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In the hope of trying to come up with a clear mental picture for what a transformation is in physics, I encounter some difficulties due to the variety of objects that appear in physics. While no picnic, the notion of transformations in field theory seems fairly straightforward, with the automorphisms on the bundle performing admirably in that role, this does leave the other big common object in Lagrangian mechanics out, which is extended objects. For extended objects, we have some $p$-dimensional submanifold $\Sigma$ (the $p$-brane), with some embedding

$$X : \Sigma \to \mathcal{M}$$

and then your action is either performed on $\Sigma$ or $\mathcal{M}$, ie for instance, for the Nambu-Goto action

$$S[X] = -T \int_{\Sigma} d\mu[X_* g]$$

or, if we expect $g$ to be dynamical

$$S[X, g] = -T \int_{\mathcal M} d^p\sigma d^nx \delta^{(n)}(x^\alpha - X^\alpha(\sigma)) \sqrt{\det\left[ g_{\mu\nu}(x)\partial_a X^\mu(\sigma)\partial_b X^\nu(\sigma) \right]}.$$

In the first case, we may get lucky and have the target space shaped in a pleasing manner in such a way that we can treat everything as a section of a vector bundle once again. For instance, for the theory of a free point particle in Minkowski space, it's entirely possible to treat it as a section of a $\mathbb{R}^3$ bundle over the line, or even as such a bundle over $\bigsqcup \mathbb{R}$ for several free point partices, in which case everything works out as before. But I suspect that this luck runs out pretty fast if we consider fairly general theories.

For instance, I don't believe that we can do much of such a unified treatment if the two collide, as in the case of a point particles in an EM or gravitational field, in which case the description as a theory on the target manifold will have to do, and I don't think it really works as a simple section of a vector bundle, as it is not defined at every point of the manifold.

Nor is the reinterpretation of the embedding as a field theory always possible, as their respective topologies may prevent that. For the Minkowski torus $\mathbb{R} \times S$, this would give rise to a sphere bundle $\pi : \mathbb{R} \times S \to \mathbb{R}$ for a point particle which would certainly work locally but then everything would have to be treated with much more care. Even worse, there may not be any proper bundle $\pi : M \to \Sigma$ with the structure of a fiber bundle at all.

Therefore I feel that these two should probably be treated in a separate manner. But then this leads me to my questions (outside of the usual "is all that I've said correct") :

  • Outside of automorphisms on the fields, diffeomorphisms on $\mathcal{M}$ and $\Sigma$ (which induce the relevant transformations of the brane embeddings), are there any relevant transformations one may perform on such a system? I'm guessing automorphism in the space of all embeddings for $\Sigma$ and $\mathcal{M}$ might be relevant to deform the object as well.
  • Are those two things the most general way we can treat a (reasonable) physical theory, at least as far as Lagrangian mechanics go?
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  • $\begingroup$ I'll just point out that if you have a theory of maps $\psi:\Sigma\rightarrow M$, then you have a theory of sections of $\pi: E=\Sigma\times M\rightarrow\Sigma$. In this case, the set of automorphisms of $E$ decompose into a vertical and a horizontal part, with the vertical being $\text{diff}(M)$ and the horizontal being $\text{diff}(\Sigma)$. So while I am not 100% confident, I suspect the diff groups of the two manifolds exhaust the set of possible transformations. $\endgroup$ – Bence Racskó Aug 29 '19 at 11:24

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