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We propose a simple criterion to identify when Nambu-Goldstone bosons (NGBs) for different symmetries are redundant.
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Since ρ \rho is real and antisymmetric, its rank is always even
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The constant part of currents can be completely fixed by requiring the commutation relation [ Q i , j j 0 ( x ) ] = i f i j k j k 0 ( x ) [Q_{i},j_{j}^{0}(x)]=if_{ij}^{\,\,\,k}j_{k}^{0}(x)
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If interested in the Bogoliubov spectrum ω 2 = c s 2 k 2 ( 1 + k 2 4 m 2 c s 2 ) \omega^{2}=c_{s}^{2}k^{2}(1+\frac{k^{2}}{4m^{2}c_{s}^{2}}) , one should not integrate n n out and leave it as a conjugate field to the phase. In this case, the quantity μ \mu in the text loses the meaning of the chemical potential. Instead, we define the chemical potential as μ ( n ) ≡ ∂ ε ( n ) ∂ n \mu(n)\equiv\frac{\partial\varepsilon(n)}{\partial n} , using the internal energy ε ( n ) = 1 2 g n 2 \varepsilon(n)=\frac{1}{2g}n^{2} . The EOM of n n describes the time-evolution of the phase, θ ˙ reg = μ ( n ) + m Ω ⋅ u → × u → ˙ + m 2 v → 2 − 1 2 m n ∇ 2 n \dot{\theta}_{\mathrm{reg}}=\mu(n)+m\Omega\cdot\vec{u}\times\dot{\vec{u}}+\frac{m}{2}\vec{v}^{2}-\frac{1}{2m\sqrt{n}}\nabla^{2}\sqrt{n} . Other EOMs remain unchanged
Cited in the paper.
In 1 + 2 1+2 D, the exception at the quadratic order in u → \vec{u} is ϵ i j u i u ˙ j \epsilon_{ij}u^{i}\dot{u}^{j} , which changes by a total derivative ∂ 0 ( ϵ i j c i u j ) \partial_{0}(\epsilon_{ij}c^{i}u^{j})
Cited in the paper.