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The Mathematics of Quantization

So our problem becomes purely algebraic and has the following form. Assume that in the Absolute space we have a congruence of lines which is well defined if we know the points P and Q in which the generic line of the congruence meets the Absolute. Therefore the two points satisfy the equation of the Absolute, given in the form of a dot product:

(1)

Here p and q denote the set of four homogeneous coordinates of the respective points. In case where p and q are functions of a parameter the congruence describes a ruled surface in the absolute space. In general a given congruence can contain many ruled surfaces.

A generic point X, indicated by a material particle on the generic line of the congruence, can be written in the form

(2)

where λ is a continuous non-homogeneous parameter on the line. For λ = 0 we have X ≡ P, for λ = ∞ we have X ≡ Q.

Our problem is to find the condition that the point X(t) describes a trajectory orthogonal to the current ray of the congruence of rays.

This problem was solved by Dan Barbilian (Barbilian, 1974) in the following manner. As we said, it can be translated into condition that the points X, dX and the conjugate of PQ with respect to Absolute, say P’Q’ are in the same plane – the polar plane of the point X. Let’s put this algebraically.

The equation of the polar plane of point X in an arbitrary position along the line PQ can be written using equation (2) as

(3)

where ξ is the image of the current point of the plane, and α is a parameter. This represents a bundle of planes intersecting along the line P’Q’. Now let’s write the condition that one of these planes, the current one, contains the points X and X + dX. Because

(4)

we have the condition that the plane (3) contains the two points as

(5)

where we used the equation (1). The dot product (p, q) is always nonzero, except for the case when P ≡ Q. Therefore the system (5) defines the differential 1-form

(6)

and we have the theorem:

Equation (6) gives the necessary and sufficient condition that the trajectory of the current point (2) is orthogonal to the ray PQ while this ray is variable.

Based on this we can prove a relation of “quantization”: the orthogonal trajectories to a certain line are equidistant. Indeed, if for one trajectory the parameter is λ₁ and for the other is λ₂, while everything in equation (6) remains the same, we have

(7)

But this logarithm represents the Cayleyan distance between the two points of the trajectories situated on our line. Indeed, the ratio λ₁/λ₂ is the cross ratio of the four points (X₁, X₂, P, Q). By Laguerre’s formula, the logarithm of that ratio is the Cayleyan distance between X₁ and X₂, whence our conclusion.

In general the motion of the internal particle is not perpendicular to the generic line of the congruence. However, Barbilian proves that the differential form

(8)

is invariant with respect to all linear transformations of the points in space. It represents the component of motion of the material particle along the direction PQ. Therefore the component of motion perpendicular to the ray PQ, is also an invariant, and can be calculated from geometrical considerations. We are not interested here in these, but in some historical facts which, apparently dissimilar, can be brought together under the same concept by the present mathematical philosophy.

The differential form

(9)

is the non-Euclidean generalization of a differential form defined by Carathéodory (Carathéodory, 1937) for a bundle of light rays, in order to characterize the propagation of the wave surface with respect to the light rays. Remember that Fresnel had to do quite a tour of imagination when defining the light ray, which is the key of the modern theory of light. The light ray, however, is a fundamental concept that can be extracted from our experience without any physical considerations. But then we have to explain the propagation according to Huygens Principle in some other manner, and this involves the progress of similar points along different rays. The Carathéodory differential form (9) is just the adequate tool of the necessary mathematics involved in such a problem.

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