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The answer by Black Monolith is pretty good, so let me discuss the issue in a different, more general way. We are taught that systems typically find themselves in their ground state. The implicit reasoning is that, if the system is coupled to some environment/bath/a bunch of other systems, then they can lose energy via that coupling, and by the second law of ...

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You need to think carefully about what space your quantities act on. $\phi$ is a quantum field (operator) acting on Hilbert space. However, by itself, $T^a$ is just an element of a Lie algebra in some representation. It acts on the appropriate representation space, not on Hilbert space. Therefore, the notation $T^a \lvert 0 \rangle$ is sloppy. If you go back ...

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The spontaneous symmetry break happen only in thermodynamic limit(that is, sufficiently large number of particles or system sizes). In the example you gave there are indeed many degenerate ground state -- for any constant $c$ $\phi(x)=c$ is a ground state,since $H$ depends on $\partial_x \phi$ instead of $\phi$ itself, just as what you've mentioned. However,...

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I think the key is this: This expresses the fact that a global translation of the solid as a whole does not affect the internal energy. Now, the ground state of any specific realization of the solid will be defined through a static array of atoms each located at a fixed coordinate RI=Ia⇒ϕI=0. It's contrasting moving all of the atoms together, versus how ...

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Let me point to one example of $\Bbb Z_2$ symmetry breaking. It is contained in my paper S. Shlosman: Phase transitions for two-dimensional models with isotropic short-range interactions and continuous symmetry, Comm. Math. Phys. 71(1980), 207-212. There I consider a 2D system with $O(2)$ symmetry group (which is disconnected). In agreement with the Mermin-...

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This question seems to be based on a false premise, namely, that systems that are nonlinear classically are linear when quantized. Really, the opposite tends to be the case. E.g., Maxwell's equations in vacuum are exactly linear but in QED there is nonlinearity due to interactions mediated by electron loops. Quantum mechanics is linear at the level of the ...

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What does "at tree level" mean? Look at this question. It means that you look only at diagrams that contain no "bubbles" attributable (in colloquial language) to "bubbles"", which are attributable {in colloquial language) to pairs of whatever particles-antiparticles. These particles-antiparticles are somehow connected ...

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I'll give a brief answer that is somewhat at odds with Chiral Anomaly's answer. When we do quantum field theory we expand about some 'saddle point' of the action, i.e. some place where the variation of the action vanishes and thus we can think of it as classical. When you are playing around with just the Lagrangian in quantum field theory, you are looking at ...

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The question has two parts: (1) What does "tree level" mean, and why are tree-level effects said to be classical in nature? (2) Is the Higgs mechanism classical in nature? Tree level $\to$ classical Solving linear differential equations is easy. Solving nonlinear differential equations is hard, usually too hard. If the nonlinearities are small ...

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This question is answered in detail in the paper Phonons as Goldstone Bosons. The question What is the difference between a photon and a phonon? is also closely related. Here, I'll just give some basic intuition. Consider two solids that are identical except that one of them is shifted slightly in space compared to the other. Both of them are equally stable: ...

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