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The low-energy physics of systems with spontaneous symmetry breaking is governed by the associated Nambu-Goldstone (NG) bosons.
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The power of momentum in the dispersion relation is bounded from above by a value depending on the space dimension d d though. Generalizing naively the argument of the classic paper by Coleman [ 21 ] to Lorentz-noninvariant systems, the correlation function of the NG field would diverge at large distance if the power of momentum exceeded this limit
Cited in the paper.
To avoid misunderstanding, let us emphasize that every conserved charge of course generates a symmetry transformation on the phase space of the classical theory. Nevertheless, it may not be possible to obtain this charge as a consequence of an off-shell invariance of the action in the Lagrangian formalism. This is what we mean by saying that the charge is not of the Noether type
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The last step here makes use of the assumption that the adjoint representation matrices R ( θ ) R(\theta) are orthogonal, which in turn follows from the full antisymmetry of the structure constants of the Lie algebra. This can be achieved for any compact semisimple Lie algebra by a proper choice of basis
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While the orthogonality is trivial in the case of a nonzero eigenvalue with multiplicity one, in general one has to be a bit careful. First, note that when λ k = 0 \lambda_{k}=0 , ℐ 𝒖 k ∗ \mathcal{I}\bm{u}_{k}^{*} is orthogonal to 𝒖 k \bm{u}_{k} by construction, thanks to the antisymmetry of ℐ \mathcal{I} . Second, suppose that some eigenvalues are equal, say, λ k + 1 = λ k \lambda_{k+1}=\lambda_{k} . We can then choose the eigenvector 𝒖 k + 1 \bm{u}_{k+1} to be orthogonal to both 𝒖 k \bm{u}_{k} and ℐ 𝒖 k ∗ \mathcal{I}\bm{u}^{*}_{k} . This already guarantees that ℐ 𝒖 k + 1 ∗ \mathcal{I}\bm{u}_{k+1}^{*} will be orthogonal to all three of these vectors. One can thus construct an orthonormal basis by induction
Cited in the paper.