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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 BG\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

    PB𝔾 conn P \longrightarrow \mathbf{B}\mathbb{G}_{conn}

    on a phase space PP, then a symmetry of the theory means that there is a GG acting on PP such that the prequantum bundle descends to the homotopy quotient

    P B𝔾 conn P/G \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

    P Ψ V/𝔾 B𝔾 conn B𝔾 \array{ P &\stackrel{\Psi}{\longrightarrow}& V/\mathbb{G} \\ \downarrow &\swArrow_{\simeq}& \downarrow \\ \mathbf{B}\mathbb{G}_{conn} &\longrightarrow& \mathbf{B}\mathbb{G} }

    (take V=V = \mathbb{C} and 𝔾=U(1)\mathbb{G} = U(1) for traditional quantum theory).

    So this is something defined on phase space PP. If that also descends to the homotopy quotient P/GP/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 GG-equivariant. If not, then the wavefunction “breaks” the GG-symmetry.

    Now if on top of this we have that the given Ψ\Psi is a “ground state”, then if it does not descend to the homotopy quotient we say “the GG-symmetry is spontaneously broken”.

    To axiomatize what “ground state” means: introduce another \mathbb{R}-action on PP which is Hamiltonian,i.e. with respect to which the prequantum bundle is required to be equivariant. Then ask Ψ\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.)

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