advanced_tools:group_theory:quotient_group

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advanced_tools:group_theory:quotient_group [2017/12/17 13:01] jakobadmin |
advanced_tools:group_theory:quotient_group [2018/05/15 06:58] (current) jakobadmin ↷ Links adapted because of a move operation |
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<tabbox Why is it interesting?> | <tabbox Why is it interesting?> | ||

- | Quotient groups are crucial to understand, for example, [[advanced_notions:symmetry_breaking|symmetry breaking]]. When a [[advanced_tools:group_theory|group]] $G$ breaks to a [[advanced_tools:group_theory:subgroup|subgroup]] $H$ the resulting [[advanced_notions:symmetry_breaking:goldstones_theorem|Goldstone bosons]] live in the quotient space: $G/H$. | + | Quotient groups are crucial to understand, for example, [[advanced_notions:symmetry_breaking|symmetry breaking]]. When a [[advanced_tools:group_theory|group]] $G$ breaks to a [[advanced_tools:group_theory:subgroup|subgroup]] $H$ the resulting [[theorems:goldstones_theorem|Goldstone bosons]] live in the quotient space: $G/H$. |

Moreover, quotient groups are a powerful way to understand geometry. Instead of a long list of axioms one can study geometry by treating the corresponding space as a homogeneous space (= coset space) and then study invariants of transformation groups of this homogeneous space. | Moreover, quotient groups are a powerful way to understand geometry. Instead of a long list of axioms one can study geometry by treating the corresponding space as a homogeneous space (= coset space) and then study invariants of transformation groups of this homogeneous space. | ||

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--> Goldstone Bosons# | --> Goldstone Bosons# | ||

- | The famous [[advanced_notions:symmetry_breaking:goldstones_theorem|Goldstone bosons]] that appear through the process of spontaneous [[advanced_notions:symmetry_breaking|symmetry breaking]] of a group $G$ to some subgroup $H$ live in the coset space $G/H$. | + | The famous [[theorems:goldstones_theorem|Goldstone bosons]] that appear through the process of spontaneous [[advanced_notions:symmetry_breaking|symmetry breaking]] of a group $G$ to some subgroup $H$ live in the coset space $G/H$. |

This can be understood by considering an explicit example: | This can be understood by considering an explicit example: |

advanced_tools/group_theory/quotient_group.txt · Last modified: 2018/05/15 06:58 by jakobadmin

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