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theorems:goldstones_theorem [2018/03/05 14:23] jakobadmin [Layman] |
theorems:goldstones_theorem [2020/04/12 15:05] (current) 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]] | ||
+ | |||
+ | <-- | ||
+ | |||
+ | ---- | ||
+ | |||
+ | * For a nice summary see http://pages.physics.cornell.edu/~ajd268/Notes/GoldstoneBosons.pdf | ||
- | <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> | ||