Quantum Field Theory

The interaction picture is a hybrid of the two. We split the Hamiltonian up asH = H 0 + H int (3.10)The time dependence of operators is governed by H 0 , while the time dependence ofstates is governed by H int . Although the split into H 0 and H int is arbitrary, it’s usefulwhen H 0 is soluble (for example, when H 0 is the Hamiltonian for a free field theory).The states and operators in the interaction picture will be denoted by a subscript Iand are given by,|ψ(t)〉 I= e iH 0t |ψ(t)〉 SO I (t) = e iH 0t O S e −iH 0t(3.11)This last equation also applies to H int , which is time dependent. The interactionHamiltonian in the interaction picture is,H I ≡ (H int ) I = e iHot (H int ) S e −iH 0t(3.12)The Schrödinger equation for states in the interaction picture can be derived startingfrom the Schrödinger picturei d|ψ〉 Sdt= H S |ψ〉 S⇒ i d dt⇒i d|ψ〉 Idt(e−iH 0 t |ψ〉 I)= (H0 + H int ) S e −iH 0t |ψ〉 I= e iH 0t (H int ) S e −iH 0t |ψ〉 I(3.13)So we learn thati d|ψ〉 Idt= H I (t) |ψ〉 I(3.14)3.1.1 Dyson’s Formula“Well, Birmingham has much the best theoretical physicist to work with,Peierls; Bristol has much the best experimental physicist, Powell; Cambridgehas some excellent architecture. You can make your choice.”Oppenheimer’s advice to Dyson on which university position to accept.We want to solve (3.14). Let’s write the solution as|ψ(t)〉 I= U(t, t 0 ) |ψ(t 0 )〉 I(3.15)– 51 –