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I am starting an entry spontaneously broken symmetry. But so far no conceptualization or anything, just the most basic example for sponatenously broken global symmetry.
have added sketchy paragraphs on examples of spontaneously broken symmetry in the context of gravity in general and KK-reduction in particular. Main point being to provide precise page-and-verse pointers to the literature.
expanded the Idea-section further
What would a HoTT formalisation look like? Some passage from the context of a $\mathbf{B} G$ to some other or trivial equivariant situation, I guess.
[edit: I have briefly added the following to the nLab entry here]
Yes, so given a prequantum line bundle
$P \longrightarrow \mathbf{B}\mathbb{G}_{conn}$on a phase space $P$, then a symmetry of the theory means that there is a $G$ acting on $P$ such that the prequantum bundle descends to the homotopy quotient
$\array{ P &\longrightarrow& \mathbf{B}\mathbb{G}_{conn} \\ \downarrow & \nearrow \\ P/G }$Now a state (a wavefunction) is a section of the associated line bundle, hence a horizontal morphism in
$\array{ P &\stackrel{\Psi}{\longrightarrow}& V/\mathbb{G} \\ \downarrow &\swArrow_{\simeq}& \downarrow \\ \mathbf{B}\mathbb{G}_{conn} &\longrightarrow& \mathbf{B}\mathbb{G} }$(take $V = \mathbb{C}$ and $\mathbb{G} = U(1)$ for traditional quantum theory).
So this is something defined on phase space $P$. If that also descends to the homotopy quotient $P/G$ (this is hard to draw the diagram for here, but I hope it is clear what I mean ) then that makes the wavefunction also $G$-equivariant. If not, then the wavefunction “breaks” the $G$-symmetry.
Now if on top of this we have that the given $\Psi$ is a “ground state”, then if it does not descend to the homotopy quotient we say “the $G$-symmetry is spontaneously broken”.
To axiomatize what “ground state” means: introduce another $\mathbb{R}$-action on $P$ which is Hamiltonian,i.e. with respect to which the prequantum bundle is required to be equivariant. Then ask $\Psi$ to (be polarized and) be a minimal eigenstate of the respective Hamiltonian. That makes it a “ground state”.
(In short, take the translation between traditional geometric quantization and its formalization in cohesive homotopy-type theory as in hgp and then follow the usual story through this translation.)
added pointer to today’s
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