Non-linear group representations

Different representations of the Lorentz group in infinite-dimensional vector spaces are related to the types of particles (classification by spin and mass). These representations are linear. Are non-linear representations of groups used in physics? I would like to know examples of non-linear representations.

• I thought that only linear actions of groups were considered to be representations. Are we using a different definition of "representation" here? Mar 23 at 2:46

Yes they are. In particular in Quantum Field Theories non-linear representation of groups appear in the context of spontaneous symmetry breaking.

You start from a Lagrangian with a symmetry group $$\mathcal G$$ that is then broken to a subgroup $$\mathcal H$$. The new Lagrangian after symmetry breaking is then linearly invariant under $$\mathcal H$$, but it still contains the symmetry from the full group $$\mathcal G$$, it's just more hidden. In particular some fields will transform non-linearly.

As an explicit example (not the simplest, but for which I already have the results) consider a $$SO(3)$$ invariant theory with a field $$\Phi$$ that takes a vev $$f$$ in the $$3$$ direction, breaking the symmetry to $$SO(2)$$.

As usual, you parametrize $$\Phi$$ as a radial mode $$\sigma$$ plus two Goldstones modes $$\Phi(x) = e^{i \frac{\sqrt{2}}{f} \Pi^i(x) T^i} (0,\,0,\,f+\sigma(x))$$ where $$i=1,2$$ and $$T^i$$ are the "broken" generators.

The theory you get in terms of $$\sigma$$, $$\Pi_1$$ and $$\Pi_2$$ is still invariant under the full $$SO(3)$$ group:

the SO(2) subgroup just linearly mixes the two $$\Pi$$ fields, while transformations along the broken generators act as

$$\Pi^i \to \Pi^i + |\Pi| \cot \frac{|\Pi|}{f}\alpha^i + \left( \frac{f}{|\Pi|} -\cot\frac{|\Pi|}{f} \right) (\alpha_j \Pi^j) \frac{\Pi^i}{|\Pi|}$$ where $$\alpha$$ are the parameters of the transformations and $$|\Pi| = \sqrt{(\Pi^1)^2+(\Pi^2)^2}$$

This transformation is clearly non-linear and in particular it implies a shift-symmetry under $$\Pi \to \Pi + c$$, whose consequence is also that the $$\Pi$$ must be massless.

Similar constructions can be also done for the Poincaré group. For example in 1405.7384 they show that the action for a point particle can be derived from a non-linear realization of the broken Poincaré group.

• Weinberg QFT vol 2, the chapter on spontaneous symmetry breaking
• CCWZ constructions (explained in Weinberg, but also reviewed in chapter 2 of 1506.01961 from which I took this example)
• Isn't a "non-linear representation of a group" a contradiction in terms? I thought that representations of a group were linear actions of that group by definition. Can you clarify? Mar 23 at 2:49
• @DifferentialCovariance True. A representation is linear by definition. I've heard this called a "realization", meaning that it only concretely implements the product rule on some space. With this definition a representation is a linear realization. Mar 23 at 9:21

In general, if $$G$$ is a group and $$M$$ is some set we can define a left $$G$$-action to be a map $$\rho:G\times M\to M$$ such that $$\rho(e,x)=x,\quad \forall x\in M,\\ \rho(g,\rho(h,x))=\rho(gh,x),\quad \forall g,h\in G, x\in M$$

where $$e\in G$$ is the identity. In general we just denote $$\rho(g,x)=g\cdot x$$ and then the two properties are $$e\cdot x=x,\quad \forall x\in M,\\ g\cdot(h\cdot x)=(gh)\cdot x,\quad \forall g,h\in G,x\in M.$$

A right $$G$$-action can be defined similarly. This is a very general concept in which the elements of a group are realized through the map $$\rho$$ as transformations acting in a particular set of objects. Notice there is no extra assumptions on $$G$$ or on the set of objects $$M$$.

It can be the case that $$M$$ is a vector space. In that case you could consider one specific case of this definition, in which the map $$x\mapsto \rho(g,x)$$ for $$g\in G$$ fixed is linear. In that case, the elements of $$G$$ are realized as linear transformations. This makes sense now because the set on which $$G$$ acts has more structure and we can bring that structure into play. This is nothing more than a group representation. In that case a group representation is one specific case of a group action.

And indeed there are cases in Physics where one considers actions which do not act linearly. Now suppose that $$G$$, apart from a group, is also a smooth manifold, such that the operations of multiplication and inversion are smooth. We then say that $$G$$ is a Lie group. Suppose further that $$M$$ is also a smooth manifold. In that case we may consider a left $$G$$-action $$\rho:G\times M\to M$$ which now is supposed to be smooth.

This finds its place in Physics for example in the subject of non-abelian gauge theory. In standard QFT when we have a non-abelian gauge theory with structure group $$G$$ one often just says that the gauge potential is just one adjoint-valued field $$A_\mu^a(x)$$. The fact is that there is more to it.

Indeed, given the structure group $$G$$ one may consider what one calls principal $$G$$-bundles over $$M$$. I won't dive into the technical definition here (if you want to get the details I suggest the book Modern Differential Geometry for Physicists by Isham), but I will just say that a principal $$G$$-bundle is one particular kind of smooth manifold $$P$$ with a projection map $$\pi:P\to M$$, which carries a $$G$$-action such that $$M\simeq P/G$$. Now, when we have a principal $$G$$-bundle we may talk about connections on said bundle.

When $$P$$ is trivial, meaning that $$P\simeq M\times G$$ with a trivial right $$G$$-action $$(x,g)\cdot h=(x,gh)$$ a connection can be specified by one adjoint-valued field $$A_\mu^a(x)$$. That is the standard QFT story. But when $$P$$ is non-trivial, the gauge field cannot be specified anymore by a single $$A_\mu^a(x)$$. In that case, the most accurate way of thinking of the gauge field is as a connection on a principal $$G$$-bundle. In particular, the topology of the principal bundle is in a sense part of what specifies the gauge field, inasmuch as in gravity the spacetime topology is part of what specifies the metric. Matter fields then are sections of the so-called associated bundles constructed by representations of $$G$$.

That story (which obviously cannot be told in full in a single answer) is just meant to call your attention to the fact that non-linear group actions are central to non-abelian gauge theory, since they enter the very definition of the main object in these theories: namely the principal $$G$$-bundle over which the gauge field is meant to be a connection.