Spontaneous symmetry breaking in SHO

[Shadumu]

Spontaneous symmetry breaking refers to the solution of a system loses some symmetry in its Lagrangian. Consider a Simple Harmonic Oscillator, its lagrangian is time translationally invariant but its solution is periodic in time, thus not time-translational invariant. Is this Spontaneous symmetry breaking?

 

[DrDu]

No, it is not. The concept of spontaneous symmetry breaking is a concept from quantum mechanics (QM) rather than classical mechanics. The point is that in QM, you can usually form superpositions of solutions of the equations of motion and they will be solutions again. In the case of the harmonic oscillator, you can form any superpositions of the different solutions which aren't completely symmetric. The point of spontaneous symmetry breaking is that this possibility to form superpositions of solutions of different symmetry is no longer possible. For an harmonic oscillator, this is not possible. Rather, it requires systems of infinite extent, like for example a magnet which is built up of an infinite number of spins.