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Optics to Projection

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In quantum mechanics, the essential difference is that the equations of motion of a particle are replaced by the Schr÷dinger equation for a wave. This Schr÷dinger equation is obtained from a canonical formalism, which cannot be expressed in terms of the fields alone, but which also requires the potentials. Indeed, the potentials play a role, in Schr÷dinger's equation, analogous to that of the index of refraction in optics.

Aharanov, Bohm

optics & mechanics

Least action

The discovery was amazing. Mechanics is the study of moving bodies— cannonballs moving in a parabolic arc, pendulums swinging regularly from side to side, and planets moving in ellipses around the Sun. Optics is about the geometry of light rays, reflection and refraction, rainbows and prisms and telescope lenses. That they were connected was a surprise; that they were the same was unbelievable.

It was also true. And it led directly to the formal setting used today by mathematicians and mathematical physicists, not just in mechanics and optics but in quantum theory too: Hamiltonian systems. Their main feature is that they derive the equatiuons of motion of a mechanical system from a single quantity, the total energy, now called the Hamiltonian of the system. The resulting equations involve not just the position of the parts but how fast they are moving—the momentum of the system. Finally, the equations have the beautiful feature that they do not depend on the choice of coordinates.

In the radical relation thus contemplated by Descartes, in his view of algebraical geometry, the related things are elements of position of a variable point which has for locus a curve or a surface; and the number of these related elements is either two or three. In the relation contemplated by me, in my view of algebraical optics, the related things are, in general, in number, eight: of which, six are elements of position of two variable points of space, considered as visually connected; the seventh is an index of color; and the eighth, which I call the characteristic function,—because I find that in the manner of its dependence on the seven foregoing are involved all the properties of the system,—is the action between the two variable points; the word action being used here, in the same sense as in that known law of vision which has been already mentioned. I have assigned, for the variation of this characteristic function, corresponding to any infinitesimal variations in the positions on which it depends, a fundamental formula; and I consider as reducible to the study of this one characteristic function, by the means of this one fundamental formula, all the problems of mathematical optics...



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The Proposed Generalization

The effectiveness of quantum, as compared to digital algorithms, [...] therefore suggests that spin, as the continuous spectrum of values between zero and one, with the alternative interpretation of a weighting function appropriate to neural nets, should become the Cybernetic Machine Group's principal focus for 1999. For the behaviour of spin relative to a reference spin, as in quantum entanglement, implies quantum parallelism, i.e., a superposition of all weighting possibilities simultaneously, [thus] generalizing the concept of the artificial neural net. It can [therefore] be postulated, that if such quantum neural network models [...] can be devised, (providing an understanding of their actual physics), then the key to new technology, by means of which NP complete problems can be solved, will be to hand.

The Evidence for such a Generalization

Such a generalization of a neural network, is, in principle, as Perus1 has shown, a highly valid concept, since the two formalisms can be set down in identical ways so as to express their properties, except that the neural net formalism concerns real quantities, while the quantum systems formalism concerns complex quantities. Weights, taking values from 0 to 1—the key to understanding traditional neural nets—therefore become complex quantities, expressible through unit vectors (spins) in terms of phase. Wave properties and considerations of phase, could therefore contribute additional structure and understanding, both as to how neural net parallelism works, and to an explanation of the basis of the technology by means of which this can be achieved. One can then ask, do such considerations provide a better explanation of actual neuron dynamics and morphology, and if so, attempt experimental validation advancing biological understanding.


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After all, our very definition of a particle or metastable nuclear state is based on its classification as the carrier of a definite representation of the Poincarť group [...]



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All the fifty years of conscious brooding have brought me no closer to the answer to the question, "What are light quanta?" Of course today every rascal thinks he knows the answer, but he is deluding himself.

The question now is, how does it really work? What machinery is actually producing this thing? Nobody knows any machinery. Nobody can give you a deeper explanation of this phenomenon than I have given [...]


Well, obviously the extra dimensions have to be different somehow because otherwise we would notice them.

Now it may be asked why these hidden variables should have so long remained undetected.


The aspects of things that are most important for us are hidden because of their simplicity and familiarity.


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Plane Map

Mathematics has introduced the name isomorphic representation for the relation which according to Helmholtz exists between objects and their signs. I should like to carry out the precise explanation of this notion between the points of the projective plane and the color qualities [...] the projective plane and the color continuum are isomorphic with one another. Every theorem which is correct in the one system sigma1 is transferred unchanged to the other sigma2. A science can never determine its subject matter except up to an isomorphic representation. The idea of isomorphism indicates the self-understood, insurmountable barrier of knowledge. It follows that toward the "nature" of its objects science maintains complete indifference. This for example what distinguishes the colors from the points of the projective plane one can only know in immediate alive intuition [...] 



[It] became possible to affirm that projective geometry is indeed logically prior to Euclidean geometry and that the latter can be built up as a special case. Both Klein and Arthur Cayley showed that the basic non-Euclidean geometries developed by Lobachevsky and Bolyai and the elliptic non-Euclidean geometry created by Riemann can also be derived as special cases of projective geometry. No wonder that Cayley exclaimed, "Projective geometry is all geometry."

The principle of duality in projective geometry states that we can interchange point and line in a theorem about figures lying in one plane and obtain a meaningful statement. Moreover, the new or dual statement will itself be a theorem—that is, it can be proven. On the basis of what has been presented here we cannot see why this must always be the case for the dual statement. However, it is possible to show by one proof that every rephrasing of a theorem of projective geometry in accordance with the principle of duality must be a theorem. This principle is a remarkable characteristic of projective geometry. It reveals the symmetry in the roles that point and line play in the structure of that geometry.