I want to propose a metatheorem about physics. I would be surprised if it has not been stated in some form or another elsewhere by someone else.

 

The Fundamental Metatheorem of Physics

 

Given a space of fields [math]\mathcal{C} [/math] (sections of a bundle over spacetime) with (local) coordinates [math]\phi^{a}[/math], an action [math]S = \int_{M} d^{n}x \mathcal{L}(\phi, \partial \phi)[/math], appropriate boundary conditions and the representation of the fields under the appropriate symmetry groups then all the physics is contained in the above data.

 

By this I mean that all observables, classical and quantum can be calculated from the above with no extra data needed. So, even more, I "claim" that any ghosts, quantum corrections etc can all be derived (does not need to be uniquely in general) from the above.

 

I make no assertion that it is easy, well understood (in general) or anything like that. The metatheorem above is really a formal statement. It does not give you the tools to achieve this.

 

So, does the metatheorem hold? What do you think about? Is it even worth formulating this metatheorem? All points of view welcomed.

That sounds like it could be interesting, ajb.

 

I take it that d^n x isn't intended to apply to the Hilbert space of quantum mechanics, right? --or thermodynamic phase space. Do you think you could maybe describe what you mean in simple terms for the rest of us?

[math]d^{n}x[/math] is the coordinate measure on n-dimensional space.

 

Basically, the statement is "everything is contained in the Lagrangian density".