How I understand the Higgs mechanism is that, although it is called “spontaneously breaking of the gauge symmetry”, the gauge symmetry is actually not broken in the higgs mechanism and it is the global symmetry that is broken. The gauge invariance makes the Goldstone boson disappear, but it is the interaction between the gauge boson and the higgs boson and the spontaneously breaking of the global symmetry which gives a non-zero vacuum expectation value of the higgs field that make the gauge boson acquire a mass.

We know that Mermin-Wagner theorem (Coleman theorem) says that there is no spontaneously breaking of global continuous symmetry in dimension 1+1D. So does the Mermin-Wagner theorem apply to the Higgs mechanism? Why or why not?

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    $\begingroup$ Mermin-Wagner theorem forbids SSB in 1+1D and 2+0D but not in 2+1D. $\endgroup$ – Qmechanic May 25 '18 at 8:15
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    $\begingroup$ @Qmechanic The Mermin-Wagner theorem forbids (in certain systems, not quite generally) SSB in 2+1D at nonzero temperature. The Coleman theorem forbids SSB in Lorentz-invariant systems in 1+1D. That's at least what the original papers are about. $\endgroup$ – Tomáš Brauner May 25 '18 at 19:10
  • $\begingroup$ What does a gauge theory look like in 1+1D? $\endgroup$ – Stephen Powell May 26 '18 at 15:28
  • $\begingroup$ @StephenPowell In Coleman's paper, he wrote:" At least formally, it is possible to set up the theory of gauge fields in two dimensions. The gauge fields have no dynamical degrees of freedom, but do create Coulomb interaction between charge densities. (This is most easily seen by quantizing the theory in axial gauge.)" $\endgroup$ – ymr May 27 '18 at 5:10

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