Light moves in straight lines. *Uh, no, not really.* Particles follow routes determined by the forces that act on them. *Wrong again.* The usual way to look at motion is only one possible interpretation of reality. There's an alternative approach, equally good: the **path-integral** method.

Consider all the possible ways for a beam of light to get from one point to another. *Which one takes the least time?* If the space between start and endpoint is empty, the answer is simply a straight line --- and that's the route the light takes. But what if there's a wall of clear glass set between the origin and the destination? The speed of light is slower within glass, so the winning least-time path will bend to reduce the amount of glass that the light has to go through. That zig-zag makes the total path length outside the glass a bit longer. There's a trade-off determined by the relative speed inside and outside, which is precisely what "Snell's Law" predicts with its tricky trigonometric functions of "index of refraction". That's the Least Time Principle.

Particles --- like electrons, or bowling balls, or planets --- move from Point A to Point B not to minimize time, but to minimize something else, a funny combination of kinetic and potential energy called the Lagrangian. Add up *("integrate")* the value of the Lagrangian at every point along the path and find the route that makes the answer as small as possible. That's the classical pathway that the particle will follow.

Why in the world does this work? There are two answers, one formal and one physical:

**Formally**, write down the mathematical equation that describes the sum of the Lagrangian and vary it over all possible paths. With some heavy manipulation and brute cleverness*(aka the Calculus of Variations)*you'll find that the minimal value occurs precisely when a particle moves as Newton's Law*(force = mass * acceleration)*compels it to. The same equivalence holds for light, or for systems responding to other types of constraint. So*formally*the Path Integral approach is "just" a super-subtle rewriting of the commonplace laws of motion.**Physically**, think of the particle as not a particle*(is this zen?)*but rather as a host of tiny rapidly-vibrating waves, spreading out in all directions through available space. The waves bend when they move from one medium to another, and they bounce off things that get in their way. These waves are wiggling so fast that they mostly cancel each other out --- except along a certain special route where they oscillate in phase and reinforce one another. That route is precisely the classical path that the particle takes. So*physically*the Path Integral approach is "just" a super-sneaky way to analyze the little waves.

Why call the second explanation "physical"? Because it's not merely a trick --- all particles are, at a deep level of reality, governed by quantum-mechanical laws that describe them in terms of tiny vibrations *(aka the Schrödinger Wave Equation)*. And light is even more so: it's made of electromagnetic waves that naturally interfere with one another, bend around obstacles, and bounce off barriers.

Or perhaps not. For many practical purposes it's a lot easier to let particles be particles, and to treat light as rays that fly merrily along. One of my first personal-computational-recreational projects, when I got my hands on a programmable calculator with an attached X-Y plotter ca. 1975, was to write simple programs to trace rays of light through various lenses and media. *(Yeah, I was a strange bird. And maybe "was" is the wrong verb tense, eh?!)*

I had such fun listening to the stepper-motors whirr and watching the little felt-tip pen **kachunk kachunk** across the paper, drawing a line along the path that the imitation light followed. I put simulated lenses together to make simulated telescopes. I experimented algorithmically with retroreflective road-sign material: little beads that catch and carom photons back toward the direction that they came from.

And I also played in the conjugate domain, adding up little navies of waves and seeing how they interfered, constructively or destructively. I remember watching the plotter pen spiral, around and around toward the center of the page, as it summed wiggly terms for me near a zero of the Riemmann zeta function ...

*(see also FringeOfThings (25 Jun 1999), CoherentInterference (28 Dec 1999), CollegeCollage3 (29 Sep 2001), PersonalProgrammingHistory (2 Apr 2002), FractalFeynman (30 Jan 2003), CombinatorialInterference (10 Sep 2003), PrimeObsession (4 Jan 2004), BrewsterAngle (19 Feb 2004), ... )*

TopicScience - TopicPersonalHistory - 2004-09-18

*(correlates: FringeOfThings, FreeTrope, BrewsterAngle, ...)*