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If $$\tag{1} \delta\varphi~=~\varepsilon$$ is a global shift symmetry, we can gauge the symmetry, i.e. enhance it to a local symmetry by (i) introducing a gauge field $A_{\mu}$ with gauge symmetry $$\tag{2} \delta A_{\mu} ~=~\partial_{\mu}\varepsilon,$$ and (ii) replace partial derivatives $\partial_{\mu}\varphi$ with covariant derivatives $$\tag{3} ... 1 I will try to address the first point raised by the OP, i.e. the occurrence of spontaneous symmetry breaking in Bose-Einstein condensation. The free boson gas is described by the hamiltonian:$$ H_V=\int_V\frac{d^sx}{2m}\big|\nabla\phi(x)\big|^2.  The ground state satisfies $H_V\Psi_0 = 0,\ \forall V$ and hence $\nabla \phi(x)\Psi_0=0,\ \forall x.$ ...