Chapter 5 – Interactions

Particle paths

We see distortion when looking at water in a glass. This is because light travels slower in water than air. It finds the fastest path from an object to your eye which isn’t a straight line.

How do photons know the fastest path? One way of answering this is using Feynman’s path integration and Quantum Field theory.

There are two completely different approaches give the same answer:

• Schrödinger’s equation moves forwards from the start and finds a path to the destination
• Path integration looks at all ways of reaching a destination

This is another example of wave mechanics and Quantum Field Theory taking different approaches to reach the same result.

Richard Feynman made the radical proposal called path integration: what if particles consider every possible path from A to B? Not just straight and curved but also zig-zags, loops and wild detours! Every possible path contributes.

The clever part that makes this work is how the paths get added together

Think of a spinning pointer traveling along  each path

Add the final pointers at the end of each path. Paths that differ by the ½ wavelength will cancel out. The paths that survive will have neighbors that end with similar pointers. The sum of all the final pointers represents
the probability of ending at B. In empty space, a straight line survives because paths close to it have almost the same length so add to the final pointer value.

This gets more complex because in addition to considering all paths, we need to consider all possible interactions along the paths. Luckily the more complex interactions are only needed to get results with very high precision.

Although we have calculated the probability of reaching an end point by considering all paths, the particles don’t actually take these paths. They don’t have an exact position until observed and observing them changes the wave function.