ProtoQuant.com

VII. The Harmonic Oscillator in a Gravitational Field

Let us apply the Stoka Theorem here in order to describe a harmonic oscillator in gravitational field. The only (minor) problem we have is that the theorem can be used to find the joint invariant functions between either one of the realizations (34) and (35) for the harmonic oscillator and (87) (or (88)) for the field. It is a matter of physical decision to find which ones of these joint invariant functions would be appropriate for the description of physical system field - oscillator. We will take these cases in turn separately. The discussion will be carried on as based on equations (87) and only occasionally shall we pass to the real counterpart of the result, involving u and v.

The first Stoka System involves the equations (34) and (87) and can be written as

(90)

The rank of this system is 3, so there is but only one independent integral. This is

(91)

Thus, any joint invariant function is here a regular function of the algebraic expression (91). This function does not say much by itself; however it is a valuable heuristic tool. Indeed, first of all we have by (91) the way to place our emphasis in this study: the gravitational potential is in direct relation with the fixed points of the homographic action of the symplectic matrix characteristic to a quadratic Hamiltonian (see equations (40)). Applying this philosophy to equations (53), taken for b₀ = c₀ = 0 in (51), and thus mimicking the evolution from a free particle to harmonic oscillator, we find that every symmetric transformation matrix in that parameterization has the fixed points of its homographic action given by

(92)

A comparison with equation (86) gives the conclusion noticed before, namely that the gravitational potential is explicitly related to an evolution of the physical parameters of the oscillator (equation (51)), where the time of evolution (Stoler parameter) is a solution of the Laplace equation (τ ~ Ψ), and a (~ - φ) is an arbitrary phase. This is the reason we chose the particular form (46) for the constants L, M, N. Thus the gravitational field provides an ensemble of oscillators, geographically spread in space on harmonic surfaces, making possible the measurement of a certain local oscillator: these oscillators actually provide the physical conditions of a gauge transformation representing the measurement.

Further details can be unveiled by considering the Stoka System corresponding to equations (35) and (87):

(93)

The rank of this system is 3, so we have two independent integrals. They are

(94)

We do not have here an obvious physical interpretation either. However, let us direct the ideas along the following lines: the gravitational field determines a Gaussian weight function for every oscillator in the field, with exponent as given by (91)

(95)

Suppose that for the oscillator itself - or for an ensemble of measurements in the phase plane - we have a Gaussian process, as characterized by PDF

(96)

These are requirements of Stoka Theorem. It also allows us to say that the PDF associated with measurements in gravitational field is the product of the weight function (95) and the PDF (96). Thus we can find the probability density of the parameters a, b, c given the gravitational field by considering (x, y) random and then integrating - the principle of randomization (Lavenda 1995). We have

(97)

or, in view of equations (95) and (96),

(98)

Obviously, this PDF is a solution of the Stoka System (93). We may conclude that the gravitational field determines a weight function allowing us to restrict the distinguished paths in the phase plane of the harmonic oscillator. This basic ingredient of the measurement description is usually chosen a priori (Mensky 2003). The parameters of distinguished paths too are statistical variables, with their PDF given by equation (98). As a matter of fact this PDF can be interpreted as referring to the gravitational field just as well: p (h| a, b, c).

Along the same lines we may discuss the behavior of the physical action of an ensemble of oscillators in gravitational field. As shown before, the action is a quadratic form that can conveniently written as

(99)

Now, in a world dominated by gravitational field, i.e. characterized by the weight function given in equation (97), it is only natural to think of a PDF of this very function. To find this PDF we proceed directly by first finding the characteristic function of z:

(100)

from which the PDF sought for is found by a Fourier inversion based on tabulated formulas (Gradstein, Ryzik 1995):

(101)

with I₀ - the modified Bessel function of order zero and A, B two constants to be calculated from the equations

(102)

This PDF has the nice feature to give closed form mean and standard deviation for the variable z

(103)

As expected these two parameters, which according to current physical ideas are measured quantities, do satisfy to Stoka Theorem being evidently solutions of system (93). Thus, it is theoretically confirmed that the mean of physical action of the harmonic oscillator, as well as its variance, explicitly depends on the gravitational field. Whatever we do in estimating these fundamental quantities carries the imprint of the background field. This fact is a very important, in that it gives logical foundation to some cosmological descriptions of the fundamental constants.

The series of examples can be continued with other aspects of measurement in gravitational field and even with other ‘tools’ not just the harmonic oscillator. The Stoka Theorem opens here unlimited possibilities, both by offering direct solution and as a verification method for solutions obtained otherwise. However, we shall stop here for the examples just given are enough in making the point.

Pages: 1 2 3 4 5 6 7 8 9 10