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This question refers to the effort of conceptualizing a "reverse" problem: If it is meaningful mathematically to have a group(Lie) acting on a set $M$ and mapping the element of $M$ to another element on a different set $N,N\neq M$, is it interesting from a physical perspective?

Assume as usual a $G$-set of a linear representation of a group( the set on which the group acts is a vector space and the action respects the linear structure). We can then assume that for an element $p \in V $, where $V $ is the representation, we have $g \in G | gp=q , q \in V $. The action is obviously transitive.

Now, to a more subtle subject, let' s assume that the action is still transitive but we wish to point out the fact that there exists an isotropy subgroup of $G$, $H$: $H \subset G $ which we take as closed. The isotropy group means that $\forall h \in H | hp=p , p\in M$, where $M$ now is the general manifold- G-set. This means that now, we don' t care for a representation of $G$, but if $G$ acts transitively on $M$, and $H$ is the isotropy group, it can be shown there is a unique manifold structure for $M$ coinciding with the quotient space: $M = G/H $. But, as I see it, for the group $G$ to act transitively, it means again that for $p,q \in M | gp=q, g \in G $.

Such a case is presented for example in non-linear sigma models and more particular to the low-energy effective theory of QCD for the interaction of nucleons. Beyond the immediate understanding of the usefulness of such an approach, and also it' s mathematical clarity, I' m trying to conceptualize whether or not it is meaningful to have a set M on which G acts and mapping us to elements of another set. The reason of this question is based on the fact that for the effective model to work, we still need to suppose that G acts transitively.

Question

So, is there meaningful at all to think mathematically that a group G may act on a set M and mapping the elements of M to another set N, and if so, is it at all meaningful and interesting to think like this in physics (not only for solving a particle classification problem, though not rejecting such a possibility) ? Would it have any use to think on such a basis on a problem like the effective model mentioned above, or exactly because it is without meaning, any such conceptualization is useless and so there is a straightforward way for thinking: we either find a suitable linear representation or a non-linear based on the transitive action of a group on it' s G-set.


  • quoting from Weinberg's QFT-2 :

The transformation rules make no reference to a particular linear $G$ transformation properties of the original fields. Indeed we did not need to start with a lagrangian that was invariant under linear $G$ transformations in order to deduce these transformation rules.

But, it still seems to me that we assume the group acts transitively.

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If I understand your question correctly, you are asking if it makes sense, from a physical point of view, to consider maps of the form

$$\pi:G\to\operatorname{Map}(M,N)$$

with the property that $\pi$ is a group homomorphism.

When $M=N$, there is a natural product on $\operatorname{Map}(M,M) = \operatorname{Map}(M)$, which is composition, for surely the composition of two maps from $M$ to $M$ yields a map from $M$ to $M$. But what about the case $M\neq N$? What product can we take on $\operatorname{Map}(M,N)$ to turn it into a group?

So, to me, it seems that we have problems already from a mathematical point of view. Perhaps a physical argument that could go with this difficulty is that a group represents some sort of symmetry of an object (e.g. a manifold), and there doesn't seem to be a clear meaning for the "symmetry of an object that is mapped into another object".

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  • $\begingroup$ Hi and thanks for the reply. In the OP it is consider in a general fashion a group action on a manifold( a linear or not representation) and the question regards whether or not it has any meaning to consider the image of the group action not to belong to the G-set that is acted on; do you claim that not only perhaps physically but also mathematically this is ill-defined as a group action? Thanks again. $\endgroup$ – Constantine Black Jan 11 '18 at 9:39
  • $\begingroup$ A group action is a way to associate elements of a group to transformations of some space. If you want to associate group elements to maps from one space to another, then this is not a group action. The difficulty of trying to do something like this is expressed in the answer above. $\endgroup$ – Phoenix87 Jan 13 '18 at 14:13

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