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theorems:goldstones_theorem [2017/09/29 07:29] jakobadmin [Student] |
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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> | ||
- | <note tip> | + | <tabbox Intuitive> |
- | Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party. | + | * For an intuitive explanation of Goldstone's theorem, see [[http://jakobschwichtenberg.com/understanding-goldstones-theorem-intuitively/|Understanding Goldstone’s theorem intuitively]] by J. Schwichtenberg |
- | </note> | + | |
| | ||
- | <tabbox Student> | + | <tabbox Concrete> |
<blockquote> | <blockquote> | ||
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much simpler and more elegant explanation than we had.** | much simpler and more elegant explanation than we had.** | ||
+ | (11In condensed-matter language, the | ||
+ | Goldstone mode produces a chargedensity | ||
+ | wave, whose electric fields are | ||
+ | independent of wavelength. This gives | ||
+ | it a finite frequency (the plasma frequency) | ||
+ | even at long wavelength. In | ||
+ | high-energy language the photon eats | ||
+ | the Goldstone boson, and gains a mass. | ||
+ | The Meissner effect is related to the gap | ||
+ | in the order parameter fluctuations (~ | ||
+ | times the plasma frequency), which the | ||
+ | high-energy physicists call the mass of | ||
+ | the Higgs boson.) | ||
- | <cite>https://arxiv.org/pdf/cond-mat/9204009.pdf</cite> | + | <cite>https://arxiv.org/pdf/cond-mat/9204009.pdf and http://pages.physics.cornell.edu/~sethna/StatMech/EntropyOrderParametersComplexity.pdf</cite> |
</blockquote> | </blockquote> | ||
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</blockquote> | </blockquote> | ||
- | |||
- | <tabbox Researcher> | ||
- | <note tip> | + | ---- |
- | The motto in this section is: //the higher the level of abstraction, the better//. | + | |
- | </note> | + | |
- | --> Common Question 1# | + | **Examples** |
- | + | --> Landau phonons in Bose-Einstein condensates# | |
- | <-- | + | |
- | --> Common Question 2# | + | "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 Examples> | ||
- | --> Example1# | + | ---- |
+ | |||
+ | * For a nice summary see http://pages.physics.cornell.edu/~ajd268/Notes/GoldstoneBosons.pdf | ||
- | <-- | + | <tabbox Abstract> |
+ | |||
+ | <blockquote> | ||
+ | It was known from perturbative investigations of self-interacting scalar fields | ||
+ | by Goldstone that the local current conservation may lead to a divergent global | ||
+ | charge resulting from the contribution of a massless scalar (”Goldstone”) boson | ||
+ | which impedes the large distance convergence and in this way causes a situation | ||
+ | which was appropriately referred to as spontaneous symmetry breaking (SSB). | ||
+ | Kastler, Swieca and Robinson showed that this cannot happen in the presence of a mass gap [12], and in a follow up paper (based on the use of the Jost-Lehmann-Dyson | ||
+ | representation) Swieca together with Ezawa [13] succeeded to | ||
+ | prove the Goldstone theorem in a model- and perturbation- independent way. | ||
+ | |||
+ | The Goldstone theorem states that a Noether symmetry in QFT is spontaneously broken | ||
+ | precisely if a massless scalar ”Goldstone boson” prevents the convergence of some of the global | ||
+ | charge $Q= \int j_0 = \infty.$ | ||
+ | |||
+ | This quasiclassical prescription leads to a model-defining first order interaction | ||
+ | density which maintains the conservation of the symmetry currents in | ||
+ | all orders. There are symmetry-representing unitary operators for each finite | ||
+ | spacetime region O but the global charges $Q= \int j_0$ of same symmetry generating | ||
+ | currents diverge. This is the definition of SSB whereas the shift in field | ||
+ | space procedure is a way to prepare such a situation whenever SSB is possible. | ||
+ | For the later presentation of the Higgs model it is important to be aware of a | ||
+ | fine point about SSB whose nonobservance led to a still lingering confusion. As | ||
+ | soon as scalar self-interacting fields are coupled to s = 1 potentials the physical | ||
+ | interpretation of the field shift manipulation on a Mexican hat potential as a | ||
+ | SSB is incorrect; one obtains the Higgs model for the wrong physical reasons | ||
+ | and misses the correct reasons why there can be no self-interacting massive | ||
+ | vectormesons without the presence of a H-field. Although this can be described | ||
+ | correctly in the gauge theoretic formulation, a better understanding is obtained | ||
+ | in the positivity preserving string-local setting of LQP (see section 6) | ||
+ | |||
+ | <cite>https://arxiv.org/pdf/1612.00003.pdf</cite></blockquote> | ||
+ | |||
+ | <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> | ||
+ | |||
+ | |||
- | --> Example2:# | ||
- | |||
- | <-- | ||
- | | ||
- | <tabbox History> | ||
</tabbox> | </tabbox> | ||