Interacting Quantum Fields C6, HT 2015
aRudolf Peierls Centre for Theoretical Physics University of Oxford OX1 3PN Oxford, United Kingdom
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1 Interacting Quantum Fields
In this part of the lecture we will develop the formalism that allows one to understand the basic features of perturbatively interacting quantum ﬁeld theories (QFTs). The main goal is to de- rive the Feynman rules for bosonic ﬁeld theories and to show how to calculate matrix elements and cross sections for simple scattering processes. More formal aspects of interacting QFTs (such as e.g. the exponentiation of vacuum diagrams or the Lehmann-Symanzik-Zimmermann reduction formula) are only mentioned in passing.
1.1 Toy Models In what follows we will only deal with weakly coupled QFTs, i.e., theories that can be truly considered as small perturbations of the free ﬁeld theory at all energies. We will look in more detail at two speciﬁc examples that serve the purpose of toy models. The ﬁrst example of a weakly coupled QFT we will study is the φ4 theory, 1 1 λ L = (∂ φ)2 − m2φ2 − φ4 , (1.1) 2 µ 2 4! where φ is our well-known real scalar ﬁeld. For (1.1) to be weakly-coupled we have to require λ 1. We can get a hint for what the eﬀects of the additional φ4 term will be. Expanding it in terms of ladder operators, we ﬁnd terms like
a†(p)a†(p)a†(p)a†(p) , a†(p)a†(p)a†(p)a(p) , (1.2) etc., which create and destroy particles. This signals that the φ4 Lagrangian (1.1) describes a theory in which particle number is not conserved. In fact, it is not too diﬃcult to check that the number operator N does not commute with the Hamiltonian, i.e.,[N,H] 6= 0.
1 The second example we will look at is a scalar Yukawa theory. Its Lagrangian is given by 1 1 L = (∂ ϕ∗)(∂µϕ) + (∂ φ)2 − M 2ϕ∗ϕ − m2φ2 − g ϕ∗ϕφ , (1.3) µ 2 µ 2 with g M, m. This theory couples a complex scalar ϕ to a real scalar φ. In this theory the individual particle numbers for ϕ and φ are not conserved. Yet, the Lagrangian (1.3) is invariant under global phase rotations of ϕ, which ensures that there will be a conserved charge Q obeying [Q, H] = 0. In fact, we have met this charge already in our discussion of the quantization of the complex Klein-Gordon ﬁeld (see (1.76) in the script “Canonical Quantization”). In consequence, in the scalar Yukawa theory the number of ϕ particles minus the number of ϕ anti-particles is conserved. Notice also that the potential in (1.3) has a stable minimum at ϕ = φ = 0, but it is unbounded from below, if −gφ becomes too large. This means that we should not mess to much with the scalar Yukawa theory.
1.2 Interaction Picture In quantum mechanics (QM), there is a useful viewpoint called the interaction picture, which allows one to deal with small perturbations to a well-understood Hamiltonian. Let me brieﬂy recall how this works. In the Schr¨odingerpicture, the states evolve as id/dt|ψiS = H |ψiS, while the operators OS are time independent. In contrast, in the Heisenberg picture the states do not evolve with time, but the operators change with time, namely one has |ψiH = iHt iHt −iHt e |ψiS and OH = e OS e . The interaction picture is a hybrid of the two. We split the Hamiltonian as follows H = H0 + Hint , (1.4) where in the interaction picture the time dependence of operators OI is governed by H0, while the time dependence of the states |ψiI is governed by Hint. While this split is arbitrary, things are easiest if one is able to solve the Hamiltonian H0, e.g., if H0 is the Hamiltonian of a free theory. From what I have said so far, it follows that
iH0t iH0t −iH0t |ψiI = e |ψiS , OI = e OS e . (1.5) Since the Hamiltonian is itself an operator, the latter equation also applies to the interaction Hamiltonian Hint. In consequence, one has
iH0t −iH0t HI = (Hint)I = e Hint e . (1.6) The Schr¨odingerequation in the interaction picture is readily derived starting from the Schr¨odingerpicture, d d i |ψi = H |ψi , =⇒ i e−iH0t |ψi = (H + H ) e−iH0t |ψi , dt S S dt I 0 int I d =⇒ i |ψi = eiH0t H e−iH0t|ψi , (1.7) dt I int I d =⇒ i |ψi = H |ψi . dt I I I
2 Dyson’s Formula In order to solve the system described by the Hamiltonian (1.4), we have to ﬁnd a way of how to ﬁnd a solution to the Schr¨odingerequation in the interaction basis (1.7). Let us write the solution as |ψ(t)iI = U(t, t0)|ψ(t0)iI , (1.8) where U(t, t0) is an unitary time-evolution operator satisfying U(t, t) = 1, U(t1, t2)U(t2, t3) = † U(t1, t3), and U(t1, t3) U(t2, t3) = U(t1, t2). Inserting (1.8) into the last line of (1.7), we ﬁnd d i U(t, t ) = H (t)U(t, t ) . (1.9) dt 0 I 0
If HI would be a function, the solution to the diﬀerential equation (1.9) would read Z t ? 0 0 U(t, t0) = exp −i dt HI (t ) . (1.10) t0
Yet, HI is not a function but an operator and this causes ordering issues. Let’s have a closer look at the exponential to understand where the trouble comes from. The exponential is deﬁned through its power expansion,
Z t Z t 2 Z t 2 0 0 0 0 (−i) 0 0 exp −i dt HI (t ) = 1 − i dt HI (t ) + dt HI (t ) + .... (1.11) t0 t0 2 t0 When we diﬀerentiate this with respect to t, the third term on the right-hand side gives Z t Z t 1 0 0 1 0 0 − dt HI (t ) HI (t) − HI (t) dt HI (t ) . (1.12) 2 t0 2 t0
The second term of this expression looks good since it is part of HI (t)U(t, t0) appearing on the right-hand side of (1.9), but the ﬁrst term is no good, because the HI (t) sits on the wrong 0 side of the integral, and we cannot commute it through, given that [HI (t ),HI (t)] 6= 0 when 0 t 6= t . So what is the correct expression for U(t, t0) then? The correct answer is provided by Dyson’s formula,1 which reads Z t 0 0 U(t, t0) = T exp −i dt HI (t ) . (1.13) t0 Here T denotes time ordering as deﬁned in (1.78) of my script “Canonical Quantization”. It is easy to prove the latter statement. We start by expanding out (1.13), which leads to Z t 0 0 U(t, t0) = 1 − i dt HI (t ) t0 (1.14) 2 "Z t Z t Z t Z t0 # (−i) 0 00 00 0 0 00 0 00 + dt dt HI (t )HI (t ) + dt dt HI (t )HI (t ) + .... 0 2 t0 t t0 t0
1Essentially ﬁgured out by Dirac, but in its compact notation due to Dyson.
3 In fact, the terms in the last line are actually the same, since
Z t Z t Z t Z t00 0 00 00 0 00 0 00 0 dt dt HI (t )HI (t ) = dt dt HI (t )HI (t ) t t0 t t 0 0 0 (1.15) Z t Z t0 0 00 0 00 = dt dt HI (t )HI (t ) , t0 t0 where the range of integration in the ﬁrst expression is over t00 ≥ t0, while in the second expression one integrates over t0 ≤ t00, which is, of course, the same thing. The ﬁnal expression is simply obtained by relabelling t0 and t00. In fact, it is not too diﬃcult to show that one has
Z t Z t1 Z tn−1 1 Z t dt1 dt2 ... dtn HI (t1) ...HI (tn) = dt1 . . . dtn T (HI (t1) ...HI (tn)) . (1.16) t0 t0 t0 n! t0 Putting things together this means that the power expansion of (1.12) takes the form
Z t Z t Z t0 0 0 2 0 00 0 00 U(t, t0) = 1 − i dt HI (t ) + (−i) dt dt HI (t )HI (t ) + .... (1.17) t0 t0 t0 The proof of Dyson’s formula is straightforward. First, observe that under the T operation, all operators commute, since their order is already ﬁxed by time ordering. Thus,
Z t Z t d d 0 0 0 0 i U(t, t0) = i T exp −i dt HI (t ) = T HI (t) exp −i dt HI (t ) dt dt t t 0 0 (1.18) Z t 0 0 = HI (t)T exp −i dt HI (t ) = HI (t)U(t, t0) . t0 Notice that since t, being the upper limit of the integral, is the latest time so that the factor HI (t) can be pulled out to the left. Before moving on, I have to say that Dyson’s formula is rather formal. In practice, it turns out to be very diﬃcult to compute the time-ordered exponential in (1.13). The power of (1.13) comes from the expansion (1.17) which is valid when HI is a small perturbation to H0.
Anatomy of Scattering Processes Let us now try to apply the interaction picture to QFT, starting with an easy example, namely the interaction Hamiltonian of the Yukawa theory, Z 3 ∗ Hint = g d x ϕ ϕφ . (1.19)
Unlike the free scalar theories discussed before, this interaction does not conserve the particle number of the individual ﬁelds, allowing particles of one type to morph into others. In order to see why this is the case, we look at the evolution of the state, i.e., |ψ(t)i = U(t, t0)|ψ(t0)i, in the interaction picture. If g M, m, where M and m are the masses of
4 ϕ and φ, respectively, the perturbation (1.19) is small and we can approximate the full time- evolution operator U(t, t0) in (1.13) by (1.17). Notice that (1.17) is, in fact, an expansion in powers of Hint. The interaction Hamiltonian Hint contains ladder operators for each type of particle. In particular, the φ ﬁeld contains the operators a† and a that create or destroy φ particles.2 Let’s call this particle mesons (M). On the other hand, from earlier discussions † it follows that the ﬁeld ϕ contains the operators a+ and a−, which implies that it creates ϕ anti-particles and destroys ϕ particle. We will call these particles nucleons (N).3 Finally, the † † action of ϕ is to create nucleons through a− and to destroy anti-nucleons via a+. While the individual particle number is not conserved, it is important to emphasize that Q = N+ − N− is conserved not only in the free theory, but also in the presence of Hint. At † † ﬁrst order in Hint, one will have terms of the form a+a−a which destroys a meson and creates ¯ a nucleon-antinucleon pair, M → NN. At second order in Hint, we have more complicated † † † processes. E.g., the combination of ladder operators (a+a−a)(a+a−a ) gives rise to the scat- tering process NN¯ → M → NN¯. The rest of this section is devoted to calculate the quantum amplitudes for such processes to occur. In order to calculate the amplitude, we have to make an important, but slightly dodgy, assumption. We require that the initial state |ii at t → −∞ (ﬁnal state |fi at t → ∞) is an eigenstate of the free theory described by the Hamiltonian H0. At some level, this sounds like a reasonable approximation. If at t → ∓∞ the particles are well separated they do not feel the eﬀects of each other. Moreover, we intuitively expect that the states |ii and |fi are eigenstates of the individual number operators N and N±. These operators commute with H0, but not with Hint. As the particles approach each other, they interact brieﬂy, before departing again, each going on its own way. The amplitude to go from |ii to |fi is given by
lim hf|U(t+, t−)|ii = hf|S|ii , (1.20) t∓→∓∞ where the unitary operator S is known as the scattering or simply S-matrix. There are a number of reason why the assumption of non-interacting initial and ﬁnal states |ii and |fi is shaky. First, one cannot describe bound states. E.g., naively this formalism cannot deal with the scattering of an electron (e−) and proton (p) which collide, bind, and leave as a hydrogen atom. Second, and more importantly, a single particle, a long way from its neighbors, is never alone in ﬁeld theory. This is true even in classical electrodynamics, where the electron sources the electromagnetic ﬁeld from which it can never escape. In quantum electrodynamics (QED), a related fact is that there is a cloud of virtual photons surrounding the electron. This line of thought gets us into the issues of renormalization. Although both problems can be circumvent, let me, for the time being, simply use the assumption of non- interacting asymptotic states. After developing the basics of scattering theory, we will brieﬂy revisit the latter problem.
2The argument p of the ladder operators a† and a etc. is dropped hereafter in the text. 3Of course, in reality nucleons are spin-1/2 particles, and do not arise from the quantization of a scalar ﬁeld. Our scalar Yukawa theory is therefore only a toy model for nucleons interacting with mesons.
5 1.3 Wick’s Theorem Using Dyson’s formulas (1.13) and (1.17), we can now in principle4 calculate all matrix ele- ments of the form hf|T HI (x1) ...HI (xn) |ii , (1.21) where |ii and |fi are assumed to be asymptotically free states. The ordering of the operators HI is ﬁxed by time ordering. However, since the interaction Hamiltonian contains certain creation and annihilation operators, it would be convenient if we could start to move all annihilation operators to the right, where they can start eliminating particles in |ii (recall that this is the deﬁnition of normal ordering). Wick’s theorem tells us how to go from time- ordered to normal-ordered products. Before stating Wick’s theorem in its full generality, let’s keep it simple and try to rederive something that we have seen already before. This is always a very good idea.
Case of Two Fields The most simple matrix element of the form (1.21) is
h0|T (φI (x)φI (y))|0i . (1.22) We already calculated this object in Chapter 1.4 of the script “Canonical Qunatization” and gave it the name Feynman propagator. What we want to do now is to rewrite it in such a way that it is easy to evaluate and to generalize the obtained result to the case with more than two ﬁelds. We start by decomposing the real scalar ﬁeld in the interaction picture as
+ − φI (x) = φI (x) + φI (x) , (1.23)
5 with Z Z + 3 −ipx − 3 † ipx φI (x) = d p˜ a(p) e , φI (x) = d p˜ a (p) e . (1.24) This decomposition can be done for any free ﬁeld. It is useful since
+ − φI (x)|0i = 0 , h0|φI (x) = 0 . (1.25) Now we consider the case x0 > y0 and compute the time-order product of the two scalar ﬁelds,