Notes
[Notebook: DB 62 Interpretation]
[Sunday 13 January 2008 - Saturday 19 January 2008]
[page 105]
Sunday 13 January 2008
Anatole Kaletsky Weekend Australian 12-13/1/2008 Business page 29:
'It seems that a market economy is rather like an aircraft: it needs a minimum speed to keep flying and if it moves any slower it stalls.' Inherently dynamic.
We consider that the Universe is inherently dynamic [pure act], and like a juggler or an aircraft can only pull things off if the timing (which is closely related to speed) is right. We have to arrange things so that there is no interval between the end of one action and the beginning of another, like the arrival of a ball at a certain spacetime point and the arrival of a bat (or a hand) at the same point. As in so many things, timing is everything. Too late or too early reduced the probability of success. In the simple world of quantum physics timing (phase) and probability of success are linked by the Born equation pk = |< sk || psi >| 2. Interpretation of quantum mechanics - Wikipedia
The Cantor Universe is in effect a digital Hilbert space. This space, in the continuous paradigm, assumes Cantor's hypothesis, that ℵ1 is the cardinal of the continuum and uses all the technology of limits and analysis in general to arrive at the definition of Hilbert space. A fundamental assumption is the definition of a metric which allows the definition of convergent
[page 106]
series etc.
We hold the digital Hilbert space together not with proximity but with logical connections estabished by Turing machines. We assume that these connections are observables and we see a Turing machine as the coupling between a state and an observer, or more generally, between two states,
We can identify two methods of making larger sets out of smaller ones, by subsets (a standard modern proof) and by permutations, akin to Cantor;s diagonal argument. Gellert We are inclined to identify the subset method with bosons, each subset representing any number of bosons, eg photons with the same frequency [state] and permutations representing fermions, particles each of which has a distinct state.
No cloning is built into the symmetric Universe since if two symbols are identical there is only one of them. Symmetric network
MEANING DIFFERENTIATES
Logical connection = containment, as symbolized by the connectives a contains b or b is an a.
A digital Hilbert space needs a metric (computational distance) and a norm (a complete (halted)) computation.
Monday 14 January 2008
The normed length of computation is ℵ0 ?The number of different computations is ℵ0.
The execution of a Turing machine is a proof that the terminal condition follows from the starting condition given the algorithm represented by the machine.
Tuesday 15 January 2008
Computational distance is measured by proper time. The advent of space allows for parallel processing. The growth of space (expansion of the Universe) points to some advantage of parallel over serial processing. Or perhaps we should look at space as memory, in which case we see some selective advantage of memory over processing power. We may think of memory as trivial processing in the sense that a memory simply copies itself through time going refresh, refresh, refresh, which is equivalent to looking up a one item lookup table. Maybe this is what a photon does as it propagates through space [or any particle].
If memory is very cheap, is the lookup table is the fastest way to get results? Provided that the table is well organized we can select one of n objects in log n time, working through the log n letters in the position significant addressing system.
On the digital analogue of Hilbert space.
phys04Quantum mechanics
phys05 Hilbert spaces
[page 108]
Wednesday 16 January 2008
How does the inner product work in digital Hilbert space, if at all? In ordinary Hilbert space it provides a norm, so that the length of a vector is its inner product with itself. Each vector represents a function which is an ordered and complex entity which, since it is a point in a large (high entropy) space is capable of carrying much information and so is in effect a message. The inner product (effectively an integration) destroys all this information and yields a single (complex) number which represents the 'closeness' of two vectors. In a normed space the closeness ranges from 0 (orthogonal) to 1 (parallel). In quantum mechanics this is interpreted as the overlap between two vectors and gives the probability that a measurement using one of the vectors as a ruler will give the other.
How do we interpret this in digital space? First, both vectors (data packets) must occupy the same Hilbert space, that is be sets of the same cardinal number of symbols. Second we must find a meaning for the pairwise multiplication of elements of the packets. If the symbols are [binary] digital this is equivalent to a pairwise and. Finally, when we add up the result, it tells us how many places are both 1's. Not very informative, and completely overlooking the phase relationships between the elements of vectors and the use of complex conjugation z.zbar in the individual products.
The inner product gives us the probability that two nodes
[page 109]
in a network, represented by two vectors in a Hilbert space are communicating, that is in the same state (at least insofar as they are described by the vectors in question?).
Do we need to change the formalism at all, or simply the interpretation of the formalism? In the network, things communicate when they are in phase (as with driven oscillators) and so propagate themselves and increase their amplitudes. On the contrary, when they are out of phase they tend to communicate destructively and decrease their amplitude [excitation vs inhibition].
Is there any value in getting caught up in all this physical detail? I seem to be motivated by my own desire to understand quantum mechanics and quantum field theory through a network paradigm which is based on logical continuity rather than mere closeness, thus opening the way to understanding the Universe as an intelligent information processing system.
Continuity is a symmetry that works because things remain the same as we go along. So the ruled lines in this book which guide my writing are continuous but the writing itself i a discontinuous string of symbols. The string has a certain probabilistic structure which is determined by my language, but the only real explanation of this structure lies in understanding the meaning of what I write. This understanding explains why I used the world 'write' rather than 'wrinkle' in the previous sentence. So we see quantum mechanics as a way of computing the traffic in various channels but we need more detailed theories like quantum field theory, chemistry and biology to tell us what the traffic actually means.
[page 110]
So something of an impasse. But this is where the complexity invariant heuristic comes in, since in this picture I am a partile like any other and so I might gain insight into particles bigger and smaller than myself by considering my own thoughts and feelings.
The basic thought and feeling is that I enjoy doing this even though it sometimes seems hopeless (the post Christmas let down? - this is the first day for a month when I have not had five or ten house guests), and keep going. In a way my deepest wish is to explain this 'lust for life' that keeps me (and the rest of the Universe) going. Stone From an evolutionary point of view lust is a tautological property since the ones that exist are the in effect the ones that want to exist. Some have a settled lifestyle like a permanent job in the bank and others are inclined to set out on their camels to cross trackless deserts hoping to find Shangri-La on the other side. Shangri-La - Wikipedia The payoff in both cases is survival, which can in a sense be normalized to a probability of continued existence of 1. In the first case this probability is achieved by a high probability of a small reward (a weekly pay packet) or a low probability of a large reward, like finding gold. The shares on the stock market reflect this spectrum, the blue chips paying a regular dividend every year while the dreadfuls continue to raise money on the strength of the treasure that is just out of their reach. I am in the second category, contemplating the small probability that I will reap the immense reward of global scientific theological unity.
[page 111]
A fully free Universe would fill the Cantor Universe in the sense that every point in the Cantor Universe (at a given peer level) would be equiprobable. Instead as quantum mechanics shows, the Universe is constrained to exist in Hilbert spaces of varying cardinality. Whence does this constraint arise? Our general principle (built into wave mechanics by the requirement that phase = 0 at boundaries) is that constraints breed structure. So we ask what are the constraints that confine function space to Hilbert space: metric, inner product, normalization, eigenvalue equation: what do these mean?
Thursday 17 January 2008
Why is the Cantor Universe cut down to Hilbert space? The answer (at the moment) seems to lie in the difference between formal and dynamic systems. The Cantor Universe is a formal structure (a physicist would say a completely unexcited field, without energy) which is considered to be eternal (tota simul - all parts existing at once). The physical world, on the other hand, is dynamic and energy is conserved and limited so only a subset of all possible states can be excited at a given moment. On the other hand, the ergodic hypothesis suggests that given time it will explore all points in its phase space. Ergodic hypothesis - Wikipedia, Phase space - Wikipedia
The probability that the Universe (or god) exists is 1. We may imagine any isolated system as a Universe and assign to it
[page 112]
also a probability of one. We may now [allow] such a Universe to split into 2, n, ℵ0 or aleph(n) parts with probabilities (respectively) of 1/2, 1/n. 1/ℵ0 or 1/aleph(n), so maintaining normalization. Given such 2, n, etc state systems, we can then examine them quantum mechanically, assigning a Hamiltonian (and energy) to their interactions.
Energy = communication rate in acts per second.
When we split a Universe (isolated system) in two we create equal and opposite potential and kinetic energies so that the total energy remains zero but the two halves of the split Universe communicate with one another and attract one another. In the case of the physical Universe, the quanta of communication are all the particles in the Universe, the attraction is gravitational and the rest mass of the communication quanta is balanced by the negative potential energy of the expanding Universe (I wish). Feynman
Friday 18 January 2008
On the relationship between Hilbert space and the symmetric Universe. The symmetric Universe as we have described it is a transfinite phase space for the Universe. We have filled it with Turing machines to connect points in the space by full duplex communication which respects the no-cloning theorem by maintaining the formal distinction of every point in the Universe. No cloning theorem - Wikipedia Two identical particles are formally the same point and therefore one, not two.
[page 113]
The wave functions of quantum mechanics inhabit Hilbert spaces and we can imagine the wave function of the Universe occupying the Hilbert space of the Universe. Everett III We can construct the Hilbert space of the Universe by tensor products of Hilbert spaces of countably infinite dimensions in a way exactly analogous to our construction of the transfinite symmetric Universe, so that at any given peer layer the Hilbert space of the Universe and the symmetric Universe have the same cardinal so that we can in principle construct a one to one mapping between them. Transfinite network
Where they differ is in their metric. We use the transfinite cardinals to count the number of different permutations or structures in the symmetric Universe and the number of dimensions in the Hilbert space of the Universe, but the Hilbert space is normalized in a way that enables us to calculate the probability of a given state.
To simplify this discussion we can invoke Cantor's principle of finitism and begin our construction with the two state system known in quantum information theory as a qubit. Hallett, Nielsen & Chuang, Qubit - Wikipedia
So after years of bashing my head on impervious formalism I can see clearly now how quantum mechanics serves to compute the traffic in a network, given a certain set of base state that describe the network. This open up a network way to introduce special and general relativity and arrive at quantum field theory by an alternate route.
Why are there so many identical particles in the Universe
[page 114]
when the no cloning theorem tells us that no two states can be the same (given that they are all descended from some ancient initial state we call the initial singularity)? This is because (for instance) the electron wave function is not a full description of an electron , but we must also take into account the 'meaning' of the electron (which is invisible to quantum mechanics and traffic analysis) [and] which is (in the simplest picture) encoded in its spacetime position; so an electron at point x in a protein (which is at point X in my body) has a different meaning from an electron at point y in an identical protein at point Y in my body, und so weiter.
-->