$$\mathbf{a}\left(\mathbf{k}\right) = \sum_{\alpha = 1}^2 \boldsymbol{\varepsilon}_\alpha\left(\mathbf{k}\right) a_\alpha\left(\mathbf{k}\right) $$ $$\mathbf{a}^\dagger\left(\mathbf{k}\right) = \sum_{\alpha = 1}^2 \boldsymbol{\varepsilon}^*_\alpha\left(\mathbf{k}\right) a^\dagger_\alpha\left(\mathbf{k}\right) $$ $$\boldsymbol{\varepsilon}_\alpha \in \mathbb{C}^3 \; \text{such that } \boldsymbol{\varepsilon}_\alpha^\dagger \boldsymbol{\varepsilon}_\beta = \delta_{\alpha\beta}\; \text{and}\; \boldsymbol{\varepsilon}_\alpha^\dagger \mathbf{k} = 0$$ That is $\boldsymbol{\varepsilon}_1\left(\mathbf{k}\right) $ and $\boldsymbol{\varepsilon}_2\left(\mathbf{k}\right) $ are orthogonal complex vectors in the plane perpendicular to $\mathbf{k} $. I am trying to get from $$ \mathbf{S} = -\frac{i}{2} \int \mathop{d^3k} \mathbf{a}^\dagger\left(\mathbf{k}\right) \times \mathbf{a}\left(\mathbf{k}\right) + \mathbf{a}^\dagger\left(-\mathbf{k}\right) \times \mathbf{a}\left(-\mathbf{k}\right) $$ to $$\mathbf{S} = -i\int \mathop{d^3k} \mathbf{a}^\dagger\left(\mathbf{k}\right) \times \mathbf{a}\left(\mathbf{k}\right)$$