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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJul 31st 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexI am starting an entry [[spontaneously broken symmetry]]. But so far no conceptualization or anything, just the most basic example for sponatenously broken _global_ symmetry.

   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.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJul 31st 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexhave 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.

   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.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJul 31st 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexexpanded the Idea-section further

   expanded the Idea-section further

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeJan 11th 2015
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexWhat 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.

   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.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeJan 11th 2015
   • (edited Jan 11th 2015)
   • PermaLink
   Author: Urs
   Format: MarkdownItex[edit: I have briefly added the following to the nLab entry [here](http://ncatlab.org/nlab/show/spontaneously+broken+symmetry#FormalizaitonInCohesion)] 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](http://ncatlab.org/schreiber/show/Higher+geometric+prequantum+theory) and then follow the usual story through this translation.)

   [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.)

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJun 26th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to today's * Jose Bernabeu, _Symmetries and their breaking in the fundamental laws of physics_ ([arXiv:2006.13996](https://arxiv.org/abs/2006.13996)) <a href="https://ncatlab.org/nlab/revision/diff/spontaneously+broken+symmetry/19">diff</a>, <a href="https://ncatlab.org/nlab/revision/spontaneously+broken+symmetry/19">v19</a>, <a href="https://ncatlab.org/nlab/show/spontaneously+broken+symmetry">current</a>

   added pointer to today’s

   • Jose Bernabeu, Symmetries and their breaking in the fundamental laws of physics (arXiv:2006.13996)

   diff, v19, current

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeApr 24th 2024
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to today's * Tomas Brauner, *Effective Field Theory for Spontaneously Broken Symmetry* &lbrack;[arXiv:2404.14518](https://arxiv.org/abs/2404.14518)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/spontaneously+broken+symmetry/20">diff</a>, <a href="https://ncatlab.org/nlab/revision/spontaneously+broken+symmetry/20">v20</a>, <a href="https://ncatlab.org/nlab/show/spontaneously+broken+symmetry">current</a>

   added pointer to today’s

   • Tomas Brauner, Effective Field Theory for Spontaneously Broken Symmetry [arXiv:2404.14518]

   diff, v20, current