The inner product between coherent states is ⟨α2|α1⟩= exp h − 1 2 α∗ 2 ·α2 − 1 2 α∗ 1 ·α1 +α ∗ 2 ·α1 i ,(C2) whereα ∗ ·β≡ P i α∗ i βi

Liouville coherent states To begin with, we define the coherent states for both bosons, fermions as the displacement of the vacuum, |α⟩=D(α)|0⟩ ⊗L