_{1}

The aim in this paper is to construct an affine transformation using the classical physics analogy between the fields of optics and mechanics. Since optics and mechanics both have symplectic structures, the concept of optics can be replaced by that of mechanics and vice versa. We list the four types of eikonal (generating functions). We also introduce a unitary operator for the affine transformation. Using the unitary operator, the kernel (propagator) is calculated and the wavization (quantization) of the Gabor function is discussed. The dynamic properties of the affine transformed Wigner function are also discussed.

Geometrical optics serves as a powerful tool for investigating optical systems. The path of a light ray is described by an eikonal. When the light rays are paraxial rays, this is classified as linear optics. In this approximation, the propagation of the light ray is described by the product of the refraction and the transfer matrices [

In general, the ABCD-matrix is specified by a three parameter (A, B, C, D with AD ? BC = 1) class of linear transformations [

In this article, we develop the mathematical properties of an affine transformation from the optical and mechanical points of view. Since the affine transformation has a displacement part, we are able to discuss the translation in phase space. Thus, we show that the affine transformation not only distorts but also displaces the Wigner function. Because this displacement can have time dependency, the Wigner function moves dynamically in phase space.

This paper is organized in the following way. In Section 2, we define the affine transformation and show the eikonals which generate this transformation. In Section 3, we turn to the quantum mechanical case for the affine transformation. We show that the operator of the affine transformation is obtained from the product of the displacement operator and the unitary operator of the ABCD-transformation. We also calculate the kernels of the affine transformation. In Section 4, we treat the wavization by referring to the Gabor function. In Section 5, we discuss the affine transformation of the Wigner function. We give an explicit form of the affine transformed Wigner function and examine the change in its configuration and the displacement of the Wigner function. Section 6 is devoted to a summary.

The general affine transformation is defined by a linear combination of position q and momentum p with the four parameters A, B, C and D and the displacements for position E and momentum F. We define the affine transformed position Q and momentum P as

with the lossless (area-preserving or power-preserving) condition

In classical mechanics, this condition comes from which affine transformation

that is, Q and P are canonical variables [

In geometrical optics, the path of the light ray is described by an eikonal. In the following discussion, we restrict ourselves to Gaussian optics, so each q and p is one- dimensional variable. There are four types of eikonal in Gaussian optics. We list the four types below;

By substituting (1) into (4), we rewrite these eikonals in terms of two of the four canonical variables q, p, Q and P,

These four functions

We listed four types of the generating functions in (5). From the theoretical and experimental points of view, it sometimes happens that we cannot describe the affine transformation via one of them. For example, the affine transformation in (20) below has zero component in

Substituting this relation into (5a) and

In this section, we consider the quantum mechanical version of the affine transformation.

Corresponding to the canonical transformation in classical mechanics, the unitary transformation plays a central role in quantum mechanics. Analogous to the classical affine transformation (1), we define the quantum mechanical affine transformation as follows,

where

To obtain the unitary operator

which generates the displacements in position and momentum,

The other one is the unitary transformation

which generates the ABCD-transformation [

where

and

then

We consider the unitary operator

Indeed, we obtain

which is the quantum mechanical affine transformation (7) as we expected.

Now, we calculate the kernel of the affine transformation. The kernel is just the transition amplitude from the position q at an initial time to the position Q at a later time given by

Using the formulae

we obtain

Substituting the transition amplitude in terms of

We include the “irrelevant” constant phase factor which has often been neglected in the literature [

Substituting these parameters into (19), we obtain

which is the same equation as that obtained from the path integral [

The other kernels are derived in the same manner. We list all four types of transition amplitude below:

where the W’s in the exponentials are the generating functions (5) which generate the canonical transformation (1).

It is worth commenting here that it is well known in classical mechanics [

Quantum mechanics is obtained by the “quantization” of classical mechanics. Similarly, physical optics is constructed by the “wavization” of geometrical optics [

where

width is obtained from

To make the calculation easier, this wave packet (23) can be rewritten in the form,

with

of the coherent state wave function

Using this expression, we obtain

This result with (24) gives

that is, the Gabor function satisfies the minimum uncertainty relation.

We obtain the affine transformation of the Gabor wave packet by using the kernel (22),

and

where we introduce two complex variables,

Having the probability density from (30) and (31), we obtain

The center of the Gabor function propagates along the affine transformation (1). From these Equation (33), we obtain the variances

and the uncertainty relation

Since the only constraint for the parameters

Let us show two examples here. As we saw in (25), the Gabor function with

which coincides with the uncertainty relation of the squeezed state [

which coincides with the uncertainty relation [

The Wigner function [

When we take a Gabor function (23) for any wave function

Now we apply the unitary operator

To cast the right hand side, we use the coordinate identity operator

Substituting the kernel (22a) into (41) and integrating over u, we obtain

where

where we use the formula

where we use

As an example of a wave function

This equation is a generalization of (39), that is, in its initial state, the Wigner function of the Gabor function is represented by (39). Once the affine transformation switches on, the Wigner function changes along with (45). Note that integrating (45) over P and Q respectively, we recover (33);

which is the correct character of the Wigner function.

We have developed the mathematical properties of an affine transformation from the optical and mechanical points of view. The kernels of the affine transformation were clearly derived and comprise the eikonals (generating functions) which generated the affine transformation in optics (mechanics).

Using the kernel, we discussed the wavization of the Gabor function. The Gabor function has a Gaussian profile and is symmetric in position and momentum. We found the time development of the uncertainty relation, according to the affine transformation.

We also discussed the affine transformation of the Wigner function and showed not only the distortion but also the dynamic movement of the Wigner function in phase space.

Ogura, A. (2016) Affine Eikonal, Wavization and Wigner Function. Journal of Modern Physics, 7, 1738-1748. http://dx.doi.org/10.4236/jmp.2016.713156