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Is this the kind of symmetry to which Noether's theorem applies? If so it should come with a conservation of something. Any idea of what?
by Aardwolf 5y ago
Is this the kind of symmetry to which Noether's theorem applies? If so it should come with a conservation of something. Any idea of what?
- nautilius 5y agoA rotational symmetry implies conservation of angular momentum.
- ganzuul 5y agoWhat would scale symmetry conserve? Entropy?
- scarmig 5y agoA relevant discussion that I'm digesting: https://physics.stackexchange.com/questions/2670/what-is-the-conserved-quantity-of-a-scale-invariant-universe https://physics.stackexchange.com/questions/2670/what-is-the...
- codethief 5y agoThe link doesn't seem very relevant. A scale-invariant or even conformally invariant universe is something completely different from some substance exhibiting (approximate[0]) conformal invariance during a phase transition. [0]: The paper from the article explicitly refers to "rotational invariance at large scales".
- Aardwolf 5y agoBut we already know which symmetry is related to angular momentum, or do you mean a different kind of angular momentum related to phase transitions?
- codethief 5y agoI don't think it is. Physicists often summarize Noether's Theorem as "A symmetry implies the existence of a conserved quantity" but really what Noether's Theorem refers to are symmetries of a physical system's action[0]. I don't know a lot about phase transitions but judging from the article and the linked references, the symmetry we're talking about here doesn't seem to be a symmetry of any action. [0]: https://en.wikipedia.org/wiki/Action_(physics) https://en.wikipedia.org/wiki/Action_(physics)
- miles7 5y agoConformally symmetric critical systems are often, though not always, studied by modeling them by conformal field theories, which do involve an action obeying a conformal symmetry. A good reference about conformal field theories is any book or paper by John Cardy who is exceptionally clear. Here is a good one where he shows an action of a CFT at the beginning of section 3.0.1: https://arxiv.org/abs/0807.3472 https://arxiv.org/abs/0807.3472 For some reason the Wikipedia page on CFT’s says they are quantum which can be true but is certainly not necessarily the case, and a huge amount of the beautiful mathematics of CFTs was worked out for purely classical systems (classical statistical mechanics). The whole subject is a triumph of the 20th century and deserves to be better known. I’m pretty sure Noether’s theorem does apply to CFTs and is used quite heavily to analyze them. See for example these notes: https://www.google.com/amp/s/tracingcurves.wordpress.com/2019/10/02/notes-on-string-theory-conformal-field-theory-ward-identities-and-noethers-theorem/amp/ https://www.google.com/amp/s/tracingcurves.wordpress.com/201...
- codethief 5y agoI stand corrected then! Thank you! > I’m pretty sure Noether’s theorem does apply to CFTs I'm pretty sure, too. :) Then again, the paper only demonstrates "rotational invariance at large scales". So if anything, any associated conserved charge would only be conserved (and defined!) approximately, i.e. in the large-scale limit.
- codethief 5y agoUpdate: Link to relevant MathOverflow question: https://mathoverflow.net/questions/257896/lagrangian-formulation-of-the-ising-model-as-a-conformal-field-theory https://mathoverflow.net/questions/257896/lagrangian-formula...