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Physicists love punchlines. The punchlines of famous theorems in physics are repeated over and over. However, almost everyone forgets that every theorem is necessarily based on assumptions. While everyone knows the punchlines of famous theorems and loves to repeat them over and over, no one cares about the assumptions.


Goldstone's Theorem

“Just to round things out, what about superconductors? They’ve got a broken gauge symmetry, and have a stiffness to decays in the superconducting current. What is the low energy excitation? It doesn’t have one. But what about Goldstone’s theorem? Well, you know about physicists and theorems . . . That’s actually quite unfair: Goldstone surely had conditions on his theorem which excluded superconductors. Actually, I believe Goldstone was studying superconductors when he came up with his theorem. It’s just that everybody forgot the extra conditions, and just remembered that you always got a low frequency mode when you broke a continuous symmetry. We of course understood all along why there isn’t a Goldstone mode for superconductors: it’s related to the Meissner effect. The high energy physicists forgot, though, and had to rediscover it for themselves. Now we all call the loophole in Goldstone’s theorem the Higgs mechanism, because (to be truthful) Higgs and his high–energy friends found a much simpler and more elegant explanation than we had. We’ll discuss Meissner effects and the Higgs mechanism in the next lecture.”

The Coleman-Mandula Theorem

“compare how many people know the conclusions of the Coleman-Mandula theorem to how many people remember its assumptions!” John Baez

Have a look at the long list of loopholes in the Coleman-Mandula theorem.

The Mermin-Wagner Theorem

In two dimensions, crystals provide another loophole in a well-known result, known as the Mermin–Wagner theorem. Hohenberg, Mermin, and Wagner, in a series of papers, proved in the 1960s that two-dimensional systems with a continuous symmetry cannot have a broken symmetry at finite temperature. At least, that is the English phrase everyone quotes when they discuss the theorem; they actually prove it for several particular systems, including superfluids, superconductors, magnets, and translational order in crystals. Indeed, crystals in two dimensions do not break the translational symmetry; at finite temperatures, the atoms wiggle enough so that the atoms do not sit in lock-step over infinite distances (their translational correlations decay slowly with distance). But the crystals do have a broken orientational symmetry: the crystal axes point in the same directions throughout space. (Mermin discusses this point in his paper on crystals.) The residual translational correlations (the local alignment into rows and columns of atoms) introduce long-range forces which force the crystalline axes to align, breaking the continuous rotational symmetry.

causes/propaganda/theorems.txt · Last modified: 2017/11/02 10:29 (external edit)