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IV. One-Parameter Groups Along and Across Distinguished Paths

Coming back to our harmonic oscillator, the quadratic form in equations (29) or (30) is the implicit form of the equation of a distinguished path, i.e. a trajectory of the action of group given by operators from equation (34). Indeed, such a path can be obtained from the condition that the most general vector tangent to the group leaves it invariant. This amounts to partial differential equation

(36)

where a, b, c, are real numbers and X’s are the operators given by equation (34). At the risk of repeating again well-known facts, it is worthwhile insisting a little upon the integration of this partial differential equation. We can integrate it in two ways: implicitly and explicitly. To this end, we write the equation in the form

(37)

having the characteristic ordinary differential equation

(38)

This differential equation can be integrated, on one hand directly, giving the implicit solution

(39)

Thus, the general solution of equation (37) is an arbitrary function of the expression from equation (39). On the other hand, we can assign a common differential value to the two ratios from equation (38), thus integrating it explicitly. We then have the system

(40)

where dt is the common value of the differentials from equation (38). Equations (40) give obviously the motion of the system, as generated by (39) as a Hamiltonian. Here the form of solution depends on the sign of expression

(41)

In order to illustrate the ideas we shall choose this quantity as positive, thus leading to the harmonic oscillator type of solution, which physically interests us most:

(42)

Consequently, in this particular case t is indeed a time parameter as we know it. It is easy to see, by eliminating the time from the above equations that the quadratic expression (39) plays the part of a conservation law for this evolution as well. Also, it can be shown that, had we chose the discriminant from equation (41) as negative, we then would have obtained an inverted harmonic oscillator case, involving hyperbolic functions instead of cosine and sine.

By the way of example, we may choose to analogously discuss the situation in the space of the parameters a, b, c - the transversal motion. This time they are group variables and, instead of equation (36) we have the following equation:

(43)

where L, M, N are real constants and X’s are the operators from equation (35). The characteristic system has two ordinary differential equations

(44)

and is better to first integrate it explicitly, by assigning a time parameter τ in the common differential value of the ratios, and then find the implicit integrals. Supposing the same conditions as before, and looking for linear integrals, the result is: the constants L, M, N are limited by the condition

(45)

and we choose this constant to be unity. The condition (45) can be realized in many different ways, but we restrict the case (for reasons which will become clear later on) to a one - parameter family of integrals as given by the particular choice

(46)

with φ an arbitrary angle. There are three integrals of system (44), and they are given by

(47)

Here A, B, C are integration constants, the first one being also a constant of motion and one of the implicit integrals. A second implicit integral is quadratic and can be obtained by eliminating the time between the last two equations (47), which results in another constant of motion:

(48)

It is interesting to note that for the particular motion for which A = 0, and only for this motion, φ is given by the orientation of the ellipse, for we have

(49)

In this instance, A’ - the quadratic integral of equation (48) - is given by

(50)

Thus, it is perhaps a nontrivial result the fact that a harmonic oscillator has a determined constant frequency only in case where the arbitrary parameter φ is given by the orientation of the distinguished trajectory of the oscillator, no matter of this orientation. Otherwise the frequency is no more determined, and we may have to determine its true value from A and A’ given by the equations (47) and (48) as depending on an arbitrary parameter. Possibly, this is the deep reason why the classical results are always reported in terms of distinguished trajectories at zero angle of orientation, being certainly the reason why the importance of this variable has been noticed so late in the last century. Besides, a nonzero angle of orientation usually means the presence of dissipation, which destroys conservation property if the evolution is not properly described.

In the general case, when φ is a free constant, the group equations are determined from equations (47) by writing A, B, C explicitly in terms of the initial conditions a₀, b₀ and c₀ say. Thus the group transformations are

(51)

It helps noticing that these transformations are of the form (32) and can be obtained by a transformation of the symplectic matrix (involution) associated with the triplet a, b, c:

(52)

with the parameters α, β, γ, δ given by equations

(53)

This is a unimodular transformation which acting in the phase plane reveals an interesting connection. Indeed, it can be written in the complex form as

(54)

In the theory of quantum measurements z can be interpreted as the eigenvalue of the annihilation operator (She, Heffner 1966). Then equation (54) exhibits the transformation introduced by Stoler (Stoler 1970, 1971) when discussing the equivalents of minimum uncertainty states. These last states are obtained for φ = 0. For arbitrary φ Stoler transformation describes a complex extension of the frequency, and the corresponding states are no more minimum uncertainty states. Finally, it is to be observed that a group evolution is not always a time evolution: while the parameter t from equations (42) can safely be taken as time according to our notion of time, not the same can be said of the parameter τ from equations (51). This last parameter is what we would like to call a Stoler parameter that may or may not be related to time and, in any case, if so related it is only through the intermediary of the physical parameters characterizing the oscillator. As a matter of fact the classical time from equation (42) is actually related to the arbitrary parameter φ from the present problem. This is a well-known fact (She, Heffner 1966), and will be reiterated here when stressing the relationship between Dynamics and Cosmology.

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