What does it mean to have a function of a field?

Question

There was a Leonard Susskind lecture on the Higgs Boson I watched the other day, and he talked about graphing the field where the domain was some sort of field space. The lecture is here at about 6 min: https://www.youtube.com/watch?v=JqNg819PiZY&t=350s

But this idea seems bizarre to me. I have some sort of conceptual difficulty imagining functions of fields. The way I've dealt with it is by imagining that the domains of functions of fields are defined over all possible field configurations. So, as if there was a space that contained the classical gravitational, E&M fields and all other variants, all "side-by-side" in this domain. Is that a good way to visualize these objects? It does seem to raise the question; Susskind treated his domain like it was continuous: really it seemed like a manifold. What does it mean for fields to be "next" to each other? How can you continuously vary all fields so that they could all live inside the same manifold?

And, what is the domain space of say the lagrangian density $\mathcal{L}(\phi, \partial\phi)$? I can treat $\mathcal{L}$ just like any other function, but I still can't see what it means physically to have a function of a field, and so my conceptual trick sort of breaks down when I have a multivariate function where one variable is the derivative of another. I'm perfectly fine treating these things as mathematical objects, and I can, say, derive the Euler-Lagrange equation using this function. But this doesn't feel great, it kind of feels like I'm cheating. I know this is a really basic and likely fundamental question, but I couldn't find anything online.

Does anyone have any good way to visualize what such functions mean?

Answer 1

If you are referring to the segment around 8:37, the plot is of field strength (not space) vs energy. It's meant to be a motivational rather than analytically correct plot. Excitations in the field are quantized so this sort of plot only makes sense at some coarse grained approximation.

The point is that excitations in the field store energy (they require energy to produce and release energy when they disappear). You can imagine this really classically as just vibrations in a pool of water. It takes energy to add them (throwing in a rock, for example), but energy can also be extracted from them--this is practically employed in offshore electrical generators.