the arithmetic of energy (cardinal) reflecting the logic of process (ordinal).
[page 92]
'The atoms of Newton and Boyle are hard little balls, Euler's atoms like musical instruments.' Park, quoted in Sacks Tungsten p 82 n. Sacks
