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theorems:goldstones_theorem [2018/05/15 04:58]
jakobadmin ↷ Page moved from advanced_notions:symmetry_breaking:goldstones_theorem to theorems:goldstones_theorem
theorems:goldstones_theorem [2018/05/15 05:00]
jakobadmin
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 ====== Goldstone'​s theorem ====== ====== Goldstone'​s theorem ======
  
-<tabbox Why is it interesting?> ​ 
  
-<​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</​cite>​ 
-</​blockquote>​ 
  
-<​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>​
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 <​cite>​http://​www.jstor.org/​stable/​pdf/​10.1086/​518324.pdf</​cite>​ <​cite>​http://​www.jstor.org/​stable/​pdf/​10.1086/​518324.pdf</​cite>​
 </​blockquote>​ </​blockquote>​
 +
 +
 +----
 +
 +**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]] ​
 +
 +<--
  
 ---- ----
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-<​tabbox ​Researcher+<​tabbox ​Abstract
  
 <​blockquote>​ <​blockquote>​
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 <​cite>​https://​arxiv.org/​pdf/​1612.00003.pdf</​cite></​blockquote>​ <​cite>​https://​arxiv.org/​pdf/​1612.00003.pdf</​cite></​blockquote>​
  
---Common Question 1#+<tabbox Why is it interesting?​
  
-  +<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.
  
---Common Question 2#+<​cite>​https://​arxiv.org/​pdf/​1703.05448.pdf</​cite>​ 
 +</​blockquote>
  
-  
-<-- 
-  ​ 
-<tabbox 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]] ​ 
-<-- 
  
  
-  ​ 
-<tabbox History> ​ 
  
 </​tabbox>​ </​tabbox>​
  
  
theorems/goldstones_theorem.txt · Last modified: 2018/05/15 05:00 by jakobadmin