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For reference, I come from a mathematics background (mostly differential geometry). I have a very limited understanding of upper-level physics (I'm currently trying to fix this).

This is my current understanding of symmetry breaking:

Suppose we have a system-describing thingamabob (e.g. a Lagrangian) for a given space and a (Lie) group $G$ acting on that space such that the system-describing thingamabob is invariant under that action. If a particular instance described by the system-describing thingamabob is only invariant under the action of a (closed?) subgroup $H$ of $G$, then we say that the symmetry was broken from $G$ to $H$.

I've been led to believe that this concept gives a particularly nice way to describe how gravity works, but this does not seem readily apparent to me. Thus, I'm led to two questions:

  1. Is my understanding of "symmetry breaking" correct?

  2. How would one go about using this idea?

Please keep in mind that I will understand terms like "Lie group", "manifold", and "curvature", but would be floundered by terms like "boson", "fermion", or "a nice, clean idea expressed in just about a million indices in Einstein notation".

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Your general idea is correct. As an explicit example, consider a field theory of $n$ scalar fields that can be placed in an $n$-component column vector $\phi$, with a potential, $V(\phi)$. In addition, suppose a symmetry group, $G$ of the Lagrangian.

The potential $V(\phi)$ has a family of degenerate minima that form a manifold, $\mathcal M$. Now, for a theory with a vacuum expectation value, $\phi_0$, we expect $D(g)\phi_0$ for an appropriate representation, for an element $g\in G$, to be a valid minimum, if $\phi_0$ is a minimum.

However, if $G$ is spontaneously broken to a subgroup, $H$, then we expect that only $D(h)\phi_0$ for $h\in H$ to also be a valid minimum, rather than the full group, $D(g)\phi_0$.

Therefore, $D(gh)\phi_0 = D(g)\phi_0$ and we can define equivalence classes by the fact that $g_1 \sim g_2$ if and only if $g_2 = g_1 h$ for $h \in H$. Such equivalence classes you will recognise as being in the coset, $G/H$.

The vacuum manifold then is simply, $\mathcal M = G/H$. The global properties of $\mathcal M$ are particularly important in physics. For example, in three spatial dimensions, magnetic monopoles only arise if the group $\pi_2 (G/H)$ is non-trivial.

Your second question, how one would go about 'using this idea' is extremely broad; spontaneous symmetry breaking arises in nearly every branch of mathematical physics.


Application

Spontaneous symmetry breaking can be used as a mechanism to endow massless fields with mass, as is the case with the Higgs mechanism in the Standard Model.

As an example, consider an $SO(3)$ gauge theory, with three real scalars, $\phi_i$, with Lagrangian,

$$\mathcal L = -\frac14 (F^a_{\mu\nu})^2 + \frac12(\partial_\mu \phi_i - ig A^a_\mu \tau^a_{ij}\phi_j)^2 + \frac12 m^2 \phi^2_i - \frac{\lambda}{4!}(\phi_i)^4.$$

The potential is minimised for $|\langle \vec \phi\rangle|^2 = v^2 = 6m^2/\lambda$. By an $SO(3)$ transformation, we can pick all the components of $\vec \phi$ to vanish except, $\langle \phi_3 \rangle = v$. You can check that $\langle \vec \phi \rangle$ is invariant under $$H = SO(2) \subset G = SO(3).$$

$SO(2)$ and $SO(3)$ differ in the number of generators by two, giving rise to two Goldstone bosons. We can expand the Lagrangian to see,

$$\mathcal L = -\frac14 (F^a_{\mu\nu})^2 + \frac{1}{4}g^2 A^a_\mu A^b_\mu \vec{v}^T \{\tau^a, \tau^b\}\vec{v}$$

and inserting the explicit generators $\tau^a$ one finds,

$$\mathcal L = -\frac14 (F^a_{\mu\nu})^2 + \frac12 m_A^2(A^1_\mu A^1_\mu + A^2_\mu A^2_\mu)$$

where we have identified a mass, $m_A^2 = g^2 v^2$. Thus, we find the theory possesses a massless gauge boson, $A^3$ as well as two massive gauge bosons, $A^{1,2}$. As you come from a mathematics background, remember to think of $A$ as a connection, taking values in the Lie algebra of the gauge group.


A final word: Since you state your own interests lie in Lie groups and smooth manifolds, if symmetry breaking is of interest to you, then you will probably find things like solitons, soliton moduli spaces and vacuum manifolds of interest mostly, and a good book is,

  • J. Weinberg, Classical Solutions in Quantum Field Theory, Cambridge University Press.
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  • $\begingroup$ If you would give a more specific answer to my second question, then I'm definitely ready to accept this answer. Perhaps giving an example of how it is used, or a general idea of how it is used ("you start with A, ...[symmetry breaking stuff]..., and end up with B")? $\endgroup$ – Robin Goodfellow Dec 29 '16 at 20:10
  • $\begingroup$ @RobinGoodfellow See the updated answer. $\endgroup$ – JamalS Dec 29 '16 at 22:06
  • $\begingroup$ Excellent! Just for my curiosity, what are $m$ and $\lambda$ in the Lagrangian? From looking around, I'm almost certain that $F$ is the "curvature" of $A$... $\endgroup$ – Robin Goodfellow Dec 30 '16 at 22:57
  • $\begingroup$ @RobinGoodfellow Yes, $F$ is the curvature of $A$, and we can think of $A$ as the section of a principal bundle. From the physics side, $A$ is a potential and $F$ is the field strength. We interpret $\lambda$ as a coupling constant describing an interaction between the $\phi$ fields, and $m$ is the mass of the particles which are excitations of the $A^{1,2}$ fields. $\endgroup$ – JamalS Dec 30 '16 at 23:02
  • $\begingroup$ @RobinGoodfellow Actually, to be more correct, the field $A$ is to be thought of as the section of the bundle, $P \times_G \pi$ where $\pi$ is the appropriate representation and $P$ is the principal bundle. $\endgroup$ – JamalS Dec 31 '16 at 12:57

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