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III. Phase Plane of the Harmonic Oscillator

For geometrical visualization of conservation laws (when such laws exist) it is always pertinent to have a proper description of the oscillator in its phase plane. Here the variables are q and p - coordinate and momentum - and the algebraic expression of energy gives what we would like to call distinguished paths in the phase plane that may be used to frame the set of proper (true) paths, these last being paths taken freely in the phase plane and pinned down by events of measurement. Such an algebraic expression, when quadratic, generally depends on three parameters a, b, c say, and if it is conserved we can write:

Note in passing that the algebraic expression of energy is not the only one that can give us distinguished paths serving to frame the set of true paths in the phase plane. We will meet such things here shortly. Restricting the situation, for the sake of illustration, to the case depicted in equation (28) for distinguished paths, let us briefly review the history of the phase plane of 1D oscillator.

In usual physical conditions the equation (28) represents ellipses in the phase plane. The importance of these distinguished paths has been realized mainly with the occasion of the first quantization at the beginning of the century past. In the beginning was an ellipse referred to its proper axes, i.e. always depicted for b = 0 in equation (28). It was only in the mid-eighties of the past century that the importance of the general position of the ellipse, still referred to its center, was recognized (Hannay, 1985). Only as late as early nineties (Mensky, 1993) it has been realized that adding a random walk in the phase plane may answer to the problem of measurements, much in the manner of initial approach of Feynman (Feynman, Hibbs, 1965), i.e. with an apriori given measure of the paths. However on this occasion it has also been realized that, in order to function properly, the formalism needs a basic ingredient, namely a weight function to restrict the contribution of paths - thus extending the measurement concept to that of continuous measurement. Even though this weight function can be, in principle, of any type, fact it is always chosen to be of Gaussian type, i.e. invariant by Hamiltonian motion of the oscillator in the phase space in a sense that will be made precise shortly.

Geometry shows that the distinguished paths can even be taken as framing the whole phase plane, for instance by defining a geometrical Absolute of the plane. The big problem here is that almost everything in a phase plane is a figment of our imagination: we imagine that a distinguished path joins two points in the phase plane, that there is at least a true path pinned down by events of measurements, that nothing happens between two events and thus declare them neighbors and, to pinpoint this quality, we imagine that between neighbors the path is a straight line. We have almost nothing to limit this imagination. What Physics did here is to let the imagination go free, and then externally set a measure of the things it invented. This is e.g. the case of Feynman paths integrals from which the idea of continuous measurement springs: two points are correlated by a virtual infinity of possible paths on which we have to set a measure and this measure is accessible to Physics. What we mean to say here is that we have no control but on the correlation of these imaginary things. Then it is this correlation that is accessible to physical description. However we have to pay attention to the way of physically describing the correlations, for these creations of imagination emerge with strings attached and an apriori view - no matter if classical or quantum - cannot handle the situation properly. For instance, a point in the phase plane can be considered as an event due to measurement just as well as the result of an evolution. Is it possible to simultaneously describe this situation? What is its correspondence with the physical reality? First of all we have to insure the ability to describe geometrically the sets of points and curves we need in support of our imagination. An idea then would be to build ‘realizations’ of the harmonic oscillator that can be put in a stochastic form (Feinsilver, Schott 1989) - the largest imagination freedom here - associated with a true path, but for this we need to characterize both variables and parameters from a statistical point of view. We do not know the group that possibly takes a proper path into another. We don’t even know if there can be such a group, which we cannot take for granted. However, we are certainly in position to know the group that takes a distinguished path into another. At the risk of repeating well-known concepts, this simple fact will be now illustrated for the specific case of equation (28) above, and the mechanism of introducing the SL (2,R) group into considerations will then be apparent. We insist upon keeping the physical parameters in their utmost generality regarding the notation, primarily for the sake of Geometry, but also due to the fact that their physical meaning is not as direct as it may seem at the first sight.

We are taking now into consideration the case (ac-b²) > 0, a, c > 0 when, as pointed out before, the equation (28) gives ellipses characterizing either a damped harmonic oscillator - if dynamics is considered - or equal probability contours - if statistics is considered. The ellipse is in a general position with respect to its center. The group that possibly transforms these distinguished paths into one another - if there is one at all - must be a group acting in three variables a, b, c. Suppose now that we measure the position and the momentum with the outcome (x, y), and we have a set of possible alternatives for each measurement. The group acting upon these alternatives - if there is one at all - must be acting in two variables, not three as before. Certainly, the two actions cannot be identical, but they can be realizations of the same algebraic structure. Their closest relation we may think of can only be an isomorphism. This can be easily seen by supposing two alternatives situated on two distinguished paths: the alternative (x, y) situated on [Q]_a,b,c,

and the alternative (x’, y’) situated on [Q]_{a’,b’,c’}

Now suppose, as we certainly always can do, that the two alternatives are related by the real homogeneous transformation - a SL (2, R) transformation

It is easy to see that between the distinguished paths we have the following linear transformation:

This shows that a, b, c are the components of a mixed tensor of SL (2, R). However, we look at the two sets of equations (31) and (32) as characterizing two different isomorphic groups. They are both representatives of the known SL (2, R) structure. As continuous groups they always have three parameters no matter of the number of variables, and the structure equations are:

where X_k are the infinitesimal generators. Specifically, in the case of group (31) we have the differential realization

while in the case of group (32) we have the differential realization

These differential realizations do characterize geometries describing the corresponding actions of SL (2, R). The geometries are related to classical aspects of the problems in phase plane. Then a measurement in the phase plane would have to be described by the fact that, x, y as well as a, b, c are stochastic variables and, moreover, they are somehow correlated. Although it may seem sufficient that a, b, c are deterministic variables, there are cases where they have to be taken statistically as well. An obvious reason is the fact that one can imagine a measurement result as being situated on an infinite number of distinguished paths. Thus, we need to take into consideration cases where the components of the vectors (34) and (35) are simultaneously stochastic variables. Consequently, it seems important that we are able, on one hand, to give a functional structure to those coefficients as related to some physical situations and, on the other hand, to construct ‘boundaries’ like the above distinguished paths, which serve exclusively to the purpose of framing and measuring the extent of proper paths in the phase plane.

The dynamical model intimately related to the distinguished paths (28) is, as we have already mentioned, a Hamiltonian model closely connected to one-dimensional harmonic oscillator. For the case of measurements we have to find the way the parameters of the apparatus are determined, because certainly they are results of some interaction processes. The theoretical physics explored almost every aspect of the problem, by and large along the lines of a priori selecting the functional form of the parameters and then analyzing the theoretical consequences. It seems to us that there is not much left unexplored that would allow for a rational approach of the problem of measurement. Yet the only rational fact to start with is that everything in this problem bounces back and forth between the two realizations (34) and (35) of the algebraic structure (33). Then the question is: do we have a geometrical instrument capable of handling some other realizations of that structure with the chance to offer insight into the details of the measurement process? The answer is in the affirmative: yes, we have, it is the Stoka Theorem. But first let us show how the group theory allows us to overcome the lack of a conservation law.

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