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 theorems:goldstones_theorem [2018/05/15 06:58]jakobadmin ↷ Page moved from advanced_notions:symmetry_breaking:goldstones_theorem to theorems:goldstones_theorem theorems:goldstones_theorem [2018/05/15 07:00]jakobadmin Both sides previous revision Previous revision 2020/04/12 15:05 jakobadmin 2018/05/15 07:00 jakobadmin 2018/05/15 06:59 jakobadmin 2018/05/15 06:58 jakobadmin ↷ Page moved from advanced_notions:symmetry_breaking:goldstones_theorem to theorems:goldstones_theorem2018/03/28 17:15 jakobadmin [Student] 2018/03/05 14:23 jakobadmin [Layman] 2017/12/04 08:01 external edit2017/11/15 17:07 jakobadmin 2017/11/15 17:06 jakobadmin [Examples] 2017/11/15 17:06 jakobadmin [Examples] 2017/11/06 09:49 jakobadmin [Researcher] 2017/10/27 13:07 jakobadmin ↷ Page moved from symmetry_breaking:goldstones_theorem to advanced_notions:symmetry_breaking:goldstones_theorem2017/09/29 07:32 jakobadmin [Student] 2017/09/29 07:31 jakobadmin [Student] 2017/09/29 07:29 jakobadmin [Student] 2017/09/29 07:28 jakobadmin created Next revision Previous revision 2020/04/12 15:05 jakobadmin 2018/05/15 07:00 jakobadmin 2018/05/15 06:59 jakobadmin 2018/05/15 06:58 jakobadmin ↷ Page moved from advanced_notions:symmetry_breaking:goldstones_theorem to theorems:goldstones_theorem2018/03/28 17:15 jakobadmin [Student] 2018/03/05 14:23 jakobadmin [Layman] 2017/12/04 08:01 external edit2017/11/15 17:07 jakobadmin 2017/11/15 17:06 jakobadmin [Examples] 2017/11/15 17:06 jakobadmin [Examples] 2017/11/06 09:49 jakobadmin [Researcher] 2017/10/27 13:07 jakobadmin ↷ Page moved from symmetry_breaking:goldstones_theorem to advanced_notions:symmetry_breaking:goldstones_theorem2017/09/29 07:32 jakobadmin [Student] 2017/09/29 07:31 jakobadmin [Student] 2017/09/29 07:29 jakobadmin [Student] 2017/09/29 07:28 jakobadmin created Last revision Both sides next revision Line 1: Line 1: ====== Goldstone'​s theorem ====== ====== Goldstone'​s theorem ====== -  ​ - <​blockquote>​ - Goldstone'​s theorem states that whenever a continuous global symmetry is spontaneously broken, there exists a massless excitation about the spontaneously broken vacuum. Decomposing $\Phi(x)=|\Phi(x) |e^{i\rho(x)}$,​ $\rho$ transforms as $\rho(x) \to \rho(x) + \theta$. Hence the Lagrangian can depend on $\rho$ only via the derivative = $\partial_\mu \rho$; there cannot be any mass term for $\rho$, and it is a massless field. $\rho$ --- identified as the field which transforms inhomogeneously under the broken symmetry --- is referred to as the Goldstone boson. - - <​cite>​https://​arxiv.org/​pdf/​1703.05448.pdf​ - ​ - <​tabbox ​Layman> + <​tabbox ​Intuitive> * For an intuitive explanation of Goldstone'​s theorem, see [[http://​jakobschwichtenberg.com/​understanding-goldstones-theorem-intuitively/​|Understanding Goldstone’s theorem intuitively]] by J. Schwichtenberg * For an intuitive explanation of Goldstone'​s theorem, see [[http://​jakobschwichtenberg.com/​understanding-goldstones-theorem-intuitively/​|Understanding Goldstone’s theorem intuitively]] by J. Schwichtenberg ​ ​ - <​tabbox ​Student> + <​tabbox ​Concrete> <​blockquote>​ <​blockquote>​ Line 117: Line 111: <​cite>​http://​www.jstor.org/​stable/​pdf/​10.1086/​518324.pdf​ <​cite>​http://​www.jstor.org/​stable/​pdf/​10.1086/​518324.pdf​ + + + ---- + + **Examples** + + --> Landau phonons in Bose-Einstein condensates#​ + + "The Bose-Einstein condensation is characterized by the + breaking of a global U(1) gauge group (acting on the Bose particle field + as the U(1) group of Example 1), as very clearly displayed by the free + Bose gas.5 The U(1) breaking leads to the existence of Goldstone + modes, the so-called Landau phonons, and the existence of such excitations + may in turn indicate the presence of a broken U(1) symmetry"​ [[https://​arxiv.org/​pdf/​1502.06540.pdf |Source]] ​ + + <-- ---- ---- Line 123: Line 133: - <​tabbox ​Researcher> + <​tabbox ​Abstract> <​blockquote>​ <​blockquote>​ Line 157: Line 167: <​cite>​https://​arxiv.org/​pdf/​1612.00003.pdf​ <​cite>​https://​arxiv.org/​pdf/​1612.00003.pdf​ - --> Common Question 1# + - +
​ - <-- + Goldstone'​s theorem states that whenever a continuous global symmetry is spontaneously broken, there exists a massless excitation about the spontaneously broken vacuum. Decomposing $\Phi(x)=|\Phi(x) |e^{i\rho(x)}$,​ $\rho$ transforms as $\rho(x) \to \rho(x) + \theta$. Hence the Lagrangian can depend on $\rho$ only via the derivative = $\partial_\mu \rho$; there cannot be any mass term for $\rho$, and it is a massless field. $\rho$ --- identified as the field which transforms inhomogeneously under the broken symmetry --- is referred to as the Goldstone boson. - --> Common Question 2# + <​cite>​https://​arxiv.org/​pdf/​1703.05448.pdf​ + - - <-- - ​ -  ​ - --> Landau phonons in Bose-Einstein condensates#​ - "The Bose-Einstein condensation is characterized by the - breaking of a global U(1) gauge group (acting on the Bose particle field - as the U(1) group of Example 1), as very clearly displayed by the free - Bose gas.5 The U(1) breaking leads to the existence of Goldstone - modes, the so-called Landau phonons, and the existence of such excitations - may in turn indicate the presence of a broken U(1) symmetry"​ [[https://​arxiv.org/​pdf/​1502.06540.pdf |Source]] ​ - <-- - ​ -  ​
theorems/goldstones_theorem.txt · Last modified: 2020/04/12 15:05 by jakobadmin