Homework 1.

Consider a time dependent one body operator

H(t) = V_{ij}(t)C_{i}^{†}C_{j}

Define

B(t, t′) = T exp [− ∫_{t′}^{t} V (t′′)dt′′]
U(t, t′) = T exp [− ∫_{t′}^{t} H(t′′)dt′′]

and

U(t, t′)_{ij} = ⟨0 |C_{i} U(t, t′) C_{j}^{†}| 0⟩

a) Prove that

B(t, t′)_{ij} = U(t, t′)_{ij}

b) Define “Heisenberg operators”

C_{Hi}(τ) ≡ U(0, τ)C_{i}U(τ, 0)

and prove that

C_{Hi}(τ) = ∑_{j} B_{ij}(τ, τ′)C_{j}(τ′)