Mathematicians prove symmetry of phase transitions at critical moments | Hacker News Reader

Mathematicians prove symmetry of phase transitions at critical moments

quantamagazine.org

90 pointspseudolus18 comments

No, they don't.

> in a proof posted in December, a team of five mathematicians has come closer than ever before to proving that conformal invariance is a necessary feature of these physical systems as they transition between phases. The work establishes that rotational invariance — one of the three symmetries contained within conformal invariance — is present at the boundary between states in a wide range of physical systems

Can we stop inflating titles, especially for mathematics?

Ok, we've added critical moments to the title above. That seems to be the hedge in the subtitle.

Anyone know if this has implications for P=NP, where they’ve found parallels between phase transitions and the “danger zone” of NP-complete problems?

Background: for the NP-complete 3SAT (Boolean satisfiability where the clauses all have up to 3 terms OR’d together), there are ratios of clauses to variables that make instances trivially satisfiable or or trivially unsatisfiable. In between them are the hardest cases, where you run into exponential time blow-up. That transition is analogized to phase transitions in matter.

Is it implied that the different phase transitions of - say water - are related to the various ways symmetry can break? This makes sense intuitively, I think: Ice -> liquid you are changing size a bit but liquid water still has some transient structure at the molecular level; liquid -> gas you are changing size a lot and gaining rotational + translational invariance? These symmetries I think make sense if you think about water the individual water molecules are doing during the various phases.

Not necessarily: https://physics.stackexchange.com/questions/105166/symmetry-...

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?

A rotational symmetry implies conservation of angular momentum.

What would scale symmetry conserve? Entropy?

A relevant discussion that I'm digesting:

https://physics.stackexchange.com/questions/2670/what-is-the...

The 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".

But we already know which symmetry is related to angular momentum, or do you mean a different kind of angular momentum related to phase transitions?

I 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)

Conformally 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

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/201...

I 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.

Update: Link to relevant MathOverflow question: https://mathoverflow.net/questions/257896/lagrangian-formula...

Formal maths is not my strong point these days, so I’m reading this from a more abstract position, but it sounds like yet more pointers to systems being more than a sum of their parts.

Am I understanding it correctly?

This might just be my ignorance of the domain talking, but why would there be invariance across the form?

What’s the proposed rule beneath the rule of conformal invariance?

Am I just whimsy to be reminded of Armani-Hamed’s theories about particle scattering?

I'm with you not being an expert and being unable to understand most of the lingo and the article. The conformal invariance example in the pictures spoke to me about the fractal nature in the universe. Fractality just keeps popping up in my readings and that may have colored my read of the article.

Yes, these when these models become critical (eg. where the distinction between liquid & gas goes away [1]) they form clouds of condensation at all scale sizes, so it is like a fractal.

[1] https://twitter.com/johncarlosbaez/status/127591543287712563...

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