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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.

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

2 Answers 2

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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.


For more infos:

  • 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)
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  • $\begingroup$ 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? $\endgroup$
    – Galen
    Mar 23 at 2:49
  • $\begingroup$ @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. $\endgroup$
    – FrodCube
    Mar 23 at 9:21
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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.

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