Fermat's Principle, Particles with Rest Mass and Quantum Mechanics

Addendum Oct. 26, 2022  In ((3)) one should formally subtract i.e. d/dx{ -E1 t + (p1x) x +E2t + (p2x) x) = 0. Then p1x-p2x=0 and -E1 dt1/dx + E2 dt2/dx = 0. Thus conservation of momentum implies Fermat's principle. The minus sign between dt1/dx and dt2/dx follows from t1 being a variable of x and t2 of (L-x) where L is the length of the mirror such that initial and final y values have the same magnitude. (This does not change the conclusions made in the note we argue.)

In a previous note (1) we argued that Fermat’s least time principle applied to light involved finding an expression for time in terms of spatial variables and the taking of d/dx where x is an invariant spatial variable in the problem. For example, for reflection from a stationary or constantly moving mirror in the y direction or refraction with different n(i) (indices of refraction in the y direction) x is invariant because there is no medium discontinuity in the x direction. We argued that Fermat’s principle is equivalent to a conservation law in the direction of the invariant variable and that this conservation is in fact that of momentum in this direction. In (2) we argued that E=pc with c=x/t i.e. the wave dispersion equation for light contains Fermat’s principle because one may write it as 0= -Et+px and apply d/dx using t(x) with x being the invariant variable. With conservation of momentum in the x direction and E1 = E2 for the two rays: dt1/dx + dt2/dx = 0 for a two ray problem.

    Fermat’s principle and E=pc pertain to light. The latter may be written as 0= -Et+px ((1)) for light with x/t=c which as we argued above leads to Fermat’s principle if one takes d/dx for two rays. One may note, however, that one also obtains Fermat’s principle for constant= -Et+px with momentum conservation in the x direction. Furthermore a particle with rest mass bouncing off a floor may behave like light. This we argue leads to the idea of considering two different frames. For light  -E1 t1 + p1 x1 = 0 = -Et + px with x/t = x1/t1 = c.

For a particle with rest mass  -mo t1 + 0 = -Et+px with x/t=v  where c=1 and mo is energy in a rest frame. (This is in fact the idea of a Lorentz invariant.)  Thus Fermat’s principle for a free particle seems to appear in the form -Et+px = constant. The constant is 0 for light. 

   The form -Et+px is consistent with a wavelength proportional to 1/p and a frequency to 1/E not only for light, but for the particle with rest mass. We argue that light moving in a medium is a complicated statistical process of scattering, but that the problem is treated in a “thermodynamical” manner by characterizing the medium by a single number i.e. single piece of information, namely the index of refraction i.e.  c→c/n and p→pn for light. We argue that movement within the medium is like moving in a kind of constant potential for light.  -Et+ (pn) x still holds and there is still conservation of momentum in the x direction for two media. Thus the idea of d/dx → p is already present in the photon form -Et+px which follows from Maxwell’s equations, but could also be derived from Fermat’s principle as argued in (2). In other words, ideas of quantum mechanics for a particle with rest mass could ultimately follow from Fermat’s principle, we argue.

   A free particle with rest mass is linked with constant= -Et+px and if one wishes to have an eigenfunction of momentum then:  -id/dx exp(ipx) = p exp(ipx). For a constant potential,  just like with light, p would change, but the form exp(ipx) would remain. Thus it seems that even in average situations (like refraction) one still wants to have a periodic type spatial function W(x) such that  -d/dx d/dx W(x) /W(x) = <pp> such that  <pp>/2m + V(x) = E for a particle with rest mass. This is the Schrodinger equation and it is found mathematically that only certain En are allowed if one wishes to have specific boundary conditions like W(x) →0 at +/- infinite for say an oscillator problem. Thus we argue that the fundamental idea of taking a derivative of an invariant spatial variable in Fermat’s principle which automatically yields conservation of momentum is a basic idea found in quantum mechanics even for a particle with rest mass, namely that d/dx is associated with momentum. Mathematically d/dx is a translation operator and so the link with Fermat’s principle shows how translation is linked with a spatially invariant spatial variable e.g. x.

 Furthermore Fermat’s principle shows that even complicated statistical scattering problems such as refraction (in which the photon scatters  off of different molecules in different media) can be reduced to a periodic average problem with p→pn. This suggests that for a particle with a rest mass in a potential V(x) one should also have a periodic spatial variable such that d/dx d/dx is linked to average momentum, we argue.