Second Quantisation - Changing Basis

Question

If you have an Interacting Hamiltonian \begin{equation}H_{int}= \sum_{\sigma,\sigma'}\sum_{\mathbf{p},\mathbf{k}} a^\dagger_{\mathbf{p},\sigma}a^\dagger_{\mathbf{k},\sigma'}a_{\mathbf{k},\sigma'}a_{\mathbf{k},\sigma}\end{equation}

Is there a standard procedure for changing basis in Second Quantisation? say like moving to the Helicity basis $|\mathbf{p},s\rangle$ I cant see how best to undertake such a procedure

Answer 1

Let me first note that oen can speak about:

• changing the single-particle basis (used for perfrming the second quantization)
• changing the basis in the Fock space (e.g., when performing bosonization, introducing elementary excitations, etc.)

I assume the question is about the first option. If we base ourselves on the prescription where the creating/annihilation operators are expansion coefficients $$ \hat{\Psi}(\mathbf{x})=\sum_na_n\phi_n(\mathbf{x}), \\ \hat{\Psi}^\dagger(\mathbf{x})=\sum_na_n^\dagger\phi_n^*(\mathbf{x}), $$ then the answer is readily obvious, as $\hat{\Psi}(\mathbf{x}), \hat{\Psi}^\dagger(\mathbf{x})$ should remain the same regardless of the basis $\{\phi_n(\mathbf{x})\}$ that one uses.