Quantum Field Theory I II - Heidelberg University
Quantum Field Theory I IIInstitute for Theoretical Physics, Heidelberg UniversityTimo Weigand
LiteratureThis is a writeup of my Master programme course on Quantum Field Theory I (Chapters 1-6) andQuantum Field Theory II. The primary source for this course has been Peskin, Schröder: An introduction to Quantum Field Theory, ABP 1995, Itzykson, Zuber: Quantum Field Theory, Dover 1980, Kugo: Eichtheorie, Springer 1997,which I urgently recommend for more details and for the many topics which time constraints haveforced me to abbreviate or even to omit. Among the many other excellent textbooks on QuantumField Theory I particularly recommend Weinberg: Quantum Field Theory I II, Cambridge 1995, Srednicki: Quantum Field Theory, Cambridge 2007, Banks: Modern Quantum Field Theory, Cambridge 2008as further reading. All three of them oftentimes take an approach different to the one of this course.Excellent lecture notes available online include A. Hebecker: Quantum Field Theory, D. Tong: Quantum Field Theory.Special thanks to Robert Reischke1 for his fantastic work in typing these notes.1 Forcorrections and improvement suggestions please send a mail to reischke@stud.uni-heidelberg.de.
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Contents12The free scalar field91.1Why Quantum Field Theory? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .91.2Classical scalar field: Lagrangian formulation . . . . . . . . . . . . . . . . . . . . .111.3Noether’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .141.4Quantisation in the Schrödinger Picture . . . . . . . . . . . . . . . . . . . . . . . .171.5Mode expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .181.6The Fock space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .211.7Some important technicalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .231.7.1Normalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .231.7.2The identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .241.7.3Position-space representation . . . . . . . . . . . . . . . . . . . . . . . . . .241.8On the vacuum energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .251.9The complex scalar field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .281.10 Quantisation in the Heisenberg picture . . . . . . . . . . . . . . . . . . . . . . . . .301.11 Causality and Propagators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .331.11.1 Commutators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .331.11.2 Propagators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .351.11.3 The Feynman-propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . .361.11.4 Propagators as Green’s functions . . . . . . . . . . . . . . . . . . . . . . . .39Interacting scalar theory412.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .412.2Källén-Lehmann spectral representation . . . . . . . . . . . . . . . . . . . . . . . .422.3S-matrix and asymptotic in/out-states . . . . . . . . . . . . . . . . . . . . . . . . .462.4The LSZ reduction formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .482.5Correlators in the interaction picture . . . . . . . . . . . . . . . . . . . . . . . . . .532.5.1Time evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .552.5.2From the interacting to the free vacuum . . . . . . . . . . . . . . . . . . . .562.6Wick’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .592.7Feynman diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .615
6CONTENTS2.7.1Position space Feynman-rules . . . . . . . . . . . . . . . . . . . . . . . . .632.7.2Momentum space Feynman-rules . . . . . . . . . . . . . . . . . . . . . . .63Disconnected diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .652.8.1Vacuum bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .671-particle-irreducible diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . .672.10 Scattering amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .702.10.1 Feynman-rules for the S -matrix . . . . . . . . . . . . . . . . . . . . . . . .722.11 Cross-sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73Quantising spin 12 -fields772.82.9343.1The Lorentz algebra so(1, 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .773.2The Dirac spinor representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .813.3The Dirac action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .843.4Chirality and Weyl spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .863.5Classical plane-wave solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .893.6Quantisation of the Dirac field . . . . . . . . . . . . . . . . . . . . . . . . . . . . .913.6.1Using the commutator . . . . . . . . . . . . . . . . . . . . . . . . . . . . .913.6.2Using the anti-commutator . . . . . . . . . . . . . . . . . . . . . . . . . . .943.7Propagators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .973.8Wick’s theorem and Feynman diagrams . . . . . . . . . . . . . . . . . . . . . . . .993.9LSZ and Feynman rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100Quantising spin 1-fields4.1Classical Maxwell-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.2Canonical quantisation of the free field . . . . . . . . . . . . . . . . . . . . . . . . . 1054.3Gupta-Bleuler quantisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.4Massive vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124.5Coupling vector fields to matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1134.5.1Coupling to Dirac fermions . . . . . . . . . . . . . . . . . . . . . . . . . . 1144.5.2Coupling to scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1164.6Feynman rules for QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1174.7Recovering Coulomb’s potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1214.7.15103Massless and massive vector fields . . . . . . . . . . . . . . . . . . . . . . . 124Quantum Electrodynamics5.1127QED process at tree-level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1275.1.1Feynman rules for in/out-states of definite polarisation . . . . . . . . . . . . 1275.1.2Sum over all spin and polarisation states . . . . . . . . . . . . . . . . . . . . 1285.1.3Trace identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1295.1.4Centre-of-mass frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
CONTENTS75.1.5 Cross-section . . . . . . . . . . . . . . . . . . . . . . . . .5.2 The Ward-Takahashi identity . . . . . . . . . . . . . . . . . . . . .5.2.1 Relation between current conservation and gauge invariance5.2.2 Photon polarisation sums in QED . . . . . . . . . . . . . .5.2.3 Decoupling of potential ghosts . . . . . . . . . . . . . . . .5.3 Radiative corrections in QED - Overview . . . . . . . . . . . . . .5.4 Self-energy of the electron at 1-loop . . . . . . . . . . . . . . . . .5.4.1 Feynman parameters . . . . . . . . . . . . . . . . . . . . .5.4.2 Wick rotation . . . . . . . . . . . . . . . . . . . . . . . . .5.4.3 Regularisation of the integral . . . . . . . . . . . . . . . . .5.5 Bare mass m0 versus physical mass m . . . . . . . . . . . . . . . .5.5.1 Mass renormalisation . . . . . . . . . . . . . . . . . . . . .5.6 The photon propagator . . . . . . . . . . . . . . . . . . . . . . . .5.7 The running coupling . . . . . . . . . . . . . . . . . . . . . . . . .5.8 The resummed QED vertex . . . . . . . . . . . . . . . . . . . . . .5.8.1 Physical charge revisited . . . . . . . . . . . . . . . . . . .5.8.2 Anomalous magnetic moment . . . . . . . . . . . . . . . .5.9 Renormalised perturbation theory of QED . . . . . . . . . . . . . .5.9.1 Bare perturbation theory . . . . . . . . . . . . . . . . . . .5.9.2 Renormalised Perturbation theory . . . . . . . . . . . . . .5.10 Infrared divergences . . . . . . . . . . . . . . . . . . . . . . . . . 531541551581646Classical non-abelian gauge theory1656.1 Geometric perspective on abelian gauge theory . . . . . . . . . . . . . . . . . . . . 1656.2 Non-abelian gauge symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1676.3 The Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1717Path integral quantisation7.1 Path integral in Quantum Mechanics . . . . . . . . . . . . . . .7.1.1 Transition amplitudes . . . . . . . . . . . . . . . . . . .7.1.2 Correlation functions . . . . . . . . . . . . . . . . . . .7.2 The path integral for scalar fields . . . . . . . . . . . . . . . . .7.3 Generating functional for correlation functions . . . . . . . . .7.3.1 Functional calculus . . . . . . . . . . . . . . . . . . . .7.4 Free scalar field theory . . . . . . . . . . . . . . . . . . . . . .7.5 Perturbative expansion in interacting theory . . . . . . . . . . .7.6 The Schwinger-Dyson equation . . . . . . . . . . . . . . . . .7.7 Connected diagrams . . . . . . . . . . . . . . . . . . . . . . . .7.8 The 1PI effective action . . . . . . . . . . . . . . . . . . . . . .7.9 Γ(ϕ) as a quantum effective action and background field method.173173173178179183183185188191195196200
8CONTENTS7.10 Euclidean QFT and statistical field theory . . . . . . . . . . . . . . . . . . . . . . . 2037.11 Grassman algebra calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2067.12 The fermionic path integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21189Renormalisation of Quantum Field Theory8.1 Superficial divergence and power counting . . . . . . . . .8.2 Renormalisability and BPHZ theorem . . . . . . . . . . .8.3 Renormalisation of φ4 theory up to 2-loops . . . . . . . .8.3.1 1-loop renormalisation . . . . . . . . . . . . . . .8.3.2 Renormalisation at 2-loop . . . . . . . . . . . . .8.4 Renormalisation of QED revisited . . . . . . . . . . . . .8.5 The renormalisation scale . . . . . . . . . . . . . . . . . .8.6 The Callan-Symanzyk (CS) equation . . . . . . . . . . . .8.7 Computation of β-functions in massless theories . . . . . .8.8 The running coupling . . . . . . . . . . . . . . . . . . . .8.9 RG flow of dimensionful operators . . . . . . . . . . . . .8.10 Wilsonian effective action & Renormalisation Semi-GroupQuantisation of Yang-Mills-Theory9.1 Recap of classical YM-Theory . . . . . . . . . . .9.2 Gauge fixing the path integral . . . . . . . . . . . .9.3 Faddeev-Popov ghosts . . . . . . . . . . . . . . .9.4 Canonical quantisation and asymptotic Fock space9.5 BRST symmetry and the physical Hilbert space . 259262264
Chapter 1The free scalar field1.1Why Quantum Field Theory?In (non-relativistic) Quantum Mechanics, the dynamics of a particle is described by the time-evolutionof its associated wave-function ψ(t, x) with respect to the non-relativistic Schrödinger equationih̄ ψ(t, x) Hψ(t, x), t(1.1)ˆ2 pwith the Hamilitonian given by H 2m V ( x̂). In order to achieve a Lorentz invariant framework, anaive approach would start by replacing this non-relativistic form of the Hamiltonian by a relativisticexpression such asq(1.2)H c2 pˆ 2 m2 c4or, even better, by modifying the Schrödinger equation altogether such as to make it symmetric in t and the spatial derivative . However, the central insight underlying the formulation of QuantumField Theory is that this is not sufficient. Rather, combining the principles of Lorentz invariance andQuantum Theory requires abandoning the single-particle approach of Quantum Mechanics. In any relativistic Quantum Theory, particle number need not be conserved, since the relativisticdispersion relation E 2 c2 p2 m2 c4 implies that energy can be converted into particles andvice versa. This requires a multi-particle framework. Unitarity and causality cannot be combined in a single-particle approach: In Quantum Mechanics, the probability amplitude for a particle to propagate from position x to y isiG ( x, y) h y e h̄ Ht xi .(1.3)2p̂One can show that e.g. for the free non-relativistic Hamiltonian H 2mthis is non-zero evenµ0µ0if x ( x , x) and y (y , y) are at a spacelike distance. The problem persists if we replaceH by a relativistic expression such as (1.2).Quantum Field Theory (QFT) solves both these problems by a radical change of perspective:9
10CHAPTER 1. THE FREE SCALAR FIELD The fundamental entities are not the particles, but the field, an abstract object that penetratesspacetime. Particles are the excitations of the field.Before developing the notion of an abstract field let us try to gain some intuition in terms of a mechanical model of a field. To this end we consider a mechanical string of length L and tension T along thex-axis and excite this string in the transverse direction. Let φ( x, t ) denote the transverse excitation ofthe string. In this simple picture φ( x, t ) is our model for the field. This system arises as the continuumlimit of N mass points of mass m coupled by a mechanical spring to each other. Let the distance ofthe mass points from each other projected to the x-axis be and introduce the transverse coordinatesqr (t ), r 1, . . . , N of the mass points. In the limit 0 with L fixed, the profile qr (t ) asymptotesto the field φ( x, t ). In this sense the field variable x is the continuous label for infinitely many degreesof freedom.We can now linearise the force between the mass points due to the spring. As a result of a simpleexercise in classical mechanics the energy at leading order is found to beE N X1r 0dqr (t )m2dt!2 k(q2r qr qr 1 ) O(q3 ),k T.L(1.4)In the continuum limit this becomes!2!2 ZL 1 2 φ( x, t ) 1 φ( x, t ) ρcE dx ρ2 t2 x(1.5)0in terms of the mass density ρ of the string and a suitably defined characteristic velocity c. Note thatthe second term indeed includes the nearest neighbour interaction because φ( x, t ) x!2φ( x δx, t ) φ( x, t ))' limδx 0δx!2(1.6)contains the off-diagonal terms φ( x δx, t )φ( x, t ).The nearest-neighbour interaction implies that the equation of motion for the mass points qi obey coupled linear differential equations. This feature persists in the continuum limit. To solve the dynamicsit is essential that we are able to diagonalise the interaction in terms of the Fourier modes, X!kπx,φ( x, t ) Ak (t ) sinLk 1! LX 1 2 1 2 2E ρȦ ρω A ,2 k 1 2 k 2 k(1.7)where ωk kπc/L. We are now dealing with a collection of infinitely many, decoupled harmonicoscillators Ak (t ).
1.2. CLASSICAL SCALAR FIELD: LAGRANGIAN FORMULATION11In a final step, we quantise this collection of harmonic oscillators. According to Quantum Mechanics,each mode Ak (t ) can take energy valuesEk h̄ωk (nk 1/2) nk 0, 1, 2, ., .(1.8)PThe total energy is given by summing over the energy associated with all the modes, E Ek . Astate of definite energy E corresponds to mode numbers (n1 , n2 , ., n ), where we think of nr as anexcitation of the string or of the field φ, i.e. as a quantum. In condensed matter physics, these quantised excitations in terms of harmonic modes are called quasi-particles, e.g. phonons for mechanicalvibrations of a solid. Note that the above decoupling of the degrees of freedom rested on the quadraticform of the potential. Including higher terms will destroy this and induce interactions between modes.The idea of Quantum Field Theory is to adapt this logic to particle physics and to describe a particleas the quantum of oscillation of an abstract field - just like in solid state physics we think of aquasi-particle as the vibrational excitation of a solid. The only difference is that the fields are nowmore abstract objects defined all over spacetime as opposed to concrete mechanical fields of the typeabove.As a familiar example for a field we can think of the Maxwell field Aµ ( x, t ) in classical electrodynamics. A photon is the quantum excitation of this. It has spin 1. Similarly we assign one field toeach particle species, e.g. an electron is the elementary excitation of the electron field (Spin 1/2). Wewill interpret the sum over harmonic oscillator energies as an integral over possible energies for givenmomentum,Z XE h̄ωk (nk 1/2) dp h̄ω p (n p 1/2).(1.9)k 1A single particle with momentum p corresponds to n p 1 while all others vanish, but this is just aspecial example of a more multi-particle state with several n pi , 0. In particular, in agreement withthe requirements of a multi-particle framework, at fixed E transitions between various multi-particlestates are in principle possible. Such transitions are induced by interactions corresponding to thehigher order terms in the Hamiltonian that we have discarded so far. As a triumph this formalism alsosolves the problem of causality, as we will see.1.2Classical scalar field: Lagrangian formulationWe now formalise the outlined transition from a classical system with a finite number of degreesof freedom qi (t ) to a classical field theory in terms of a scalar field φ(t, x) φ( xµ ). In classicalmechanics we start from an actionZt2S dt L(qi (t ), q̇i (t )) with L t11X(q̇i (t))2 V (q1 , , qN ),2 i(1.10)
12CHAPTER 1. THE FREE SCALAR FIELDwhere we have included the mass m in the definition of qi (t ). In a first step replaceqi φ( xµ ) φ( x), φ( x)q̇i (t ) , t(1.11)(1.12)thereby substituting the label i 1, .N by a continous coordinate x xi with i 1, 2, 3. For themoment we consider a real scalar field i.e. φ( x) φ ( x) which takes values in R, i.e.φ : xµ φ( xµ ) R.(1.13)We will see that such a field describes spin-zero particles. Examples of scalar particles in nature arethe Higgs boson or the inflaton, which cosmologists believe to be responsible for the exponential expansion of the universe during in inflation.To set up the Lagrange function we first note that in a relativistic theory the partial time derivative canonly appear as part of (1.14) µ φ ( x ) µ φ ( x ) . xThus the Lagrange function can be written asZL d3 x L(φ( x), µ φ( x)),(1.15)where L is the Lagrange density. The action therefore isZS d4 x L(φ( x), µ φ( x)).(1.16)While, especially in condensed matter physics, also non-relativistic field theories are relevant, we focus on relativistic theories in this course.Note furthermore that throughout this course we use conventions whereh̄ c 1.(1.17)Then L has the dimension mass4 , i.e. [L] 4, since [S ] 0 and [d4 x] 4.The next goal is to find the Lagrangian: In a relativistic setting L can contain powers of φ and1 µ φ µ φ ηµν µ φ ν φ, which is the simplest scalar which can be built from µ φ. The action in thiscase is"#Z14µnmS d x µ φ φ V (φ) O(φ ( φ) ) ,(1.18)2where111 µ φ µ φ φ̇2 ( φ)2222(1.19)1 Note that the only remaining option µ φ is a total derivative and will therefore not alter the equations of motion underµthe usual assumptions on
This is a writeup of my Master programme course on Quantum Field Theory I (Chapters 1-6) and Quantum Field Theory II. The primary source for this course has been ‹ Peskin, Schröder: An introduction to Quantum Field Theory, ABP 1995, ‹ Itzykson, Zuber: Quantum Field Theory, Dover 1980, ‹ Kugo: Eichtheorie, Springer 1997,
Texts of Wow Rosh Hashana II 5780 - Congregation Shearith Israel, Atlanta Georgia Wow ׳ג ׳א:׳א תישארב (א) ׃ץרֶָֽאָּהָּ תאֵֵ֥וְּ םִימִַׁ֖שַָּה תאֵֵ֥ םיקִִ֑לֹאֱ ארָָּ֣ Îָּ תישִִׁ֖ארֵ Îְּ(ב) חַורְָּ֣ו ם
This is a writeup of my Master programme course on Quantum Field Theory I (Chapters 1-6) and Quantum Field Theory II. The primary source for this course has been Peskin, Schröder: An introduction to Quantum Field Theory, ABP 1995, Itzykson, Zuber: Quantum Field Theory
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