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IV. The Nature of Information We Receive and Its Carrier

If the Kleinian thesis that Geometry is simply the study of properties of figures that are invariant with respect to a group of transformation is true, then it never found a better illustration than in the case of limited Space isotropy as described above. Indeed, the above way of conceiving the isotropy of Space, implicitly contains the basic information on the agent allowing us to pass judgments on the Space itself. We alluded to light before, as setting an alternative length unit for displacements. Well, even the electromagnetic properties of this agent come out as group invariance properties.

Indeed, according to Jaynes’ idea we must have implicitly, in stating the problem of isotropy of Space, the fundamental properties of the agent carrying the information from places Space, for this agent is a “circumstance left unspecified”. In this connection, we notice one thing of tremendous importance of the Bromwich’s characterization of isotropy, first said to be realized by Lie and then in a close form by Campbell (Campbell, 1903), viz. the following. The maximal group of isotropy of Space can be also obtained as the group of invariance of the partial differential equation

or, equivalently, of the quadratic differential form

Equation (12) – which, as Campbell shows, is closely related to condition (13) – appeared for the first time in history as representing a Maxwell stress system, with zero resultant of the electric forces and charge density given by (∆V), where ∆ is the Laplace operator (Love, 1944). As known, the Maxwell stress system represents electromagnetic perturbations, and the fact that electromagnetic field in free space can be completely characterized by a single complex function is a reality (Green, Wolf, 1953). In this connection we need to mention, first of all that between the three vectors from equation (4) and the stress-energy tensor of an electric (or magnetic) field the only difference is just a scalar gauging factor. Secondly, given such a gauging, the equation (13) is liable to represent a free electromagnetic field (Bateman, 1915). Thirdly these may seem only opportunity conjectures, but they have a fundamental connection in the very theory of stresses and strains. For the condition (13) can be realized for instance on a special surface having null normal, and evolving along this normal. The components of that normal can be then represented by three mutually harmonic quadratic polynomials (l, m, n) in two variables – surface coordinates (Burnside, Panton, 1960). Now consider a symmetric tensor (stress or strain for instance) in Space, as referred to its own directions (eigendirections). The component of this physical quantity along the normal to our surface is given by

where r_1,2,3 are the eigenvalues. Equation (14) represents a polynomial of fourth degree having the roots either all complex or all real. It turns out that we can also calculate the component of this physical quantity along the surface (tangential component), which we call τ, from equation

If for that surface the normal component p is zero, then on the surface our stress tensor does not have but tangential component.

    All these algebraic properties of tensors are mimicking facts related to our ideas about the measurement of light: it is known that light has a pressure, that in certain situations it is a transversal perturbation, etc. The above observations suggest that the scale property of isotropy is intimately related to the agent compelling us to make up theories about Universe. However, here we can say even more: the spin of particles is just a local reflection of the limited isotropy of the Universe. Indeed, the equation

represents physically a half-spin abstract space (Yamamoto, 1952) warning us that a too direct approach of the problem of interaction between the light and matter might not be a good idea in describing the interactions. The spin is, indeed, related to certain statistics. The sure thing here can only be the fact that statistics is related to a special kind of surface in Space. Thus, we need to see what the theory of surfaces has to say about these things. Meanwhile we only can state that the first of the “three laws of Cosmology”, as Sciama calls them (Sciama, 1961) should be amended to a more precise form: instead of “L’Univers en entier exerce des forces appréciables sur la matière locale” one must have “L’Univers en entier determine la structure de la matière locale au moyen des champs qu’elle peut voir”. This statement has a precise meaning related to the definition of isotropy of the Universe!

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