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  More general invariance of the action functional

+ 7 like - 0 dislike
1084 views

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:PR, where P is the space of time evolutions, ie. differentiable paths in M. The idea is that, given some initial configuration (x0,v0)TM, there is a path in P passing through x0 with velocity v0 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 Px0,v0 as the set of paths in M passing through x0 with velocity v0. A diffeomorphism ϕ:MM is a symmetry of the theory S:PR iff for each (x0,v0)TM, γPx0,v0 is a critical point of S|Px0,v0 iff ϕγ is a critical point of S|Pϕ(x0),ϕ(v0).

It is not obvious to me that this implies S=Sϕ1, where ϕ1 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 H1(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.

This post has been migrated from (A51.SE)
asked Dec 30, 2011 in Theoretical Physics by Ryan Thorngren (1,925 points) [ no revision ]

2 Answers

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Let ϕs:PP be the induced diffeomorphism on the space of paths. You are assuming that the zero set Zero(dS) coincides with the zero set Zero(ϕsdS). This does not even imply that dS=ϕsdS, let alone S=ϕsS.

An example would be a free particle on R. Let S=˙x(t)2dt and consider the scaling transformation xexp(s)x. Then the critical points are straight lines x(t)=x0+v0t 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(ϕsdS) and dS=ϕsdS, consider the graph of dS in TP. The first condition only fixes the intersection points with the zero section, while the second condition fixes the graph itself. Clearly, in the C 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=ϕsdS does not imply that S=ϕsS: you only know that S=ϕsS+c(s), where c is a locally-constant function on P which vanishes at s=0.

This post has been migrated from (A51.SE)
answered Dec 30, 2011 by Pavel Safronov (1,120 points) [ no revision ]
Thanks for the simple counterexample.

This post has been migrated from (A51.SE)
+ 2 like - 0 dislike

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

This post has been migrated from (A51.SE)
answered Dec 30, 2011 by Squark (1,725 points) [ no revision ]
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.

This post has been migrated from (A51.SE)
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

This post has been migrated from (A51.SE)
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?

This post has been migrated from (A51.SE)
If your transformation does not even preserve dS, there is no hope of getting any conservation laws.

This post has been migrated from (A51.SE)

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