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I. Introduction

The modern problem of field measurements is, in its essentials, entirely analogous to the old problem of radio reception: there is a local oscillator with adjustable parameters in order to allow tuning to the different frequencies of the environment. In order to illustrate the philosophy of measurement in its utmost generality, we will discuss here exclusively the 1D classical harmonic oscillator and related problems. The classical harmonic oscillator is the established model for the intimate structure of the local apparatus in a problem of field measurement. We will reformulate the problem of measurement in the following fashion in order to convey its necessary generality: one has a harmonic oscillator physically characterized by three parameters, one related to inertial properties (mass), the other related to local environment properties (stiffness) and the third related to transfer and induction properties (damping). These parameters vary in a certain way, and it seems only natural to suppose that the recorded motion of the oscillator is somehow adjusted to this variation. The problem is to find a way to describe this adjustment.

    We can translate this problem in a particularly manageable form, because the motion of harmonic oscillator is related to a one-parameter subgroup of SL (2, R). The reason we are doing this can be best illustrated by the words of E. T. Jaynes (Jaynes 1973). That particular reference aimed more than what we presently do, namely the foundation of the principle of indifference in defining probabilities. The method is to construct invariant measures on continuous groups, and it has everything in common with our present discussion. The question was, why use groups? And Jaynes said: “However, the principle of indifference may, in our view, be applied legitimately at the more abstract level of indifference between problems; because that is a matter that is definitely determined by the statement of a problem, independently of our intuition. Every circumstance left unspecified in the statement of a problem defines an invariance property, which the solution must have if there is to be any definite solution at all. The transformation group which expresses these invariances mathematically imposes definite restrictions on the form of the solution, and in many cases fully determines it.” (Italics ours). The latest experience with continuous groups in Theoretical Physics changes these last words in a program - the Jaynes program. The only problem remaining now can only refer to some details of this program. The chief detail seems to be the very applicability of a group to a certain problem - a fact that Jaynes seems to take for granted - and this is decided by our experience. It turns out that a group isomorphic to SL (2, R) describes the apriori variation of the physical parameters of harmonic oscillator. Likewise, a group isomorphic to SL (2,R) describes the gravitational field. The manners of these descriptions are different, however the isomorphism is an attractive feature from the outset: could not one take it into account in order to show how the harmonic oscillator in gravitational field comes across and thus add something to the problem of measurement? We show here that this is possible. In order to do it, let us notice in the spirit of Jaynes’ program, that in describing what we think is a free harmonic oscillator first of all we do not specify the parameters, for the description should be valid at any scale and certainly the field is left unspecified also, for we are talking about the free harmonic oscillator. Thus, if there are invariants in the description of the harmonic oscillator, they certainly are related to the missing scale and field, left unspecified when describing the free harmonic oscillator. Likewise, if there are invariants in the description of the field they are then related to the missing information on the local oscillator left unspecified in the description of field. The problem is, on one hand, to build such invariants and, on the other hand, to find how they can be related to one another. The solution to such a problem seems to present itself in the form of joint invariants of isomorphic groups. We will base our analysis on a theorem of M. Stoka (Stoka 1968) extracted here in a little modernized form as presented by M. Leuci and A. M. Pastore (Leuci, Pastore 1994).

Let G_r (x₁, x₂, … , x_n) and H_r ( α₁, α₂, …, α_q ) be two isomorphic r - parameter Lie groups, with the infinitesimal generators

(1)

Then the following statements are equivalent:

1^o. There is a family of p - dimensional manifolds with equations

(2)

admitting G_r() as invariance group.

2^o.The matrix obtained by joining the components of X’s and A’s from equation (1):

(3)

has rank s < n + q.

If one of these statements holds true, then the family of invariant manifolds is given by the system of equations

(4)

where are the independent integrals of the system of partial differential equations

(5)

An arrow over a letter has the meaning of a vector, and we used the summation convention over repeated indices of different variance. One word of caution is now in order: from geometrical point of view H_r is the group of parameters associated to the group of variables G_r, indicating a possible subordination. From physical point of view it does not seem worth maintaining such a hierarchy: in different physical situations the variables may play the role of parameters as well. It is in order to avoid such a misunderstanding that we use here the above term joint invariant functions of two isomorphic groups and not invariant manifolds - the preferred geometrical term.

This theorem, referred to as the Stoka Theorem from now on, is the instrument we are seeking for in the given theoretical environment as presented. Likewise, the partial differential system (5) will be referred to as Stoka System throughout the present work. In order to profitably use it we need, on one hand to find physical realizations of SL (2, R) structure and, on the other hand to find the physical situations where the theorem may apply, i.e. to find SL (2, R) related problems of Physics. These are our next points of immediate concern.

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