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James Mac Cullagh can be safely credited with the first attempt to introduce a gauge in the case of the light phenomenon. He is also the only one, at least as far as we are aware, having an attempt to explicitly introduce the Newtonian point of view regarding forces in the case of light (Mac Cullagh, 1831). The Fresnelian approach might have seemed to him too uneasy, due to its identification of the displacement with the difference of positions. Mac Cullagh’s theory of light, introduced explicitly as a theoretical task the “elliptical motion of ether molecules”. That this seems to be the first gauge theory, should be also obvious from the fact that it is lately recognized as the legal precursor of the Maxwellian theory (Darrigol, 2002, 2010), and by this of the modern Yang-Mills gauge theory, of course. However this recognition is mainly based upon later works of Mac Cullagh’s, more along to the idea of space-time continuity (Mac Cullagh, 1836, 1839). The Mac Cullagh’s original idea of gauge from 1831, is illustrated in the following picture.

The light phenomenon is explained here in the following way: choose two harmonic linear oscillators having the same frequency in the plane, corresponding to the segments AA′ and BB′ which represent the double of their amplitudes. These segments intersect each other in an arbitrary point O, which does not necessarily correspond to their centers. If we take these segments as axes, their center would correspond to the five-point star from our figure. Then choose two instantaneous positions of the two oscillators, say those corresponding to the point P, and construct the parallelogram OAEBO. The ellipse through P tangent to the sides EA and EB of the parallelogram, then represents the light, in the sense that it can properly represent the intensity of light as defined by the integral over a period from the square of the instantaneous velocity of the point P. Just for the art’s sake let us do some calculations, omitted in the original work of Mac Cullagh, in order to illustrate the different issues of the representation.

First we need to specify what Mac Cullagh understands by “simultaneous positions of the moving molecule”. Simultaneity does not mean here but the fact that one takes into consideration the actual positions of the two motions, admitting that they are results of two motions that did not even start simultaneously. This fact should be reflected in the phase lag between the two motions. Therefore Mac Cullagh’s simultaneity is not defined by the fact that the phase lag of the two motions should be zero. Generally, either the motions did not start at the same moment of time, or the matter itself intervenes to delay them with respect to each other. But the general idea, reflected mostly in the modern theory of stochastic phase, can be illustrated if we choose another point of the ellipse corresponding to the point P: such a point will not have necessarily a moment of time continuously deriving from the time of P through a motion along the ellipse. One might say that the motion along the ellipse is ‘virtual’, and what one can use as a continuous parameter is only a fictitious phase along the ellipse. However, the chosen moment is by no means the time of the motions of the two oscillators. The moments of time of the points along the ellipse corresponding to P are randomly distributed, and given by the different special positions of the two oscillators. On the other hand, had we have chosen another point P, we would have had another ellipse in an entirely different position with respect to the one depicted in our figure. So the ellipse itself is the object of a random process. These are for us today just customary notions, but at the time of Mac Cullagh the continuity played the tune of the day in physics. It is however worth considering the history from the modern viewpoint, for we can clarify the very concept we have today with respect to light and matter in general.

Now, if we take the equations of the two motions in the form

(1)

then the parallelogram rule gives:

(2)

The velocity of this motion will be given by

(3)

so that the square of velocity is

(4)

Here the round parentheses represent terms that will disappear when integrating on time over a period, due to the trigonometric time factors accompanying them. Indeed, applying here the Mac Cullagh’s own recipe for the calculation of the intensity, i.e. the integral over a period from the square of instantaneous velocity, the last two terms disappear, and we have

(5)

Let us find now the parameters of the ellipse of these simultaneous motions. Writing the simultaneous positions (the coordinates of P) by components we have

(6)

where the amplitudes are given by the vectors

(7)

in an arbitrary orthonormal reference frame. Eliminating the time between the two equations (6) we have the ellipse

(8)

Obviously, the axes and orientation of this ellipse depend essentially on the phase lag of the two motions. If the phase lag is zero, the equations (6) show that the motion is linear, not elliptical. Therefore in order that the motion should be elliptical, it is absolutely necessary that the phase lag should not be zero. One has to show now that the expression (5) is indeed the sum of the squares of the semiaxes of ellipse. Let us calculate the elements of this ellipse.

For convenience we will take the equation of the ellipse (8) in the form:

(9)

where we denoted

(10)

We should call these vectors Mac Cullagh’s vectors, for they are particular to this very theory and play a very special part in the development of the gauge theories. The semiaxes are thus given by the eigenvalues of the matrix

(11)

divided by the square of area of the parallelogram constructed on the vectors (10), and then inverted. The eigenvalue equation of the matrix (11) is

(12)

where θ is the angle of the two vectors. Therefore the matrix has eigenvalues

(13)

so that dividing by the area of the parallelogram and taking the inverses of results, gives the squares of the semiaxes:

(14)

The sum of their squares is then

(15)

Using now the identifications from equation (10) we have for this sum the value

(16)

which is exactly the magnitude proposed by Mac Cullagh for the intensity of light, as can be seen from equation (5). Notice that the phase lag complements the geometrical angle between the two amplitude vectors, in the sense that it does not appear but only as a factor of the dot product of the two vectors, as equation (16) shows. Thus, if the two vibrations are perpendicular in their plane of motion, the intensity as defined by Mac Cullagh does not depend anymore on the phase lag of the two motions defining the process of light. As Mac Cullagh expresses it, in this special case “the intensity is independent of the difference between the origins of the two motions”.

Now we can imagine how the action of matter upon light operates. In fact, the Mac Cullagh’s construction reveals it plainly, when considered as a Newtonian construction, but with the elliptic trajectory ‘accidental’, so to speak, in the sense we mentioned above. In the point E one can find, for instance, an external accidental center of force – a material point say – acting in such a way that the Mac Cullagh’s ellipse is the trajectory of another ‘material point’ under the action of that Newtonian force. Here it becomes obvious that there are a bunch of such Keplerian ellipses, all of them characterized by the choice of the simultaneous positions of the two Mac Cullagh’s motions describing the light, exactly as in the case assumed by Newton in his Corollary 3. One can see that as light progresses through matter there are different centers of force E, thus making the interaction of light with matter a plain stochastic process.

It is here the case to say that the very Newtonian prescription for finding the force acting toward an arbitrary point from the plane of orbit, can itself be improved based on Mac Cullagh’s construction, therefore based on the optical theory of light. Indeed Newton’s prescription contains a ratio of forces, while the MacCullagh’s prescription for the calculation of intensity contains both their ratio and their angle. True, this last construction is referred to the center of ellipse and does not admit – not even suspects, one might say – that the external point might be a center of force. Nevertheless, it is equally true that, in order to get palpable and mathematically tractable results, the Newtonian prescription must make explicitly and exclusively use of the center of the orbit. If this option is not used, the equations derived for forces are not quite so neat and clean – they become extremely complicated – and it is even doubtful that the dynamics could be written analytically just as clear as it was written until now. In a word, the Newtonian theory was forced to choose, for comparison, exclusively the center of the Keplerian orbit.

Now, the point E is external to a trajectory which, this time is literally virtual. The center of force E should be simply a local material center. There are always, in the interaction of the light with matter, a multitude of such material centers, and for each one of them we have a construction like that of Mac Cullagh with a virtual ellipse, for which only one point is real. Therefore the theory of probabilities must enter from this very moment in the explanation of the light phenomenon. It is thus necessary to take this point in connection with the fact that the theory of probabilities cuts in the atomic theory only by the fact that the leaps from a given orbit, or on a given orbit, take place with equal probability for the whole orbit. The main difference here is that in the case of light the probability refers to the orbit itself, while in the case of atom it refers to the place on the orbit.

One interesting fact related to Mac Cullagh’s prescription is that the intensity should be proportional with the frequency. According to Mac Cullagh this is indeed to be expected. It pays to recall that the Planck’s quantization, as well as its descendant, the De Broglie’s relation, require proportionality of the energy with the frequency. It is only our claim, that what is measured is the average energy over a complete period, the one which gives the result independent of frequency, used exclusively today. Actually, according to Mac Cullagh what is measured is not the mean, but simply the integral of the energy over a complete period. Another interesting fact related this time to Mac Cullagh’s vectors (10) is that they can be considered as gauge vectors characterizing the ether and matter respectively in a continuous constitutive theory of matter. The ellipse (9) can then be defined as a joint invariant function of two continuous groups properly describing the light phenomenon.

 

References

Darrigol, O. (2002): Electrodynamics from Ampère to Einstein, Oxford University Press, New York

Darrigol, O. (2010): James Mac Cullagh Ether: An Optical Route to Maxwell’s Equations?, The European Physical Journal, Vol. H35, pp. 133 – 172

Mac Cullagh, J (1831): On the Intensity of Light when the Vibrations are Elliptical, Edinburgh Journal of Science, pp. 86 – 88; reprinted in The Collected Works of James Mac Cullagh, pp. 14–16

Mac Cullagh, J (1836): On the Laws of Double Refraction of Quartz, Transactions of the Royal Irish Academy, Vol. 17, pp. 461 – 469; reprinted in The Collected Works of James Mac Cullagh, pp. 63 – 74

Mac Cullagh, J (1839): An Essay Towards a Dynamical Theory of Crystalline Reflexion and Refraction, Transactions of the Royal Irish Academy, Vol. 21, pp. 17 – 50; reprinted in The Collected Works of James Mac Cullagh, pp. 145 – 184

Mac Cullagh, J (1840): On the Optical Laws of Rock–Crystals, Proceedings of the Royal Irish Academy, Vol. 1, pp. 385 – 386; reprinted in The Collected Works of James Mac Cullagh, pp. 185 – 186

Mac Cullagh, J (1880): The Collected Works of James Mac Cullagh, J. H. Jellett, S. Haughton Editors, Hodges, Figgis & Co., Dublin; Longmans, Green & C., London