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QFT II Lecture NotesNabil IqbalNovember 11, 2020Contents1 Orientation31.1Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31.2Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41.3Quantum mechanics review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .42 Path Integrals in Free Quantum Field Theory52.1The generating functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62.2Calculating the Feynman propagator: poles, analytic continuation and i . . . . . . . . . . . .72.3Higher-point functions in the free theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .83 Interacting theories103.1LSZ Reduction Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .113.2Perturbation theory and Feynman diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . .133.2.1Combinatorics and symmetry factors . . . . . . . . . . . . . . . . . . . . . . . . . . . .163.3Connected diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .163.4Scattering amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .184 Loops and Renormalization214.1Loops in λφ4 theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .214.2Coming to terms with divergences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .234.3Renormalized perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .254.3.1Counterterms and renormalization conditions . . . . . . . . . . . . . . . . . . . . . . .254.3.2Determining the counterterms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26What theories are renormalizable and what does this mean? . . . . . . . . . . . . . . . . . . .294.41

4.5A few non-renormalizable theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .314.5.1Pions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .324.5.2Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .325 Global symmetries in the functional formalism345.1Classical Noether’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .345.2Quantum Ward identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .356 Fermions376.1Fermions in canonical quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .376.2Grassman variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .386.2.1Anticommuting numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .386.2.2Integrating many anticommuting numbers . . . . . . . . . . . . . . . . . . . . . . . . .406.3Fermions in the path integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .416.4Feynman rules for fermions: an example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .427 Abelian Gauge Theories457.1Gauge invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .457.2Some classical aspects of Abelian gauge theory . . . . . . . . . . . . . . . . . . . . . . . . . .477.3Quantizing QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .497.4Things that I did not do: canonical quantization and LSZ . . . . . . . . . . . . . . . . . . . .528 Non-Abelian gauge theories548.1Non-Abelian gauge invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .548.2The Yang-Mills field-strength and action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .568.3Quantizing non-Abelian gauge theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .598.4Qualitative discussion: Non-Abelian gauge theory at long distances . . . . . . . . . . . . . . .638.4.165What are the gauge fields doing in a confined phase? . . . . . . . . . . . . . . . . . . .2

1OrientationThese lecture notes were written for QFT II in the Particles, Strings and Cosmology MSc course at DurhamUniversity. They are a basic introduction to the path integral approach to quantum field theory. Most of thematerial follows the exposition in the well-known textbooks [2–4]. Please send all typos to [email protected]’s assumed the reader has some exposure to quantum field theory in canonical quantization, and also knowshow to apply path integrals to quantum mechanics.1.1MotivationI want to begin with some philosophical words about quantum field theory. Why do we study quantum fieldtheory? What is it that makes us struggle through this confusing, technically difficult, at times seemingly1ill-defined sea of ideas?Presumably this question has many answers, but perhaps I’ll start making an extremely incomplete list of afew things that QFT is good for:1. Particle physics and the Standard Model: this is the main focus of this course (both the QFTcourse and the larger MSc course of which it is a part). The Standard Model is a wildly successfulphysical theory, it is perhaps the most precise microscopic description of nature ever, and quantumfield theory is really at its very core.Most textbooks lean heavily on this point of view, but it gives you the idea that QFT is only important ifyou care about high-energy physics, and that you need to build a giant particle accelerator to appreciateit. This is not true.2. Metals and superconductors: this is maybe not obvious, but both metals and superconductors –real-life low-energy objects that you can find in your home (or, well, nearby physics lab) – are describedby quantum field theories; they are just not relativistic. The low energy fluctuations of electronsin metals are described by something called Fermi liquid theory, and superconductors are actuallydescribed by an analogue of the Higgs mechanism that you will study in the SM part of this course.3. Boiling water: the phase diagram of water looks like picture on board. The critical point at 374 C isactually described by a three-dimensional quantum field theory called the 3d Ising model. I don’t havetime to explain why this is, but in general continuous phase transitions (.of any sort) are describedby quantum field theories.4. Quantum gravity: if you don’t care about boiling water, perhaps you like quantum gravity. The socalled AdS/CFT correspondence tells us that quantum gravity in D dimensions is precisely equivalentto a quantum field theory in D 1 dimensions (in its most well-studied incarnation it is dual to SU (N )gauge theory). By the end of this course you will know what SU (N ) gauge theory is.One may ask what all of these phenomena have in common that makes them describable by quantum fieldtheory. In essence, if you ever have a situation where there are both fluctuations – thermal, quantum, etc.and locality – in space, in time, etc. – in some sense, it is likely that some sort of quantum field theorydescribes your system.Having said all of that, in this course we will describe only relativistic quantum field theory; as always, extrasymmetries (such as those associated with relativistic invariance) simplify our lives.1 Butonly seemingly.3

1.2ConventionsIn agreement with most (but not all) books on quantum field theory, I will use the “mostly-minus” metric:ηµν η µν diag(1, 1, 1, 1)(1.1)Sadly this is the opposite convention to the one that I’m familiar with; thus everyone will need to be vigilantfor sign errors. We will mostly work in four spacetime dimensions (the “physical” value), but it is nice tokeep the spacetime dimension d arbitrary where possible.Note that Greek indices µ, ν will run over time and space both, whereas i, j will run only over the threespatial coordinates, and thusxµ (x0 , xi )(1.2)I will sometimes use x0 to denote the time component and sometimes xt , depending on what I feel looksbetter in that particular formula.I will always set c 1.2 They can be restored if required from dimensional analysis.1.3Quantum mechanics reviewIn QFT I so far, you have understood in great detail how to compute things in quantum mechanics using thepath-integral; in other words you studied quantum mechanical systems with a classical real-time LagrangianZ1 2S dtL(q, q̇)(1.3)L(q, q̇) q̇ V (q)2You then inserted this into a path-integral to define the generating functional, which was ZZZ[J] N [Dq] exp iS[q] i dtJ(t)q(t)(1.4)Rwhere [Dq] denotes the philosophically soothing but mathematically somewhat distressing “integral over allfunctions”. In the previous QFT1 course, you learned that to compute the time-ordered Green’s function,you simply had to bring down factors of q(t) when doing the average, i.e.Zh0 q̂(tN )q̂(tN 1 ) · · · q̂(t1 ) 0i N [Dq]q(tN )q(tN 1 ) · · · q(t1 ) exp (iS[q])tN tN 1 · · · t1(1.5)Note it is absolutely crucial here that on the left-hand side tN tN 1 · · · t1 . On the right-hand sideit doesn’t seem to matter; on the left-hand side it does, because these are quantum operators that do notcommute. It was explained in QFT I that path integrals only give you correlation functions that are timeordered; to emphasize this, I will often put a T around the correlation function, i.e. I may write the left-handside ash0 T (q̂(tN )q̂(tN 1 ) · · · q̂(t1 )) 0i(1.6)In fact, this quantity was typically somewhat ill-defined, even by physicist standards. However you couldmake it better by Wick-rotating, i.e. you wrotet iτ(1.7)where τ is real. This is sometimes called “Euclidean time”; note that the metric becomesds2 dt2 d x2 (dτ 2 d x2 ).2kBmakes an appearance in a homework problem (or rather it would, if I hadn’t set it to 1).4(1.8)

The overall minus sign here is of no importance; we are working in a spacetime with Euclidean signature.Note that the action also becomes! 2Z1 dq V (q) SEiS dτ(1.9)2 dτwhere SE is positive-definite (if the potential is positive-definite). This Wick-rotated path integral is ZZZ[J] N [Dq] exp SE dτ J(τ )q(τ )(1.10)Note that the object in the exponential is now real rather than imaginary, and this makes the integralconvergent3 . In general, many confusions about path integrals can be made to go away by imagining aWick-rotation to Euclidean signature. As explained in the first part, this prescription also guarantees thatyou are in the vacuum of the theory by suppressing the contribution of all states with energy E by a factorof exp( Eτ ).2Path Integrals in Free Quantum Field TheoryWith this under our belt, we will now move on to quantum field theory. You have studied the basic ideas inIFT; let me just remind you. We will begin with a study of the free real scalar field φ. This has actionZZ 114222d x ( φ(x)) m φ(x) d4 x φ(x)( 2 m2 )φ(x)(2.1)S[φ] 22The classical equations of motion arising from the variation of this action are( 2 m2 )φ(x) 0(2.2)This is philosophically the same as the quantum mechanics problem studied earlier. To make the transitionimagine going from the single quantum-mechanical variable q to a large vector qa where a runs (say) from 1to N . Now imagine formally that a runs over all the sites of a lattice that is a discretization of space, andnow qa (t) is basically the same thing as φ(xi ; t) which is exactly the system we are studying above.Now we would like to study the quantum theory. First, I point out that we can define a path integral inprecisely the same way as before, i.e. we can consider the following path-integral:ZZ [Dφ] exp (iS[φ])(2.3)Where S[φ] is the action written down above, and [Dφ] now represents the functional integral over all fieldsand not just particle trajectories.There are two main things that are nice about doing quantum field theory from path integrals the waydiscussed above. One of them is honest, the other is a bit “secret”.1. The honest one: all of the symmetries of the problem are manifest. The action S is Lorentz-invariant,and it is fairly easy to see how these symmetries manifest themselves in a particular computation.Compare this to the Hamiltonian methods used in IFT, where you have to pick a time-slice and italways seems like a miracle when final answers are Lorentz-invariant.2. The secret one: the path integral allows one to be quite cavalier about subtle issues like “what is thestructure of the Hilbert space exactly”. This is very convenient when we get to gauge fields, wherethere are subtle constraints in the Hilbert space (google “Dirac bracket”) that you can more or less notworry about when using the path integral (i.e. one can go quite far in life without knowing exactlywhat a “Dirac bracket” is).3 Well,more convergent than before, at least.5

2.1The generating functionalNow that the philosophy is out of the way, let us do a computation. We will begin by computing the followingtwo-point function:h0 T (φ(x)φ(y)) 0i(2.4)This object is called the Feynman propagator. It is quite important for many reasons; I will discuss themlater, for now let’s just calculate it. By arguments identical to those leading to (1.5), we see that we wantZh0 T (φ(x)φ(y)) 0i Z0 1 [Dφ]φ(x)φ(y) exp (iS[φ])(2.5)To calculate this, it is convenient to define the same generating functional as we used for quantum mechanics ZZ4Z[J] [Dφ] exp iS[φ] i d xJ(x)φ(x)(2.6)And we then see from functional differentiation that the two-point function is δδ1 iZ[J] ih0 T (φ(x)φ(y)) 0i Z0δJ(x)δJ(y)(2.7)J 0Each functional derivative brings down a φ(x). Now we will evaluate this function. We first note the identityderived in QFT I for doing Gaussian integrals in Section 7.1 of [1], which I have embellished with a few i’shere and there:s ZN1(2π)1dx1 dx2 · · · dxN exp xa Aab xb iJa xa exp Ja (A 1 )ab Jb(2.8)2detA2We note from the form of the action (2.1) that the path integral Z[J] we want to do is of precisely this form,where we do our usual “many integrals” limit and where a labels points in space and the operator A is A i 2 m2(2.9)We conclude that the answer for Z[J] is Z[J] det 2 m2 2πi 21 Z1exp d4 xd4 yJ(x)DF (x, y)J(y)2(2.10)where I have given the object playing the role of A 1 a prescient name DF . It is the inverse of the differentialoperator defined in (2.9) and thus satisfies i 2 m2 DF (x, y) δ (4) (x y)(2.11)This is an important result. Let us first note that the path integral is asking us to compute the functionaldeterminant of a differential operator. This is a product over infinitely many eigenvalues; it is quite a beautifulthing but we will not really need it here, so we will return to it later.The next thing to note is that the dependence on J is quite simple; the exponential of a quadratic. Indeed,inserting this into (2.7) we geth0 T (φ(x)φ(y)) 0i DF (x, y)(2.12)Thus, we have derived that the time-ordered correlation function of φ(x) is given by the inverse of i( 2 m2 ).You have already encountered this phenomenon in IFT.6

2.2Calculating the Feynman propagator: poles, analytic continuation and i Let us now actually calculate this object. I note that you have performed a similar computation in IFT, butI will repeat some elements of it to explain what the path integral is doing for us. We first go to Fourierspace:Zd4 p ip·(x y)eD̃F (p)(2.13)DF (x, y) (2π)4Inserting this into (2.11) we findZ d4 p p2 m2 e ip·(x y) D̃F (p) iδ (4) (x y)4(2π)(2.14)We now see that we Rwant the object in momentum space D̃F (p) to be proportional to p2 m2 so that wecan use the identity d4 peip·x (2π)4 δ (4) (x). Getting the factors right, we find the following expression forthe propagator in Fourier space:i(2.15)D̃F (p) 2p m2This looks nice, but actually, this expression is not yet complete, in that specifying this Fourier transformdoes not yet completely specify a function DF (x, y) in position space. To understand this, let us actuallyattempt to Fourier transform this thing back.We break the integral into time p0 and spatial p and do theptime integral first. Let me define ωp p 2 m2 :Z00dp0d3 piDF (x, y) e ip0 (x y ) i p·( x y)2π(2π)3 p20 p2 m2ZZ00d3 p11dp0 ie ip0 (x y ) i p·( x y)2π(2π)3 p0 ωp p0 ωp Z(2.16)(2.17)Now if we look at the integral, we see that there are poles when p0 ωp . The correct way to think aboutthis is to imagine it as an integral in the complex p0 plane, and we then have a pole. To actually perform theintegral, we need to specify how we go around the poles; different ways of going around the poles will give usdifferent answers.There are two steps; first let’s say that x0 y 0 0; in the case we must complete the contour below forthe integral to converge. We are not done yet; to get the second bit of information, we should recall thatpath integrals only make since if we formulate them in Euclidean signature. This helps; let’s imagine that weformulated the whole thing in Euclidean time all along. Recall from (1.7) that the Wick rotation to Euclideantime and frequency ist iτp0 ipE(2.18)0This means that if we were working in Euclidean time all along, we would have done the p0 integral up anddown the imaginary axis. This tells us which way to complete the integral, as only one pole prescription (oneup, one down) is compatible with this.Now we can do the integral. We pick up the contribution from the ωp when x0 y 0 0, and the answer is( 2πi)iDF (x, y) 2πZd3 p 1 iωp (x0 y0 ) i p·( x y)e(2π)3 2ωp Similarly, if x0 y0 we close the integral the other way and findZ00(2πi)id3 p1DF (x, y) e iωp (x y ) i p·( x y)2π(2π)3 2ωp 7x0 y0(2.19)x0 y0(2.20)

Figure 1: Circling poles in the complex frequency plane.This is it: we have done the hard work in calculating the time-ordered correlation function, also called theFeynman propagator. The spatial integral can be done, but it involves a bit of work with Bessel functionsand I may assign it as a homework.A convenient way to summarize this business with the pole is to write the Green’s function in momentumspace asiD̃F (p) 2(2.21)p m2 i where is a tiny number that tips the contour slightly upwards. This is called the Feynman prescription.To summarize: path integrals give us time-ordered correlation functions. If ever we are confused about howto go around poles, then we should remember that the path integral secretly only makes sense in Euclideansignature; this typically helps.2.3Higher-point functions in the free theoryWe just computed the two-point functions. However, one can of course compute higher point functions usingjust the same idea: e.g. say you wanth0 T φ(x1 )φ(x2 )φ(x3 )φ(x4 ) 0i(2.22)Just as in (2.7), we determine this by taking functional derivatives of Z[J]:h0 T φ(x1 )φ(x2 )φ(x3 )φ(x4 ) 0i ( i)441 Y δZ[J]Z0 i 1 δJ(xi )(2.23)However, because Z[J] is Gaussian, the resulting answer is just different combinations of DF (x, y), e.g. youcan work out:h0 T φ(x1 )φ(x2 )φ(x3 )φ(x4 ) 0i ( i)4 (DF (x1 , x2 )DF (x3 , x4 ) DF (x1 , x3 )DF (x2 , x4 ) DF (x1 , x4 )DF (x2 , x3 ))(2.24)8

Figure 2: Different ways to draw propagators in Wick’s theorem.In other words, you draw propagators between all possible pairs of points. It should be clear that this worksfor any number of insertions; this is called Wick’s theorem.9

Figure 3: Schematic plots of the spectral representation in interacting theories.3Interacting theoriesNext, we would like to study interacting quantum field theories. You have encountered an example of onesuch already, the λφ4 theory: Zm2 2 λ 4124φ φ(3.1)S[φ] d x ( φ) 224!This describes a field theory where the particles that are captured by φ can interact or scatter off of eachother, and where roughly speaking the probability for this scattering to occur is given by λ.The first thing to note is that in general we simply cannot solve interacting field theories. In this course wewill instead use the fact that we can solve free quantum field

by quantum eld theories. 4. Quantum gravity: if you don’t care about boiling water, perhaps you like quantum gravity. The so-called AdS/CFT correspondence tells us that quantum gravity in Ddimensions is precisely equivalent to a quantum eld theory in D 1 dimensions (in its most we