How would you explain Functional Integration to an 8 year old?

Question

I get the definition of the Functional Integral, but what heuristic interpretations are available to better understand the integral?

For instance, what motivates the definition? How is it related to the Functional Derivative?

Also, how would you explain this concept in a way an 8 year old would understand. For instance, with regular integration, the interpretation is area. Is there a geometric/probabilistic interpretation that can be readily explained with a picture/analogy?

Answer 1

The integral gives the area under the graph of a function $f(x)$. However, the integral can also give the average value of a function. Reviewing the definition of the summation definition of the integral, which cuts up the function into rectangles and then sums them, we see that there is a relationship between the summation and the average value formula. Namely,

$$A(f(x))={1 \over {b-a}} \cdot \int_a^b f(x) \ dx$$

Where $A(f(x))$ is the average value of $f(x)$ over the interval $[a,b]$.

Now, the Functional Integral sums up the tiny contributions from an infinite number of functions, in much the same way a regular integral sums up rectangles. The difference is that the regular integral sums up differences between a single function over a tiny interval. The functional integral sums up the difference in a functional over tiny differences between different functions. What's a Functional? A functional is very similar to a function, except for one key difference. A functional takes a function as it's input where as a function takes a single value as its input. We denote a functional by $F[f]$. An example would be the number that represents the change in $f(t)$ over an interval.

$$F[f]=f(b)-f(a)=\Delta f$$

The Functional Integral simply gives the average value of a Functional.

Here's an example. If you have a functional that returns the average value of an arbitrary function, that is bounded to $[0,1]$, over the region $[0,1]$, the functional integral will give the average value of the functional. In this case it happens to be $1/2$.

That means the average of the average value of an arbitrary function confined to the unit square is ${1 \over 2}$. Pretty cool if you think about it!

Answer 2

Functional integration is a collection of results in mathematics and physics where the domain of an integral is no longer a region of space, but a space of functions.

Answer 3

One of motivations is in physics (quantum mechanics), where you sum over all possible trajectories that a system could follow to get a probability of finding it in specific place (knowing probability distribution at earlier time). I've heard this approach is also used in probability theory to describe brownian motions.

It really is just a sum, only with bigger amount of summands. Just like regular integral. Except it's more difficult to visualize, calculate and even precisely define.