theorems:goldstones_theorem
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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> | ||
| Line 34: | Line 24: | ||
| ness and spin waves (figure 9a). In nematic liquid crystals, | ness and spin waves (figure 9a). In nematic liquid crystals, | ||
| the broken rotational symmetry introduces an orientational | the broken rotational symmetry introduces an orientational | ||
| - | elastic stiffness (it pours, but resists bending!) | + | elastic stiffness (it pours, but resists bending!) and rotational waves (figure 9b). |
| - | + | ||
| - | and rotational waves (figure 9b). | + | |
| In superfluids, the broken gauge symmetry leads to a | In superfluids, the broken gauge symmetry leads to a | ||
| stiffness which results in the superfluidity. Superfluidity | stiffness which results in the superfluidity. Superfluidity | ||
| Line 77: | Line 65: | ||
| 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> | ||
| Line 110: | Line 111: | ||
| </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> | ||
theorems/goldstones_theorem.1506662912.txt.gz · Last modified: 2017/12/04 08:01 (external edit)
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