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I recently read an interesting discussion, between Carlo Rovelli and Steven Weinberg which happened at the "Conceptual Foundations of Quantum Field Theory" conference in 1994:

Rovelli: Steve, your synthesis of the foundations of quantum field theory is remarkable. You clearly like it very much; you even conclude, with a possible irony, that we might go back to such a quantum field theory in the 'final theory.' But there is a problem. One of the four or five founding stones of your foundation of quantum field theory is Poincare invariance. Now, if I may advertise another book of yours, I learned general relativity from your 'Gravitation and Cosmology' - great book. I learned from your book that the world in which we happen to live is not Poincare invariant, and is not described by Poincare invariant theory. There is no sense in which general relativity is Poincare invariant. (If it were, what would be the Poincare transform of the closed Friedman-Robertson-Walker solution of the Einstein equation?) Thus, Poincare invariance is neither a symmetry of our universe, nor a symmetry of the laws governing our universe. Don't you find it a bit disturbing basing the foundation of our understanding of the world on a symmetry which is not a symmetry of the world nor of its laws?

To which Weinberg replied:

Weinberg: Well, I think there's always been a distinction that we have to make between the symmetries of laws and the symmetries of things. You look at a chair; it's not rotationally invariant. Do you conclude that there's something wrong with rotation invariance? Actually, it's fairly subtle why the chair breaks rotational invariance: it's because the chair is big. In fact an isolated chair in its ground state in empty space, without any external perturbations, will not violate rotational invariance. It will be spinning in a state with zero rotational quantum numbers, and be rotationally invariant. But because it's big, the states of different angular momentum of the chair are incredibly close together (since the rotational energy differences go inversely with the moment of inertia), so that any tiny perturbation will make the chair line up in a certain direction. That's why chairs break rotational invariance. That's why the universe breaks symmetries like chiral invariance; it is very big, even bigger than a chair. This doesn't seem to me to be relevant to what we take as our fundamental principles. You can still talk about Lorentz invariance as a fundamental law of nature and live in a Lorentz non-invariant universe, and in fact sit on a Lorentz non-invariant chair, as you are doing. [Added note: Lorentz invariance is incorporated in general relativity, as the holonomy group, or in other words, the symmetry group in locally inertial frames.]

I don't really understand Weinberg's "subtle" reason here. Why does that "the states of different angular momentum of the chair are incredibly close together" mean that the "chairs break rotational invariance"? Who or what is responsible for this symmetry breaking?

EDIT: After posting the question I found this previous question, which cites from Weinberg's second QFT book, where he makes a very similar argument. Unfortunately, even with this second explanation by Weinberg himself I don't really understand it.

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    $\begingroup$ Great question, and now I have to read up on it! $\endgroup$ – R. Rankin Jun 23 '17 at 8:40
  • $\begingroup$ I'm not sure what one could say beyond repeating the quote you already give, in particular, "It will be spinning in a state with zero rotational quantum numbers, and be rotationally invariant. But because it's big, the states of different angular momentum of the chair are incredibly close together (since the rotational energy differences go inversely with the moment of inertia), so that any tiny perturbation will make the chair line up in a certain direction. That's why chairs break rotational invariance." Can you be more specific what about that part is unclear? $\endgroup$ – ACuriousMind Jun 23 '17 at 9:57
  • $\begingroup$ @ACuriousMind My problem is that I don't understand what causes this asymmetric perturbation? If the laws are symmetry, where does the asymmetry for macroscopic object come from in the first place? $\endgroup$ – jak Jun 27 '17 at 11:23
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    $\begingroup$ Closely related question: physics.stackexchange.com/questions/243291/… $\endgroup$ – knzhou Oct 3 '18 at 10:18
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I think that Weinberg is simply wrong. The point he makes is the obvious one that there are non-symmetric states in a theory with a symmetry. But this is irrelevant to the question. In a theory with symmetry, you can have non-symmetric states (a chair facing East), but you have the corresponding symmetry-transformed states (the chair facing South). In general relativity, there is no "Poincaré-transformed" of, say, a closed Friedman universe. In fact, perhaps unhappy of his first answer, Weinberg has later added the "Added note" about Lorentz invariance being incorporated in general relativity as the symmetry in locally inertial frames. But this refers to the APPROXIMATE dynamics of matter in a tiny spacetime region, not to the dynamics of the full metric. There is no action of the Poincaré group on the space of solutions of the Einstein equations.

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