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When we say spontaneous symmetry breaking what do we mean by "spontaneous"?

The dictionary, for a non native speaker, relates it to the absence of an external stimulus.

In QFT and SM it simply says that Lagrangian remains symmetric but the quantum vacuum state breaks such an invariance.

Are we considering the field operator as something external?

Are we considering an explicit breaking something external?

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Spontaneous in this context really just means that the symmetric state is known to be unstable with respect to perturbations. Technically, if you could maintain a perfectly symmetric state, the system under investigation would not be able to collapse, because its lower energy states are asymmetric. But in practice all that this means is that the decision of what state to collapse to comes from some sort of noise, maybe quantum noise: some usually-ignorable inhomogeneity that we cannot ignore here. It is that the world is messy and our perfectly symmetric model at some level fails to be a perfect description of the world, which allows the symmetry to be broken to transition to the ground state: more accurately one might say that the symmetry was approximate in the first place because these "noise" sources, whatever they are, did not respect it.

The classic example is classical, it is a configuration at a potential energy maximum point, like a pencil balanced on its point, or a satellite at the right place between the Earth and the Sun where gravity balances out.

The gradient at this maximum is 0, so there is no force. But it doesn't stay there, because any perturbation from that maximum takes you to a point slightly lower in energy: not lower to first-order, but lower to second-order or higher even order. But the key thing, is that continuity says that those points themselves, then have nonzero forces pointing away from the maximum. If the world were ideal, the symmetric state were perfect, then you would not see any motion from the equilibrium, but the world is not ideal, the symmetry is not respected by everything, and so in practice the pencil falls down, or the satellite needs to do station keeping in order to stay where it is.

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Here's one intuitive way to understand symmetry breaking. Let's say some element E, in solid form, makes a orthorhombic lattice, similar to cubic, but different spacing in x, y and z. Out in space, we have a box of E, at high temperature, so that E is in gas form. Then there is no preferred direction in space. Next lower the temperature, so that a orthorhombic crystal starts to form. As E cools, more and more of it's atoms attach to the orthorhombic crystal. This crystal now has picked out three directions in space. Which ones? That's random, spontaneous. The symmetry of a gas having no preferred direction has been broken and we now have a solid with tree preferred directions to which any additional atoms adjust.

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