More general invariance of the action functional

Question

I will formulate my question in the classical case, where things are simplest.

Usually when one discusses a continuous symmetry of a theory, one means a one-parameter group of diffeomorphisms of the configuration space $M$ which fix the action functional $S:P\rightarrow \mathbb{R}$, where $P$ is the space of time evolutions, ie. differentiable paths in $M$. The idea is that, given some initial configuration $(x_0,v_0)\in TM$, there is a path in $P$ passing through $x_0$ with velocity $v_0$ and minimizing $S$ among all such paths. I will assume that this path is unique, which is almost always the case. Thus, if a diffeomorphism fixes $S$, it commutes with determining this path. One says that the physics is unchanged by taking the diffeomorphism.

Now here's the question: are there other diffeomorphisms which leave the physics unaltered? All one needs to do is ensure that the structure of the critical points of $S$ are unchanged by the diffeomorphism.

I'll be more particular. Write $P_{x_0,v_0}$ as the set of paths in $M$ passing through $x_0$ with velocity $v_0$. A diffeomorphism $\phi:M\rightarrow M$ is a symmetry of the theory $S:P\rightarrow \mathbb{R}$ iff for each $(x_0,v_0) \in TM$, $\gamma \in P_{x_0,v_0}$ is a critical point of $S|_{P_{x_0,v_0}}$ iff $\phi \circ \gamma$ is a critical point of $S|_{P_{\phi (x_0),\phi^* (v_0)}}$.

It is not obvious to me that this implies $S = S \circ \phi^{-1}$, where $\phi^{-1}$ is the induced map by postcomposition on $P$. If there are such symmetries, what can we say about Noether's theorem?

A perhaps analogous situation in the Hamiltonian formalism is in the correspondence between Hamiltonian flows and infinitesimal canonical transformations. Here, a vector field $X$ can be shown to be an infinitesimal canonical transformation iff its contraction with the Hamiltonian 2-form is closed. This contraction can be written as $df$ for some function $f$ (and hence $X$ as the Hamiltonian flow of $f$) in general iff $H^1(M)=0$. Is this analogous? What is the connection? It's been pointed out that this obstruction does not depend on the Hamiltonian, so is likely unrelated.

Thanks!

PS. If someone has more graffitichismo, tag away.

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Answer 1

Let $\phi_s:P\rightarrow P$ be the induced diffeomorphism on the space of paths. You are assuming that the zero set $Zero(dS)$ coincides with the zero set $Zero(\phi_s^* dS)$. This does not even imply that $dS = \phi_s^* dS$, let alone $S = \phi_s^* S$.

An example would be a free particle on $\mathbf{R}$. Let $S=\int \dot{x}(t)^2dt$ and consider the scaling transformation $x\rightarrow\exp(s) x$. Then the critical points are straight lines $x(t)=x_0+v_0t$ and the transformation clearly preserves them. On the other hand, the action gets multiplied by $\exp(2s)$.

To understand the differences between $Zero(dS)=Zero(\phi_s^* dS)$ and $dS=\phi_s^* dS$, consider the graph of $dS$ in $T^*P$. The first condition only fixes the intersection points with the zero section, while the second condition fixes the graph itself. Clearly, in the $C^\infty$ world you can adjust the behavior of $dS$ away from the intersection points as much as you like. In the holomorphic world it would be enough to remember the Taylor expansions around the critical points.

Finally, $dS=\phi_s^*dS$ does not imply that $S=\phi_s^*S$: you only know that $S=\phi_s^*S + c(s)$, where $c$ is a locally-constant function on $P$ which vanishes at $s=0$.

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Comments

Thanks for the simple counterexample.

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Answer 2

Firstly, its enough for the variation of the action to be a total divergence (in the more general field theory case), i.e. in the case of mechanics - a time derivative. The classic example would be boost symmetry - transitions between frames of reference. Only problem it doesn't quite fit your framework since it depends explicitly on the time coordinate

Secondly, it's enough for this to hold on-shell, i.e. when the equations of motion are satisfied. In the field theory case the classic example for this is supersymmetry. Probably a mechanical (1-dimensional) analogue exists. However, this example lives in the slightly more general world of supermanifolds. Of course it's possible to construct artificial examples of this kind which fit your setting precisely - you can tweak the action functional almost any way you like away from critical points (just take care to avoid creating new critical points)

Thirdly, as the examples above show the statement "usually when one discusses a continuous symmetry of a theory, one means a one-parameter group of diffeomorphisms of the configuration space..." is not correct. Instead we can consider any time-local transformation on the history space. Btw, this entire discussion is equally relevant to discrete symmetries. Also one often considers multi-parameter groups but this is already semantics

I don't think your Hamiltonian analogy is correct since my examples above don't involve any topological obstructions. Btw, an example of a flow which is symplectic but not Hamiltonian is the time evolution of a particle on a circle under the influence of a constant force driving it e.g. clockwise everywhere, which is a system without Lagrangian formulation

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Comments

Ah, thanks. You've clarified quite a few of my thoughts. I wanted to focus on continuous symmetries because underlying all this I was thinking about its ramifications for Noether's theorem. Could you clarify what you mean by "time-local transformation on the history space"? I'm thinking some sort of automorphism of the space of possible on-shell worldlines, but I'm not sure what structure you'd want to preserve for each line.

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Time-local means the result of the transformation at each point of time depends only on a small time-neighbourhood. This includes all transformations which can be expressed using time itself, configuration space coordinates and a finite number of derivatives of the configuration space coordinates, which is a popular definition of locality in physics texts. However, I believe the correct mathematical definition is the following. A "time local" transformation is a smooth automorphism of the _sheaf_ of (off-shell) histories. Some technicalities are involved in defining smoothness for infinite dim

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Okay, that makes some sense. Now, returning to the (simplistic) formulation I had above, supposing the existence of a symmetry which preserves the critical structure of the action but not necessarily the action, what can we say about any conserved quantities arising therefrom?

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If your transformation does not even preserve $dS$, there is no hope of getting any conservation laws.

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