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 advanced_tools:group_theory:quotient_group [2017/12/17 12:01]jakobadmin advanced_tools:group_theory:quotient_group [2018/05/15 04:58] (current)jakobadmin ↷ Links adapted because of a move operation Both sides previous revision Previous revision 2018/05/15 04:58 jakobadmin ↷ Links adapted because of a move operation2017/12/17 12:01 jakobadmin 2017/12/17 12:01 jakobadmin [Examples] 2017/12/17 11:59 jakobadmin [Student] 2017/12/17 11:58 jakobadmin [Student] 2017/12/17 11:57 jakobadmin [Student] 2017/12/17 11:57 jakobadmin [Student] 2017/12/17 11:57 jakobadmin created 2018/05/15 04:58 jakobadmin ↷ Links adapted because of a move operation2017/12/17 12:01 jakobadmin 2017/12/17 12:01 jakobadmin [Examples] 2017/12/17 11:59 jakobadmin [Student] 2017/12/17 11:58 jakobadmin [Student] 2017/12/17 11:57 jakobadmin [Student] 2017/12/17 11:57 jakobadmin [Student] 2017/12/17 11:57 jakobadmin created Line 3: Line 3:  ​  ​ - 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. Line 64: Line 64: --> 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: