Notes
[Sunday 20 July 2008 - Saturday 26 July 2008]
[Notebook: DB 64 Gravitation]
Sunday 20 July 2008
Monday 21 July 2008
[page 70]
Tuesday 22 July 2008
In our quantum mechanical world anything can happen so miracles are not so much miraculous as natural.
Pagans, cynics and practical people who have to deal with the world as it is, not some prophetic fantasy.
The Pope has joined the apologizers, but there is yet no acknowledgement that the evil done . . .
[page 71]
may have been made more probable by Catholic Sexual policy/hypothesis.
One has to find some principles of human behaviour to incorporate into a religion. Physical theology takes the view that if we take care of the physics, the rest will take care of itself, which [is] in effect how the world is built. Everything has to obey the laws of physics but what it makes from this constraint is not constrained, any more than an alphabet can constrain the words constructed with it. Any definite alphabets it adequate, when properly ordered, to express any reality.
Wednesday 23 July 2008
What is the difference between a quantum mechanical superposition and an ordered sequence? When we interrogate repeated copies of a given superposition, we get a random sequence of results, each corresponding to an eigenvalue of the operator we use to observe the superposition. If we observe with a different operator, we see a diaphaneity set of eigenvalues.
On the meaning of quantum mechanics.
Quantum mechanics predicts traffic in different edges of a network. Given any node (measurement operator) quantum mechanics enables us to compute the probability that this node will communicate with each of the other nodes in the network. This is equivalent to predicting the traffic on the edge joining the observing node to each of the others.
[page 72]
Thursday 24 July 2008
Rain after a dry period. Both meteorological and mystical. Mystics too seem to get upset when they cannot get it up, their prayer goes dry, their visions lose resolution.
Agatha my epistemological queen: 'She wished with all her heart that she had met the dead man even if only once. It was so hard to get an idea of people you had never seen. You had to rely on other people's judgment, and Emily had never acknowledged that any other person's judgment was superior to her own. Other people's impressions were no good to you. They might be just as true as yours, but you couldn't act on them. You couldn't as it were, use another person's angle of attack.'
Point of view = angle of attack = measurement operator.
Impotence: cannot establish the physical conditions for the desired outcome, eg insufficient troops, weapons, or tactics to win the battle, not enough strength to carry the weight.
Apply the model to colonization: destructive communication. New software (English culture) taking over the hardware (the land) from Aboriginal culture. We can say in hindsight that it was as inevitable as one species driving another to extinction by occupying its niche.
Friday 25 July 2008
[page 73]
Each basis (set of basis states) in a given Hilbert space (network) represents the point of view of one node in the network? Transforming to a new basis thus becomes equivalent to moving to a new point of view. The search for understanding is the search for bases (points of view) which make things look simple. This is what I am trying to do here. One is heading for trouble, however, if one simplifies simply by refusing to look at certain aspects of the system. So the whole relationship between the British government and the original people of Australia was both simplified and falsified by the fiction that Australia was an empty land owned by the Crown, which Crown had complete dominion over the land and its people. This fiction, since it contradicted reality, led to the war between Britain and the Aborigines that ultimately destroyed their way of life. Most of the actual killing in the war appears to have been the work of the graziers and farmers to whom the Crown sold, leased and licensed the land which it fictionally possessed. The Old Testament describes a similar situation, where God gave the promised land to the Israelites leading to fighting and killing which continue in a pretty direct line to this day, approximately 3000 years later. . . .
A two state system seen as a network. The abstract qubit |q > is represented in the basis |0 >, |1 > by the vector expression |q > = a |0 > + b |1>, where the amplitudes a and b are complex numbers constrained by the equation a 2 + b2 = 1, normalization. Considered as an isolated system, |q > evolves unitarily at a frequency determined by its energy. This evolution is represented by an energy operator or Hamiltonian. It is customary to represent unitary operators by U and Hamiltonians
[page 74]
by H. In physics the two symbols are equivalent because (by assumption and experiment) the Hamiltonians of all physically realized quantum states are unitary. Mathematically this is equivalent to requiring that all Hamiltonians have real diagonals and elements reflected in the diagonal are complex conjugates of one another, ie Hermitean matrices. By considering infinitesimal changes in a quantum system we arrive at differential equations for their evolution whose solutions are generally superpositions of complex exponentials, that is periodic or wave functions. The frequency of each element of the superposition is determined by its energy. So we represent the Hamiltonian on a generic qubit by the generic Hamiltonian
Since there is but one abstract Hilbert space for each dimension (for the qubit = 2), the abstract Hamiltonian applies to all 2 state systems.
So much for an isolated qubit. We now explore what happens when two isolated systems are brought into contact, that is communicate, with one another.
In its current state, quantum theory holds that there are two general classes of particles (or events in the world known as fermions and bosons. In general fermions . . . (eg electrons) communicate with one anther by the exchange of bosons (photons) and bosons by the exchange of fermions (?).
We represent the interaction of two to state systems by a four state system whose Hilbert space is the tensor product of
[page 75]
of the spaces of the interacting qubits. States in this space may be represented by a vector |qq > = a |00 > + b |01 > + c |10 > + d |11 >. Considered as an isolated system, |qq > evolves in a continuous unitary manner. Within this system, however, we have a discontinuous event often called the 'collapse of the wavefunction'. This 'collapse' is analogous to the roll of a die, where the superposition of six possible outcomes of the roll is 'collapsed' into the face showing when the die comes to rest. In both cases we may consider the outcome as a snapshot of a stationary point in the dynamic system. The dimension of the Hilbert space of the quantum system is equal to the number of distinct stationary points observable in the dynamics. So, although the coefficients a and b in the expression |q > = a |0 > + b |1> are continuous complex numbers, observations of |q > always yield either |0 > or |1 > with probabilities determined by the value of a (or b) (axiom of observation)
We might say that the superposition describes a system (like a rolling die) which moves too fast for the eye (observer) to see, and a stationary state as one that persists for long enough to be observed. In the limit where a state is completely stationary its rate of change (and therefore energy) is zero and it may be considered as a fixed point in the dynamics. Since dynamics maps systems onto themselves, Brouwer's fixed point theorem guarantees at least one fixed point (and maybe more in higher dimensions).
In classical quantum mechanics the collapse of the wave function is a quantum event. The mathematical formalism defines both the for and the frequency of such events. So in a two state system, the form is either p or not-p and the relative frequency of the events is determined by some constraint on these events. If there is no constraint (as in a fair coin) the outcomes p[and not-p are equiprobable.
The collapse of the wave function is represented in the quantum formalism by the eigenvalue equation U |q > = u |q > where U is a matrix representing a unitary operator and u is a scalar. |q > is then said to be an eigenvector of U and u is the corresponding value. In words, if we observe a qubit |q > with an operator U, we only see outcomes whose form is represented by eigenvectors of U, and whose frequency depends upon the corresponding eigenvalue u. Mathematically the set of eigenvectors of U form an orthogonal basis in which the matrix representing U is diagonal.
From an abstract point of view, the interaction of two qubits is perfectly symmetrical. In physics, however, one of the qubits (the laboratory qubit) is reporting to a physicist and the other (the unknown qubit) is what the physicist wishes to measure. Clearly the choice of the laboratory qubit is a factor in the results of actual measurements on the unknown qubit. This is completely consistent with the general idea [that] what we see depends on how we look. Quantum mechanical systems, like people, are essentially blind to things they are not looking for. Nevertheless, the number of possible outcomes is limited by the complexity of the system. Classical continuous physics envisages a continuum of measurements in a continuous system. Quantum mechanics can be consistently expanded to infinite dimensional Hilbert space to allow such an infinity of outcomes, but practical laboratory experiments are easiest if an observing system is used that yields a binary outcome. Most of physics comes down to
[page 77]
measuring the presence or absence of certain outcomes (stationary states) at certain pints in spacetime.
Neither the laboratory nor the mystery qubits are in reality isolated, the former being connected to a physicist and the latter to the rest of the world, but here we consider them as parts of a system isolated from the rest of the world, a network of two. Since the quantum no-cloning theorem suggests that all quantum interactions occur pairwise, a network of two is sufficient to construct a network of any size through time division multiplexing, each element of the constructed network communicating with different elements at different times. Each communication event is thus assigned a lifetime (birth + death) and a location in the Universe of parallel and sequential events (though we can only detect parallelism by using the special theory of relativity to transform sequences of events back to their parallel source.
Zurek Zurek
The physicist introduces himself into the global network, as we all do. The absorption of a photon may simple change the state of an atom, but if that atom is coupled to a larger system like the eye of a mammal, all sorts of other things may happen as a consequence.
Positive feedback leads to events, negative to non-events.
