What's the right Fourier transformation definition of creation or annihilation operators in lattice model?
Answers
Very often one can view the solid as lattice model and then use the the language of second quantization,namely taking the occupation number representation, to express the Hamiltonian of the system. In this representation the creation and annihilation operators play the fundamental role. But in recently I just found some discrepancies for the definition of these operators for lattice model: Vijay Shenoy Version (starting at 13:10): For 3D lattice model the Fourier transformation of the creation operator between K space and real space will be: c†iσ=1N−−√∑k∈BZe−ik⃗ ⋅r⃗ c†σ(k⃗ )(1) here the subindices σ and i are the spin index and site index, respectively and N stands for all K points in first Brillouin zone. And Vijay didn't show the inverse transformation.