# Defining a metric on the space of paths

Imagine the following path integral

$$\int_{x(0)=x_i}^{x(T)=x_f} \mathcal{D}x \, e^{\frac{i}{\hbar}S[x]}.$$

This integral is defined over the space of all paths that satisfy the boundary conditions $$x(0)=x_i$$ and $$x(T)=x_f$$. I am interested in defining a metric on this space in order to be able to quantify how similar two different paths are. I appreciate any suggestions on how to proceed.

• The path integral measure is often defined by first considering a lattice with finite spacing $\epsilon$, integrating over the value at every lattice point separately. Then one takes the limit $\epsilon \to 0$ to end up with the path integral. Maybe a similar approach could be used to define how "close" two functions are to each other. – scaphys Apr 21 at 22:58
• Alternatively, have a look at $L^p$-distance. – scaphys Apr 21 at 23:07

In $$\mathbb{R}^n$$ the typical notion of distance is $$|x| = \sqrt{\sum_i x_i^2}$$. This naturally generalizes to square-integrable functions on $$[a,b]$$, $$|f| = \sqrt{\int_a^b |f(x)|^2 dx}.$$ Thus, one notion of the distance between $$f,g$$ on $$[a,b]$$ is $$|f-g|=\sqrt{\int_a^b |f(x)-g(x)|^2 dx}.$$
• Thank you so much. In my case $f,g$ are complex functions; then I guess I have to modify this to $$|f-g|= \sqrt{\int_a^b (f(x)-g(x))(f(x)-g(x))^{\ast} \, dx.$$ Correct? – B. T. Apr 22 at 7:14
• Yes, but you can use the same notation nevertheless, since for $z \in \mathbb{C}$, the typical norm is defined as $|z| \equiv\sqrt{zz^*}$. – scaphys Apr 22 at 10:23