On Perturbation Theory, Dyson Series, and Feynman Diagrams

On Perturbation Theory, Dyson Series, and Feynman Diagrams The Interaction Picture of QM and the Dyson Series Many quantum systems have Hamiltonians of the form Hˆ = Hˆ0 + Vˆ where Hˆ0 is a free Hamiltonian with known spectrum — which is used to classify the states of the full theory — while Vˆ is a perturbation which causes transitions between the eigenstates of the Hˆ0. For example, in scattering theory ˆ 2 Preduced Hˆ0 = , Vˆ = potential V (xˆrelative). (1) 2Mreduced Similarly, for the quantum scalar field Φˆ with a (λ/24)Φˆ 4 self-interaction we have 2 m2 d3p 1 ˆ 3x 1 ˆ 2 x 1 ˆ x ˆ 2 † H0 = d 2 Π ( ) + 2 ∇Φ( ) + Φ = 3 Epaˆpaˆp + const 2 (2π) 2Ep Z Z λ while Vˆ = d3x Φˆ 4(x). 24 Z (2) To study the transitions (scattering, making new particles, decays, etc.) caused by Vˆ we want to use a fixed basis of Hˆ0 eigenstates, but we want to keep the transitions separate from the ordinary wave-packet motion and spreading — which are due to the Schrödinger phases e−iEt of the Hˆ0 itself. The picture of QM which separates these effects is the interaction picture. In the Schrödinger picture, the operators are time-independent while the quantum states −iHtˆ evolve with time as |ψiS(t)= e |ψi(0). In the Heisenberg picture it’s the other way around — the quantum states are time independent while the operators evolve with time — and the two pictures are related by the time-dependent unitary operator eiHtˆ , +iHtˆ ˆ +iHtˆ ˆ −iHtˆ |ΨiH (t) = e |ΨiS(t) ≡ |ΨiS(0) ∀t, AH (t) = e ASe . (3) The interaction picture has a similar relation to the Schrödinger’s, but using the exp(iHˆ0t) instead of the exp(iHtˆ ), +iHˆ0t −iHˆ0t AÎ (t) = e AˆSe , (4) +iHˆ0t +iHˆ0t −iHtˆ |ψiI(t) = e |ψiS(t) = e e |ψiH =6 const. 1 In the interaction picture, quantum fields depend on time as if they were free fields, for example 3 0 d p p =+Ep ˆ x −ipx +ipx † ΦI ( , t) = 3 e aˆp + e aˆp , (5) 16π Ep Z regardless of the interactions. This is different from the Heisenberg picture where interacting fields depend on time in a much more complicated way. The time-dependence of quantum states in the interaction picture is governed by the perturbation Vˆ according to Schrödinger-like equation d i |ψi (t) = Vˆ (t) |Ψi (t). (6) dt I I I The problem with this equation is that the VÎ operator here is itself in the interaction picture, +iHˆ0t −Hˆ0t so it depends on time as VÎ(t)= e VˆSe . Consequently, the evolution operator for the interaction picture ˆ ˆ UI (t, t0): |ψiI(t) = UI(t, t0) |ψiI (t0) (7) −iVˆ (t−t0) is much more complicated than simply e . Specifically, UÎ (t, t0) satisfies ∂ i Uˆ (t, t ) = Vˆ (t) Uˆ (t, t ), Uˆ (t = t )=1, (8) ∂t I 0 I I 0 I 0 and the formal solution to these equations is the Dyson series t t t2 UÎ(t, t0) = 1 − i dt1 VÎ(t1) − dt2 VÎ(t2) dt1 VÎ(t1) tZ0 tZ0 tZ0 t t3 t2 + i dt3 VÎ (t3) dt2 VÎ(t2) dt1 VÎ(t1) + ··· (9) tZ0 tZ0 tZ0 ∞ n = 1+ (−i) ··· dtn ··· dt1 VÎ(tn) ··· VÎ (t1). n=1 X t0<tZ1<···Z<tn<t Note the time ordering of operators VÎ (tn) ··· VÎ(t1) in each term. 2 The Dyson series obviously obeys the initial condition UÎ (t = t0) = 1. To see that is also satisfies the Schrödinger-like differential equation (8), note that in each term of the series, the only thing which depends on t is the upper limit of the leftmost dtn integral. Thus, taking ∂/∂t of the term amounts to skipping that integral and letting tn = t, t tn t2 ∂ n i (−i) dt Vˆ (t ) dt − Vˆ (t − ) ··· dt Vˆ (t ) = (10) ∂t n I n n 1 I n 1 1 I 1 tZ0 tZ0 tZ0 tn=t t2 n−1 = Vˆ (t = t) × (−i) dt − Vˆ (t − ) ··· dt Vˆ (t ) . I n n 1 I n 1 1 I 1 tZ0 tZ0 th st In other words, i∂/∂t of the n term is VÎ (t) × the (n − 1) term. Consequently, the whole series satisfies eq. (8). Thanks to the time ordering of the VÎ (t) operators in each term of the Dyson series — the earliest operator being rightmost so it acts first, the second earliest being second from the right, etc., etc., until the latest operator stands to the left of everything so it acts last — we may rewrite the integrals in a more compact form using the time-orderer T. Earlier in class, I have defined T of an operator product, but now I would like to extend this by linearity to any sum of operators products. Similarly, we may time-order integrals of operator products and hence products of integrals such as t 2 t t def def T dt′ Vˆ (t′) = T dt dt Vˆ (t )Vˆ (t ) = dt dt TVˆ (t )Vˆ (t ) I 1 2 I 1 I 2 1 2 I 1 I 2 tZ0 ZZt0 ZZt0 (11) = dt1 dt2 VÎ(t2)VÎ(t1) + dt1 dt2 VÎ (t1)VÎ(t2) t0<tZZ1<t2<t t0<tZZ2<t1<t where the domain of each dt1 dt2 integral is color-coded on the diagram below: 3 t2 t t2 > t1 t1 > t2 (12) t0 t1 t0 t Integral over the blue triangle t0 < t1 < t2 < t is what appears in the Dyson series. But there is t1 ↔ t2 symmetry between the blue and red triangles, and the corresponding integrals on the bottom line of eq. (11) are equal to each other. Hence the triangular integral in the Dyson series may be written in a more compact form as t 2 1 dt dt Vˆ (t )Vˆ (t ) = T dt′ Vˆ (t′) . (13) 1 2 I 2 I 1 2 1 t0<tZZ1<t2<t tZ0 Similar procedure applies to the higher-order terms in the Dyson series. The nth order term is an integral over a simplex t0 < t1 < t2 < ··· < tn < t in the (t1,...,tn) space. A hypercube t0 < t1,...,tn < t contains n! such simplexes, and after time-ordering the Vˆ operators, integrals over all simplexes become equal by permutation symmetry. Thus, t 1 ··· dt ··· dt Vˆ (t ) ··· Vˆ (t ) = ··· dt ··· dt TVˆ (t ) ··· Vˆ (t ) n 1 I n I 1 n! n 1 I n I 1 t0<tZ1<···Z<tn<t Z t0 Z t n 1 = T dt′ Vˆ (t′) . (14) n! 1 tZ0 Altogether, the Dyson series becomes a time-ordered exponential n ∞ t t (−i)n Uˆ (t, t )=1+ T dt′ Vˆ (t′) ≡ T-exp −i dt′ Vˆ (t′) . (15) I 0 n! I I n=1 X tZ0 tZ0 4 Of particular interest is the evolution operator from the distant past to the distant future, +∞ def Sˆ = Uˆ (+∞, −∞) = T-exp −i dt′ Vˆ (t′) . (16) I I −∞Z This operator is properly called ‘the scattering operator’ or ‘the S–operator’, but everybody calls it ‘the S–matrix’. In the scalar field theory where λ Vˆ (t) = d3x Φˆ 4(x, t), (17) I 24 I Z the S–matrix has a Lorentz-invariant form −iλ Sˆ = T-exp d4x Φˆ 4(x) . (18) 24 I wholeZ spacetime Note that the Φˆ I (x) here is the free scalar field as in eq. (5). Similar Lorentz-invariant expressions exist for other quantum field theories. For example, in QED ˆ 4 ˆµ ˆ ˆ S = T-exp +ie d x AI (x)ΨI(x)γµΨI(x) (19) wholeZ spacetime ˆµ where both the EM field AI (x) and the electron field ΨI(x) are in the interaction picture so they evolve with time as free fields. S–Matrix Elements The S-matrix elements hout| Sˆ |ini should be evaluated between physical incoming and outgoing 2-particle (or n-particle) states. In the potential scattering theory, the potential Vˆ becomes irrelevant when all particles are far away from each other, so the asymptotic states ⋆ |ini and hout| are eigenstates of the free Hamiltonian Hˆ0. But in the quantum field theory, ⋆ Strictly speaking, the asymptotic states are wave packets moving in space due to Hˆ0. But once we take the limit of infinitely large distance between wave packets for different particles in the asymptotic past and the asymptotic future, we may then make the wave packets themselves very thick in space but having very narrow ranges of momenta and energies. In this limit, the asymptotic states become eigenvalues of the free Hamiltonian Hˆ0. 5 the perturbation Vˆ is always present, and the asymptotic n–particle states |k1,...,kni of the interacting field theory are quite different from the free theory’s n–particle statesa ˆ† ··· aˆ† |0i. kn k1 Even the physical vacuum state |Ωi is different from the free theory’s vacuum |0i. In the spring semester, we shall learn how to obtain the physical S–matrix elements hout| Sˆ |ini from the correlation functions of fully-interacting quantum fields, Fn(x1,...,xn) = hΩ| TΦˆ H (x1) ··· Φˆ H (xn) |Ωi (20) — and also how to calculate those correlation functions in perturbation theory. But this semester, I am taking a short-cut: I will explain the perturbation theory for the naive S- matrix elements ′ ′ † † free : k ,k ... Sˆ free : k1,k2,... = h0|··· aˆ ′ aˆ ′ Sˆ aˆ aˆ ···|0i (21) 1 2 k2 k1 k1 k2 between the 2-particle (or n -particle) states of the free Hamiltonian Hˆ0, and then I will tell you — without proof — how to modify the perturbative expansion to obtain the S- matrix elements between the physical asymptotic states.

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