I’ve been only tenuously on the internet this month, so I missed Anshul’s posts about emergence (here and here). I’m closer on this Q. to Pines and Laughlin than to Anshul and Wilczek (see e.g. here — I still stand by most of that I think). What’s weird about that Wilczek article is that he identifies the main question, but then suavely ignores it.

There are various things I want to say in response to these articles (none of which I entirely agree with), but this is the gist: **1. The thermodynamic limit destroys upward heritability. 2. “Emergence” is a result of this breakage.**

According to Wilczek, the reason that particle physics concepts move up into the infrared is that microscopic laws, when “applied to large bodies, retain their character.” Let’s try to unpack that. Obviously, inferences from an approximate microscopic theory will generally not scale up to the macroscopic level (b’se of how errors propagate) but one might reasonably expect some *structural* properties of the microscopic theory to provide useful deductive guidance at higher levels — e.g., the idea that if the microscopic theory is invariant under some symmetry, then so will any higher level; or if (let’s get nonrelativistic at this point) the microscopic theory only has particles of charge e, then the macroscopic theory is constrained to have particles of charge ne where n is an integer. In fact, of course, neither of these arguments is true: spontaneous symmetry breaking is a thing, as is fractionalization, as is the presence of “emergent” symmetries at critical points that were never there in the underlying model.

These inferences are false, not at any intermediate step, but because of pathologies that arise when you take limits. In the present case, the relevant limit is the thermodynamic limit (more precisely, it is the way the thermodynamic limit commutes, or fails to commute, with other limits such as the quasistatic and linear response limits). Virtually *all* nontrivial emergent phenomena are due to these pathologies. For instance:

- In the transverse field Ising model, a finite-size system at T = 0 in the ferromagnetic phase flips between the all-up and all-down ground states at a rate 1/t that is exponential in N, the number of spins. The magnetization measured on times short compared with t is large and finite, but on much longer times it is zero. Depending on whether you take the averaging time or N to zero first, you either find a phase transition or not. These noncommuting limits are in some sense the
*opposite*of upward heritability, if you interpret this as saying that the laws do not change their character: the microscopic dynamics obeys certain symmetries, but the large-scale behavior does not. - Thouless and Gefen explained how the fractional quantum Hall effect similarly allows large systems to defy microscopic symmetries. The Byers-Yang theorem requires the ground state of a system with “fundamental” charge e to be periodic in 1/e. However, you can get around this by having multiple different ground state branches that don’t mix (in the thermodynamic limit) — if you want particles with a charge m then you just need m ground states. The switchings between these ground states guarantee the validity of “the letter of” the underlying microscopic theorem while permitting its “spirit” to be violated.

In all such cases, there remains some literal sense in which a “deductive path” exists from the microscopic to the macroscopic world. However, this deductive path is emphatically **not** a local path: the standard cond. mat. phenomena illustrate that inductive reasoning from particles of N systems to particles of N+1 (or 2N) systems will not help identify emergent phenomena. You need to know where you are going to end up there.

(I think Wilczek might say that “upward heritability” is really about the fact that both cond-mat and hep-th are about symmetry arguments although the symmetries are different in the two cases. I don’t buy this at all. If there is a puzzle here, and for L&P-type reasons I don’t think there is one, it can be resolved by arguing that “the unreasonable effectiveness of mathematics” separately explains both.)

A flip side is that a large family of microscopic possibilities end up at the same macroscopic model. (For instance the 1/3 Laughlin state, or minor deformations of it, is a true ground state for an enormous range of electron-electron interaction strengths and profiles.) When you change scales, some information is lost and other information is amplified; whether a particular piece of information is going to be lost or amplified is a property of the coarse-graining and not a property of the underlying microscopic theory.

Hi, I just wanted to drop by and say that my vote goes for Laughlin and Pines!

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If someone ever writes a comprehensive history of physics, “The Unreasonable Effectiveness of Mathematics” should be its title.

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