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V. Joint Invariant Functions Related to Fundamental Theoretical Physics

After this brief description of the main aspects of the two SL (2, R) actions we are considering here, let us come back to our main problem: the joint invariant functions. It is noteworthy that the joint invariant functions related to the two realizations (34) and (35) do correspond to some fundamental aspects of the physical knowledge. The Stoka System (5) for the actions as given by the operators (34) and (35) is:

(55)

The rank of the matrix of coefficients is 3, so we have two independent integrals. They are

(56)

Consequently we can construct physical quantities related to harmonic oscillator only as functions of both algebraic expressions above. This is a net gain of application of the Stoka Theorem, having far reaching implications. The first and most important among these is that if one of expressions (56) is missing in the final equations of our theory, then this is indication of the fact that it is a constant.

A well-known particular function, explicitly containing both expressions (56), is the Gaussian probability density function (PDF), as given by the equation

(57)

This appears to be the objective reason for the fact that the Gaussian is so important for human knowledge in general and, regarding the present subject, it is actually the one giving reasonable results as a weight function in the problem of continuous measurements (Mensky, 1993).

Regarding the joint invariant functions of the two SL (2, R) actions it is worth making an observation supporting the quantization process. The presence of two algebraic expressions in (56), which in the simple physical situation of the classical constant mass harmonic oscillator have the precise meaning of energy and frequency, invites us to observe that the law of conservation of energy should not be considered alone when taken as a fundamental physical fact. Apparently this is the deep reason for the fact that it was a combination of the two expressions (56) that played a major part in the Physics of last century. Let us discuss this fact in a little more detail.

To start with, notice that the second expression from the equation (56) is the manifold of transitivity of the action given by the operators (35), for that action is intransitive. This geometrical fact can be interpreted in many ways, at least two of which we believe are fundamental in understanding the possibility of Modern Physics. From statistical point of view the quantity (ac - b²) is a measure of dispersion of the ensemble of measurements (x, y) as characterized by the PDF from equation (57). In fact, if we calculate the Information Energy (Onicescu 1966) - an indicator of the width of a distribution - for the PDF (57) according to

(58)

the result is

(59)

Thus, it may be worth noticing the fact that the manifolds of transitivity of the action (35) are surfaces of constant Information Energy of the Gaussian PDF.

This fact can be made accountable for the Uncertainty Principle as follows. Like we said before, the coefficients a, b, c can have a pure statistical meaning, as well as a pure mechanical meaning. In a pure statistical evaluation of the coefficients a, b, c we can use the principle of Maximum Information Entropy. This fact is known to work quantum mechanically (She, Heffner 1966), and no doubt it works also classically as it can be readily verified. In terms of the statistical estimators of the measurements (x, y) we then have

(60)

where VAR and COV denote the variance and covariance of the ensemble of measurements. Now, constant Information Energy entails a constant D. If we take this constant as where is Planck’s constant, we have automatically

(61)

with Δx and Δy are the standard deviations of the ensemble of measurements. Thus is the minimum of √(D) and this minimum occurs for the case of uncorrelated ensemble of measurement results.

Along the same line of reasoning, we have another fundamental theoretical case: purely mechanically - the case of damped harmonic oscillator - the parameters a, b, c are predetermined by mass, damping and stiffness. The existence of a transitivity surface in the space of these parameters appears to be the logical reason for the uniqueness of the frequency of harmonic oscillator. Apart from this, the Gaussian PDF still has a physical interpretation but in this case the parameters a, b, c do not need to be statistically estimated. Among the well-known fundamental physical functions having a long history in this case we have the ratio between energy and frequency - the physical action:

(62)

This function is, again, a joint invariant function of the actions (34) and (35). The role it played in the quantization process is well known. Suffice it to say that the quantization itself appears to be not only possible, but also sort of necessary, only because it is a byproduct of group transitivity and, as Stoka Theorem shows, the conservation of energy alone is not a fundamental fact: it needs to be supplemented with constancy of frequency. Deeper considerations on the subject would take us too far away from the main subject of the present work, so we choose to elaborate on this in a future work.

The last observations point towards the necessity of considering the connection between the two aspects of the measurement problem - purely statistical and purely mechanical. In the present context, this connection is given by the joint invariant functions of two actions of the type (35), one in variables a, b, c and the other in like variables X, Y, Z say. The Stoka System here is

(63)

The rank of the matrix of this system is in general 3. Consequently there are three independent integrals of this system. They are

(64)

The first two of them express the intransitivity of the two actions, as expected. The third one is then a measure of interconnection between pure Mechanics and pure Statistics. The general solution of the system (63) is a function of all three expressions (64). We will present some examples in this work later on. For the moment, only notice that a joint invariant function of Mechanics and Statistics of the harmonic oscillator does not involve only the Planck’s constant and frequency as requested by the group transitivity. It asks for something more: this something should be a relation of compatibility between the statistical estimators in the phase plane and the classical parameters of the oscillator. The fact is well known in the practice of measuring the frequency spectrum of an ensemble of field oscillators: we need a criterion that gives the best frequency window of measurements. The well-known classical case is the Nyquist criterion.

Incidentally, the same problem could have been also considered from another point if view, connecting the evolution with the measurement. The equation (60) illustrates what we like to call ’second order statistical hypotheses‘. Should we consider ‘first order statistical hypotheses‘ i.e. equations giving the estimates of means of classical coordinates:

(65)

we would then have to write the Stoka System between two actions of type (34), one in (x, y), the other in (p, q). The system would then be

(66)

The rank here is 3, so we have just one first integral. This is the bilinear expression:

(67)

This simple fact legitimates the choice of straight lines between otherwise uncorrelated events in the phase plane. Indeed, the joint invariant (67) can be constructed with reference to any two points of the phase plane either results of measurements or of evolution. In case it is constant - a linear conservation law - it represents a straight line. This fact is the logical reason of perfect square Lagrangian describing the measurements of harmonic oscillator: the measurement (x, y) is done on a state representing an evolution (p, q). One can further add that these two states are identical only if the Lagrangian of measurement is zero.

So much for now about the fundamental aspects of the Physics related to the harmonic oscillator. We insisted a little more on stressing the importance of Stoka Theorem to this simple situation first of all because this is being done for the first time ever and needs to be shown in detail and, secondly, because this theorem is capable of casting light on things concealed to routine consideration. The frequency - energy case above is a case in point. We will come later on with another example, in which the application of Stoka Theorem is instrumental. However, it is quite clear even by now that we have through this theorem the weaponry to deal with the fundamental problems of Physics, including the big one - the measurement problem. Nothing is said nevertheless regarding the structure of the field that may be involved in a measurement. We will introduce now a physical field specifically describing gravitation.

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