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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJul 31st 2011

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.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJul 31st 2011

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.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeJul 31st 2011

expanded the Idea-section further

• CommentRowNumber4.
• CommentAuthorDavid_Corfield
• CommentTimeJan 11th 2015

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.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJan 11th 2015
• (edited Jan 11th 2015)

[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.)

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeJun 26th 2020