To be used in future
 Section 5.1: Sum of paths Up Chapter 5: Path Integrals Section 5.3: Free particle revisited 

5.2 Interpretation

After we have broken up the evolution in these small time steps, it pays to think a bit more about the interpretation of the integrals. For each value of , we do an integration over all allowed values of , see Fig. 5.1↓. If we think about these integrals another way, we can also argue that each possible contribution in the integral corresponds to one choice of the position ; in other words each term is a “path” . So we can interpret the path integral as the sum over all possible paths between the two points and --Hence the name!
figure Figures/paths.png
Figure 5.1 A picture of a number of possible paths in the path integral
What types of paths contribute to the path integral? If we take the limiting process seriously, we get an interesting picture, see the illustrations in Fig. 5.1↑. Here the blue curve goes to a smooth path in the limit that goes to zero, but the green and especially purple paths don’t--they seem to show some kind of almost fractal behaviour. Such fractal paths are key to the underlying Wiener process (see below). It also suggest that any path between and contributes, no matter how crazy--but not equally! From this simple analysis we can conclude that by construction the paths are continuous, but that they are not necessarily differentiable in the limit . Mathematicians have studied such curves for a long time, and show that they are reasonably straightforward to deal with, so they should not be as daunting as they seem in first instance.
 Section 5.1: Sum of paths Up Chapter 5: Path Integrals Section 5.3: Free particle revisited