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Since it seems that you're dealing with this in a statistical mechanics context, I'll use the Ising model as an example. Spontaneous symmetry breaking is a phenomenon in which the Hamiltonian or Lagrangian of your system has a certain symmetry, but some relevant state of your system does not. Take the nearest neighbor 2D Ising model $$ H=-J\sum_{\langle i,j\...


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I will try a simple explanation. Try putting a cylindrical pencil standing upright on a ball. Eventually, for a second, you will succeed, but then the pencil will probably fall. What tells the pencil in which direction to fall? At the start (pencil upright on the ball) there is cylindrical symmetry to the whole thing. The answer would be that some ...


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This statement is called in the literature Adler zero. Nambu-Goldstone boson couples to the associated Noether current with a strength parametrized by the decay constant $f$ : $$ \langle 0 | J_\mu (x)| \phi(p) \rangle = -i p_\mu f e^{-i p x} $$ The matrix element between physical states has a pole for $p^2 \rightarrow 0$ and the residue corresponds to the ...


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There are many examples of symmetry breaking outside of field theory and particle physics. Crystallisation involves symmetry breaking in the transition from a highly symmetric liquid phase to crystals with a smaller discrete set of symmetries. The formation of magnetic domains in a ferromagnetic substance is another example. In biology a symmetric single-...


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Symmetry is a much larger concept than symmetry breaking. The latter is specific to the field theory and the condensed matter physics (in the context of phase transitions, which can often be characterized as states with different symmetries). Symmetry is however a much larger subject: the other applications in physics that readily come to mind are the ...


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In the most formal sense (contrasting with the informal usage), a symmetry means that if you know something about one state, you can infer information about another state. The most trivial example is that if you've seen that gravity works in the past, you may assume it will work in the future. This is an example of a time-translation symmetry. The laws of ...


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I think the questions in this post are "philosophy of science" material. What Importance Does "Breaking of Symmetry" plays in our daily observable universe A great importance in vehicles : think of an asymmetric in mass car. When the weight is not symmetric to the direction of motion one continually compensates. It is important that ...


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Symmetry breaking is, in fact a topic within field theory. To understand it means understanding the mathematics of field theory. Certainly we can handwave and speak in analogies, but this does not convey accurately what the topic is really about or what it does for us. I will say this though. There is a difference between what we call symmetry and what might ...


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The simplest way to prove this statement (actually, it gives you more, namely an upper bound on the 2-point function that decays with a power law) is the argument by McBryan and Spencer. You can actually prove much more, namely the full rotation invariance of the limiting Gibbs state. In my opinion, the simplest proofs are due to Dobrushin and Shlosman and ...


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