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Hilbert spaces and the symmetric network

Phase space

We may think of a space as a domain of possibility. A room, for instance, is a space where we may arrange ourselves and our goods in a practically infinite number of ways. Newton founded modern physics by postulating an infinite Euclidian space inhabited by all the particles of the universe. His system of the world showed how the movements of these particles are constrained by the laws which now bear his name. Newton.

The development of mathematical spaces is limited only by imagination and logical consistency. In addition to the 'real' spaces used by Newton and Einstein, we can construct abstract spaces ('data structures') whose value lies in helping us to visualize complex systems. The first such 'phase spaces' were used to model the dynamics of large numbers of particles obeying Newtonian physics. Phase space - Wikipedia

The state of motion each particle in ordinary space can be specified by six numbers, three specifying its position and three specifying its momentum relative to a stationary set of Cartesian axes. The state of motion of single particle can thus be represented by a point in a six dimensional phase space. We can represent the joint state of two particles in twelve dimensions, and of n particles in 6n dimensions.

The evolution of the whole ensemble of n particles then become the path of the representative point through the phase space. The motion of this point may be constrained by functions representing the conservation of energy, momentum and other parameters. Given time, a system may visit all the points in its phase space or be restricted to a certain subsets of its phase space

Function space

A function is a mapping from some set of points (its domain) into another set (its range). Function (mathematics) - Wikipedia A typical function may be represented by a series which is a map from the domain of natural numbers (0, 1, 2, 3 . . .) to some range where we find the value of the function corresponding to each natural number.

Any function can be represented by a point or vector in a phase space with the same number of dimensions as the cardinal of its domain. Function space - Wikipedia Since there is a countable infinity of natural numbers (cardinal aleph(0)), an infinite series may be represented in its entirety by one point in a function space of countably infinite dimension.

From an abstract point of view, there is one function space for every finite, infinite or transfinite domain. All we require of the objects in these domains and their corresponding ranges is that they be uniquely identifiable so that the functional mapping can be specified unequivocally.

Abstract function spaces so defined are identical to the mathematical universe described by Georg Cantor. Cantor, Reid, Dauben

Hilbert space

The symmetric universe as we have described it serves as a transfinite phase space for the universe. Immensity, Transfinite network We have filled it with Turing machines (which are themselves points in this phase space) to communicate between points in the space. The whole structure respects the quantum no cloning theorem by maintaining the formal distinction of every point in the universe. No cloning theorem - Wikipedia In this model, two completely identical particles are formally the same point and therefore one, not two.

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, Quantum mechanics

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. Tensor product - Wikipedia The number of dimensions in such tensor products grow exponentially just like the transfinite cardinal numbers. At any given peer layer the Hilbert space of the universe and the symmetric universe have the same cardinal so that we may establish a one to one correspondence between them.

Where they differ is in their metrics. 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. Hilbert spaces have another metric defined by the 'inner product' of vectors in the space, which can be used to predict the frequency of quantum events. Inner product space - Wikipedia

The details of this calculation derive from postulate (6) of quantum mechanics:

(the probability of finding a given outcome s_o is p_i = |<s_o || q >|^2, where |q > is the preexisting state of the system.

Since the probability of some outcome from an observation must be 1, we require that [sigma]_k p_i = [sigma]_k |<s_o || q >|^2, = 1. This equation is independent of the complexity of the system, measured by k, the number of dimensions in its Hilbert space. From this it follows that vectors in Hilbert space must be normalized to 1.

This means, in general, that the higher the cardinal of the Hilbert space in terms of dimensions, the lower the probability of observing the state corresponding to any given dimension. The exchange of electrons, which can be described adequately in four dimensional Hilbert space, is much more frequent that the exchange of people like myself, whose phase space has a high transfinite dimension.

Since the normalization of an isolated system (universe) must be maintained through all transformations of the system, the operators transforming the state vectors in the Hilbert space must preserve normalization, that is, be unitary, as required by postulate (3) the evolution of isolated quantum systems is unitary. Unitarity (physics) - Wikipedia

Network traffic

In a network, the probability of communication between two nodes measures the rate of communication between them. From this point of view we see that quantum mechanics provides us with a general algorithm for computing the rate of communication between different entities in the universe. It allows us to calculate, for instance, the rates at which to states of an atom communicate by emitting or absorbing photons.

In the physical universe, the rate of communication is measured by quanta of action per unit of time which is equivalent to energy. Action (physics) - Wikipedia The traffic on a link is measured by the energy of the link. Energy - Wikipedia

(revised 23 October 2008)

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