theorems:goldstones_theorem

This shows you the differences between two versions of the page.

Both sides previous revision Previous revision Next revision | Previous revision Last revision Both sides next revision | ||

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 |
||
---|---|---|---|

Line 1: | Line 1: | ||

====== 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> | ||

Line 117: | Line 111: | ||

<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]] | ||

+ | |||

+ | <-- | ||

---- | ---- | ||

Line 123: | Line 133: | ||

- | <tabbox Researcher> | + | <tabbox Abstract> |

<blockquote> | <blockquote> | ||

Line 157: | Line 167: | ||

<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: 2020/04/12 15:05 by jakobadmin

Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Share Alike 4.0 International