Introduction to Relativistic Quantum Field Theory

Prof. Dr. Stefan Dittmaier, Dr. Heidi Rzehak and Dr. Christian Schwinn

Albert-Ludwigs-Universit¨at Freiburg, Physikalisches Institut D-79104 Freiburg, Germany

Summer-Semester 2014

Draft: October 1, 2014 2 Contents

1 Introduction 7

I Quantization of Scalar Fields 9

2 Recapitulation of Special Relativity 11 2.1 Lorentz transformations, four-vectors, tensors ...... 11 2.1.1 Minkowskispace ...... 11 2.1.2 Lorentztransformations ...... 12 2.1.3 Differentialoperators...... 14 2.1.4 Tensors ...... 15 2.2 LorentzgroupandLorentzalgebra ...... 15 2.2.1 Classification of Lorentz transformations ...... 15 2.2.2 Infinitesimal transformations and group generators ...... 17 2.3 PoincaregroupandPoincarealgebra ...... 19 2.3.1 Thebasics...... 19 2.3.2 Generatorsasdifferentialoperators ...... 20 2.4 Relativistic point particles ...... 22

3 The Klein–Gordon equation 25 3.1 Relativistic wave equation ...... 25 3.2 Solutions of the Klein–Gordon equation ...... 26 3.3 Conservedcurrent...... 28 3.4 Interpretation of the Klein–Gordon equation ...... 29

4 Classical Field Theory 31 4.1 Lagrangian and Hamiltonian formalism ...... 31 4.1.1 Lagrangianfieldtheory...... 31 4.1.2 Hamiltonian field theory ...... 35 4.2 Actionsforscalarfields...... 36 4.2.1 Freerealscalarfield ...... 36 4.2.2 Freecomplexscalarfield ...... 37 4.3 Interactingfields ...... 38

3 4 CONTENTS

4.3.1 Scalar self-interactions ...... 38 4.3.2 Explicit calculation of the Green function (propagator) ...... 40 4.4 SymmetriesandtheNoetherTheorem ...... 43 4.4.1 Continuoussymmetries...... 43 4.4.2 DerivationoftheNoethertheorem ...... 45 4.4.3 Internal symmetries and conserved currents ...... 46 4.4.4 Translation invariance and energy-momentum tensor ...... 47

5 Canonical quantization of free scalar ﬁelds 49 5.1 Canonical commutation relations ...... 49 5.2 FreeKlein–Gordonﬁeld ...... 51 5.3 ParticlestatesandFockspace ...... 55 5.4 Fieldoperatorandwavefunction ...... 58 5.5 Propagatorandtimeordering ...... 59

6 Interacting scalar ﬁelds and scattering theory 61 6.1 AsymptoticstatesandS-matrix ...... 61 6.2 PerturbationTheory ...... 63 6.3 Feynmandiagrams ...... 68 6.3.1 Wick’stheorem ...... 68 6.3.2 FeynmanrulesfortheS-operator ...... 69 6.3.3 FeynmanrulesforS-matrixelements ...... 70 6.4 Crosssectionsanddecaywidths ...... 76

II Quantization of fermion ﬁelds 81

7 Representations of the Lorentz group 83 7.1 Liegroupsandalgebras ...... 83 7.1.1 Deﬁnitions...... 83 7.1.2 Liealgebras ...... 85 7.1.3 Irreducible representations ...... 87 7.1.4 Constructingrepresentations...... 89 7.2 Irreducible representations of the Lorentz group ...... 90 7.3 Fundamental spinor representations ...... 91 7.4 Productrepresentations ...... 93 7.5 Relativistic wave equations ...... 95 7.5.1 Relativistic ﬁelds ...... 95 7.5.2 Relativistic wave equations for free particles ...... 96 7.5.3 TheDiracequation...... 97 CONTENTS 5

8 Free Dirac fermions 99 8.1 Solutions of the classical Dirac equation ...... 99 8.2 Quantization of free Dirac ﬁelds ...... 102 8.2.1 Quantizationprocedure...... 102 8.2.2 ParticlestatesandFockspace ...... 103 8.2.3 Fermionpropagator...... 107 8.2.4 Connection between spins and statistics ...... 108

9 Interaction of scalar and fermion ﬁelds 109 9.1 Interactingfermionﬁelds...... 109 9.2 Yukawatheory ...... 111 9.2.1 FeynmanrulesfortheS-operator ...... 111 9.2.2 Feynman rules for S-matrix elements ...... 113

III Quantization of vector-boson ﬁelds 117

10 Free vector-boson ﬁelds 119 10.1 Classical Maxwell equations ...... 119 10.2Procaequation ...... 121 10.3 Quantization of the elmg. ﬁeld ...... 122 10.3.1 Preliminaries ...... 122 10.3.2 Gupta–Bleuler quantization: ...... 122 10.4Photonpropagator ...... 127

11 Interacting vector-boson ﬁelds 129 11.1 Electromagnetic interaction ...... 129 11.2 Perturbation theory for spinor electrodynamics ...... 132 11.2.1 ExpansionoftheS-operator ...... 132 11.2.2 Feynman rules for S-matrix elements ...... 134 11.3 Importantprocessesof(spinor)QED ...... 137 11.3.1 Elasticepscattering ...... 137 11.3.2 OtherimportantprocessesinQED ...... 140 6 CONTENTS Chapter 1

Introduction

Relativistic quantum ﬁeld theory = mathematical framework for description of elementary particles and their interactions

Guiding principles for the construction of ﬁeld theory and of speciﬁc models of interactions:

relativistic structure of space-time • principles of quantum mechanics • + empirical knowledge collected in colliders experiments (mainly e e−, e±p, pp, p¯p) •

Some empirical facts on particle collisions:

Particle creation and annihilation is possible in collisions. • Relativistic kinematics (four-momentum conservation, conversion of mass and en- • ergy) is extremely well conﬁrmed.

The spectrum of observed particles is very rich, but only very few are really elemen- • tary:

– Leptons (spin 1/2): e, νe, µ, νµ, τ, ντ – Quarks (spin 1/2): u, d, s, c, b, t

confined in hadrons, i.e. mesons (qq¯) or baryons (qqq) – Gauge bosons (spin 1):| {z } force carriers of the strong and electroweak interactions ± gluons (confined) γ, Z0, W bosons – Higgs boson (spin 0): | {z } | {z } lends mass to all elementary particles, recently discovered (full identification ongoing)

7 8 CHAPTER 1. INTRODUCTION

Four fundamental interactions can be distinguished: • – electromagnetic interaction electroweak interaction – weak interaction – strong interaction – gravity (not accessible by collider experiments)

Features of interaction models (special) relativistic covariance • description of particles by states in Hilbert space •  relativistic quantum field theory description of particle dynamics by local fields  •  field quantization •   internal symmetries between particles not only global,  gauge theories • but also local o The role of symmetries space-time symmetry: • Lorentz/Poincarécovariant formulation of field theory mass and spin as fundamental properties of particles → internal symmetries: • – unification of different particles into multiplets of symmetry groups further quantum numbers (charge, isospin, etc.) → – connection between symmetry and dynamics by gauging the symmetry:

introduction of gauge bosons global symmetry local symmetry with own dynamics and couplings to matter ﬁelds

The role of ﬁeld quantization resolution of various inconsistencies in relativistic wave equations • (negative-energy solutions, probability interpretation, etc.) wave-particle dualism • creation and annihilation of particles • connection between spin and statistics • (bosons: spin = 0,1,...; fermions: spin = 1/2, 3/2, ...) Part I

Quantization of Scalar Fields

Chapter 2

Recapitulation of Special Relativity

2.1 Lorentz transformations, four-vectors, tensors

2.1.1 Minkowski space Deﬁnitions and notation: 3-vectors: ~a =(ai), Latin indices: i =1, ..., 3 span 3-dim. position space ֒ • → Contravariant 4-vectors: aµ =(a0,~a), Greek indices: µ =0, ..., 3 • span 4-dim. Minkowski space ֒ → Space-time points (events): xµ =(x0, ~x)=(c t, ~x) • Natural units used in the following: c 1, ~ 1 → → Metric tensor: (g )=(gµν) = diag(+1, 1, 1, 1) • µν − − − Comment: µν Equations like g = gµν —although correct for each coeﬃcient—should be

avoided, since the two sides correspond to two diﬀerent geometrical objects.

Covariant 4-vectors: a =(a0, ~a)= g aν, aµ = gµνa • µ − µν ν Note: Einstein’s convention used, i.e. summation over pairs of equal upper and lower indices Scalar product: • a b = a0b0 ~a ~b = aµb = a bµ = g aµbν = gµνa b (2.1) · − · µ µ µν µ ν Length of 4-vectors: aµa =(a0)2 ~a2 = g aµaν = ... • µ − µν :”space-time distance s2 of two events “a” and “b ֒ → s2 =(x x )µ(x x ) =(t t )2 (~x ~x )2 (2.2) a − b a − b µ a − b − a − b

11 12 CHAPTER 2. RECAPITULATION OF SPECIAL RELATIVITY

Basic principles of special relativity relativity principle laws of physics equivalent in all frames of inertia • → constancy of speed of light value of c is equal in all frames • → Scalar products (space-time distances, etc.) independent of frame of reference ! ⇒ Classiﬁcation of space-time distances: (independent of reference frame!)

time-like > 0 > 0 (x x )2 = light-like if (x x )2 = 0 , i.e. t t ~x ~x = 0 a − b   a − b   | a − b|−| a − b|   space-like   < 0   < 0 (2.3)      time-like: Signals with velocity < speed of light c can be sent from xµ to xµ. • a b light-like: If t > t , a light ray can be sent from xµ to xµ. • b a a b space-like: There is a frame with t = t , i.e. “a” and “b” happen simultaneously. • a b All events “b” with (x x )2 = 0 form the light-cone of x . a − b a The light cone of x separates events causally connected/disconnected to “a”. ⇒ a 2.1.2 Lorentz transformations = all coordinate transformations of Minkowski space that leave the space-time distances (2.2) invariant

Deﬁnitions: Homogenous Lorentz transformations = all linear transformations • characterized by 4 4 matrix Λ: × µ µ ν a′ =Λ ν a , matrix notation (contravariant vectors!): a′ =Λa (2.4) Invariance property of Λ:

µ ν µ ν ρ σ ! ρ σ µ ν ! T gµνa′ a′ = gµνΛ ρ Λ σ a a = gρσa a gµν Λ ρ Λ σ = gρσ, Λ gΛ= g ⇒ (2.5) All scalar products invariant: a′ b′ = a b ⇒ · · Inhomogenous Lorentz transformations (Poincare transformations) • = all aﬃne transformations of space-time characterized by 4 4 matrix Λ and 4-vector a: × µ µ ν µ x′ =Λ ν x + a , x′ =Λx + a (2.6) 2 2 At least all space-time distances invariant: (x′ x′ ) =(x x ) ⇒ a − b a − b 2.1. LORENTZ TRANSFORMATIONS, FOUR-VECTORS, TENSORS 13

Examples:

Rotations: • 1 0 Λ = with DT D =1, sothat ΛT gΛ = g (2.7) D 0 D D D Rotation around the x3 axis: cos ϕ sin ϕ 0 − D (ϕ)= sin ϕ cos ϕ 0 (2.8) 3   0 0 1   Boosts (relating inertial frames moving with a relative velocity ~v): • 3 Boost B3 in the x direction:

3 2 t′ = γ(t + v x ), γ =1/√1 v 1 1 − x′ = x , 2 2 (2.9) x′ = x , 3 3 x′ = γ(x + v t)

Convenient parametrization of ΛB3 by rapidity ν, where v = tanh ν:

γ 0 0 γv cosh ν 0 0 sinh ν 0 10 0 0 10 0 ΛB =   =   = ΛB (ν) (2.10) 3 0 01 0 0 01 0 3 γv 0 0 γ  sinh ν 0 0 cosh ν          Comment: The angles ϕ~e (0 ϕ π, ~e = 1) that parametrize all rotations deﬁne a compact ≤ ≤ | | set (the angles π~e correspond to the same rotation). ± The rapidities ν~e are contained in a non-compact set of numbers ( <ν< ). non-compact Lie group −∞ ∞ The Lorentz transformations form a . See also below. ⇒

Inverse Lorentz transformations µ µ ν µ 1 µ ν a′ =Λ ν a , i.e. a = (Λ− ) ν a′ Proposition: 1 µ α βµ µ (Λ− ) = g Λ g Λ (2.11) ν να β ≡ ν Proof: 1 1 Verify Λ− Λ= 1 (ΛΛ− = 1 analogously):

µ ν α βµ ν α ν βµ (2.5) βµ µ µ Λν Λ ρ = gνα Λ β g Λ ρ =(gνα Λ β Λ ρ) g = gβρ g = gρ = δρ (2.12) 14 CHAPTER 2. RECAPITULATION OF SPECIAL RELATIVITY

q.e.d.

Application: Lorentz transformation of covariant 4-vectors

ν ν ρ ν ρσ σ xµ′ = gµνx′ = gµν Λ ρ x = gµν Λ ρ g xσ =Λµ xσ. (2.13)

2.1.3 Diﬀerential operators

Deﬁnitions:

covariant 4-gradiant: • ∂ ∂ ∂ = , ~ (2.14) µ ≡ ∂xµ ∂t ∇

contravariant 4-gradiant: • ∂ ∂ ∂µ = gµν∂ = , ~ (2.15) ≡ ∂x ν ∂t −∇ µ

wave (d’Alembert) operator: • ∂2 ∂ ∂µ = ~ 2 (2.16) ≡ µ ∂t2 − ∇

µ µ ν µ Lorentz/Poincare transformation properties: x′ =Λ ν x + a

4-gradients: • ν ∂ ∂x ∂ ν µ µ ν ′ ∂µ = µ = µ ν =Λµ ∂ν , ∂ = ... =Λ ν ∂ (2.17) ∂x′ ∂x′ ∂x

wave operator: •

µ ν µ ρ ν ρ ν ′ = ∂µ′ ∂′ =Λµ Λ ρ ∂ν∂ = gρ ∂ν∂ = ∂ν ∂ = = invariant (2.18)

4-divegence of a vector ﬁeld V µ: •

∂ V µ(x)= ∂ V 0(x)+ ∂ V i(x)= V˙ 0(x)+ ~ V~ = invariant (2.19) µ 0 i ∇· 2.2. LORENTZ GROUP AND LORENTZ ALGEBRA 15

2.1.4 Tensors Deﬁnitions

Contravariant tensor T µ1...µn of rank n = object that transforms like the direct prod- • uct aµ1 ...aµn under the change of coordinate frames, i.e.

µ1...µn µ1 µn ρ1...ρn T ′ =Λ ρ1 ... Λ ρn T (2.20)

A covariant tensor T of rank n transforms like x ...x . • µ1...µn µ1 µn A mixed-rank (n, m) tensor transforms as •

µ1...µn µ1 µn σ1 σm ρ1...ρn T ′ ν1...νm =Λ ρ1 ... Λ ρn Λν1 ... Λνm T σ1...σm (2.21)

Invariant tensors:

µν µν Metric tensor: g′ = g , g′ = g • µν µν Totally antisymmetric tensor: • +1 if (µνρσ) = even permutation of (0123) ǫµνρσ = 1 if (µνρσ) = odd permutation of (0123) (2.22)  −  0 otherwise Transformation: 

µνρσ µνρσ µνρσ ǫ′ = ǫ det Λ = ǫ = invariant pseudo-tensor (2.23) × ± (see Exercise 1.1)

2.2 Lorentz group and Lorentz algebra

2.2.1 Classiﬁcation of Lorentz transformations Deﬁnition: The set of Lorentz transformations forms the Lorentz group L:

closure: Λ Λ = Λ = Lorentz transformation (prove!) • 1 2 associativity: Λ (Λ Λ )=(Λ Λ )Λ • 1 2 3 1 2 3 unit element: (Λ )µ = δµ • e ν ν 1 µ µ inverse elements: (Λ− ) = Λ • ν ν 16 CHAPTER 2. RECAPITULATION OF SPECIAL RELATIVITY

Important discrete Lorentz tansformations:

parity P (space inversion): • 0 µ µ µ x x x′ =Λ = , Λ = diag(+1, 1, 1, 1) (2.24) → P ν ~x P − − − −

time reversal T : • 0 µ µ µ x x x′ =ΛT = − , ΛT = diag( 1, +1, +1, +1) (2.25) → ν ~x −

Invariant properties of Λ matrices and classiﬁcation:

det Λ = 1, since ΛT gΛ= g Def.: L Λ det Λ = 1 • ± ⇒ ± ≡{ | ± } L+ = subgroup of proper Lorentz transformations (L = subgroup) − 6 Λ0 1, since g =1= g Λµ Λν = (Λ0 )2 (Λi )2 • | 0| ≥ 00 µν 0 0 0 − 0 0 0 Def.: L↑ Λ Λ 1 , L↓ Λ Λ 1 ≡{ | 0 ≥ } ≡{ | 0 ≤ − } L↑ = subgroup of orthochronous Lorentz transformations (L↓ = subgroup) 6 Consequence: break-up of the Lorentz group into four disconnected subsets • 0 det Λ : Λ 0 : Example:

L+↑ + > 1 Λ= 1

L↑ 1 > 1 Λ=ΛP (2.26) − − L↓ 1 < 1 Λ=ΛT − − − L↓ +1 < 1 Λ= 1 =Λ Λ + − − P T

0 Def.: L↑ Λ det Λ = 1, Λ 1 + ≡{ | ± 0 ≥ } = group of proper, orthochronous (special) Lorentz transformations

Decomposition of Λ (non-trivial!): • Each Λ L↑ can be written as a product of a rotation and a boost: ∈ +

Λ=ΛBΛD. (2.27)

The rotations form a subgroup of L+↑ , while the boosts do not. 2.2. LORENTZ GROUP AND LORENTZ ALGEBRA 17

2.2.2 Infinitesimal transformations and group generators Infinitesimal rotations and boosts: Consider the rotation (2.8) and the boost (2.10) for infinitesimal parameters δϕ and δν: 10 0 0 0 1 δϕ 0 2 3 2 ΛD (δϕ)=  −  + (δϕ ) 1 iδϕ J + (δϕ ), (2.28) 3 0 δϕ 1 0 O ≡ − O 00 0 1     1 00 δν 0 10 0 2 3 2 ΛB (δν)=   + (δν ) 1 iδν K + (δν ), (2.29) 3 0 01 0 O ≡ − O δν 00 1      General infinitesimal rotations or boosts parametrized by six parameters δφi and δνi: Λ (δϕ )= 1 iδ~ϕ J~ = 1 iδϕ J i, Λ (δν )= 1 iδ~ν K~ = 1 iδν Ki (2.30) D i − · − i B i − · − i Definitions: J i = generator of infinitesimal rotations around the xi axis (angular momentum) Ki = generator of infinitesimal boosts in the xi direction Properties of the generators: Explicitly: • 000 0 0 0 00 00 0 0 000 0 0 0 0i 0 0 i 0 J 1 =   , J 2 =   , J 3 =  −  , (2.31) 0 0 0 i 0 0 00 0i 0 0 − 00i 0  0 i 0 0 00 0 0    −          0 i 00 00 i 0 000 i i 000 0 00 0 0 00 0 K1 =   , K2 =   , K3 =   (2.32) 00 00 i 000 0 00 0 00 00 0 00 0 i 000             Hermiticity: i i i i • J † = J = hermitian, K † = K = anti-hermitian (2.33) − Commutation relations: • [J i,J j] = iǫijkJ k (relations of angular momentum) (2.34) [J i,Kj] = iǫijkKk (K~ transforms as 3-vector operator) (2.35) [Ki,Kj]= iǫijkJ k. (2.36) − Comment: The third equation expresses the fact that boosts do not form a subgroup of L, but

that the product of two boosts in general involves a rotation (the so-called Wigner rotation ).

18 CHAPTER 2. RECAPITULATION OF SPECIAL RELATIVITY

General inﬁnitesimal Lorentz transformations: General form: Λµ (δω)= δµ + δωµ + ... • ν ν ν :condition for Lorentz transformation ֒ → µ ν µ µ ν ν gµνΛ ρΛ σ = gµν(δρ + δω ρ)(δσ + δω σ)+ ... ! (2.37) = gρσ + δωσρ + δωρσ + ... = gρσ, δω are antisymmetric, ⇒ δω = δω , (2.38) σρ − ρσ and comprise six independent entries corresponding to δϕi and δνi. Generators M αβ: • i Λµ (δω)= δµ + δω gαµ δβ δµ δω (M αβ)µ (2.39) ν ν αβ ν ≡ ν − 2 αβ ν (M αβ)µ = i(gαµδβ gβµδα) (2.40) ⇒ ν ν − ν

Matrix notation: Λ(δω)= 1 i δω M αβ − 2 αβ Connection between J i, Ki and M αβ: •

j 0j j µ 0µ j jµ 0 i , (µ, ν)=(0, j) or (j, 0), K = M , (K ) ν = i(g δν g δν )= (2.41) − (0 otherwise, 1 i J k = ǫijkM ij , (J k)mn = ǫijk(gimδ gjmδ )= iǫmnk (2.42) 2 2 jn − in − (e.g. J 3 = M 12)

Commutators: (Lorentz algebra) • [M µν , M ρσ]= i(gµρM νσ gµσM νρ gνρM µσ + gνσM µρ) (2.43) − − − Comment: Proof straightforward, easiest based on commutators for J i, Ki.

Finite Lorentz transformations: :result from inﬁnitesimal transformations upon (matrix) exponentiation ֒ → i Λ (~ϕ) = exp iϕ J i , Λ (~ν) = exp iν Ki , Λ(ω) = exp ω M αβ D − i B − i −2 αβ (2.44) Comment: N ϕi i The iterative limit ΛD(ϕi) = lim 1 i J is demonstrative, but not of much N − N →∞ practical use.

2.3. POINCARE GROUP AND POINCARE ALGEBRA 19 2.3 Poincare group and Poincare algebra

2.3.1 The basics Deﬁnition: The Poincare group P is the group of all inhomogeneous Lorentz transformations (Λ, a) with x′ =Λx + a, where Λ L, a = any 4-vector. Obvious restrictions are ∈ P ↑ with Λ L↑ , etc. + ∈ +

P , P ↑ , etc. are non-compact Lie groups with 10 independent parameters. ⇒ + Subgroups:

(Λ, 0): groups L, L↑ , etc. • + (1, a): (abelian) group T of 4-dim. translations • 4

Composition law: (Λ2, a2)(Λ1, a1) = (Λ2Λ1, Λ2a1 + a2) (2.45) as in L P is semi-direct product: P = T ⋊ L ⇒ 4 | {z } Inﬁnitesimal transformations and generators: General transformation: • i (Λ, a) = exp ω M αβ + ia P µ (2.46) −2 αβ µ generators for translations (4-momentum) |{z} Inﬁnitesimal transformation: • i (1 + δω,δa) = 1 δω M αβ + iδa P µ + ... (2.47) − 2 αβ µ

Poincare algebra: • µν ρσ [M , M ] = ..., as in L+↑ , (2.48) µ ν [P ,P ] = 0, since T4 abelian, (2.49) [P µ, M ρσ] = i(gµρP σ gµσP ρ) (2.50) − Proof of (2.50):

1 i αβ µ i αβ (1 + δω, 0)− (1, δa) (1 + δω, 0) = 1 + δω M (1 + iδa P ) 1 δω M + . . . 2 αβ µ − 2 αβ 1 = 1 + iδa P µ δω δa [M αβ, P µ]+ . . . µ − 2 αβ µ 20 CHAPTER 2. RECAPITULATION OF SPECIAL RELATIVITY

1 (1 + δω, 0)− (1, δa) (1 + δω, 0) = (1 δω, 0) (1 + δω, δa) − = (1, δaµ δωµν δa ) − ν = 1 + i(δaµ δωµν δa )P + . . . − ν µ i = 1 + iδa P µ δω (δaν P µ δaµP ν)+ . . . µ − 2 µν − . follows upon comparing coeﬃcients for arbitrary δω δa (2.50) ֒ → αβ µ q.e.d.

2.3.2 Generators as diﬀerential operators Inspect operation of transformations (Λ, a) on some scalar function φ(x):

! 1 φ(x) φ′(x′) = φ′(Λx + a) = φ(x), i.e. φ′(x) = φ Λ− (x a) (2.51) −→(Λ,a) − Translations: • – infinitesimal: µ φ′(x) = φ(x δa) = φ(x) δa ∂ φ(x)+ ... − − µ = [1 + iδaµ(i∂ )+ ] φ(x) µ ··· [1 + iδaµP + ] φ(x), (2.52) ≡ µ ··· P µ = i∂µ = (i∂ , i~ ) = 4-momentum operator as differential operator ⇒ t − ∇ – finite transformations:

µ φ′(x) = φ(x a) = exp ia P φ(x) (2.53) − { µ } Homogeneous Lorentz transformations: • – inﬁnitesimal:

1 µ i αβ µ ν φ′(x) = φ Λ− x = φ x + δω (M ) x + 2 αβ ν ··· i = φ(x)+ δω (M αβ)µ xν ∂ φ(x)+ 2 αβ ν µ ··· = i(gαµδβ gβµδα) ν − ν i | {z } = φ(x)+ δω i(xβ∂α xα∂β)φ(x)+ 2 αβ − ··· i φ(x) δω Lαβφ(x)+ (2.54) ≡ − 2 αβ ··· Lαβ = i(xβ∂α xα∂β) = xαP β xβP α ⇒ − − = generalized angular momentum operator as diﬀerential operator 2.3. POINCARE GROUP AND POINCARE ALGEBRA 21

– ﬁnite transformations:

1 i αβ φ′(x) = φ Λ− x = exp ω L φ(x) (2.55) −2 αβ n o General case: • 1 µ i αβ φ′(x) = φ Λ− (x a) = exp ia P ω L φ(x) (2.56) − µ − 2 αβ n o 22 CHAPTER 2. RECAPITULATION OF SPECIAL RELATIVITY 2.4 Relativistic point particles

Point particle in non-relativistic mechanics: momentum = m~x˙

dxµ But: = (1, ~x˙) = 4-vector, since dt = invariant dt 6 6 ! m~x˙ is not part of a 4-vector ֒ → (m = mass = invariant particle property = constant !)

Correct relativistic generalization with 4-velocity: dxµ uµ = , dτ = dt √1 v2, v = ~x˙ dτ − | | = proper time of the particle = time in particle rest frame (“intrinsic clock”) dt d~x dt dt = , = , ~x˙ =(γ,γ~x˙), γ = 1/√1 v2 (2.57) dτ dτ dτ dτ −

Relativistic 4-momentum:

pµ = muµ = (mγ, mγ~x˙)=4-vector (2.58) invariant square: p2 = m2γ2 m2γ2v2 = m2 ֒ → − ~p2 p0 = p = m2 + ~p2 = m 1+ + ... for ~p m ⇒ 0 2m2 | | ≪ p = relativistic (total) energy of particle with mass m (E = mc2 = rest energy (restoring c = 1 here ֒ → 0 6 T = p m = kinetic energy (2.59) 0 − Comments: pµ = muµ can be directly derived from ansatz ~p = m~x˙ f(v), • · demanding momentum conservation in collisions and f(0) = 1

~p follows from invariance of action S = dt L for point particle • Z  p0 is Hamilton function of free particle = conserved  see exercises ! •   4-momentum conservation directly follows from translational  • invariance of Lagrange function / action     2.4. RELATIVISTIC POINT PARTICLES 23

Comment: c = 1, ~ = 1 All kinematical quantities are measured in the same unit: ⇒ 1 1 [E] = [m] = [p] = [x− ] = [t− ]. (2.60)

Useful units in high-energy elementary particle physics:

energy unit:

• 2 24 [E] = Giga electron Volt (GeV) , GeV/c = 1.8 10− g. (2.61) ×

unit of length: • [x] = Fermi = fm. (2.62)

Relation between units: ~c = 0.197GeV fm

Example: particle decay π− µ− ν → µ masses: m↑m ↑↑0 π µ ≈ Rest frame of π−:

α ~ 2 2 pπ =(mπ, 0), pπ = mπ, (2.63) pα =(E , ~p ), p2 = E2 ~p 2 = m2 , (2.64) µ µ µ µ µ − µ µ pα =(E , ~p ), p2 = E2 ~p 2 = 0 (2.65) ν ν ν ν ν − ν

energy conservation: mπ = Eµ + Eν (2.66)

momentum conservation: ~0= ~pµ + ~pν (2.67)

(2.67) in (2.64) – (2.65): m2 = E2 E2 =(E E )(E + E ) (2.68) µ µ − ν µ − ν µ ν = (Eµ Eν )mπ (2.69) (2.66) −

2 2 mπ mµ mπ mµ Eµ = + , Eν = (2.70) ⇒ 2 2mπ 2 − 2mπ

Note: The direction of the decay is not ﬁxed. π− (= spin 0, no polarization!) decay isotropically in their rest frame. 24 CHAPTER 2. RECAPITULATION OF SPECIAL RELATIVITY Chapter 3

The Klein–Gordon equation

3.1 Relativistic wave equation

Non-relativistic quantum mechanics:

Spinless particle (scalar) ,state vector ψ(t) Hilbert space ֒ • → ψ(t, ~x)= ~x ψ| (t) i= ∈ (complex) wave function in position representation h | i Observables Hermitian operators • → Examples: position ~xˆ and momentum ~pˆ in position space ~ ~xˆ = ~x = multiplicative, ~pˆ = ~ i ∇ Correspondence principle: H (x ,p ) Hˆ (ˆx , pˆ ) with [ˆx , pˆ ] = i~δ • classical i i → i i i j ij ∂ Time evolution by Schr¨odinger equation: i~ ψ = Hˆ ψ • ∂t | i | i ~pˆ 2 Free, spinless particle: Hˆ = • 2m :Schr¨odinger equation is wave equation in position space ֒ → ∂ ~ 2 i~ ψ(t, ~x)= ~2 ∇ ψ(t, ~x) (3.1) ∂t − 2m

Note: wave equation invariant under Galilei trafo ~x′ = R~x + ~a, t′ = t + ∆t 1 ψ′(t, ~x)= ψ t ∆t, R− (~x ~a) is solution if ψ(t, ~x) is. ⇒ − −

25 26 CHAPTER 3. THE KLEIN–GORDON EQUATION

Relativistic generalization: ∂ ~ Idea: E i~ and ~p ~ in energy-momentum relation → ∂t → i ∇ ~p 2 E = Schr¨odinger equation • 2m ⇒ E = c ~p 2 +(mc)2 problem with arbitrarily high spatial derivatives, • ⇒ no covariance ! p 1 ∂2 ˆ mc E2 = c2~p 2 +(mc2)2 φ = ~ 2 + 2 φ 2 ~ • ⇒ −c ∂t −∇ ! 1 / (reduced Compton wave length)

~,c 1 | {z } =→ ∂ ∂µ + m2 φ = + m2 φ =0 Klein–Gordon equation (3.2) ⇒ µ ansatz as wave equation for complex wave function φ ֒ →

Relativistic covariance:

φ in two diﬀerent frames of reference (x′ =Λx + a): φ′(x′)= φ′(Λx + a)= φ(x)

2 2 0 = + m φ(x)=(′ + m ) φ′(x′) form invariance of KG eq. • ⇒ invariant 2 2 0=(| +{zm )}φ(x)= ( + m ) φ′(Λx + a) • 1 φ′(x) = φ Λ− (x a) obeys KG eq. as well. ⇒ −

3.2 Solutions of the Klein–Gordon equation

Fourier ansatz in momentum space: (KG eq. = linear!) 4 d p ipx φ(x)= φ(t, ~x)= e− φ˜(p) (2π)4 Z 4 2 d p ipx 2 2 m φ(x) = e− ( p + m ) φ˜(p)=0 + ֒ → (2π)4 − Z All φ˜(p) with p2 = m2 are solutions with p0 = ω , where ω =+ ~p 2 + m2. ⇒ ± p p p 3.2. SOLUTIONS OF THE KLEIN–GORDON EQUATION 27

Comment: The sign ambiguity is due to the square of E in energy–momentum relation (genuine

relat. feature!).

General solution:

4 d p 2 2 ipx ipx φ(t, ~x)= (2π) δ(p m ) θ(+p )e− a(~p) +θ( p )e− b∗( ~p) (2π)4 −  0 − 0 −  Z ensures p2 = m2 arbitrary complex functions   4 | {z } |{z} | {z } d p 2 2 ipx +ipx = (2π)δ(p m ) θ(+p ) e− a(~p) + e b∗(~p) (2π)4 − 0 Z h i d3p = 3 d˜p = invariant phase-space volume | 2ωp(2π){z ≡ } Z Z

ipx +ipx = d˜p e− a(~p) + e b∗(~p) (3.3) Z h i Comment: 4 4 4 The integral d p is Lorentz invariant, since d p′ = d p det(Λ) with det(Λ) = 1. | | | | Z Z2 Z Moreover, sgn(p0′ ) = sgn(p0) for p′ = Λp and p 0. ≥

Some explicit formulas: 4 3 0 d p d p dp 4 = 3 , (2π) (2π) 2π Z Z Z 1 dp0 δ(p2 m2)= dp0 δ (p0)2 ~p 2 m2 = dp0 δ p0 + ω + δ p0 ω − − − 2ω p − p Z Z Z p h i

Note: Negative-energy solutions b∗ raise problems.

Energy spectrum p0 ( , m] [m, ) not bounded from below. .(Particle can emit∈ an−∞ inﬁnite− amount∪ ∞ of energy (by perturbations ֒ • → System is unstable (no ground state) ! ⇒ Conversely, redeﬁning p0 0 leads to solutions with “wrong” time-evolution phase • 0 ≥ factor e+ip t from non-relat. QM point of view.

Setting b(~p) 0 is not consistent in presence of interactions. .No solution≡ of stability problem ֒ • →

QFT solves problem upon intepreting b∗ solutions as antiparticles. • .a, a∗ (b∗, b) become annihilation/creation operators for (anti)particles ֒ → 28 CHAPTER 3. THE KLEIN–GORDON EQUATION 3.3 Conserved current

Interpretation of φ as quantum mechanical wave function? ,Requirement: conserved “probability current” ~j with probability density ρ ֒ → obeyingρ ˙ + ~ ~j = 0, so that d3x ρ(t, ~x) = const. ∇ Z Recall non-relat. QM: ~j, ρ derived from Schr¨odinger equation and its conjugate:

∂ 1 2 ∂ 1 2 i ψ = ~ ψ , i ψ∗ = ~ ψ∗ , ∂t −2m∇ − ∂t −2m∇

∂ ∂ 1 2 2 i ψ∗ ψ + ψ ψ∗ = ψ∗ ~ ψ ψ ~ ψ∗ ⇒ ∂t ∂t −2m ∇ − ∇ h i ∂ 2 i ψ = ~ ψ∗ ~ ψ ψ ~ ψ∗ . (3.4) ⇒ ∂t | | ∇· 2m ∇ − ∇ =ρ = ~j |{z} − Analogous manipulation with KG equation:| {z }

µ 2 µ 2 (∂µ∂ + m )φ =0 , (∂µ∂ + m )φ∗ =0 , µ 2 µ 2 0= φ∗(∂ ∂ + m )φ φ(∂ ∂ + m )φ∗ ⇒ µ − µ µ µ = ∂ [φ∗∂ φ φ∂ φ∗], continuity equation X (3.5) µ − = 2mi jµ − | {z } µ i µ µ µ Conserved current: j = [φ∗∂ φ φ∂ φ∗], with ∂ j =0 ⇒ 2m − µ

? 0 i 0 0 ρ = j = φ∗∂ φ φ∂ φ∗ = acceptable probability density ? 2m − Problem: ρ can become negative, i.e. ρ = probability density ֒ → 6 QFT solution: Conserved current jµ = (ρ,~j) interpreted as charge (ρ) and current (~j) density of electric (or generalized)∝ charge. 3.4. INTERPRETATION OF THE KLEIN–GORDON EQUATION 29 3.4 Interpretation of the Klein–Gordon equation

Clashes between principles of non-relat. QM and KG eq. / relat. covariance:

Special relativity: space and time should be treated on equal footing. • QM: position is treated as an operator, time as a parameter.

Negative energy solutions of the KG eq. • Conserved density j0 cannot be interpreted as probability density. • Fields φ transforming as 4-vectors under Lorentz transformation cannot be inter- • µ preted as wave functions as the matrices Λ for boosts are not unitary.

Non-vanishing probability for propagation over space-like distances: • Assume resolution of momentum ~p: ∆pi mc ≤ ~ ~c & localization of a particle only within ∆xi ֒ → ∼ ∆pi mc2 probability = 0 for propagation between space-like separated events a and b ֒ → if (x x )26 (~/mc)2 a − b ∼ Violation of causality due to quantum ﬂuctuations ? ⇒

Outlook to solutions by relat. QFT:

Field φ(t, ~x) satisﬁes the (covariant!) KG equation. • ~x and t are both treated as parameters. .Elimination of the asymmetry of space and time ֒ • → For a quantum mechanical description, the ﬁeld is promoted to an operator: • φ(t, ~x) φˆ(t, ~x) → acting on particle states Hilbert space: ∈

– action of φˆ†(x) creates a particle / annihilates an antiparticle at point x; – action of φˆ(x) creates an antiparticle / annihilates a particle at point x.

QFT naturally becomes a many-particle theory. ⇒ Formalization best done within Lagrangian approach to continuum mechanics ֒ → 30 CHAPTER 3. THE KLEIN–GORDON EQUATION Chapter 4

Classical Field Theory

4.1 Lagrangian and Hamiltonian formalism

4.1.1 Lagrangian ﬁeld theory Recapitulation of classical mechanics of point particle generalized coordinates q = q (t) and velocitiesq ˙ = dqi • i i i dt tb action S = dt L(qi, q˙i, t) • ta Z Lagrange function Hamilton’s principle:| {zS =} extremal, i.e. δS = 0, with respect to the variations • d q (t) q (t)+ δq (t), q˙ (t) q˙ (t)+ δq˙ (t), δq˙ (t) δq (t), i → i i i → i i i ≡ dt i with the boundary conditions: δqi(ta/b)=0 d ∂L ∂L = 0, Euler–Lagrange equations of motion ⇒ dt ∂q˙i − ∂qi From discrete to continuous systems: Example (see e.g. Ref. [9]): chain of equal mass points with mass m connected with massless uniform springs with force constant D (coupled harmonic oscillators for small qi): i 1 i i + 1 j − equilibrium < a > displacement >< > from equilibrium qi−1 qi qi+1 − 31 32 CHAPTER 4. CLASSICAL FIELD THEORY

1 kinetic energy: T = mq˙2 • 2 i i X 1 potential energy: V = D(q q )2, q q = extension length • 2 i+1 − i | i+1 − i| i X m a q q 2 Lagrangian: L = T V = a q˙2 D i+1 − i • − 2a i − 2 a i " # X = Li

Euler–Lagrange equations for coordinate| qi: {z } • m qi+1 qi qi qi 1 q¨ Da − + Da − − = 0 a i − a a = Y = µ = si = mass / length|{z} = extension / length |{z} | {z } Note: The force / length on an elastic rod is f = Ys with Y = Young modulus (constant!). Continuum limit: • discrete i continuous x, a dx → → i X Z qi+1(t) qi(t) q(t, x + a) q(t, x) a dx ∂q − = − → a a −−−−−→ ∂x

1 ∂q 2 L = dx µ q˙(t, x)2 Y ⇒ 2 − ∂x Z " # = = Lagrangian density L Comments: | {z } ∂q – does not only depend on q(x) andq ˙(x), but also on due to nearest neigh- L ∂x bour interaction (2nd spatial derivative next-to-nearest neighbour interaction, etc.). → – In more dimensions q(t, x) is generalized to the ﬁeld φ(t, ~x).

Generalization to ﬁelds: generalized coordinates: • continuum limit µ qi(t) φ(t, ~x)= φ(x) = dyn. degree of freedom at x =(t, ~x) discrete −−−−−−−−−−→ continous index i “label” ~x 4.1. LAGRANGIAN AND HAMILTONIAN FORMALISM 33

Note: φ may carry more indices (for spin or other d.o.f.). Lagrange function: • L[φ, φ˙]= d3x (φ, φ,˙ ~ φ, . . . ) = functional of φ and φ˙, L ∇ Z Lagrangian density, “Lagrangian” i.e. L maps φ, φ˙ to a number (dependent on t, but not on ~x) | {z } action S in relativistic theories: • – S = S[φ] = dt L[φ, φ˙] = functional of ﬁeld φ, i.e. S maps φ(x) to a constant. Z – S =! Lorentz invariant (= scalar)

= dt L[φ, φ˙]= d4x (φ, φ,˙ ~ φ, . . . ) L ∇ Z Z Lorentz invariant = (φ, φ,˙ ~ φ, . . . ) = (φ, ∂φ, . . . ) = Lorentz scalar ⇒ L L ∇ | {z }L – d4x extends over complete Minkowski space with φ 0 sufficiently fast for | |→ Zxµ . | |→∞ – S[φ] is invariant under the transformation + ∂µF (φ, ∂φ, . . . ), since L→L µ surface terms vanish. .is unique up to partial integration ֒ → L Hamilton’s principle: • δS = 0 under variation φ(x) φ(x) + δφ(x) with arbitrary infinitesimal δφ(x) vanishing at infinity: → ∂ ∂ 0= δS = d4x L δφ(x)+ L ∂ δφ(x) ∂φ(x) ∂(∂ φ(x)) µ Z µ part. int. ∂ ∂ = d4x L ∂ L δφ(x), (4.1) ∂φ(x) − µ ∂(∂ φ(x)) Z µ ∂ ∂ L ∂µ L =0 Euler–Lagrange equations for fields (4.2) ⇒ ∂φ − ∂(∂µφ) Generalization to higher derivatives: recall variational derivative • δ ∂ ∂ ∂ L = L ∂µ L + ∂µ∂ν L + ... (4.3) δφ ∂φ − ∂(∂µφ) ∂(∂µ∂ν φ)

only relevant for higher-order derivatives δ Equation of motion (EOM): L =0 | {z } ⇒ δφ 34 CHAPTER 4. CLASSICAL FIELD THEORY

Comment: Derivation of the EOM via functional derivative:

δF [φ] 1 = lim (F [φ(y)+ ǫδ(x y)] F [φ(y)]) (4.4) δφ(x) ǫ 0 ǫ − − →

Application to action functional:

S[φ] = d4y (φ(y), ∂φ(y)) (4.5) L Z δS[φ] ∂ ∂ = d4y L δ(x y)+ L ∂yδ(x y) ⇒ δφ(x) ∂φ(y) − ∂(∂yφ(y)) µ − Z µ ∂ ∂ = L ∂x L (4.6) ∂φ(x) − µ ∂(∂xφ(x)) µ

δS[φ] EOM: = 0 (4.7) ⇒ δφ(x)

Rules for functional derivatives:

F , G = functionals; a, b, c = functions

δ δ φ(y)= δ(x y) , a(y) = 0 • δφ(x) − δφ(x)

δ δF [φ] δG[φ] (aF [φ]+ bG[φ]) = a + b , • δφ(x) δφ(x) δφ(x)

δ δF [φ] δG[φ] (F [φ]G[φ]) = G[φ]+ F [φ] . • δφ(x) δφ(x) δφ(x)

Relation between variational and functional derivative consider function f(φ(x)) of ﬁeld φ(x) as speciﬁc type of functional ֒ →

δ δf δf f (φ(y)) = (φ(x)) δ(x y) δ(x y) (4.8)

δφ(x) δφ − ≡ δφ −

δf variational derivative δφ | {z } as function of φ(x)

4.1. LAGRANGIAN AND HAMILTONIAN FORMALISM 35

4.1.2 Hamiltonian ﬁeld theory canonically conjugated momenta: • ∂L ∂L pi = π(x)= ∂q˙i −→ ∂φ˙(x) (4.9) point particles ﬁelds

Hamilton function and density: • H(q ,p )= p q˙ L H = H(t)= d3x (φ, ~ φ, π) (4.10) i i i i − −→ H ∇ i X Z Hamilton density, Hamiltonian | {z } with (φ, ~ φ, π)= πφ˙ (4.11) H ∇ −L Hamiltonian EOMs: • ∂H ∂H ˙ δ δ q˙i = , p˙i = φ = H, π˙ = H (4.12) ∂pi − ∂qi −→ δπ − δφ

Note: and φ˙ depend on φ, ~ φ, and π (but not on derivatives of π). H ∇ Derivation of Hamiltonian EOMs:

δ ∂ ∂φ˙ ∂ ∂φ˙ H = H = φ˙ + π L = φ˙ , (4.13) δπ ∂π ∂π − ∂φ˙ ∂π =π δ ∂ ∂ ∂φ˙ ∂ ∂ ∂φ˙ ∂ H = H ~ H = π |{z} L L ~ H δφ ∂φ − ∇∂ ~ φ ∂φ − ∂φ − ∂φ˙ ∂φ − ∇∂ ~ φ ∇ ∇ =π ∂ ∂ Lag. EOM ∂ ∂ = L ~ H = ∂µ |{z}L ~ H − ∂φ − ∇∂ ~ φ − ∂(∂µφ) − ∇∂ ~ φ ∇ ∇ ∂ ∂ ∂ ∂ = L ~ L ~ H = π,˙ −∂t ∂φ˙ −∇∂ ~ φ − ∇∂ ~ φ − ∇ ∇ =π ∂ ∂φ˙ ∂ ∂ ∂φ˙ ∂ because H |{z} = π L L = L . q.e.d. ~ ~ − ~ − ˙ ~ − ~ ∂ φ π,φ ﬁxed ∂ φ ∂ φ ∂φ ∂ φ ∂ φ φ,φ˙ ﬁxed ∇ ∇ ∇ ∇ ∇ =π

|{z} 36 CHAPTER 4. CLASSICAL FIELD THEORY 4.2 Actions for scalar ﬁelds

µ 2 Question: Which Lagrangian leads to the Klein–Gordon eq. (∂µ∂ + m ) φ = 0, where φ(x) = real or complex scalar ﬁeld ?

4.2.1 Free real scalar ﬁeld The Lagrangian Requirements on : L KG eq. is linear in φ; EOMs reduce by one power in φ. • L .is bilinear in φ and its derivatives ∂ φ, etc ֒ → L µ KG involves only derivatives up to 2nd order. • .contains only derivatives up to 2nd order ֒ → L = Lorentz invariant. •L .is linear combination of bilinear, Lorentz-invariant terms formed by φ, ∂ φ, etc ֒ → L µ All possible terms: φ2, φφ, (∂ φ)(∂µφ). ⇒ µ But: φφ = ∂ (φ∂µφ) (∂ φ)(∂µφ) can be omitted. µ − µ irrelevant surface term = real. | {z } •L Most general ansatz: ⇒ = A(∂ φ)(∂µφ)+ Bφ2 = Agµν(∂ φ)(∂ φ)+ Bφ2 with A, B = real L µ µ ν Determine A, B by EOM:

∂ ∂ B 0= ∂ L L = ∂ (Agµνδρ∂ φ + Agµνδρ∂ φ) 2Bφ =2A ∂ ∂ρφ φ (4.14) ρ ∂(∂ φ) − ∂φ ρ µ ν ν µ − ρ − A ρ (B/A =! m2, A =1/4 (=convention, see H below ֒ → − Lagrangian for free scalar ﬁeld: ⇒ 1 1 1 = (∂ φ)(∂µφ) m2φ2, alternatively: = φ( + m2)φ (4.15) L 2 µ − 2 L −2 4.2. ACTIONS FOR SCALAR FIELDS 37

Hamiltonian formalism: ∂ Canonically conjugated ﬁeld: π = L = φ˙ • ∂φ˙ 1 Hamiltonian: = πφ˙ = π2 +(~ φ)2 + m2φ2 ֒ → H −L 2 ∇ h i Explicit solution: • ipx +ipx φ(x)= d˜p e− a(~p) + e b∗(~p) 2 2 p0=+√~p +m Z h i

ipx +ipx π(x)= i d˜pp0 e− a(~p) e b∗(~p) (4.16) 2 2 − − p0=+√~p +m Z h i

Note: b(~p)= a(~p) for a real scalar ﬁeld. :Hamiltonian ֒ → 3 1 3 2 i(p+q) x H = d x = d x d˜p d˜q p q ~p ~q + m a(~p)a(~q)e− · + c.c. H 2 − 0 0 − · Z Z Z Z n 2 i(p q) x + p q + ~p ~q + m a(~p)a∗(~q)e− − · + c.c. 0 0 · 3 i~k ~x 3 o Use identity d x e · = (2π) δ(~k)

3 Z 1 d p 2 2 2 2ip0x0 = p + ~p + m a(~p)a( ~p)e− + c.c. 2 (2π)3 (2p )2 − 0 − Z 0 n =0 2 2 2 + p + ~p + m a(~p)a∗(~p) + c.c. | 0 {z } 2 2 p0=√~p +m o 2 = d˜p ~p 2 + m2 a(~p) = const > 0 X (4.17) | | Z p Hamiltonian fulfills requirement on kinetic energy (constant, non-negative). ⇒ 4.2.2 Free complex scalar field Complex scalar field φ can be decomposed: φ = 1 (φ + iφ ) with φ ,φ = real • √2 1 2 1 2 2 2 2 ( + m )φ =0, i =1, 2 ( + m )φ =0, ( + m )φ∗ = 0 (4.18) i ⇔ Lagrangian (2 free real fields ˆ=1 complex field): • 1 1 = (∂ φ )(∂µφ ) m2φ φ + (∂ φ )(∂µφ ) m2φ φ L 2 µ 1 1 − 1 1 2 µ 2 2 − 2 2

µ 2 2 =(∂ φ∗)(∂ φ) m φ∗φ, alternatively: = φ∗( + m )φ (4.19) µ − L − 38 CHAPTER 4. CLASSICAL FIELD THEORY

( ) EOMs from variations of φ ∗ : • ∂ ∂ µ 2 0= ∂µ L L = ∂µ∂ + m φ, (4.20) ∂(∂µφ∗) − ∂φ∗

∂ ∂ µ 2 0= ∂µ L L = ∂µ∂ + m φ∗ (4.21) ∂(∂µφ) − ∂φ Conjugate ﬁelds and Hamiltonian: • ∂ ∂ π = L = φ˙∗ , π∗ = L = φ˙ (4.22) ˙ ˙ ∂φ ∂φ∗ 2 2 2 2 (πφ˙ + π∗φ˙∗ = π + ~ φ + m φ (4.23 = ֒ → H −L | | |∇ | | | Explicit solutions as in (4.16), but with a(~p) = b(~p) • 6 :Hamiltonian ֒ → 2 2 H = ... = d˜p ~p + m a(~p)a∗(~p)+ b∗(~p)b(~p) = const > 0 X (4.24) Z p 4.3 Interacting ﬁelds

4.3.1 Scalar self-interactions Extension analogous to L = T V in point mechanics: (real φ as example) − 1 m2 = (∂ φ)(∂µφ) φ2 V (φ) (4.25) L 2 µ − 2 − kinetic term potential term

| {z } | {z 3 } 4 with V = c0 + c1 φ + c3φ + c4φ + ... (4.26) Comments: =0 =0 c = 0: arbitrary definition of energy|{z} offset|{z} • 0 c = 0: arbitrary offset in φ, such that V is minimal in ground state φ 0 • 1 ≡ (V + for φ , otherwise system unstable. V has minimum.) → ∞ | |→∞ → c = 1 m2 0: otherwise no minimum of V at φ 0 • 2 2 ≥ ≡ Dim. analysis: action [S] = [~] = 1, dim[d4x]= 4, dim[∂] = +1 = dim. of mass • − dim[ ] = dim[V ] = 4, dim[φ] = 1, dim[c ] = 1, dim[c ] = 0, dim[c ]= 1, . . . ⇒ L 3 4 5 − Convenient convention: • 4 n cn = gn/Λ − with gn = dimensionless and Λ = common mass scale 4.3. INTERACTING FIELDS 39

QFT: Theories with [couplings] < 0 are non-renormalizable, • i.e. some observables diverge for short-distance interactions (UV limit). But: Such theories can still be useful as low-energy eﬀective ﬁeld theories, where momenta > Λ (distances < 1/Λ) are excluded. Non-renormalizable interactions can also involve derivatives of order > 2. •

EOM and its Green function: ∂ ∂V ∂ L = m2φ , L = ∂µφ ∂φ − − ∂φ ∂(∂µφ) KG eq. with interaction: ⇒

∂ ∂ 2 ∂V 0= ∂µ L L = φ + m φ + , (4.27) ∂(∂µφ) − ∂φ ∂φ Non-linear 2nd order partial differential equation :(define Green function D(x, y ֒ → + m2 D(x, y)= δ4(x y). (4.28) x − − :(integral equation equivalent to (4.27 ֒ → ∂V (φ(y)) φ(x)= φ (x) + d4y D(x, y) (4.29) ⇒ 0 ∂φ Z solution of free KG eq., 2 ( + m| {z)φ}0 =0 Check: ∂V (φ(y)) ∂V (φ(x)) ( + m2)φ(x)=( + m2)φ (x) + d4 y ( + m2)D(x, y) = x x 0 x ∂φ − ∂φ =0 Z δ4(x y) − − | {z } | {z } Iterative solution for sufficiently weak interaction: (perturbation theory)

0th approximation: free motion •

φ(x)= φ0(x) (4.30)

1st approximation: insert φ = φ in r.h.s. of (4.29) Born approximation • 0 → ∂V (φ (y)) φ (x)= φ (x)+ d4y D(x, y) 0 (4.31) 1 0 ∂φ Z 40 CHAPTER 4. CLASSICAL FIELD THEORY

2nd approximation: inserting φ = φ in r.h.s. of (4.29) • 1 ∂V (φ (y)) φ (x)= φ (x)+ d4y D(x, y) 1 2 0 ∂φ Z ∂V (φ (y)) = φ (x)+ d4y D(x, y) 0 0 ∂φ Z ∂2V (φ (y )) ∂V (φ (y )) + d4y 0 1 D(x, y ) d4y D(y ,y ) 0 2 + ..., (4.32) 1 ∂φ2 1 2 1 2 ∂φ Z Z (Expansion to 2nd order in potential terms singles out correction to Born approx.) →

n-th approximation: insert φ = φn 1 in r.h.s. of (4.29) • − Visualization of nth correction: free propagation between n local interactions with V at space-time points yi. Hope that ֒ → φn(x) φ(x). (4.33) n −−−→→∞ 4.3.2 Explicit calculation of the Green function (propagator) Deﬁning Eq. (4.28) = linear, inhomogenous diﬀ. eq. :Fourier ansatz ֒ → 4 d k ik (x y) D(x, y)= D(k) e− · − (4.34) (2π)4 Z Note: D(x, y)= D(x y) because of translational invariance − Insertion of ansatz into (4.28):

4 2 d k 2 2 ik (x y) ( + m )D(x, y)= D(k)( k + m )e− · − x (2π)4 − Z 4 ! 4 d k ik (x y) = δ (x y)= e− · − (4.35) − − − (2π)4 Z

1 1 D(k) = 2 2 = ⇒ k m k0 ~k 2 + m2 k0 + ~k 2 + m2 − − p p 1 1 1 2 2 ± ~ = 0 + 0 with k0 = k + m (4.36) ~ 2 2 k k − k k− ± 2 k + m − 0 − 0 q

p 0 i~k(~x ~y) ∞ dk 1 1 ik0(x0 y0) D(x, y) = dk˜ e − e− − (4.37) 0 + 0 ⇒ 2π k k0 − k k0− Z Z−∞ − − 0 Note: Prescriptions needed to resolve convergence problem near poles at k = k0± ! 4.3. INTERACTING FIELDS 41

Solution: Move the poles into the complex plane by an inﬁnitesimal shift iδ (δ > 0) and use identity

iκx ∞ e dκ − = 2πi θ( x) (4.38) κ iδ ∓ ± Z−∞ ±

Comment: Prove of identity with residue theorem:

Interpret dκ as line integral in complex κ-plane and close contour with half-circle

of inﬁnite radius in such a way that half-circle does not contribute: R Integrand on half-circle exp Im(κ)x damping for Im(κ)x< 0. ∝ { } ⇒

Close contour in lower (upper) half-plane for x> 0 (x< 0). ⇒

Im(κ) Im(κ)

x> 0 x< 0

+iδ +iδ

iδ Re(κ) iδ Re(κ) − −

iκx iκx e− 2πi, e− 0, dκ = − dκ = κ iδ 0. κ iδ +2πi. I ± I ± q.e.d.

Application of Eq. (4.38):

0 ik0(x0 y0) ∞ dk e− − ikσ(x0 y0) 0 0 σ + − 0 − − 0 σ = i e θ (x y ) with k0 = k0 or k0 (4.39) 2π k k0 iδ ∓ ± − Z−∞ − ± Poles at kσ iδ correspond to forward propagation in time (contribution only for x0 >y0); ⇒ 0 − σ 0 0 poles at k0 +iδ correspond to backward propagation in time (contribution only for y > x ).

4 diﬀerent types of propagators: ⇒ Poles at k± iδ: retarded propagator forward propagation of all modes • 0 − → 0 0 ik(x y) D (x, y)= iθ(x y ) dk˜ e− − + c.c. =real (4.40) ret − − Z 42 CHAPTER 4. CLASSICAL FIELD THEORY

Properties: (non-trivial!)

– Causal behaviour: D (x, y)=0for(x y)2 < 0 ret − – Lorentz invariance appropriate for causal wave propagation in classical ﬁeld theory ֒ →

Poles at k± + iδ: advanced propagator backward propagation of all modes • 0 →

Dadv(x, y)= Dret(y, x) (4.41) backward propagation in classical ﬁeld theory ֒ →

+ Poles at k0 iδ and k0− + iδ: Feynman propagator • − + k with forward, k− with backward propagation ֒ → 0 0 4 ik(x y) d k e− − 2 2 D (x, y) = , k± iδ ~k + m iǫ F 4 0 + 0 0 (2π) k k + iδ k k− iδ ∓ ≡ ± − Z − 0 − 0 − q d4k e ik(x y) = − − (inﬁnitesimal ǫ> 0) (4.42) (2π)4 k2 m2 + iǫ Z − 0 0 ik (x y) 0 0 +ik (x y) = iθ(x y ) dk˜ e− · − iθ(y x ) dk˜ e · − (4.43) − − − − Z Z = DF(y, x) = complex

Properties:

– Lorentz invariance (obvious!), ǫ> 0 acts like decay width for all modes – Causal behaviour non-trivial: 2 2 mr 2 D (x, y) = 0 for s =(x y) < 0 (exp. decay e− with r = √ s ). F 6 − ∝ − Causality restored by independence of qm. measurements at x, y with s2 < 0. – Propagator naturally appears in QFT: x0 >y0: forward propagation of particles with k0 > 0; y0 > x0: backward propagation of particles with “k0 < 0” reinterpreted as forward propagation of antiparticles ֒ → Comment: No transport of information by acausal behaviour of DF(x,y).

Phenomenon similar to Einstein–Podolsky–Rosen paradox.

+ Poles at k +iδ and k− iδ: Feynman propagator D (x, y)∗ for time-reversed QFT • 0 0 − F 4.4. SYMMETRIES AND THE NOETHER THEOREM 43

Illustration of perturbative expansion for φ(x):

Dret(x, y) DF(x, y) x0 = t x0 = t

antiparticle pair creation propagation xi xi

4.4 Symmetries and the Noether Theorem

Noether theorem in classical mechanics: Every continuous symmetry of a system leads to a conservation law, e.g. Rotational invariance conservation of angular momentum. • ⇒ Translational invariance cartesian momentum conservation. • ⇒ Now: generalization to ﬁeld theory.

4.4.1 Continuous symmetries Definition: A field theory possesses an infinitesimal continuous symmetry if a transformation

a φ φ′ φ + δω ∆ (φ), (4.44) k → k ≡ k a k leaves the action invariant, S[φ′]= S[φ] . (4.45) Notation: δω = inﬁnitesimal parameters of the transformation • a const. for a global symmetry, = function(x) for a local symmetry. 44 CHAPTER 4. CLASSICAL FIELD THEORY

∆a(φ) = functions of all fields φ and their derivatives. • k k Index a = “internal” index or Lorentz index (a may stand for multiple indices). • Index k runs over all fields φ . • k Implications for and S: L If all φ 0 sufficiently fast for xµ , the invariance (4.45) of S implies that can only change→ by a total derivative:| |→∞ L

a,µ 2 δ (φ′) (φ)= ∂ (K (φ)δω )+ (δω ), (4.46) L≡L −L µ a O since 4 a,µ S[φ′]= S[φ]+ d x ∂µ(K (φ)δωa) = S[φ]. (4.47) ZV = surface integral (Gauss!) = 0 Example for internal symmetries:| {z } U(1) symmetry of a complex scalar theory: • µ 2 =(∂ φ∗)(∂ φ) m φ∗φ V (φ∗φ) = invariant under trafo L µ − − φ′ = exp iqω φ, i.e. ∆(φ)= iqφ with q =const. (4.48) {− } −

Note: If φ describes an electrically charged particle, φ φ′ is an elmg. gauge transformation with q being the electric charge. → SU(N) symmetry of N complex scalars: Φ = (φ ,...,φ )T • 1 N µ 2 =(∂ Φ)†(∂ Φ) m Φ†Φ V (Φ†Φ) = invariant under trafo L µ − − Φ′ = U Φ, with U †U = 1, det(U)=+1, (4.49) U = “special” SU(N) = group of all special, unitary N N matrices U. | {z } × Exponential parametrization of U and inﬁnitesimal transformations:

U(ω ) = exp igT aω , T a = generators = matrices, g = const. a {− a} U(δω ) = 1 igT aδω + ..., i.e. ∆a(φ)= igT a φ (4.50) a − a k − kl l a 1 Properties of T (since U † = U − ):

1 a U(δωa)− = U( δωa)= 1 + igT δωa + ... − a a a = U(δω )† = 1 + ig(T )†δω + ... T =(T )† = hermitian, a a ⇒ 1 = det(U) = exp Tr( igT aω ) Tr(T a)= T a =0. (4.51) { − a } ⇒ kk a =1,...,n = N 2 1 = # independent traceless, hermitian N N matrices. ⇒ − × 4.4. SYMMETRIES AND THE NOETHER THEOREM 45

SO(N) symmetry of N real scalars: Φ=(φ ,...,φ )T • 1 N = 1 (∂ Φ)T(∂µΦ) 1 m2ΦTΦ V (ΦTΦ) = invariant under trafo L 2 µ − 2 − T Φ′ = R Φ, with R R = 1, det(R)=+1, (4.52) SO(N) = group of all special, orthogonal N N matrices R. × Exponential parametrization of R and inﬁnitesimal transformations: R(ω ) = exp igT aω ,... ∆a(φ)= igT a φ . (4.53) a {− a} k − kl l a T 1 Properties of T (since R = R− ): T a = T a = imaginary, T a =0, a =1,...,n = N(N 1)/2. (4.54) kl − lk kk − Space-time symmetries: µ µ µ µ µ Space-time translations x x′ = x + ω with ω = const.: • → µ 2 µ µ φ(x) φ′(x)= φ(x ω)= φ(x) ω ∂ φ(x)+ (ω ) ∆ (φ)= ∂ φ, (4.55) → − − µ O ⇒ − i.e. index a acts as Lorentz index µ. Transformation of the Lagrangian: δ = (x ω) (x)= ω ∂ν ω ∂ Kµν Kµν = gµν . (4.56) L L − −L − ν L ≡ ν µ ⇒ − L 4.4.2 Derivation of the Noether theorem Noether theorem: For each global symmetry of the action, φ φ + δω ∆a(φ) , δ = δω ∂ Ka,µ, δω = const., (4.57) k → k a k L a µ a µ there is a set of conserved currents ja , 0= ∂ jµ = ∂ j0 + ~ ~j , (4.58) µ a 0 a ∇ a if the ﬁelds φk satisfy the EOMs.

Proof: 0 = δ δω ∂ Ka,µ(φ) L − a µ ∂ a ∂ a a,µ = L δωa∆k(φ)+ L δωa∂µ∆k(φ) δωa∂µK (φ) ∂φk ∂(∂µφk) −

EOM ∂ ∂ = ∂ L δω ∆a(φ)+ L δω ∂ ∆a(φ) δω ∂ Ka,µ(φ) µ ∂(∂ φ ) a k ∂(∂ φ ) a µ k − a µ µ k µ k ∂ = δω ∂ L ∆a(φ) Ka,µ(φ) (4.59) a µ ∂(∂ φ ) k − µ k ja,µ ≡ Note:| A sum over repeated{z labels} k is implied. q.e.d. 46 CHAPTER 4. CLASSICAL FIELD THEORY

Implications: Noether currents: • a,µ ∂ a a,µ a j = L ∆k(φ) K (φ), ∂j =0. (4.60) ∂(∂µφk) − Note: ja only ﬁxed up to a constant factor. Noether charges: • Qa(t) d3x ja,0(t, ~x)= d3x π ∆a(φ) Ka,0(φ) . (4.61) ≡ k k − Z Z = ∂L h ∂(∂0φ) i Charge conservation: |{z}

a 3 a,0 3 a 2 a Q˙ (t)= d x ∂0j (t, ~x)= d x ~ ~j = d A~ ~j =0, (4.62) − ∇· Gauss − · ZV =const. ZV IA(V ) if the currents ja vanish suﬃciently fast for ~x . | |→∞ 4.4.3 Internal symmetries and conserved currents Reconsider examples from Sect. 4.4.1 ֒ → U(1) symmetry of the complex scalar theory: • µ 2 =(∂ φ∗)(∂ φ) m φ∗φ V (φ∗φ), φ′ = exp iqω φ, ∆(φ)= iqφ. (4.63) L µ − − {− } − Noether current:

µ ∂ ∂ µ µ j = iq L φ L φ∗ = iq [(∂ φ∗)φ φ∗(∂ φ)] (4.64) − ∂(∂ φ) − ∂(∂ φ ) − − µ µ ∗

µ = iqφ∗←→∂ φ with f(x)←→∂ g(x) f(x)∂ g(x) (∂ f(x))g(x). µ ≡ µ − µ Note: Result agrees (up to prefactor) with conserved current of Sect. 3.3.

Conserved charge:

3 0 3 0 3 Q = d x j (x) = iq d xφ∗←→∂ φ = iq d x (π∗φ∗ φπ) , (4.65) − Z Z Z with explicit solution (3.3) of free KG equation equation

1 2ip0t 2ip0t Q = q d˜p b∗(~p)b(~p) a∗(~p)a(~p) b(~p)a( ~p)e− + a∗(~p)b∗( ~p)e + c.c. − 2 − − − − Z h i = q d˜p a(~p) 2 b(~p) 2 . (4.66) | | −| | Z Positive- and negative-frequency modes carry opposite charges, ⇒ in line with the antiparticle interpretation of the negative-frequency modes b ! 4.4. SYMMETRIES AND THE NOETHER THEOREM 47

SU(N) symmetry of N complex scalars: • n Noether currents:

a,µ ∂ a ∂ a j = L ∆k(φ)+ L ∆k(φ∗) ∂(∂µφk) ∂(∂µφk∗) µ a µ a = ig [(∂ φ∗)T φ (∂ φ )T φ∗] , n =1, . . . , n. (4.67) − k kl l − k kl l

4.4.4 Translation invariance and energy-momentum tensor Recall: space-time translations of ﬁelds and : L ∆ν (φ)= ∂ν φ , δ = δω ∂ Kµν with Kµν = gµν . (4.68) k − k L − ν µ − L “Current” j ˆ= energy-momentum tensor θ: (sign prefactor = convention) ⇒ ∂ ∂ θµν L ∆ν(φ) Kµν (φ) = L ∂ν φ gµν . (4.69) ≡ − ∂(∂ φ ) k − ∂(∂ φ ) k − L µ k µ k Conserved “charges” Q form a 4-vector: ⇒

P µ d3x θ0µ = d3x π ∂µφ g0µ . (4.70) ≡ k k − L Z Z Individual components:

P 0 = d3x π φ˙ = d3x = H = Hamilton function, (4.71) k k −L H Z h i Z P~ = d3x π ~ φ = ﬁeld momentum. (4.72) − k ∇ k Z “Charge” conservation ˆ= conservation of energy and momentum of the ﬁelds. ⇒ P˙ µ =0. (4.73)

Comment: The “current” of Lorentz transformations forms a rank-3 tensor, and the associated

“charge” the ﬁelds’ angular momentum.

Example: free complex scalar ﬁeld µ 2 =(∂ φ∗)(∂ φ) m φ∗φ. L µ − Energy-momentum tensor: • µν µ ν ν µ µν θ =(∂ φ∗)(∂ φ)+(∂ φ∗)(∂ φ) g . (4.74) − L 48 CHAPTER 4. CLASSICAL FIELD THEORY

Hamiltonian: • 2 2 2 2 0 3 = πφ˙ + π∗φ˙∗ = π + ~ φ + m φ , P = d x . (4.75) H −L | | |∇ | | | H Z Field momentum: • i 3 0i 3 i i P = d x θ = d x (φ˙∗ ∂ φ + φ˙ ∂ φ∗). (4.76) Z Z Insertion of plane-wave solutions yields • P µ = d˜ppµ a(~p) 2 + b(~p) 2 , (4.77) | | | | Z i.e. each mode characterized by a(~p), b(~p) carries energy p0 = ~p 2 + m2, momentum ~p. ∝ ∝ p Non-uniqueness issue of θµν Possible redeﬁnition of θµν: µν µν ρµν θ˜ = θ + ∂ρΣ , (4.78) with any rank-3 tensor Σρµν = Σµρν (antisymmetry in the ﬁrst two indices). − Features of θ˜µν :

Conservation: • ˜µν µν ρµν ∂µθ = ∂µθ + ∂µ∂ρΣ =0. (4.79)

=0 =0 by symmetry

Field momentum: | {z } | {z } • µ 3 0µ µ 3 ρ0µ P˜ d x θ˜ = P + d x ∂ρΣ ≡ V V i0µ 00µ Z Z = ∂iΣ , since Σ = 0 = P µ + d2Ai Σi0µ = P µ,| {z } (4.80) Gauss IA(V ) =0 if Σ vanishes fast enough for xµ | |→∞ | {z } Both tensors are equally suited as energy-momentum tensors. ⇒ Comment: This freedom can be used to construct conserved tensors with desired properties

(e.g. symmetry, gauge invariance) that are not automatically satisﬁed by the form

directly obtained from the Noether theorem.

Chapter 5

Canonical quantization of free scalar ﬁelds

5.1 Canonical commutation relations

Consider discrete system with coordinates qk(t) and canonical conjugate momenta pk(t) and its quantization in the Heisenberg picture: qk(t),pk(t) are hermitian operators obeying the classical EOMs; • Heisenberg’s commutation relations. • !(Take continuum limit q (t) φ(t, ~x ֒ → k → Comment: Heisenberg picture is more appropriate for field quantization than Schrödinger picture, because the EOM for the field (which becomes an operator) is known. Recall

the connection of the two pictures by the unitary transformation for time evolution:

1 ψ H ψ(t0) = U − (t,t0) ψ(t) = const. for some ﬁxed t0, (5.1) | i ≡ | i | i qm. state in H picture qm. state in S picture = t-dependent

|{z} | {z } ˆ 1 ˆ OH(t) U − (t,t0) O U(t,t0), (5.2) ≡ qm. operator qm. operator in H picture in S picture |{z} | {z } where the time evolution operator U satisﬁes the diﬀerential equation

dU(t,t ) 0 ˆ 1 i = HU(t,t0) with U(t0,t0)= . (5.3) dt

49 50 CHAPTER 5. CANONICAL QUANTIZATION OF FREE SCALAR FIELDS

Quantization procedure:

Discrete system: Continuous system:

Canonical variables q (t), p (t) Canonical ﬁeld operators φ (x), • k k • k obey commutators: πk(x) obey [q (t),p (t)] = i~δ , [φ (t, ~x), π (t, ~y)] = i~δ δ(~x ~y), k l kl k l kl −

[qk(t), ql(t)] = 0, [φk(t, ~x),φl(t, ~y)] = 0,

[pk(t),pl(t)]=0 [πk(t, ~x), πl(t, ~y)]=0 Note: The commutator relations Note: The commutators only hold only hold for equal times. for equal times t = x0 = y0.

H and L are hermitian operators The hermitian operators = πφ˙ • obtained from classical quantities • and are obtained analogously:H − via the correspondence principle: L L

unique up to reordering cl cl cl cl H(qk ,pl ) H(qk,pl) (φ ,∂φ ) (φ ,∂φ ) −−−−−−−−−−−→ L k k → L k k The operators fulﬁll the EOMs: EOMs: • • dq (t) 1 ∂φ (t) 1 k = [q (t),H], k = [φ (t),H], dt i~ k ∂t i~ k dp (t) 1 ∂π (t) 1 k = [p (t),H] k = [π (t),H] dt i~ k ∂t i~ k ( ˆ=classical EOM: 1 [.,.] ˆ= .,. ) i~ { } States Ψ are time independent. States Ψ are time independent. • | i • | i Microcausality for space-like dis- • tances is demanded:

[φk(x), πl(y)] = 0,

[φk(x),φl(y)] = 0, etc., for (x y)2 < 0, also for x0 = y0. − 6 5.2. FREE KLEIN–GORDON FIELD 51

Comment: In the canonical formalism Lorentz covariance is not manifest as

⋆ the commutator relations are imposed at equal times,

⋆ the Hamilton operator is not a Lorentz scalar,

⋆ time is singled out in the EOMs.

Observables are, however, Lorentz invariant which can be shown, for example, via

the functional integral formalism.

5.2 Free Klein–Gordon ﬁeld Classical: QFT:

Real KG ﬁeld φ(x) hermitian ﬁeld operator φ†(x)= φ(x) −→ = 1 (∂ φ)(∂µφ) 1 m2φ2 = 1 (∂ φ)(∂µφ) 1 m2φ2 L 2 µ − 2 −→ L 2 µ − 2 ( + m2)φ =0 ( + m2)φ = 0 (operator equation) ⇒ ⇒

Complex KG ﬁelds φ(x), φ(x)∗ non-hermitian ﬁeld operators φ(x), φ†(x) µ 2 −→ µ 2 =(∂ φ∗)(∂ φ) m φ∗φ =(∂ φ†)(∂ φ) m φ†φ L µ − −→ L µ − Solution of KG equation:

ipx ipx ipx ipx φ(x)= d˜p a(~p)e− + b∗(~p)e φ(x)= d˜p a(~p)e− + b†(~p)e −→ Z Z functions a(~p), b(~p) operators a(~p), b(~p) −→ real case: b(~p)= a(~p) real case: b(~p)= a(~p)

Canonical momentum: ∂ ∂ ∂ ∂ π = L = φ˙∗, π∗ = L = φ˙ π = L = φ˙†, π† = L = φ˙ ˙ ˙ ˙ ˙ ∂φ ∂φ∗ −→ ∂φ ∂φ†

Meaning of the operators a, a†, b, b† ?

1. Calculate a, a†, b, b† from φ(x), φ†(x) (inverse Fourier transformation):

3 i~q ~x ip0x0 3 ip0x0 3 d x e− · φ(x)= d˜p a(~p)e− (2π) δ(~p ~q)+ b†(~p)e (2π) δ(~p + ~q) − Z Z h i 1 iq0x0 iq0x0 = a(~q)e− + b†( ~q)e , (5.4) 2q0 − q0=√~q 2+m2 h i 3 i~q ~x i iq0x0 iq0x0 d x e− · φ˙(x)= a(~q)e− b†( ~q)e . (5.5) −2 − − q0=√~q 2+m2 Z h i 52 CHAPTER 5. CANONICAL QUANTIZATION OF FREE SCALAR FIELDS

3 iq0x0 i~q ~x 0 3 iqx a(~q) = i d x e e− · iq φ(x)+ φ˙(x) = i d x e ←→∂ φ(x), (5.6) ⇒ − 0 Z = eiqx h i Z 3 iqx b(~q) = i d x e| ←→∂{z0 φ†(x}) (derived analogously) (5.7) Z with

f←→∂ g = f(∂ g) (∂ f)g. (5.8) µ µ − µ

2. Commutator relations: (choose x0 = y0)

3 3 iqx ipy 0 0 a(~q), a†(~p) = d x d y e e− iq φ(x)+ φ˙(x), ip φ†(y)+ φ˙†(y) 0 0 • − x = y Z Z q0 = ~q 2 + m2 h 0 0 i p0 = p~p 2 + m2 = iq φ(x)+ π†(x), ip φ†(y)+ π(y) − p | 0 {z 0 } = iq [φ(x), π(y)] + ip π†(x),φ†(y) − = iq0iδ(~x ~y) + ip0( i)δ(~x ~y) − − − − 3 i(q p)x 0 0 = d x e − (q + p ) q0 = ~q 2 + m2 Z 0 p 2 2 p = ~p + m p = (2π)32 ~p 2 + m2δ(~q ~p), (5.9) −

p 3 2 2 b(~q), b†(~p) = ... = (2π) 2 ~p + m δ(~q ~p), (5.10) • − p [a(~q), a(~p)] = ... = b†(~q), b†(~p) =0 forallothercommutators. (5.11) • 3. Energy and momentum 4-vector:

µ 3 µ µ µ0 P = d x π(∂ φ)+ π†(∂ φ†) g (5.12) − L Z Comment: Ordering issue of operators solved later (normal ordering).

Energy:

0 3 µ 2 H = P = d x 2π†π (∂ φ†)∂ φ + m φ†φ (5.13) − µ Z 3 2 = d x π†π +( φ†)( φ)+ m φ†φ (5.14) ∇ ∇ Z 1 0 = ... = d˜p p a(~p)a†(~p)+ a†(~p)a(~p)+ b(~p)b†(~p)+ b†(~p)b(~p) . (5.15) 2 Z 5.2. FREE KLEIN–GORDON FIELD 53

3-momentum:

3 P~ = d x π φ + π† φ† − ∇ ∇ Z 1 = ... = d˜p ~p a(~p)a†(~p)+ a†(~p)a(~p)+ b(~p)b†(~p)+ b†(~p)b(~p) . (5.16) 2 Z Commutation relations:

1 0 H, a†(~p) = d˜q q a(~q), a†(~p) a†(~q)+ a†(~q) a(~q), a†(~p) 2 Z n=(2π)32p0δ(~q ~p) =(2π)32p0δ(~q ~p)o − − 0 = p a†(~p), | {z } | {z } (5.17) 0 [H, a(~p)] = H, a†(~p) † = p a(~p), (5.18) − − P~ , a†(~p) = ~pa†(~p), P~ , a(~p) = ~pa(~p), (5.19) − h i h i µ µ µ µ P , a†(~p) = p a†(~p), [P , a(~p)] = p a(~p), (5.20) ⇒ − µ µ µ µ P , b†(~p) = p b†(~p), [P , b(~p)] = p b(~p). (derived analogously) − (5.21) 4. Comparison with system of independent harmonic oscillators of quantum machanics:

2 ~ pk 1 2 2 ω H = + mω q = (a a† + a† a ) 2m 2 k 2 k k k k Xk Xk

with shift operators ak, ak† obeying

[a , a†]= δ , [a , a ] = [a† , a†] = 0 [H, a† ]= ~ωa† , [H, a ]= ~ωa . k l kl k l k l k k k − k Illustration for energy eigenstate E : | i

H a† E = [H, a† ] E + a† H E =(~ω + E)a† E , k| i k | i k | i k| i i.e. a† E is energy eigenstate to energy E + ~ω (a† “creates” energy ~ω). k| i k

Interpretation of a(~p), a†(~p), b(~p), b†(~p) as creation and annihilation operators for ⇒ﬁeld modes (=particle excitations):

( ) ( ) a † and b † correspond to two independent, free particle types X and X¯, re- • spectively, both with mass m: 0 2 2 a(~p) / a†(~p) annihilates / creates particle X with energy ~ω = p = ~p + m • and 3-momentum ~p (de Broglie momentum). p 54 CHAPTER 5. CANONICAL QUANTIZATION OF FREE SCALAR FIELDS

0 2 2 b(~p) / b†(~p) annihilates / creates particle X¯ with energy ~ω = p = ~p + m • and 3-momentum ~p. p 5. Electric current density and charge operators [cf. Eq. (4.64)]:

µ µ µ µ j = iq (∂ φ†)φ φ†(∂ φ) = iqφ†←→∂ φ, (5.22) − − 3 0 Q = iq d x φ†π† πφ = iqφ†←→∂ φ (5.23) − Z = ... = q d˜p a†(~p)a(~p) b(~p)b†(~p) . (5.24) − Z Comment: Ordering issue of operators solved later (normal ordering).

Commutation relations: ⇒

Q, a†(~p) =+qa†(~p), [Q, a(~p)] = qa(~p), (5.25) − Q, b†(~p) = qb†(~p), [Q, b(~p)]=+qb(~p), (5.26) − i.e. a† and b increase charge by amount q, while a and b† reduce charge by amount q. Particle X carries charge +q, particle X¯ carries charge q (X¯ = antiparticle). ⇒ − Def.: Charge conjugation C

C ipx ipx C φ (x) φ†(x)= d˜p b(~p)e− + a†(~p)e , (φ†) (x) φ(x), (5.27) ≡ ≡ Z i.e. C interchanges particle and antiparticle.

Real KG ﬁeld:

C Hermitian ﬁeld operator φ(x)= φ†(x)= φ (x), i.e. a(~p)= b(~p). • X X¯ (Particle is its own antiparticle.) ⇒ ≡ Factor 1/2 in and . • H L

µ 1 µ P = d˜p p a(~p)a†(~p)+ a†(~p)a(~p) (5.28) ⇒ 2 Z with

µ µ µ µ P , a†(~p) = p a†(~p), [P , a(~p)] = p a(~p). (5.29) − 5.3. PARTICLE STATES AND FOCK SPACE 55

Electric current and charge operators: jµ = const. 1, Q = const. 1. • × ×

Q, a†(~p) = [Q, a(~p)]=0, (5.30) ⇒ i.e. particle creation / annihilation does not change overall charge. Charge q =0, X is electrically neutral. ⇒ X Comment: The problem with the (divergent) constant in Q is solved by normal ordering (=part renormalization process of ).

5.3 Particle states and Fock space

Idea: construct Hilbert space of qm. states upon applying creation operators to ground state (analogy to qm. harmonic oscillator).

Deﬁnition: Fock space

X(~p ) = a†(~p ) 0 1 particle (5.34) | 1 i 1 | i X¯(~p ) = b†(~p ) 0 1 antiparticle (5.35) | 1 i 1 | i X(~p )X(~p ) = a†(~p )a†(~p ) 0 2 particles (5.36) | 1 2 i 1 2 | i X¯(~p )X¯(~p ) = b†(~p )b†(~p ) 0 2 antiparticles (5.37) | 1 2 i 1 2 | i X(~p )X¯(~p ) = a†(~p )b†(~p ) 0 1 particle, 1 antiparticle | 1 2 i 1 2 | i . . (5.38)

X(~p ) ...X(~p ) = a†(~p ) ...a†(~p ) 0 n particles (5.39) | 1 n i 1 n | i . .

Fock space = Hilbert space spanned by all the particle and antiparticle states: • 0 , X(~p ) , X¯(~q ) ,..., X(~p ) ...X(~p )X¯(~q ) ... X¯(~q ) ,... | i | 1 i | 1 i | 1 n 1 m i 56 CHAPTER 5. CANONICAL QUANTIZATION OF FREE SCALAR FIELDS

Properties of Fock states:

The many-particle states are symmetric with respect to particle exchange: • ...X(~p ) ...X(~p ) ... = ...X(~p ) ...X(~p ) ... | i j i | j i i ... X¯(~p ) ... X¯(~p ) ... = ... X¯(~p ) ... X¯(~p ) ... (5.40) | i j i | j i i Particles X and X¯ are bosons. (Fermionic states are antisymmetric.) ⇒ Normalization of one-particle states: •

a 0 =0 X(~p) X(~q) = 0 a(~p)a†(~q) 0 | =i 0 a(~p), a†(~q) 0 h | i h | | i h | | i = (2π)32p0δ(~p ~q) 0 0 = (2π)32p0δ(~p ~q) = Lorentz invariant, − h | i − =1 (5.41) |{z} X¯(~p) X¯(~q) = ... = (2π)32p0δ(~p ~q), (5.42) h | i − X(~p) X¯(~q) =0. (5.43) h | i Note: X(~p) X(~p) for momentum eigenstates. h | i→∞ .Wave packets needed to obtain normalized one-particle states Ψ with Ψ Ψ = 1 ֒ → | i h | i

Vacuum state and normal ordering: Vacuum state 0 : no particle, i.e. 0 P µ 0 =! 0, 0 Q 0 =! 0, etc. | i h | | i h | | i But: These conditions are not fulﬁlled automatically. Enforce condition “by hand” (concept of normal ordering), since ֒ → usually only changes in energy, momentum, etc., are measurable. Example: vacuum energy of real scalar ﬁeld

1 0 0 H 0 = 0 d˜p p [a(~p)a†(~p)+ a†(~p)a(~p)] 0 h | | i h | 2 | i Z p0 = d˜p 0 [a(~p), a†(~p)] +2 0 a†(~p)a(~p) 0 0 2 h | h | | i | i Z h = (2π)32p0δ(~p ~p), =0 i − with (2π)3δ(~p ~p) V = space volume | {z −} → | {z } d3p p0 = V (5.44) (2π)3 2 → ∞ Z number of states in V | {z } 5.3. PARTICLE STATES AND FOCK SPACE 57

H for a state f = d˜p f(~p) ~p (wave packet), ⇒ h i | i | i Z f being a square-integrable function, d˜p f(~p) 2 < : | | ∞ Z f H f = d˜p f(~p) 2 p0 + 0 H 0 (5.45) h | | i | | h | | i Z , finite, observable non-observable→∞ for time-independent 0 | {z } | i | {z } Redefinition of H upon subtracting 0 H 0 ֒ → h | | i Definition: normal ordering

: A : A 0 : A : 0 = 0 (5.46) ≡ all annihilation operators shifted to the right ⇒ h | | i

Examples: : a(~p) a†(~p):= a†(~p)a(~p),

: a(~k)a†(~k)a(~p)a†(~p):= a†(~k)a†(~p)a(~k)a(~p).

Redeﬁnition of all operators, also of , , etc.: L H Deﬁnitions: •

particle number density operator: NX (~p)= a†(~p)a(~p), (5.47)

particle number operator: NX = d˜p NX (~p), (5.48) Z antiparticle number density operator: NX¯ (~p)= b†(~p)b(~p), (5.49)

antiparticle number operator: NX¯ = d˜p NX¯ (~p). (5.50) Z 4-momentum operator: •

µ µ P = d˜pp [NX (~p)+ NX¯ (~p)] . (5.51) Z electric charge operator: •

Q = q d˜p [N (~p) N ¯ (~p)] . (5.52) X − X Z 58 CHAPTER 5. CANONICAL QUANTIZATION OF FREE SCALAR FIELDS 5.4 Field operator and wave function

Interpretation of ﬁeld operator Consider one-particle wave packet for particle X (charged scalar):

X = d˜p f(~p) X(~p) = d˜p f(~p)a†(~p) 0 (5.53) | f i | i | i Z Z Interference with state φ†(x) 0 : | i

0 φ(x) f = d˜p f(~p) 0 φ(x)a†(~p) 0 (5.54) h | | i h | | i Z ikx = d˜p f(~p) dk˜ e− 0 a(~k)a†(~p) 0 (5.55) h | | i Z Z = (2π)3(2k0)δ(~k ~p) − ipx = d˜p e− f(~p) ϕ (|x). {z } (5.56) ≡ f Z Compare with wave packet of non-relat. QM: d3p ψ = ψ (0) = f(~p) ~p , | f i | f i (2π)3 | i Z 3 d p ip0t ψf (t) = U(t, 0) ψf (0) = exp iHt ψf = f(~p) e− ~p 2 , 3 0 ~p | i | i {− }| i (2π) | i p = 2m 3 Z 3 d p ip0t d p ipx ψ (t, ~x)= ~x ψ (t) = f(~p) e− ~x ~p = e− f(~p) f h | f i (2π)3 h | i (2π)3 Z = ei~p~x Z = ~x exp iHt ψ (5.57) h | {− }| f i | {z } ϕ(x) is analogue of one-particle wave function ψ(t, ~x)= ~x ψ(t) in QM. ⇒ h | i φ†(x) 0 is analogue of exp +iHt ~x = Heisenberg state (t = 0) corresponding to | i { }| i position eigenstate ~x at time t, i.e. describes particle created at position ~x at time t. | i

Space–time transformations: x x′ =Λx + a → Qm. states: • f f ′ = U(Λ, a) f with U =unitaryoperator. (5.58) | i→| i | i

.Transition amplitudes f ′ g′ = f U †U g = f g = invariant ֒ → h | i h | | i h | i Field operator: • φ(x′)= U(Λ, a) φ(x) U †(Λ, a), (5.59)

so that scalar products f ...φ(x)... g = f ′ ...φ(x′)... g′ = invariant. h | | i h | | i 5.5. PROPAGATOR AND TIME ORDERING 59

Wave function: •

ϕ ′(x′)= 0 φ(x′) f ′ = 0 U(Λ, a) φ(x) U †(Λ, a)U(Λ, a) f h | | i h | | i = 0 = invariant = 1 h | = 0 φ(x) f = ϕ(x). (5.60) h | | i | {z } | {z } 1 ϕ ′(x)= ϕ Λ− (x a) . (5.61) ⇒ − 5.5 Propagator and time ordering

Concept of time ordering of operators is very important in QFT. Deﬁnition: Time-ordering operator T

T φ (x ) ...φ (x ) φ (x )φ (x ) ...φ (x ) for x0 > x0 >...x0 , (5.62) 1 1 n n ≡ i1 i1 i2 i2 in in i1 i1 in i.e. “operators with earlier times are applied ﬁrst”.

(Feynman) Propagator of complex scalar ﬁeld (cf. Sect. 4.3.2)

iD (x, y)= 0 T φ(x)φ†(y) 0 . (5.63) F h i Proof: Time-ordered product of two ﬁelds:

0 0 0 0 T φ(x)φ†(y) = φ(x)φ†(y)θ(x y )+ φ†(y)φ(x)θ(y x ). (5.64) − −

ipx ipx 0 T φ(x)φ†(y) 0 = 0 T d˜p a(~p)e− + b†(~p)e ⇒ h i h Z iqy iqy d˜q a†(~q)e + b(~q)e− 0 × i Z 0 0 i(px qy) = θ(x y ) d˜p d˜q e− − 0 a(~p)a†(~q) 0 − h | | i Z Z 0 0 i(px qy) + θ(y x ) d˜p d˜q e − 0 b(~q)b†(~p) 0 − h | | i Z Z x0 >y0: particle creation at y, propagation to x, annihilation at x • x0

0 0 ip(x y) 0 0 ip(x y) = θ(x y ) d˜p e− − + θ(y x ) d˜p e − − − Z Z 60 CHAPTER 5. CANONICAL QUANTIZATION OF FREE SCALAR FIELDS

d4p e ip(x y) = − − [see Eq. (4.42)] (2π)4 p2 m2 + iǫ Z − = iDF(x, y) (5.65)

q.e.d. Chapter 6

Interacting scalar ﬁelds and scattering theory

QFT with interacting ﬁelds framework for formulating theories of fundamental interactions ֒ → But: evaluation extremely complicated systematic approximative methods needed ֒ → Exact solutions: only for some lower-dimensional models • Perturbation theory: expansion in small coupling constants g most useful for scattering problems ֒ • → Lattice calculations: numerical simulations by discretizing space–time (useful for static problems (e.g. for bound states in strong interaction ֒ • → 6.1 Asymptotic states and S-matrix

Asymptotic states in particle scattering Scattering process: i f (6.1) | i→| i with i = prepared momentum eigenstate before scattering, | i evolving into a complicated mixed many-particle state i′ after scattering, | i f = speciﬁc ﬁnal state that is contained in i′ after scattering. | i | i Relevant cases: 2-particle scattering: i = ~k ,~k • | i | 1 2i particle decay: i = ~k • | i | 1i

61 62 CHAPTER 6. INTERACTING SCALAR FIELDS AND SCATTERING THEORY

Technical description:

interaction in ﬁnite time interval [ T, T ] with T any relevant time scale • − ≫ initial state i : t (t< T ), no interaction • | iin → −∞ − ﬁnal state f : t + (t> +T ), no interaction • | iout → ∞ Subtleties:

Behaviour of φ(x) for t = x0 : • → ∓∞ 1/2 φ(x) t] Z φin/out(x), (6.2) →∓∞ wave-function renormalization constant|{z} where the asymptotics holds in the “weak” sense (for matrix elements only).

Origin of Z: φ(x) and “free” ﬁelds φin/out(x) are canonically normalized (commutators!), but

– Free ﬁelds only have non-vanishing matrix elements with one-particle states:

ikx 0 φ (x) ~k = e− . (6.3) h | in | iin – Interacting ﬁelds interfere also with multiparticle states:

0 φ(x) ~k ...~k =0. (6.4) h | | 1 ni 6 – Relation between wave functions:

0 φ(x) ~k = Z1/2 0 φ (x) ~k (6.5) h | | i h | in/out | i Comment: Z can be calculated from the vacuum expectation value of [φ[x), φ(y)] (see e.g.

[2, 3]). Under some assumption (ﬁniteness), one can show that 0

Interaction changes mass value (mass renormalization): • Asymptotic ﬁelds satisfy the free KG equation,

2 ( +m ¯ )φin/out(x)=0. (6.6)

but massm ¯ = m = mass in original Lagrangian. 6 Note: in lowest order of perturbation theory: Z =1andm ¯ = m. • Higher orders deserve care, i.e. a proper renormalization. 6.2. PERTURBATION THEORY 63

φin(x)= Sφout(x) S†, (6.9)

so that α φ β = α Sφ S† β = α φ β , etc. (6.10) in h | in| iin in h | out | iin out h | out| iout Poincar´einvariance of matrix elements requires: • S = U(Λ, a) S U †(Λ, a), (6.11)

Aim: perturbative expansion for Sfi derive relation between S, time evolution, and H ֒ → 6.2 Perturbation Theory

Recapitulation of qm. time evolution pictures: 1. Schr¨odinger picture: States carry time evolution, described by the time evolution operator U:

ψ(t) = U(t, t ) ψ(t ) , (6.13) | i 0 | 0 i dU(t, t ) i 0 = H(t)U(t, t ), U(t , t )=1. (6.14) dt 0 0 0 Properties of U: 64 CHAPTER 6. INTERACTING SCALAR FIELDS AND SCATTERING THEORY

unitarity: • 1 U(t, t0)= U − (t0, t)= U †(t0, t), (6.15) group composition law: •

U(t, t′)U(t′, t0)= U(t, t0). (6.16)

Iterative solution: t U(t, t )=1 i dt′ H(t′)U(t′, t ) 0 − 0 Zt0 t t t′ 2 =1 i dt′ H(t′)+( i) dt′ dt′′ H(t′)H(t′′)+ ... − t0 − t0 t0 Z Z Z ′ t ( i)2 t t t =1 i dt′ H(t′)+ − dt′ dt′′ H(t′)H(t′′)+ dt′′ H(t′′)H(t′) − 2 ′ Zt0 Zt0 Zt0 Zt interchange integration variables t′ t′′ + ... ↔ | {z } t t t 1 2 =1 i dt′ H(t′)+ ( i) dt′ dt′′ T [H(t′) H(t′′)] + ... − t0 2 − t0 t0 Z t Z Z T exp i dt′H(t′) . ≡ − Zt0 2. Heisenberg picture:

States are time-independent and tied to the S picture at some time t0: ψ = ψ(t ) = U(t , t) ψ(t) . (6.17) | iH | 0 i 0 | i Operators are transformed in such a way that matrix elements remain the same:

OH(t)= U †(t, t0) O(t) U(t, t0). (6.18) EOM for operators: ⇒ dOH(t) ∂O(t) i = i U˙ †O(t)U + U †O(t)U˙ + U †i U dt ∂t ∂O(t) = [O (t),H (t)] + i (6.19) H H ∂t H

with HH(t)= U †(t, t0) H(t) U(t, t0). 3. Interaction picture: Hamilton operator split into a free and an interacting part, •

H(t)= H0(t)+ Hint(t) (spectrum of H0(t) known) (6.20) 6.2. PERTURBATION THEORY 65

Time evolution from H removed from states (as in H picture): • 0 ψ(t) = U †(t, t ) ψ(t) ,O (t)= U †(t, t ) O U (t, t ) (6.21) | iI 0 0 | i I 0 0 0 0

with the time evolution operator U0 of H0, dU (t, t ) i 0 0 = H (t)U (t, t ). (6.22) dt 0 0 0 EOM of the states: • d i ψ(t) = H (t) ψ(t) with H (t)= U †(t, t )H (t)U (t, t ) , (6.23) dt | iI I | iI I 0 0 int 0 0

i.e. states evolve in time with the interaction Hamiltonian HI. EOM of the operators: •

dOI(t) ∂O(t) i = [O (t),H (t)]+i with H (t)= U †(t, t )H (t)U (t, t ), dt I 0,I ∂t 0,I 0 0 0 0 0 I (6.24) i.e. operators evolve in time with the free Hamiltonian H0,I. Time evolution operator U in the I picture: • I ψ(t) = U †(t, t )U(t, t ) ψ U (t, t ) ψ , (6.25) | iI 0 0 0 | iH ≡ I 0 | iH OI(t)= UI(t, t0) OH(t) UI†(t, t0) , (6.26) dUI(t, t0) i = U †(t, t )H (t)U(t, t )= H (t)U (t, t ). (6.27) dt 0 0 int 0 I I 0 Formal solution: t U (t, t )= T exp i dt′ H (t′) . (6.28) I 0 − I Zt0 66 CHAPTER 6. INTERACTING SCALAR FIELDS AND SCATTERING THEORY

Application to scattering in QFT: Comment: Here we assume Z = 1 andm ¯ = m, which is suﬃcient for the lowest perturbative

order and, thus, for this lecture.

Initial states: take t, t < T and subsequently T . • 0 − − → −∞ H picture and I picture are identical (no interaction yet). ⋄ ψ(t) = ψ(t ) = i = const. ⋄ | iI | 0 iH | iin U (t, t )= 1, φ (x)= φ(x). ⋄ I 0 in Interaction period: t < T

Final| states:{z } t < T

φ (x)= U (t, T ) φ(x) U †(t, T )= U (T, T ) φ (x) U †(T, T ) ⋄ in I − I − I − out I − T →∞ Sφ (x) S†. −−−→ out f(t) I UI(t, +T ) f in = test ﬁnal state, where f in typically is some measured ⋄ |free momentumi ≡ eigenstate.| i | i

S-matrix element: • f(t) ψ(t) = f U †(t, T )U (t, T ) i I h | iI in h | I I − | iin = U (T, T ) S (6.30) I − → f S i = f i = S . −→ in h || | iin {zout h | i}in fi Note: i and f are Heisenberg states corresponding to t . | iin | iout → −∞ Operators in the I picture: • Field operators (for all times t, see above) for free particle propagation: ⋄

φin(x)= UI(t) φ(x) UI†(t). (6.31) 6.2. PERTURBATION THEORY 67

Hamiltonian (needed for time evolution): ⋄ H (t) = U (t)H (t)U †(t), U (t) U (t, ) I I int I I ≡ I −∞ 3 = d x U (t) φ(x), π(x) U †(t). I Hint I Z = d3x φ (x), π (x) . (6.32) Hint in in Z Example: scalar ﬁeld theory with interaction potential, (φ)= V (φ) Lint −

3 3 H (t) = d x U (t) φ(x) U †(t)= d x φ (x) (6.33) I I Hint I Hint in Z Z with (φ )= V (φ )= (φ ). (6.34) Hint in in −Lint in Comment: This transition is more complicated if the interaction V (φ) involves derivatives, i.e.

if int involves canonical momenta. Then, in general, int(φI) = int(φI), as e.g. H H 6 −L in scalar QED.

Perturbative expansion of S-matrix: ( i i , f f ) • | i≡| iin | i≡| iin

6.3.1 Wick’s theorem Task: Computation of S-matrix elements, i.e. of

f T [ (x ) ... (x )] i , (6.37) h | Hint 1 Hint n | i where (x) = : φ (x)φ† (x) : with free “in”-ﬁelds φ , etc. Hint in in ··· in Translate time-ordered products into normal-ordered products, ⇒ so that f a†(p)...a(k) i can be evaluated explicitly. h | | i Example used for illustration: one real scalar ﬁeld φ with g (x)= : φ3(x): . (6.38) Hint −3! Note: Subscript “in” suppressed in the following.

Deﬁnition: Contraction of free, bosonic real ﬁeld operators:

: φi φj := 0 T [φiφj] 0 : ...φi 1φi+1 φj 1 φj+1 : (6.39) ··· ··· ··· h | | i· − ··· − ··· ··· Example:

φ(x)φ(y) = 0 T [φ(x)φ(y)] 0 = iD (x, y), Feynman propagator (6.40) h | | i F = complex number

Identity for time-ordered| product{z of} two ﬁeld operators (see Exercise 5.2):

T [φ(x)φ(y)] = : φ(x)φ(y): + 0 T [φ(x)φ(y)] 0 h | | i = : φ(x)φ(y): + φ(x)φ(y). (6.41)

General case of an arbitrary number of free, bosonic, real ﬁeld operators ruled by

Wick’s theorem: (Discussion in Exercise 6.2)

T [φ φ ] = : φ φ : + : φ φ φ φ : 1 ··· n 1 ··· n 1 ··· i ··· j ··· n pairs ij X + : φ φ φ φ φ φ : + ... (6.42) 1 ··· i ··· k ··· j ··· l ··· n doubleX pairs ij,kl Extension: If some of the ﬁelds in the argument of the T -product in (6.42) are within the same normal-ordered product, they will not be contracted. 6.3. FEYNMAN DIAGRAMS 69

Examples:

four ﬁelds: • T [φ φ ] = : φ φ : + : φ φ : φ φ + : φ φ : φ φ + : φ φ : φ φ 1 ··· 4 1 ··· 4 1 2 3 4 1 3 2 4 1 4 2 3

+ : φ2φ3 : φ1φ4 + : φ2φ4 : φ1φ3 + : φ3φ4 : φ1φ2 (6.43)

+ φ1φ2φ3φ4 + φ1φ3φ2φ4 + φ1φ4φ2φ3,

pair of normal-ordered products: •

T [: φ1φ2 :: φ3φ4 :] = : φ1 φ4 : + : φ1φ3 : φ2φ4 + : φ1φ4 : φ2φ3 ··· (6.44) + : φ2φ3 : φ1φ4 + : φ2φ4 : φ1φ3 + φ1φ3φ2φ4 + φ1φ4φ2φ3.

6.3.2 Feynman rules for the S-operator Application of Wick theorem expands S operator in terms of propagators and normal- ordered ﬁelds: ig S = U ( , )= T exp d4x : φ3(x): I ∞ −∞ 3! Z ig = 1 + d4x : φ3(x): 3! Z

1 ig 2 + d4x d4x : φ3(x )φ3(x ): 2 3! 1 2 1 2 Z

x1 x2

2 2 2 2 2 + 3 : φ (x1)φ (x2): φ(x1)φ(x2) + 2(3) : φ(x1)φ(x2): φ(x1)φ(x2)

x1 x2 x1 x2 70 CHAPTER 6. INTERACTING SCALAR FIELDS AND SCATTERING THEORY

+ 3! φ(x1)φ(x2) + ... (6.45)

(6.46) x1 x2

Feynman rules for graphical representation of the terms gn: ∝ 1. Draw all possible diagrams with n 3-point vertices, connected in all possible ways by lines (including disconnected diagrams).

2. Translate graphs into analytical expressions as follows:

External lines for non-contracted ﬁelds: • φ(x) = (6.47) x

Internal lines for contracted ﬁelds (=propagators): •

φ(x1)φ(x2) = (6.48) x1 x2 Vertices for interaction terms: •

ig = (6.49) 3! x

3. Include a combinatorial factor (symmetry factor) for each diagram (more detail as explained below).

4. Integrate the sum of all terms according to 1 d4x ... d4x : ... : . (6.50) n! 1 n Z 6.3.3 Feynman rules for S-matrix elements Final task: Evaluate f S i upon sandwiching expansion of S-operator between states h | | i i = ~k ,...~k , f = ~p , . . .~p . (6.51) | i | 1 mi | i | 1 ni 6.3. FEYNMAN DIAGRAMS 71

Deﬁnitions: T -matrix • f S i = f i + f S 1 i . (6.52) h | | i h | i h | − | i = 0 for i = f , f T i , T-matrix, | i 6 | i ≡ h | | i unscattered|{z} part only| scattered{z part} . Only the T -matrix contributes to a non-trivial scattering with i = f ֒ → | i 6 | i Transition matrix element (transition amplitude) • f T i = i(2π)4 δ k p . (6.53) h | | i i − j Mfi i j X X transition matrix expresses momentum element conversation, due to |{z} |translational{z invariance} Example: 2 2 scattering → i = ~k ,~k , f = ~p , ~p . (6.54) | i | 1 2i | i | 1 2i Use the expansion of the S matrix (6.45). :f T i involves expectation values of normal-ordered operator products ֒ → h | | i ~p , ~p : φ3(x): ~k ,~k , ~p , ~p : φn(x )φn(x ): ~k ,~k , n =0, 1, 2, 3. (6.55) h 1 2| | 1 2i h 1 2| 1 2 | 1 2i

Recall: φ = φin = free ﬁeld. :(Use plane-wave decomposition with creation/annihilation operators a†(~p)/a(~p ֒ → ipx ipx φ(x)= d˜p a(~p) e− + a†(~p) e . (6.56) Z Only operator combinations with two a’s and two a†’s contributes in ~p , ~p ... ~k ,~k , ⇒ h 1 2| | 1 2i i.e. ~p , ~p : φ2(x )φ2(x ): ~k ,~k . (6.57) h 1 2| 1 2 | 1 2i

Typical manipulation:

iq1xi iq1xj : ...φ(x )φ(x ): ~k ,~k = d˜q d˜q e− e− a(~q )a(~q ) ~k ,~k + ... i j | 1 2i 1 2 1 2 | 1 2i Z † † = a (~k1)a (~k2) 0 | i

3 3 iq1xi iq1xj = d q d q e− e− δ(~q ~k ) δ(~q ~k )+ δ(~q |~k {z) δ(}~q ~k ) 0 + ... 1 2 2 − 1 1 − 2 2 − 2 1 − 1 | i Z ik1xi ik2xj ik1xj ihk2xi i = e− e− + e− e− 0 + ... (6.58) | i 72 CHAPTER 6. INTERACTING SCALAR FIELDS AND SCATTERING THEORY

General cases: Define contractions of fields with external states: • ikx : ...φ(x)... : ...~k . . . = e− , (6.59) | i . . . ~p. . . : ...φ(x) : = eipx. (6.60) h | ··· Perform all possible contractions of fields in normal-ordered products with external • states. Application to example ~p , ~p : φ2(x )φ2(x ): ~k ,~k : h 1 2| 1 2 | 1 2i Three types of contractions: •

2 2 2 i(p1+p2)x1 i(k1+k2)x2 ~p , ~p : φ (x )φ (x ): ~k ,~k = 2 e e− , (6.61) h 1 2| 1 2 | 1 2i

2 2 2 i(k1 p1)x1 i(k2 p2)x2 ~p , ~p : φ (x )φ (x ): ~k ,~k = 2 e− − e− − , (6.62) h 1 2| 1 2 | 1 2i

2 2 2 i(k1 p2)x1 i(k2 p1)x2 ~p , ~p : φ (x )φ (x ): ~k ,~k = 2 e− − e− − , (6.63) h 1 2| 1 2 | 1 2i

plus the identical contributions with x x . Factor of 2. 1 ↔ 2 ⇒ Apply remaining factors and integrals for T -matrix element: • 1 (ig)2 32 d4x d4x ~p , ~p : φ2(x )φ2(x ): ~k ,~k φ(x )φ(x ) (6.64) 2 (3!)2 1 2 h 1 2| 1 2 | 1 2i 1 2 Z and use the momentum-space representation of propagator,

4 d q i iq(x1 x2) φ(x )φ(x )= 0 T [φ(x )φ(x )] 0 = e− − . (6.65) 1 2 h | 1 2 | i (2π)4 q2 m2 + iǫ Z − Example: Explicit evaluation of contribution (6.63):

2 4 1 2 (ig) 4 4 d q i iq(x1 x2) 3 i(k1 p2)x1 i(k2 p1)x2 3 d x d x e− − 2 e− − e− − 2 (3!)2 1 2 (2π)4 q2 m2 + iǫ Z Z − d4q i = (ig)2 (2π)4δ(q + k p ) (2π)4δ(q + p k ) (2π)4 1 − 2 q2 m2 + iǫ 1 − 2 Z − i = (2π)4δ(k + k p p ) (ig)2 , (6.66) 1 2 − 1 − 2 (k p )2 m2 + iǫ 1 − 2 − ;[(δ-function for momentum conservation appears explicitly [cf. (6.53 ֒ → all combinatorial prefactors have canceled. 6.3. FEYNMAN DIAGRAMS 73

Sum of the three diﬀerent contractions (6.61)–(6.63): • i i i i = (ig)2 + + Mfi (k + k )2 m2 + iǫ (k p )2 m2 + iǫ (k p )2 m2 + iǫ 1 2 − 1 − 1 − 1 − 2 −

k1 p1 k1 p1 k1 p1

= + k1 p1 + k1 p2 k1 + k2 − −

k2 p2 k2 p2 k2 p2

i i i = (ig)2 + + , (6.67) s m2 + iǫ t m2 + iǫ u m2 + iǫ − − − with the three Mandelstam variables (see Exercise 2.3):

s =(k + k )2 , t =(k p )2 , u =(k p )2 . (6.68) 1 2 1 − 1 1 − 2 Note: One diagram of the expansion (6.45) of the S-operator produces three diagrams in the expansion of the transition matrix element.

Generalization of the 2 2 example to arbitrary processes leads to → Feynman rules transition matrix elements in momentum space for contributions proportional to gN term in i for an n m scattering process: Mfi → 1. Draw all possible diagrams with N three-point vertices and n + m external legs.

2. Impose momentum conservation at each vertex.

3. Insert the following expressions:

external lines: • = 1 (6.69)

internal lines: • i = (6.70) q q2 m2 + iǫ − vertices: •

= ig (6.71)

4. Apply a symmetry factor 1/SG for each diagram (see below). 74 CHAPTER 6. INTERACTING SCALAR FIELDS AND SCATTERING THEORY

5. Integrate over (loop) momenta qi not ﬁxed by momentum conservation according to

d4q i . (6.72) (2π)4 Z

Note: In our 2 2 example S = 1, and all momenta were ﬁxed by external momenta. → G Comments to the general case:

Loop momenta (not ﬁxed by external states): • Systematic counting:

– 1 momentum integral per propagator. – 1 space–time integral per vertex (yields momentum conservation at vertex).

4 i(q+k1 p2)x1 4 Above example: d x e− − = (2π) δ(q + k p ). 1 1 − 2 Z – δ-function for overall momentum conservation split oﬀ from . Mfi remaining # momentum integrals is given by ⇒ L = #propagators #vertices + 1 = number of loops in a diagram. (6.73) −

Perturbation series for is an expansion in # loops: ⇒ Mfi – L = 0, leading order, Born approximation, – L = 1, next-to-leading order, one-loop approximation, ...

Symmetry factor S : • G S = 1 results from two sources: G 6 g – Incomplete cancellation of factor 1/3! in (x)= : φ3(x):, Hint −3! because some contractions to propagators are diagrammatically equivalent. – Incomplete cancellation of factor 1/n! in nth term of S = T exp ... , { } because some permutations of vertices xi are diagrammatically equivalent.

SGi = # permutations of internal lines or vertices that leaves the diagram unchanged = the order of the symmetry group of graph G. 6.3. FEYNMAN DIAGRAMS 75

Examples:

2 2 1 1 SG = 2, since etc. 1 1 ≡ 2 2

.only (3!)2/2 diﬀerent contractions ֒ →

y x

S = 2, since etc. G ≡ x y .only 4!/2 diﬀerent permutations ֒ → Disconnected diagrams: • Momentum conservation at each vertex .overall momentum conservation” in each connected part of a diagram“ ֒ → Example: p2

k1 p1 δ(k1 + p1 + p2) δ(k2 + p3 + p4). p4 ∝

k2 p3

Contributions only for exceptional momenta, corresponding to diﬀerent scattering proceeding in parallel. Contributions irrelevant for single scattering reaction. ⇒ Vacuum diagrams: • = (sub)diagrams with no external legs, e.g.:

– Vacuum graphs always factor from remaining graphs of the diagram. .They modify each S-matrix element S by the same factor ֒ → fi – Vacuum correction factor calculable as

0 S 0 = 0 U ( , ) 0 = phase factor, (6.74) h | | i h | I ∞ −∞ | i contradicting the initial assumption (6.8) 0 = S 0 . | i | i 76 CHAPTER 6. INTERACTING SCALAR FIELDS AND SCATTERING THEORY

Redeﬁne ⇒ U ( , ) S = I ∞ −∞ , 0 U ( , ) 0 h | I ∞ −∞ | i so that 0 S 0 = 1 and all vacuum graphs cancel in observables (i.e. they can be ignored inh | practice).| i

6.4 Cross sections and decay widths

Cross section:

Definition: dN = dσ F , s × # scattering events / time differential incoming particle flux with particle fl carrying cross section #interactions = area×time momentum|{z}~pfl |{z} |{z} where

N N v F = 1 2 rel with N = # incoming particles type 1, 2 • V 1,2 and vrel = relative velocity between incoming particles.

W n d3p dN = fi V fl N N , • s T (2π)3 1 2 l=1 transition rate Y |{z}d3p with V fl = # states of stationary waves in a box with volume V . (2π)3

Transition probability W for i f : • fi | i→| i f T i 2 W = |h | | i| (6.75) fi f f i i h | ih | i which is not well deﬁned as f T i 2 [δ(p p )]2 and f f , i i . |h | | i| ∝ i − f h | i h | i→∞ Solution: Box with ﬁnite extension in space–time, volume = V T : · (2π)8 [δ(p p )]2 V T (2π)4δ(p p ), i − f −→ · i − f ~p ′ ~p 3 0 0 ~p ~p ′ = (2π) 2p δ(~p ~p ′) → 2p V. (6.76) h | i − −−−−−→ 6.4. CROSS SECTIONS AND DECAY WIDTHS 77

1 n 1 W = V T (2π)4δ(p p ) 2, ⇒ fi · i − f (2p0 V )(2p0 V ) (2p0 V ) |Mfi | i1 i2 l=1 fl ! Y transition matrix element 1 n d3p |{z} dσ = 2 fl (2π)4δ(p p ) . (6.77) 4p0 p0 v fi (2π)3 i f i2 i2 rel |M | ! − Yl=1 ﬂux factor = dΦf , invariant phase space volume | {z } | {z } Using the Lorentz-invariant form of F , p0 p0 v = (p p )2 (m m )2, with p2 = m2 , p2 = m2 , i2 i2 rel i1 i2 − i1 i2 i1 i1 i2 i2 dσ takes the ﬁnalp form:

1 2 dσ = fi dΦf . (6.78) 4 (p p )2 (m m )2 |M | i1 i2 − i1 i2 Total cross section: p ⇒ 1 2 σtot = dσ = dΦf fi . (6.79) 4 (p p )2 (m m )2 |M | Z i1 i2 − i1 i2 Z Comments: p

dσ is Lorentz invariant. • For polarizable particles (= scalars): • 6 – initial state: take specific polarization or average over incoming spin states, – final state: analyze specific polarization or sum over outgoing spin states.

Identical particles in final state: exclude identical configurations in dΦ . • f Z .Factor 1/(n !) for n identical particles of type X in full integral ֒ → x X Differential cross sections: • Leave one or more kinematical variables in dΦf open. Z Example: Distribution in scattering angle θ for the particle f1: dσ = dσ δ(θ θ ). dθ − f1 Z 78 CHAPTER 6. INTERACTING SCALAR FIELDS AND SCATTERING THEORY

Decay width:

Particle decay: X f + + f → 1 ··· n !Treatment analogous to scattering ֒ → Results: Partial decay width: • 1 2 ΓX f = dΦf fX . (6.80) → 2m |M | X Z Total decay width: • Γtot = ΓX f , (6.81) → Xf sum over all decay channels ~ |{z} Particle lifetime: τX = . ⇒ Γtot Note: Treatment of polarizations, identical particles, and diﬀerential distributions analogous to cross section.

Example: φφ scattering in φ3 theory in lowest order Process: φ(k ) φ(k ) φ(p ) φ(p ), momentum conservation: k + k = p + p . (6.82) 1 2 → 1 2 1 2 1 2 Momenta in centre-of-mass frame: • kµ =(E, 0, 0, βE), k2 = m2, (6.83) 1,2 ± 1,2 m2 with E = beam energy, β = 1 2 = velocity, r − E pµ =(E, Eβ sin θ cos ϕ, Eβ sin θ sin ϕ, Eβ cos θ), p2 = m2, (6.84) 1,2 ± ± ± 1,2 with θ = scattering angle = angle between ~p1 and ~k1.

Mandelstam variables: • 2 2 s =(k1 + k2) =4E , (6.85) θ t =(k p )2 = p2 + k2 2p k = 2β2E2(1 cos θ)= 4β2E2 sin2 , (6.86) 1 − 1 1 1 − 1 · 1 − − − 2 θ u =(k p )2 = = 4β2E2 cos2 , (6.87) 1 − 2 ··· − 2 s + t + u =4m2. (6.88) 6.4. CROSS SECTIONS AND DECAY WIDTHS 79

Born diagrams: • k1 p1 k1 p1 k1 p1

i fi = + k1 p1 + k1 p2 (6.89) M k1 + k2 − −

k2 p2 k2 p2 k2 p2 s-channel t-channel u-channel i i i = (ig)2 + + (6.90) s m2 t m2 u m2 − − − = dependent on E and θ, but not on ϕ (rotational invariance wrt beam axis!).

Cross section: • 1 1 1 – ﬂux = = = , 4 (k k )2 m4 4 E4(1 + β2)2 m4 8E2β 1 2 − − – phase space:p (see Exercisep 7.1)

1 λ(s, m2, m2) β dΦ = dΩ = dϕ d cos θ. (6.91) 2 (2π)2 8s 1 32π2 Z p Z Z Z

σ = ﬂux dΦ 2 ⇒ 2 |Mfi| Z g4 1 1 1 2 = dϕ d cos θ + + . (6.92) 256π2E2 s m2 t m2 u m2 Z Z − − − 2π expressible in terms of logarithms → | {z } | {z } 80 CHAPTER 6. INTERACTING SCALAR FIELDS AND SCATTERING THEORY Part II

Quantization of fermion ﬁelds

Chapter 7

Representations of the Lorentz group

7.1 Lie groups and algebras

(Continuous groups (e.g. Lorentz / Poincar´egroups, many internal symmetry groups ֒ → 7.1.1 Deﬁnitions A Lie group is a group whose elements g are parametrized by a set of continuous parameters ωa , a =1,...n: g(ω)= g(ω1,...,ωn). Group-multiplication law: •

g(ω)g(ω′)= g(ω′′) with ωa′′ = fa(ωb,ωc′ ), (7.1)

diﬀerentiable functions of ωb,ωc′ | {z } n = dimension of the Lie group, • identity e corresponds to ω = 0 by convention: • a g(0) = e. (7.2)

A Lie group is called compact if the set of all ω is compact. • a A Lie group is called connected if every element g(ω) is connected to the identity e • by a continuous path in the set of the parameters ω.

Example: Lorentz group L L = non-compact (space of boosts is non-compact). • 0 > L = connected (disconnected parts characterized by det Λ = 1 and Λ < 0); • 6 ± 0 L+↑ = connected.

83 84 CHAPTER 7. REPRESENTATIONS OF THE LORENTZ GROUP

Comment: Examples:

C : GL(N, ) general linear group 2 The 2N -dimensional group of invertible complex matrices A (det A = 0). 6 C : SL(N, ) special linear group 2 The (2N 2)-dimensional group of complex matrices A with det A = 1. − : O(N) orthogonal group The N(N 1)/2-dimensional group of real orthogonal matrices M − (M T M = 1, so that det M = 1).

± : SO(N) special orthogonal group The N(N 1)/2-dimensional group of orthogonal matrices M with det M = 1.

− : U(N) unitary group 2 The N -dimensional group of unitary matrices U, U †U = 1.

: SU(N) special unitary group 2 The (N 1)-dimensional group of unitary matrices U with det(U) = 1. −

Deﬁnition: A representation D of a group G on a vector space V is a mapping of all g G to linear ∈ transformations D(g) on V that is compatible with group multiplication:

f g = h D(f)D(g)= D(f g)= D(h). (7.3) · ⇒ · V = representation space. • dimV = dimension of the representation • (if n = dimV < , D(g) are n n-matrices). ∞ × Elements v V are called multiplets (at least in physics). • ∈

Two representations D and D′ are called equivalent (D D′) if there is an invertible • ∼ transformation S so that

1 D′(g)= SD(g)S− , g G. ∀ ∈ A representation is called unitary if the matrices D(g) are unitary for all g. • 7.1. LIE GROUPS AND ALGEBRAS 85

7.1.2 Lie algebras Note: Neighborhood of identity carries almost full information about Lie group G.

Deﬁnition: Lie algebra g set of inﬁnitesimal deviations from identity e ≡ .vector space with product structure ֒ →

Properties of g in a speciﬁc representation D(g): Inﬁnitesimal group elements g = g(δω) in representation D(G): • ∂D(ω) D(g) D(δω)= 1 + δω + (δω2) = 1 iT a δω + (δω2). (7.4) ≡ a ∂ω O − D a O a ω=0 a iTD ≡− a TD = generators of the Lie| group{z in}D representation { } = basis for representation D(g) of g. Finite transformations (connected to the unit element) via exponentiation: • n ωa a a D(ω) = lim 1 i T = exp ( iωaT ) , (7.5) n − n − →∞ Composition law of G implies product in g: • Ansatz for composition functions of Eq. (7.1):

1 abc f (ω ,ω′ ) = ω + ω′ + f ω ω′ + ..., (7.6) a b c a a 2 b c i.e. fa(ωb, 0) = fa(0,ωb)= ωa. :Insertion into composition law ֒ → a b D(ω)D(ω′) = exp ( iω T ) exp iω′ T − a − b a b 1 a 2 1 b 2 a b = 1 iω T iω′ T (ω T ) (ω′ T ) ω ω′ T T + ... − a − b − 2 a − 2 b − a b ! a = exp ( if (ω,ω′)T ) − a 1 abc a 1 a b = 1 i ω + ω′ + f ω ω′ T (ω + ω′ )(ω + ω′ )T T + ... − a a 2 b c − 2 a a b b

a b i abc a 1 a b 1 a b i.e. ω ω′ T T = f ω ω′ T ω ω′ T T ω′ ω T T − a b −2 b c − 2 a b − 2 a b Basic Lie algebra relation: ⇒ T bT c T cT b [T b, T c] = i f abc T a, where f abc = f acb. (7.7) − ≡ − structure constants of g = independent of representation ! |{z} 86 CHAPTER 7. REPRESENTATIONS OF THE LORENTZ GROUP

Jacobi identity of commutators, • [T a, [T b, T c]] + [T c, [T a, T b]] + [T b, [T c, T a]] = 0, (7.8)

implies f abkf kcd + f ackf kdb + f adkf kbc =0. (7.9)

Adjoint representation • (T a ) if bca (7.10) adj bc ≡ − exists for each Lie algebra. Commutator relation (7.7) satisﬁed due to the Jacobi identiy (7.9).

Important special case: algebra of a compact Lie group

Matrix tr(T aT b) is positive deﬁnite. • :Convention ֒ → 1 tr(T aT b)= δ . (7.11) 2 ab f abc are totally antisymmetric, since ⇒ f abc = 2itr (T bT c T cT b)T a = 2itr T bT cT a T cT bT a = cyclic in abc. − − − − Finite-dimensional representations are unitary: • a a ! 1 a a a a D(ω)† = exp iω T † = D(ω)− = exp (iω T ) , i.e. T = T †. (7.12)

Generators T a are hermitian. ⇒ Comment: Deﬁning representations of matrix Lie algebras: a T = N N matrices with special properties: × C GL(N, ): complex, no restriction • SL(N, C): complex, traceless •

SO(N): imaginary, antisymmetric • SU(N): hermitian, traceless •

7.1. LIE GROUPS AND ALGEBRAS 87

Example: SU(2) = relevant group for angular momentum in QM Group elements in the deﬁning (fundamental, i.e. lowest-dimensional) representation: • U = unitary 2 2 matrix with det U = 1. × Generators in the fundamental representation: • T a = traceless, hermitian 2 2 matrices (a =1, 2, 3). × σa Usual convention: T a = , σa = Pauli matrices 2 0 1 0 i 1 0 σ1 = , σ2 = − , σ3 = . (7.13) 1 0 i 0 0 1 − Lie algebra:

[σa, σb] = 2iǫabcσc, structure constants = ǫabc = totally antisym. ǫ-tensor (7.14)

Finite group elements in fundamental representation: • i ω ω U(ω) = exp ~ω ~σ = cos i~e ~σ sin , (7.15) −2 · 2 − · 2 where ~ω = ω~e (~e 2 = 1).

Adjoint representation: • 00 0 0 0 i 0 i 0 1 2 3 − Tadj = 0 0 i , Tadj = 0 00 , Tadj = i 0 0 . (7.16) 0i− 0   i 0 0 0 0 0 −       Finite group elements: R(~ω) = exp( iT a ωa) = 3dim. rotation matrices with angle ω around axis ~e. − adj Any representation: T a = J a = components of angular momentum. • 7.1.3 Irreducible representations Deﬁnitions: A representation is called reducible if there is a subspace H of V that is invariant • under all matrices D(g), i.e. if all D(g) can be written in the block form

D (g) E(g) D(g)= 1 . (7.17) 0 D (g) 2 If there is no invariant subspace, the representation is called irreducible. • 88 CHAPTER 7. REPRESENTATIONS OF THE LORENTZ GROUP

A representation is called fully reducible if all D(g) can be written in block-diagonal • form, D1(g) 0 0 ... 0 D2(g) 0 ... D(g)=  .  , (7.18) ..    0 ...Dn(g)    where the Dn are irreducible, i.e. a fully reducible representation is the direct sum of irreducible representations: D = D D D . (7.19) 1 ⊕ 2 ⊕···⊕ n Deﬁnitions of (ir)reducibility for Lie algebras analogously. • An operator C commuting with all elements of the Lie algebra is called Casimir • operator: [C, T a]=0. (7.20)

Some facts about (ir)reducibility: Irreducible representations of abelian groups are one-dimensional. • All unitary reducible group representations are fully reducible. • Schur’s Lemma: • If a linear mapping A on a vector space V commutes with all matrices D(g) of an irreducible representation of the group G on V , i.e. AD(g)= D(g)A (7.21) for all g G, then A is a multiple of the identity: ∈ A = λD1, (7.22)

where λD depends on the representation. Schur’s Lemma applied to Lie algebras: • Casimir operators C 1 in an irreducible representation. ∝ In Lie algebras with f abc = totally antisymmetric (e.g. for compact Lie groups) there • is always the quadratic Casimir operator,

a a C2 = T T . (7.23) Comment: Proof that C2 satisﬁes (7.20):

b a a b a b a abc a c c a [C2, T ]= T [T , T ] + [T , T ]T = if (T T + T T ) = 0. (7.24)

7.1. LIE GROUPS AND ALGEBRAS 89

Example: irreducible representations of SU(2) known from the angular momentum in QM ֒ → Quadratic Casimir operator: • J~ 2 = J aJ a = total angular momentum operator, [J~ 2,J a]=0. (7.25) a X Diagonalization of J~ 2 and one component J a possible, usual choice J 3. ⇒ Irreducible representation D(j) for each ﬁxed value of j =0, 1 , 1, 3 ,... : • 2 2 J~ 2 j, m = j(j + 1) j, m , | i | i (7.26) J 3 j, m = m j, m , m = j, 1+1, . . . j. | i | i − − j, m = (2j + 1)-dimensional multiplet for each ﬁxed j. {| i} 7.1.4 Constructing representations

New group representations from two representations Di (i = 1, 2) on vector spaces Vi (dimVi = ni): D D on the direct sum V V of vector spaces (dim = n + n ): • 1 ⊕ 2 1 ⊕ 2 1 2

D1(g) 0 v1 (D1 D2)(g)= , v1 v2 = , vi Vi, ⊕ 0 D2(g) ⊕ v2 ∈ i.e. (D D )(g)(v v )=(D (g)v ) (D (g)v ). (7.27) 1 ⊕ 2 1 ⊕ 2 1 1 ⊕ 2 2 Representation is reducible by construction. D D on the direct product V V of vector spaces (dim = n n ): • 1 ⊗ 2 1 ⊗ 2 1 2 (D D )(g)(v v )=(D (g)v ) (D (g)v ). (7.28) 1 ⊗ 2 1 ⊗ 2 1 1 ⊗ 2 2 Representation is in general reducible, but decomposible into irreducible blocks D(i):

D D = D(i1) D(in). (7.29) 1 ⊗ 2 ⊕···⊕ Deﬁnitions carry over to Lie algebras: D(g)= 1 iω T a + ... − a D Direct sum representation on V V : • 1 ⊕ 2 a a TD1 0 a a a TD1 D2 = a , TD1 D2 (v1 v2)=(TD1 v1) (TD2 v2). (7.30) ⊕ 0 T ⊕ ⊕ ⊕ D2 Direct product representation on V V : • 1 ⊗ 2 a a a TD1 D2 (v1 v2)=(TD1 v1) v2 + v1 (TD2 v2). (7.31) ⊗ ⊗ ⊗ ⊗ 90 CHAPTER 7. REPRESENTATIONS OF THE LORENTZ GROUP

Example: product representations of SU(2) : addition of two angular momenta J~ (i =1, 2) with respective multiplets j , m ֒ → i | i ii J~ j , m j , m =(J~ j , m ) j , m + j , m (J~ j , m ). (7.32) | 1 1i⊗| 2 2i 1 | 1 1i ⊗| 2 2i | 1 1i ⊗ 2 | 2 2i Decomposition into irreducible blocks: (Clebsch–Gordan series)

j , m j , m = c j, m = m + m with j j j j + j , (7.33) | 1 1i⊗| 2 2i j,m | 1 2i | 1 − 2| ≤ ≤ 1 2 j X in terms of representation spaces:

(j1) (j2) ( j1 j2 ) ( j1 j2 +1) (j1+j2) D D = D | − | D | − | D . (7.34) ⊗ ⊕ ⊕···⊕ Speciﬁcally: ( 1 ) ( 1 ) (0) (1) D 2 D 2 = D D , ⊗ ⊕ (7.35) D(1) D(1) = D(0) D(1) D(2), etc. ⊗ ⊕ ⊕ 7.2 Irreducible representations of the Lorentz group

Recall: Lorentz transformations and generators (see Chap. 2)

Λ = exp i(νkKk + ϕkJ k) . (7.36) − n boost rotationo Lie algebra of generators: | {z } | {z } [J i,J j] = iǫijkJ k, (7.37) [J i,Kj] = iǫijkKk, (7.38) [Ki,Kj]= iǫijkJ k. (7.39) − Simpliﬁcation by change of basis: 1 T k = (J k iKk). [T i, T j] = iǫijk T k δ . (7.40) 1,2 2 ∓ ⇒ a b a ab

Lie algebras of L↑ and SU(2) SU(2) closely related (complex versions are identical). ⇒ + ⊗

Construction of irreducible representations of L+↑ : (analogy to SU(2) case) 3 3 ~ 2 ~ 2 Two commuting generators: T1 , T2 ; two Casimir operators: T1 , T2 .

Multiplets j1, m1; j2, m2 j1, m1 1 j2, m2 2 span (2j1 + 1)(2j2 + 1)-dimensional ֒ → | i≡|(j1,j2) i ⊗| i irreducible representation D for ﬁxed j1, j2: T 3 j , m ; j , m = m j , m ; j , m , m = j , j +1,...j , a | 1 1 2 2i a | 1 1 2 2i a − a − a a T~ 2 j , m ; j , m = j (j + 1) j , m ; j , m , j =0, 1 , 1,.... (7.41) a | 1 1 2 2i a a | 1 1 2 2i a 2 7.3. FUNDAMENTAL SPINOR REPRESENTATIONS 91

Lorentz transformations in D(j1,j2): Λ(j1,j2) = exp i(~ϕ + i~ν)T~ (j1) exp i(~ϕ i~ν)T~ (j2) , [T k, T l]=0. (7.42) − 1 − − 2 1 2 Comments:

(ja) Hermitian SU(2) generators Ta constructed as in non-relativistic QM. • Angular momentum J~ = T~ + T~ . • 1 2 . D(j1,j2) contains angular momenta j = j j , j j +1,...,j + j ֒ → | 1 − 2| | 1 − 2| 1 2 Λ(j1,j2) is unitary only for pure rotations (~ν = 0). •

Parity transformation: Behaviour of generators: P : J~ J~ (pseudo-vector), → P : K~ K~ (vector). → − P interchanges the two SU(2) factors: ⇒ (j1,j2) (j2,j1) T~1 T~2, i.e. P : D D . (7.43) ←→P → P -invariant representations: D(j,j) and D(j1,j2) D(j2,j1) for j = j . ⇒ ⊕ 1 6 2 7.3 Fundamental spinor representations

( 1 ,0) D 2 : Right-chiral fundamental representation • i ( 1 ),i σ Generators: T 2 = , T 0,i =0. (7.44) 1 2 2

( 1 ,0) i Transformations: Λ 2 = exp (~ϕ + i~ν)~σ Λ . (7.45) −2 ≡ R χ Multiplets: χ = 1 , right-chiral Weyl spinors. (7.46) χ 2 (0, 1 ) D 2 : Left-chiral fundamental representation • i ( 1 ),i σ Generators: T 0,i =0, T 2 = . (7.47) 1 2 2

(0, 1 ) i Transformations: Λ 2 = exp (~ϕ i~ν)~σ Λ . (7.48) −2 − ≡ L χ¯ Multiplets:χ ¯ = 1 , left-chiral Weyl spinors. (7.49) χ¯ 2 92 CHAPTER 7. REPRESENTATIONS OF THE LORENTZ GROUP

Properties of ΛR,L:

Λ = complex 2 2 matrices with det Λ = 1, i.e. Λ SL(2, C). • R,L × R,L R,L ∈ Useful identities: • 1 1 ΛR† =ΛL− , ΛL† =ΛR− . (7.50)

Relation by complex conjugation: •

1 i i 2 0 1 ǫ− σ ǫ = σ ∗, ǫ = iσ = . − 1 0 − i i 1 1 Λ∗ = exp (~ϕ i~ν)~σ∗ = exp ǫ− (~ϕ i~ν)~σǫ = ǫ− Λ ǫ. (7.51) ⇒ R 2 − −2 − L

complex ( 1 ,0) ∗ (0, 1 ) ( 1 ,0) conjugation (0, 1 ) Equivalence: D 2 D 2 , i.e. D 2 D 2 . ⇒ ∼ ←−−−−−−→ Comment: Construction of left (right) spinors from a right (left) spinors:

( 1 ,0) (0, 1 ) χ D 2 : (ǫχ∗) ǫΛ∗ χ∗ = Λ (ǫχ∗), i.e. ǫχ∗ D 2 , ∈ → R L ∈ (0, 1 ) 1 1 1 1 ( 1 ,0) χ¯ D 2 : (ǫ− χ¯∗) ǫ− ΛL∗ χ¯∗ = ΛR(ǫ− χ¯∗), i.e. (ǫ− χ¯∗) D 2 . (7.52) ∈ → ∈

Dirac spinors 1 Parity-invariant representation for spin- 2 fermions: (e.g. needed for electromagnetism)

Dirac representation: • ( 1 ,0) (0, 1 ) D 2 D 2 (irreducible under L↑ P ) (7.53) ⊕ + ⊗

Dirac spinor ψ: • χ ψ = . (7.54) ξ¯ Lorentz transformation of ψ:

ΛRχ ΛR 0 ψ ψ′ = ¯ = ψ = S(Λ)ψ. (7.55) →Λ ΛLξ 0 ΛL S(Λ) ≡ | {z } 7.4. PRODUCT REPRESENTATIONS 93 7.4 Product representations

′ ′ Decomposition of products D(j1,j2) D(j1,j2) into irreducible blocks: ⊗ :(Possible upon using relations from SU(2 ֒ → From Clebsch–Gordan series of SU(2): • (j,0) (j′,0) (j+j′,0) (j+j′ 1,0) ( j j′ ,0) D D = D D − D | − | , ⊗ ′ ⊕ ⊕···⊕ D(0,j) D(0,j ) analogously. (7.56) ⊗

Independence of SU(2) factors, i.e. [T k, T l] = 0: • 1 2 D(j1,j2) = D(j1,0) D(0,j2) . (7.57) ⊗ independent factors | {z }

(j ,j ) (j′ ,j′ ) (j ,0) (0,j ) (j′ ,0) (0,j′ ) D 1 2 D 1 2 = D 1 D 2 D 1 D 2 (7.58) ⇒ ⊗ ⊗ ′ ⊗ ⊗ ′ = D(j1,0) D(j1,0) D(0,j2) D(0,j2) ⊗ ⊗ ⊗ use SU(2) relation (7.56) ′ ′ ′ ′ ′ ′ ′ ′ (j1+j ,j2+j ) (j1+j 1,j2+j ) (j1+j ,j2+j 1) ( j1 j , j2 j ) = ... = D 1 2 D 1− 2 D 1 2− D | − 1| | − 2| . | ⊕ {z } |⊕ {z } ···⊕ Note: Reduction important in construction of covariant quantities from products of multiplets (e.g. invariants for Lagrangians).

Examples:

Invariant spinor product: • ( 1 ,0) ( 1 ,0) (1,0) (0,0) D 2 D 2 = D D (7.59) ⊗ ⊕ trivial representation (objects Lorentz invariant) | {z } 2 2 matrix A =(a ) so that ⇒ ∃ × ij

! T aijχi′ ξj′ = aij(ΛR)ik(ΛR)jlχkξl = aijχiξj = invariant, i.e. ΛRAΛR = A. (7.60)

Solution: A = ǫ = totally antisymmetric tensor.

Lorentz-invariant product of Weyl spinors: (left-handed case analogously) ⇒ χξ ǫ χ ξ , χ¯ξ¯ ǫ χ¯ ξ¯ . (7.61) h i ≡ ij i j h i ≡ ij i j 94 CHAPTER 7. REPRESENTATIONS OF THE LORENTZ GROUP

4-vector representation: • Required: real, P -invariant representation that contains spin value j = 1 (vector!). Simplest candidate:

( 1 , 1 ) ( 1 ,0) (0, 1 ) ( 1 ,0) ( 1 ,0) ∗ D 2 2 = D 2 D 2 D 2 D 2 (7.62) ⊗ ∼ ⊗ To show: 2 2 matrices Cµ so that ∃ × µ µ µ ν χi†Cijξj = 4-vector, i.e. ΛR† C ΛR =Λ νC . (7.63) Solution: (see Exercise 8.3) Cµ = σµ =(1, σ1, σ2, σ3). (7.64) Analogously: µ µ ν µ 1 2 3 Λ† σ¯ Λ =Λ σ¯ , σ¯ =(1, σ , σ , σ ). (7.65) L L ν − − −

4-vectors from products of Weyl spinors: ⇒ µ µ χ†σ ξ, χ¯†σ¯ ξ.¯ (7.66)

Dirac representation: • Covariants from products of two Dirac spinors ψ1, ψ2 ?

( 1 ,0) (0, 1 ) ( 1 ,0) (0, 1 ) D 2 D 2 D 2 D 2 ⊕ ⊗ ⊕ h (0,0) i (0,0)h ( 1 , 1 ) i( 1 , 1 ) (1,0) (0,1) = D D D 2 2 D 2 2 D D . (7.67) ⊕ ⊕ ⊕ ⊕ ⊕ scalar psudo-scalar vector pseudo-vector anti-sym. rank-2 tensors Auxiliary quantities| {z for} explicit| {z } construction:| {z } | {z } | {z } 0σ ¯µ γµ Dirac matrices in chiral representation, (7.68) ≡ σµ 0 χ ψ ψ†γ =(φ¯†, χ†) adjoint Dirac spinor to ψ = , (7.69) ≡ 0 φ¯ i γ iγ0γ1γ2γ3 = ǫ γµγνγργσ. (7.70) 5 ≡ −4! µνρσ Contruction of covariants: (see Exercise 9.2) ⇒ ψ1ψ2 = scalar, (7.71)

ψ1γ5ψ2 = pseudo-scalar, (7.72) µ ψ1γ ψ2 = vector, (7.73) µ ψ1γ γ5ψ2 = pseudo-vector, (7.74) µ ν ψ1γ γ ψ2 = rank-2 tensor, (7.75) µ ν ψ1γ γ γ5ψ2 = rank-2 pseudo-tensor. (7.76) 7.5. RELATIVISTIC WAVE EQUATIONS 95

Some properties of the Dirac matrices: (see Exercise 9.1)

µ ν µν µ µ µ 5 γ ,γ =2g , γ γ γ =(γ )†, γ ,γ =0, (7.77) • { } 0 0 { } Tr[γµγν ]=4gµν, Tr[γµγν γργσ] = 4[gµνgρσ gµρgνσ + gµσgνρ], (7.78) • − Tr[γµ1 ...γµ2n+1 ]=0, n =0, 1,..., (7.79) Tr[γ ] = Tr[γµγνγ ]=0, Tr[γµγνγργσγ ]= 4iǫµνρσ, (7.80) • 5 5 5 − γαγ =4, γαγµγ = 2γµ. (7.81) • α α − 7.5 Relativistic wave equations

7.5.1 Relativistic ﬁelds Requirement by relativistic covariance:

Fields Φk(x) describing a specific particle transform in an irreducible representation of L+↑ or L↑ P (reducible case: different irreducible blocks = different particles). + × Lorentz transformation: (Φ = classical field, no operator)

Φk′ (x′) = Skl(Λ) Φl(x), x′ =Λx. (7.82) transformation matrix in irreducible ↑ ↑ representation of L+ or L+ P | {z } × 1 Φ′(x) = S(Λ) Φ(Λ− x) (7.83)

transformation of transformation of inner degrees of space–time argument freedom spin ֒ orbital| {z } angular momentum |→{z } → Transformations in exponential form: i S(Λ) = exp ω M αβ = ﬁnite-dim. representation, (7.84) −2 αβ 1 n i αβ o Φ(Λ− x) = exp ω L Φ(x). (7.85) −2 αβ diﬀerentialn operator o Lαβ = xαpˆβ xβpˆα |= generalized{z− orbital angular} mom.

i αβ αβ Φ′(x) = exp ω (M + L ) Φ(x). (7.86) ⇒ −2 αβ total angular momentum n → o | {z } 96 CHAPTER 7. REPRESENTATIONS OF THE LORENTZ GROUP

7.5.2 Relativistic wave equations for free particles Basic requirements: Qm. superposition principle linearity of diﬀerential equation • → General ansatz: µ Πkl(m, i∂ ) Φl(x)=0, (7.87)

N N-matrix-valued diﬀerential× operator where m = particle mass, N determined| {z } by particle spin. Order of diﬀerential eq. 2 (otherwise strange behaviour of solutions). ≤ Covariance: (7.87) has to imply • ! 0 = Π(m, iΛ∂) Φ′(Λx) with Φ′(Λx) = S(Λ) Φ(x). (7.88)

! 1 Π(m, iΛ∂) = S(Λ) Π(m, i∂) S(Λ− ). (7.89) ⇒ could also be in another representation Mass-shell condition: | {z } • In momentum space only ﬁeld modes with p2 = m2 should contribute to solution:

ixp Φ(x) = d˜p e− Φ(˜ ~p). (7.90) Z 4 d p 2 2 = 4 (2π)δ(p m ) θ(p0) (2|{z}π) − Spin projection: • Operator Π should project onto genuine spin-j part of representation D(j1,j2), where j j j j + j . If Π does not perform this projection, additional constraints | 1 − 2| ≤ ≤ 1 2 are needed to achieve it.

Examples: Klein–Gordon equation: j = 0. • 2 µ 2 ( + m )φ(x)=0, Π(m, i∂)= ∂ ∂µ + m , S(Λ) = 1. (7.91) i.e. requirements trivially fulﬁlled. Maxwell equation for elmg. 4-vector potential Aµ in Lorenz gauge: • µ µ A =0, Π(i∂)= ∂ ∂µ, S(Λ) = Λ. (7.92) µ ( 1 , 1 ) Lorenz condition ∂ Aµ = 0 eliminates spin-0 part in D 2 2 representation, only spin- 1 part remains. Comment: Photons do not really carry spin 1, but helicity h = 1 (=spin projected to direction ± of ﬂight), see below.

7.5. RELATIVISTIC WAVE EQUATIONS 97

7.5.3 The Dirac equation 1 Wave equations for spin- 2 ﬁelds: Minimalistic attempt: ( 1 ,0) (0, 1 ) 1 Wave equations for χ(x) D 2 and φ¯(x) D 2 (smallest representations with j = ): ∈ ∈ 2 Non-trivial transformation property of χ, φ¯ should result from wave equation (other- • wise additional constraints needed). .(Π(m, i∂) should mix ﬁeld components (i.e. KG operator not acceptable ֒ → Relevant covariant objects for wave equations: • µ ( 1 ,0) µ (0, 1 ) χ, σ¯ ∂ φ¯ D 2 , φ,¯ σ ∂ χ D 2 . (7.93) µ ∈ µ ∈ Proof:

χ(x) χ′(x′)=ΛR χ(x) (7.94) µ → µ µ ν σ ∂ χ(x) σ ∂′ χ′(x′)= σ Λ ∂ Λ χ(x) µ → µ µ ν R 1 µ ν µ =ΛL (ΛL− σ ΛR) Λµ ∂ν χ(x)=ΛL σ ∂µχ(x). (7.95)

µ µ ρ =ΛR† σ ΛR =Λ ρσ

µ | {z } σ¯ ∂µφ¯ analogously. q.e.d. Consequence: Only possibility for separate wave equations for χ, φ¯: • µ µ ¯ σ ∂µχ = 0, σ¯ ∂µφ = 0. Weyl equations (7.96)

µ Note: σ ∂µχ = cχ, etc., not compatible with relativistic covariance ! Solution of Weyl equations: • n Fourier ansatz: χ(x) = e ikx n with n = R,1 . ± R R n R,2 0 3 1 2 ! k k k + ik nR,1 (7.97) . 3 0 − 2 1− = 0 ֒ → k ik k + k nR,2 − − det(...) = (k0)2 (k1)2 (k2)2 (k3)2 = k2 − − − Non-trivial solutions only for| k2 = 0, i.e.{z Weyl fermions} are massless ! ⇒ Explicitly: cos ϕ sin θ e iϕ cos θ kµ = k0(1,~e), ~e= sin ϕ sin θ n = − 2 . (7.98)   ⇒ R sin θ cos θ 2 Analogously:   θ ikx 2 sin φ¯(x) = e± n , k =0, n = 2 . (7.99) L L eiϕ cos θ − 2 98 CHAPTER 7. REPRESENTATIONS OF THE LORENTZ GROUP

The Dirac equation: χ Combine Weyl spinors χ, φ¯ to Dirac spinor ψ = . φ¯ :Two covariant 1st-order equations possible ֒ → µ ¯ µ ¯ iσ ∂µχ = c1φ, i¯σ ∂µφ = c2χ. (7.100)

Note: By appropriate rescaling, equality c1 = c2 = m can be achieved. (Identiﬁcation of m as mass later.)

Matrix form: 0σ ¯µ χ χ i ∂ m = 0. (7.101) σµ 0 µ φ¯ − φ¯ = γµ, Dirac matrices (i|γµ∂{z m}) ψ = 0. Dirac Equation (7.102) ⇒ µ −

Notation: /a γµa = γ aµ (Feynman dagger) Dirac eq.: (i/∂ m) ψ =0. ≡ µ µ ⇒ − Covariance: (see Exercise 9.2)

ΛR 0 ψ(x) ψ′(x′) = S(Λ) ψ(x), S(Λ) = ,(7.103) → 0 ΛL µ ν 1 /a /a′ = γ Λ a = S(Λ) /aS(Λ)− . (7.104) → µ ν (i/∂ m) ψ(x)=0 (i/∂ ′ m) ψ′(x′) ⇒ − → − 1 = S(Λ) (i/∂ m) S(Λ)− S(Λ) ψ(x) − = S(Λ) (i/∂ m) ψ(x)=0. (7.105) − Chapter 8

Free Dirac fermions

8.1 Solutions of the classical Dirac equation

Dirac eq.: (i/∂ m) ψ =0. − Note: Each component of ψ obeys the KG equation:

0=( i/∂ m) (i/∂ m) ψ = ( /∂2 +m2) ψ = ( + m2) ψ. (8.1) − − − µ ν 1 µ ν µν 2 = γ γ ∂µ∂ν = γ ,γ ∂µ∂ν = g ∂µ∂ν = ∂ 2 { } |{z} Fourier ansatz:

ikx ψ(x) = e− u(k), u(k) = constant 4-component Dirac spinor. (8.2)

Eq. (8.1) implies k2 = m2. Set k = ~k 2 + m2. • → 0 Ansatz leads to Dirac eq. in momentum space:p • (/k m) u(k)=0. (8.3) − .4-dim. system of linear equations ֒ → Solution of Eq. (8.3) in two steps: µ µ ~ 1. Solve equation ﬁrst in rest frame of k , i.e. for kr =(m, 0): cos ϕ sin θ kµ = (k ,~k)=Λµ kν, ~k = ~k ~e, ~e = sin ϕ sin θ . (8.4) 0 ν r   | | cos θ Using the chiral representation for γµ yields   1 +1 0=(/kr m) u(kr) = m(γ0 1) u(kr) = m − u(kr). (8.5) − − +1 1 − n . Two independent solutions of the block form u(k )= , e.g. n = n ֒ → r n R,L 99 100 CHAPTER 8. FREE DIRAC FERMIONS

µ 2. Boost kr into original system: Λ 0 u(k) = S(Λ) u(k ) = R u(k ), (8.6) r 0 Λ r L

ν ν 1 k0 + ~k with ΛR = exp ~e ~σ , ΛL = exp ~e ~σ , ν = ln | | = rapidity. 2 − 2 2 k ~k 0 −| | 3 Simple form of ΛR,L after diagonalizing ~e ~σ = Uσ U †: ν/2 ν/2 e 0 e− 0 Λ = U U †, Λ = U U †, (8.7) R 0 e ν/2 L 0 eν/2 −

iϕ θ θ ~ e− cos sin ν/2 k0 k with U = (n , n ) = 2 2 , e± = ±| |, R L sin θ eiϕ cos θ s m 2 − 2 with nR,L deﬁned in Eqs. (7.98) and (7.99). 1 0 Using U n = and U n = we obtain: † R 0 † L 1 +ν/2 ν/2 ΛR nR = e nR, ΛR nL = e− nL, ν/2 +ν/2 ΛL nR = e− nR, ΛL nL = e nL. (8.8) n n is convenient choice for n in u(k ) of step 1. ⇒ ∝ R,L r Two independent standard solutions: ~ nR ΛR nR k0 + k nR uR(k) = S(Λ) √m = √m = | | , nR ΛL nR q ~  k0 k nR −| |   q ~ nL ΛR nL k0 k nL uL(k) = S(Λ) √m = √m = −| | . (8.9) nL ΛL nL q ~  k0 + k nL | | q  Analogous procedure for ansatz ψ(x) = e+ikx v(k) leads to (/k + m) v = 0 with the standard solutions:

uσ(k) vτ (k) = 0,

vσ(k) uτ (k) = 0, v (k) v (k) = 2m δ , (8.11) σ τ − στ 8.1. SOLUTIONS OF THE CLASSICAL DIRAC EQUATION 101

Spin orientations of the solutions: ( 1 ,0) (0, 1 ) Spin operator in Dirac representation D 2 D 2 : ⊕ ~ ~ ~ ~σ ~σ 1 ~σ 0 S = T( 1 ,0) T(0, 1 ) = 1 1 = . (8.12) 2 ⊕ 2 2 ⊗ ⊕ ⊗ 2 2 0 ~σ Deﬁnition: Helicity = spin projection onto direction of ﬂight ~e

1 ~e ~σ 0 h ~e S~ = . (8.13) ≡ 2 0 ~e ~σ

Standard solutions uR,L and vR,L are helicity eigenstates: 1 1 1 1 hu =+ u , hu = u , h v =+ v , h v = v . (8.14) R 2 R L −2 L R 2 R L −2 L

ikx Particle solutions ψ (x) = e− u (k) correspond to helicity states with h =+/ . ⇒ +,R/L R/L − +ikx Note: Helicity content of antiparticle solutions ψ ,R/L(x) = e vR/L(k) clariﬁed by QFT. −

General solution of free (classical) Dirac equation:

˜ ~ ikx ~ +ikx ψ(x) = dk aσ(k) uσ(k)e− + bσ∗ (k) vσ(k)e . (8.15) Z σ=R,L X arbitrary functions of ~k creation/annihilation operators ֒ | {z→} for (anti)particles| in{z QFT} 102 CHAPTER 8. FREE DIRAC FERMIONS 8.2 Quantization of free Dirac ﬁelds

8.2.1 Quantization procedure Classical Lagrangian: Dirac equation obviously reproduced by = ψ (i/∂ m) ψ, (8.16) L − considering ψ and ψ as independent. Euler–Lagrange equations: ∂ ∂ ∂ 0 = L ∂µ L = L = (i/∂ m) ψ, (8.17) ∂ψ − ∂(∂µψ) ∂ψ − ∂ ∂ 0 = L ∂µ L = mψ ∂µ iψγ = ψ m + i←/∂ adjoint Dirac eq.. ∂ψ − ∂(∂µψ) − − µ − (8.18) Canonical commutators: Preliminary consideration: Commutators [a(~k), a†(~p)], etc., lead to totally symmetric states ~k, ~p = a†(~p)a†(~p) 0 . | i | i But: Fermions require totally antisymmetric states. ! Use ansatz with anticommutators a(~k), a†(~p) in quantization ֒ → { } Canonical momentum variable to ﬁeld ψ(x): ∂ π(x)= L = iψγ = iψ†. (8.19) ∂(∂0ψ) 0 Canonical equal-time anticommutators for quantization:

ψ(t, ~x), iψ†(t, ~y) = i 1 δ(~x ~y), i.e. ψ(t, ~x), ψ(t, ~y) = γ δ(~x ~y). − 0 − n o n o ψα(t, ~x), ψβ(t, ~y) = ψα(t, ~x), ψβ(t, ~y) = 0, (8.20) Insertionn of Fourier decompositionso n o

˜ ~ ikx ~ +ikx ψ(x) = dk aσ(k)uσ(k)e− + bσ† (k)vσ(k)e , (8.21) Z σX=R,L ˜ ~ +ikx ~ ikx ψ(x) = dk aσ† (k)uσ(k)e + bσ(k)vσ(k)e− (8.22) Z σX=R,L yields anticommutators for creation/annihilation operators:

3 a (~k), a† (~p) = b (~k), b† (~p) = (2π) 2p δ(~k ~p) δ , σ τ σ τ 0 − στ n o n o aσ(~k), aτ (~p) = 0, etc. (8.23) n o 8.2. QUANTIZATION OF FREE DIRAC FIELDS 103

8.2.2 Particle states and Fock space Fock space:

Ground state 0 (vacuum, no particle excitation): 0 =( 0 )†, 0 0 =1. • | i h | | i h | i a (~p) 0 =0 , b (~p) 0 =0 ~p. (8.24) σ | i σ | i ∀ Excited states (particle states): •

Antisymmetric states Fermi–Dirac statistics ⇒ Fock space = Hilbert space spanned by all fermion and antifermion states: • 0 , f (~p ) , f¯ (~p ) , f (~p )f (~p ) ... | i | σ 1 i | τ 2 i | σ 1 τ 2 i (Anti)Fermion number operators: •

Nfσ (~p)= aσ† (~p)aσ(~p), Nf = d˜p Nfσ (~p), σX=R,L Z

Nf¯σ (~p)= bσ† (~p)bσ(~p), Nf¯ = d˜p Nf¯σ (~p). (8.30) σ=R,L X Z :Commutator relations ֒ →

[N , a† (~p)]=+a† (~p), [N , a (~p)] = a (~p), f σ σ f σ − σ [Nf , bσ† (~p)]=0, [Nf , bσ(~p)]=0, (8.31) analogously for N ¯ with a b. f ↔ 104 CHAPTER 8. FREE DIRAC FERMIONS

Field operators and wave functions: (cf. scalar ﬁelds, Sect. 5.4) One-particle wave function ϕ(x) corresponding to fermion state f (~p) : | σ i ipx ϕ (x) 0 ψ(x) f (~p) = 0 ψ(x) a† (~p) 0 = e− u (p). (8.32) fσ(~p) ≡h | | σ i h | σ | i σ

Space–time transformations of f , ψ(x), and ϕ(x): x x′ =Λx + a | i → Qm. states: •

f f ′ = U(Λ, a) f with U =unitaryoperator. (8.33) | i→| i | i

.Transition amplitudes f ′ g′ = f U †U g = f g = invariant ֒ → h | i h | | i h | i Comment: U(Λ, a) are transformations in -dimensional representation of the Poincar´e ∞ group, which is spanned by the particle states.

Field operator: •

ψ(x′)= U(Λ, a) S(Λ) ψ(x) U †(Λ, a), (8.34)

transformation of spin part of ψ(x) | {z } so that scalar products f ...ψ (x)...ψ (x)... g = f ′ ...ψ (x′)...ψ (x′)... g′ = invariant. h | 1 2 | i h | 1 2 | i Wave function: •

ϕ ′(x′)= 0 ψ(x′) f ′ = 0 U(Λ, a) S(Λ) ψ(x) U †(Λ, a)U(Λ, a) f h | | i h | | i = 0 = invariant = 1 h | = S(Λ) 0 ψ(x) f = S(Λ) ϕ(x). (8.35) h | | |i {z } | {z } 1 ϕ ′(x)= S(Λ) ϕ Λ− (x a) . (8.36) ⇒ − Wave functins transform like classical ﬁelds. 8.2. QUANTIZATION OF FREE DIRAC FIELDS 105

Properties of the particle states: Electric charge: • iqω Electric current density: (= Noether current for symmetry ψ ψ′ = e− ψ) → jµ = qψγµψ. (8.37) Operator Q for conserved electric charge: ⇒ 3 0 Q = q d x : ψγ¯ ψ := q d˜p a† (~p)a (~p) b† (~p)b (~p) = q(N N ¯). σ σ − σ σ f − f Z Z σ=R,L X (8.38) Charges of particle states:

Q f (~p) = q(N N ¯)a† (~p) 0 | σ i f − f σ | i = q [N , a† (~p)] [N ¯, a† (~p)] 0 = q f (~p) , (8.39) f σ − f σ | i | σ i † =aσ(~p) =0 ¯ ¯ Q fσ(~p) = |= {zq fσ(}~p) . (8.40) | i ··· − | i | {z } Fermion f carries charge +q, antifermion f¯ charge q. ⇒ − 4-momentum: • Energy-momentum tensor: (derived as for scalar ﬁelds) i θµν = ψ ←→∂ν γµψ. (8.41) 2 Operator P µ for 4-momentum: ⇒ µ 3 0µ µ P = d x : θ := = d˜pp N (~p)+ N ¯ (~p) . (8.42) ··· fσ fσ Z Z σ X µ µ µ µ (P , a† (~p) = p a† (~p) , P , b† (~p) = p b† (~p) . (8.43 ֒ → σ σ σ σ 4-momenta of the particle states: µ µ µ µ P f (~p) = [P , a† (~p)] 0 = p a† (~p) 0 = p f (~p) , (8.44) | σ i σ | i σ | i | σ i P µ f¯ (~p) = = pµ f¯ (~p) . (8.45) | σ i ··· | σ i Both elementary fermion and antifermion states carry 4-momentum pµ. ⇒ Alternative derivation of Eq. (8.43) via translation property of operator ψ(x):

U(Λ = 1, a)† ψ(x) U(1, a) = ψ(x a). (8.46) − µ µ =exp iPµa =exp +iPµa {− } { } { Taking aµ inﬁnitesimal| {z yields} | {z ֒ → [P , ψ(x)] = i∂ ψ(x). (8.47) µ − µ µ ( ) .(Commutators [P , aσ† (~p)], etc. from plane-wave solution for ψ(x ֒ → 106 CHAPTER 8. FREE DIRAC FERMIONS

Spin and helicity: • Make use of Lorentz transformation property of ψ:

1 U(Λ, a = 0)† ψ(x) U(Λ, 0) = S(Λ) ψ(Λ− x), (8.48)

where

U(Λ, 0) = exp i ω µν , ⋄ {− 2 µνJ } with µν = abstract operator of generalized total angular momentum, J Λ 0 S(Λ) = R = spin transformation matrix in Dirac representation, ⋄ 0 ΛL with generators M µν .

1 ~σ 0 S~ = = spin part of M µν , see Eq. (8.12). (8.49) 2 0 ~σ

1 i µν ψ(Λ− x) = exp ω L ψ(x), ⋄ {− 2 µν } where Lµν = i(xµ∂ν xν ∂µ) = generalized orbital angular momentum. − Connection between µν , M µν , and Lµν derived upon taking ω inﬁnitesimal: J µν [ µν , ψ(x)] = (M µν + Lµν ) ψ(x). (8.50) J − Restriction to rotational part of spin transformation:

J,~ ψ(x) = S~ ψ(x), where J~ = abstract generator for spin rotations. (8.51) − h i Helicities (~e= ~p/ ~p directions of J~, S~) of Fock states: ⇒ | |

3 ipx ~e J,~ a (~p) = ~e J,~ d x e u† (p) ψ(x) · σ · σ h i Z 3 ipx = d x e u† (p) ~e J,~ ψ(x) σ · Z h i = ~e S~ ψ(x)= h ψ(x) − · − 3 ipx | {z } = d x e u† (p) h ψ(x) − σ Z 1 = uσ† (p) h† = [huσ(p)]† = 2 sgn(σ)uσ(p)†, | {zwhere} sgn(R/L)=+/ , see Eq. (8.14) − 1 3 ipx = sgn(σ) d x e u† (p) ψ(x) −2 σ 1 Z = sgn(σ) a (~p), (8.52) −2 σ 8.2. QUANTIZATION OF FREE DIRAC FIELDS 107

3 ipx ~e J,~ b† (~p) = ~e J,~ d x e− v† (p) ψ(x) · σ · σ h i Z 3 ipx = d x e− v† (p)~e J,~ ψ(x) σ · Z h i 3 ipx = d x e− v† (p) h ψ(x) − σ Z 1 = [h vσ(p)]† = 2 sgn(σ)vσ(p)†, see Eq. (8.14) 1 | {z } = sgn(σ) b† (~p), (8.53) −2 σ 1 1 (e J,~ a† (~p) =+ sgn(σ) a† (~p), ~e J,~ b (~p) =+ sgn(σ) b (~p). (8.54~ ֒ → · σ 2 σ · σ 2 σ h i h i 1 1 ~e J~ f (~p) = ~e J,~ a† (~p) 0 =+ sgn(σ) a† (~p) 0 =+ sgn(σ) f (~p) , ⇒ · | σ i · σ | i 2 σ | i 2 | σ i h 1 i ~e J~ f¯ (~p) = ... = sgn(σ) f¯ (~p) , (8.55) · | σ i −2 | σ i

i.e. fermion state fR/L has helicity +/ , but antifermion state| fi¯ has helicity− /+. | R/Li − 8.2.3 Fermion propagator Deﬁnition:

0 T ψ(x)ψ¯(y) 0 = iS (x, y) fermion propagator (8.56) h | | i F x y

Note: Each (anti)commutation of two fermionic operators in : ... : and T (...) products leads to a sign change !

Calculation of SF (x, y):

ikx +ikx 0 T ψ(x)ψ(y) 0 = 0 T dk˜ a (~k)u (k)e− + b† (~k)v (k)e h | | i h | σ σ σ σ σ Z X +ipy ipy d˜p a† (~p)u (p)e + b (~p)v (p)e− 0 × τ τ τ τ | i τ Z X i(kx py) = θ(x y ) 0 dk˜ d˜p e− − u (k)u (p)a (~k)a† (~p) 0 0 − 0 h | σ τ σ τ | i Z Z σ,τ fermion propagationX from y to x ֒ → i(kx py) θ(y x ) 0 dk˜ d˜p e − v (k)v (p)b (~p)b† (~k) 0 − 0 − 0 h | σ τ τ σ | i σ,τ Z Z X (antifermion propagation from x to y (8.57 ֒ → 108 CHAPTER 8. FREE DIRAC FERMIONS

3 0 Using 0 a (~k)a† (~p) 0 = (2π) 2k δ(~k ~p)δ , etc., this yields h | σ τ | i − στ ik(x y) 0 T ψ(x)ψ(y) 0 = θ(x y ) dk˜ e− − u (k)u (k) h | | i 0 − 0 σ σ σ Z X = /k+m, completeness relation, see Exercise 10.1

ik(x y) | {z } θ(y x ) dk˜ e − v (k)v (k) − 0 − 0 σ σ σ Z X = /k m − ik(x y) ik(x y) = θ(x y ) dk˜ e− − (/k| + m){z θ(y } x ) dk˜ e − (/k m) 0 − 0 − 0 − 0 − 4 Z Z d k ik(x y) i(/k + m) = e− − , as for scalar propagator, see Sect. 4.3.2 (2π)4 k2 m2 + iǫ Z − d4k ie ik(x y) = (i/∂ + m) − − = (i/∂ + m) iD (x, y). (8.58) x (2π)4 k2 m2 + iǫ x F Z − Consequences of SF (x, y) = (i/∂x + m) DF (x, y): S (x, y) has the same causal properties as scalar propagator D (x, y). • F F Diﬀerential equation: • (i/∂ m)S (x, y) = (i/∂ m)(i/∂ + m) D (x, y)= ( + m2) D (x, y) x − F x − x F − x F = δ(x y). (8.59) − S (x, y) is inverse of the Dirac operator (i/∂ m). F − 8.2.4 Connection between spins and statistics Spin statistics theorem: Fields with integer spin (0, 1,... ) are quantized with commutators. • .States obey Bose–Einstein statistics ֒ → Fields with half-integer spin (1/2, 3/2,... ) are quantized with anticommutatorns. • .States obey Fermi–Dirac statistics ֒ →

“Proof”: otherwise several inconsistencies: Violation of causality, i.e. violation of • [Obs(x), Obs(y)] = 0 for(x y)2 < 0. (8.60) − Energy spectrum not bounded from below. System unstable. • → Statement on relation between spin and BE / FD statistics supported by experiment. • Chapter 9

Interaction of scalar and fermion ﬁelds

9.1 Interacting fermion ﬁelds

Interaction Lagrangians with a Dirac fermion:

= ψ (i/∂ m) ψ V (ψ, ψ, Φ). (9.1) L − −

Properties of interaction potential V : Each term in V contains products of at least 3 ﬁelds (2 ﬁelds free propagation). • → V = Lorentz invariant. .ψ always appears in products ψ...ψ ֒ • → V has mass dimension 4. • µ 3 (Fields: dim[φ] = dim[A ] = 1, dim[ψ]= 2 .)

V = V † = hermitian. • Examples: Yukawa interaction of a fermion and a scalar ﬁeld φ: • V = yφ ψψ, y = dimensionless coupling strength. (9.2)

.Basic interaction between fermions and the Higgs boson in the Standard Model ֒ → Electromagnetic interaction: • V = Qeψ /Aψ, e = elementary charge, Q = relative fermion charge. (9.3)

109 110 CHAPTER 9. INTERACTION OF SCALAR AND FERMION FIELDS

Comments on the perturbative machinery: :(!works as for scalar fields with few exceptions (signs ֒ → δ δ EOMs: L =0, L =0. • δψ δψ ∂ .Operators , etc., anticommute with fermionic fields ֒ → ∂ψ Symmetries if fields Φ are bosonic or fermionic: • k Φ Φ + δω ∆a(Φ) , δ = δω ∂ Ka,µ, δω = const., (9.4) k → k a k L a µ a Appropriate form of Noether currents (for signs):

∂ ja,µ = ∆a(Φ) Ka,µ(Φ), ∂ja =0. (9.5) k ∂(∂ Φ ) L − µ k Contractions for perturbative expansion of S-operator: •

Pij : Φi Φj : ( 1) 0 T [ΦiΦj] 0 : ... Φi 1Φi+1 Φj 1Φj+1 : , (9.6) ··· ··· ··· ≡ − h | | i· − ··· − ···

where Pij = number of necessary commutations for reordering fermion operators. With (9.6) Wick theorem also holds as usual. ⇒ Examples:

T [ψ ψ ] = : ψ ψ : + ψ ψ = : ψ ψ : + 0 T [ψ ψ ] 0 , ⋄ 1 2 1 2 1 2 1 2 h | 1 2 | i T [ψ ψ ψ ψ ] = : ψ ψ ψ ψ : ⋄ 1 2 3 4 1 2 3 4 + : ψ ψ : 0 T [ψ ψ ] 0 + : ψ ψ : 0 T [ψ ψ ] 0 1 2 h | 3 4 | i 3 4 h | 1 2 | i : ψ ψ : 0 T [ψ ψ ] 0 : ψ ψ : 0 T [ψ ψ ] 0 − 3 2 h | 1 4 | i − 1 4 h | 3 2 | i + 0 T [ψ ψ ] 0 0 T [ψ ψ ] 0 h | 1 2 | ih | 3 4 | i 0 T [ψ ψ ] 0 0 T [ψ ψ ] 0 . − h | 1 4 | ih | 3 2 | i 9.2. YUKAWA THEORY 111 9.2 Yukawa theory

Deﬁnition of the model: Lagrangian:

(φ, ψ, ψ) = + + , (9.7) L Lψ,0 Lφ,0 Lint (ψ, ψ) = : ψ (i/∂ m ) ψ : , Dirac fermion of mass m , Lψ,0 − f f 1 1 (φ) = : (∂ φ)(∂µφ) m2 φ2 : , neutral scalar ﬁeld of mass m , Lφ,0 2 µ − 2 φ φ (φ, ψ, ψ) = y : φ ψ ψ : , Yukawa interaction. Lint − Hamiltonian:

(φ, ψ, ψ) = (φ, ψ, ψ), since no derivative involved. (9.8) Hint −Lint 9.2.1 Feynman rules for the S-operator Expansion of the S-operator:

S = T exp d4x i (φ(x), ψ(x), ψ(x)) Lint Z = T exp d4x iy : φ(x) ψ(x) ψ(x): − Z = 1 +( iy) T d4x : φ(x) ψ(x) ψ(x): − Z 1 + ( iy)2 T d4x d4x : φ(x ) ψ(x ) ψ(x ):: φ(x ) ψ(x ) ψ(x ): + ... 2 − 1 2 1 1 1 2 2 2 Z Z

x = 1 +( iy) d4x : φ(x) ψ(x) ψ(x): − Z

1 x x + ( iy)2 d4x d4x : φ(x ) ψ(x ) ψ(x )φ(x ) ψ(x ) ψ(x ): 1 2 2 − 1 2 1 1 1 2 2 2 "Z Z

4 4 x x + d x1 d x2 : ψ(x1) ψ(x1)φ(x1)φ(x2)ψ(x2) ψ(x2): 1 2 Z Z 112 CHAPTER 9. INTERACTION OF SCALAR AND FERMION FIELDS

4 4 x x + d x1 d x2 : φ(x1) ψ(x1)ψ(x1)ψ(x2) ψ(x2) φ(x2): 1 2 Z Z

4 4 ¯ x x + d x1 d x2 : φ(x2) ψ(x2) ψ(x2)ψ(x1) ψ(x1) φ(x1): 1 2

Z Z convenient form for contractions with external states | {z }

4 4 + d x1 d x2 : ψ(x1) ψ(x1)ψ(x2) ψ(x2)φ(x1)φ(x2): x1 x2 Z Z

4 4 + d x1 d x2 : ψ¯(x2) ψ(x2)ψ(x1) ψ(x1)φ(x1)φ(x2): x1 x2 Z Z

4 4 ¯ d x1 d x2 : φ(x1) φ(x2): ψα(x1)ψ (x2)ψβ(x2)ψα(x1) − β x1 x2 Z Z Xα,β

= tr ψ(x1)ψ(x2)ψ(x2)ψ(x1) | {z }

4 4 d x1 d x2 : φ(x1)φ(x2)ψ(x1)ψ(x2)ψ(x2)ψ(x1): − x1 x2 Z Z # + ... (9.9) 9.2. YUKAWA THEORY 113

Feynman rules for graphical representation of the terms yn: ∝

1. Draw all possible diagrams with n vertices

(any number of exernal lines, including disconnected diagrams). 2. Translate graphs into analytical expressions as follows: External lines ˆ=non-contracted ﬁelds: • φ(x)= x ψ(x)= x ψ¯(x)= (9.10) x Internal lines ˆ=contracted ﬁelds (=propagators): •

φ(x1)φ(x2)= x1 x2

ψ(x1)ψ(x2)= (9.11) x1 x2 Vertices ˆ=interaction terms: •

x iy = (9.12) −

3. Order terms opposite to the fermion ﬂow indicated by the arrows. 4. For each closed fermion loop take Dirac trace and multiply by ( 1). − 5. Integrate the sum of all terms according to 1 d4x ... d4x : ... : . (9.13) n! 1 n Z 9.2.2 Feynman rules for S-matrix elements Consider n m particle process: →

i = a† ...a† 0 , f = 0 aB ...aB (9.14) | i A1 An | i h | h | m 1

where A1,...,Bm = scalar or fermion ﬁelds φ, ψ, ψ¯.

!Only contributions from terms a† ...a† aA ...aA in S ֒ → ∝ B1 Bm n 1 114 CHAPTER 9. INTERACTION OF SCALAR AND FERMION FIELDS

1. Select terms graphically: Ai, Bj ˆ=external lines in diagrams

incoming fermion/outgoing antifermion: ψ(x) contains a, b† (9.15) x outgoing fermion/incoming antifermion (9.16) x incoming/outgoing real scalar (9.17) x 2. Perform contractions of ﬁelds in normal-ordered products with external ﬁelds:

Typical manipulation:

ikx ψ(x)a† (~p) 0 = dk˜ e− u (k) a (~k)a† (~p) + ... 0 (9.18) σ | i τ τ σ | i τ Z † ~ ~ † Xh = aσ(~p)aτ (k)+ aτ (k), aσ(~p) i − { } 3 0 = (2π )2p δ(~p ~k)δστ | {z| }{z −} ipx = e− u (p) 0 + ... (9.19) σ | i Deﬁne contractions with external fermion ﬁelds:

ipx ψ(x)a† (~p) 0 = e− u (p) 0 , σ | i σ | i ipx ψ¯(x)b† (~p) 0 = e v¯ (p) 0 , σ | i σ | i 0 a (~p)ψ(x) = 0 eipxu¯ (p), h | σ h | σ ipx 0 b (~p)ψ(x) = 0 e− v (p). (9.20) h | σ h | σ

4 3. Perform the integration of d xi after inserting the propagators: Z 4 d k ik(x1 x2) i iD (x , x )= φ(x )φ(x )= e− − = (9.21) F 1 2 1 2 (2π)4 k2 m2 + iǫ x x Z − φ 1 2 4 d k ik(x1 x2) i iS (x , x )= ψ(x )ψ(x )= e− − = (9.22) F 1 2 1 2 (2π)4 /k m + iǫ x x Z − f 1 2 .Momentum conservation at each vertex ֒ → 4. Calculate symmetry factor for each graph (are all 1 in this model).

5. Determine the sign for each graph from the permutation of fermionic operators. 9.2. YUKAWA THEORY 115

Feynman rules for the transition matrix element : Mfi 1. Determine all relevant Feynman diagrams: n m scattering process n + m external lines. • → ⇒ Order of perturbation theory number of loops. • ⇒ 2. Impose momentum conservation at each vertex.

3. Insert the explicit expressions (fermionic terms ordered opposite to arrows):

f uσ(p) f u¯σ(p) ←−−p −−→p f¯ vσ(p) f¯ v¯σ(p) −−→p ←−−p φ 1 φ 1 p p ←−− i −−→ i 2 2 p /p mf + iǫ p p m + iǫ − − φ

iy − (9.23)

d4p 4. Integrate over all loop momenta p via l . l (2π)4 Z 5. For each closed fermion loop take Dirac trace and multiply by ( 1). Insert a relative sign between diagrams that result from interchanging external− fermion lines.

Example: + ( 1) − ×

6. The coherent sum of all diagrams yields i . Mfi 116 CHAPTER 9. INTERACTION OF SCALAR AND FERMION FIELDS

Example: the decay φ ff¯ in lowest order → (.Practically identical to the decays H ff¯ (f = b, τ, etc ֒ → → of the Standard Model Higgs boson. Process: φ(k) f (p ) + f¯ (p ), σ, τ = helicities. (9.24) → σ 1 τ 2

Lowest-order diagram: f

p1 φ k p2 f¯ Amplitude:

i = iy u¯ (p ) v (p ). (9.25) M − σ 1 τ 2 2 2 = y (¯u (p ) v (p )) (¯u (p ) v (p ))∗ ⇒ |M| σ 1 τ 2 σ 1 τ 2 pol σ,τ X X = (¯uσ(p1) vτ (p2))† =|v ¯τ (p2){zuσ(p1) } 2 = y uσ(p1)α u¯σ(p1)β vτ (p2)β v¯τ (p2)α σ τ Xα,β X X = (/p1+m ) = (/p2 m ) f αβ − f βα = y2 Tr (/p + m )(/p m ) { |1 f {z2 − f } } | {z } = 4y2 (p p m2 ), m2 = k2 =(p + p )2 =2m2 +2p p 1 2 − f φ 1 2 f 1 2 = 2y2 (m2 4m2 ). (9.26) φ − f Partial decay width:

1 2 Γφ ff¯ = dΦ2 , for phase space Φ2, see Exercise 7.1 → 2mφ |M| Z Xpol 4 2 2 1 1 mφ 4mφmf = − dΩ 2y2 (m2 4m2 ) 2m (2π)2 q 8m2 1 φ − f φ φ Z = 4π 3 2 2 2 y mφ 4mf | {z } = 1 2 . (9.27) 8π − mφ ! Part III

Quantization of vector-boson ﬁelds

117

Chapter 10

Free vector-boson ﬁelds

10.1 Classical Maxwell equations

Electromagnetic ﬁelds:

4-vector potential: Aµ = (Φ, A~), Φ = scalar potential (= Lorentz scalar), • 6 A~ = 3-vector potential. Field-strength tensor: • F µν = ∂µAν ∂ν Aµ, (10.1) − ˙ in components: E~ = ~ Φ A~, B~ = ~ A~. −∇ − ∇ × Elmg. gauge invariance: Field strengths do not change under • µ µ µ A A′ = A + ∂ ω, ω = ω(x) = arbitrary function of x. (10.2) → Aµ is not uniquely ﬁxed by F µν (i.e. E,~ B~ ), and a speciﬁc choice is called a gauge. Examples:

– Covariant (Lorenz) gauge: ∂A = 0. .Aµ unique up to gauge transformations with ω = 0 ֒ → – Radiation gauge: A0 = Φ = 0, ~ A~ = 0. ∇

Maxwell equations: 0= jν = ∂ F µν = Aν ∂ν (∂A). (10.3) µ − = external current density = 0 for free Aµ ﬁelds |{z}

119 120 CHAPTER 10. FREE VECTOR-BOSON FIELDS

Lagrangian density: 1 = F F µν = gauge invariant (10.4) L −4 µν 1 µ ν 1 ν µ 1 µν µ ν = (∂µAν )(∂ A )+ (∂µAν )(∂ A ) = Aµ(g ∂ ∂ )Aν. (10.5) −2 2 part. int. 2 − Check EOM: ∂ ∂ 1 1 0 = L ∂ L = ∂ δσδρ(∂µAν) 2+ δσδρ(∂ν Aµ) 2 ∂A − σ ∂(∂ A ) − σ −2 µ ν · 2 µ ν · ρ σ ρ = ∂ ( ∂σAρ + ∂ρAσ)= ∂ F σρ. − σ − σ

Solution via plane elmg. waves: µ µ ikx Ansatz: A (x)= ε (k) e− .

constant polarization vector Convenient choice:| {z covariant} gauge EOM: Aµ =0 k2 = 0, i.e. kµ light-like. • ⇒ Gauge: ∂A =0 kµε(k)=0. • ⇒ .εµ(k) is spanned by 3 independent vectors ֒ → Convenient: helicity basis µ µ µ Example: k = k0(1, 0, 0, 1), ε (k) (0, 1, i, 0)/√2 = [ε (k)]∗. ± ≡ ± ∓ Basis: εµ (k), kµ ± 2 physical “unphysical polarization”, polarizations part of the gauge degree of | {z } freedom|{z} µ Normalization: ε (k) ε ′ (k)∗ = δ ′ . λ λ ,µ − λλ General solution of Maxwell eq.: ⇒ µ ˜ ~ µ ikx ~ µ +ikx µ A (x) = dk aλ(k) ελ(k)e− + aλ∗ (k) ελ(k)∗e = A (x)∗. (10.6) λ= Z X± arbitrary functions, will become creation/annihilation | {zoperators} in QFT | {z } 10.2. PROCA EQUATION 121 10.2 Proca equation

Aim: description of massive spin-1 particles .free ﬁeld modes with momentum kµ should obey k2 = m2 = 0 ֒ → 6 Generalization of Maxwell eq.: ⇒ ( + m2)V µ ∂µ(∂V )=0, Proca equation. (10.7) −

Features of the free Proca ﬁeld V µ:

Transversality: • ∂ (...) m2(∂V ) = 0, i.e. ∂V = 0 automatically fulfilled. µ ⇒ Eq. (10.7) ( + m2)V µ = 0 and ∂V = 0. ⇒ ⇔ Lagrangian: (real V µ) • 1 1 = V V µν + m2V V µ, V µν = ∂µV ν ∂ν V µ. (10.8) L −4 µν 2 µ − .Proca eq. as EOM ֒ → µ µ µ µ Note: = invariant under V V ′ = V + ∂ ω. L 6 → Solution via plane waves: • µ ˜ ~ µ ikx ~ µ +ikx V (x) = dk aλ(k)ελ(k)e− + aλ∗ (k)ελ(k)∗e . (10.9) λ=0, Z X± Note: 3 physical polarization states exist for massive spin-1 fields for each kµ. Helicity basis for kµ =(k , 0, 0, ~k ),k = ~k 2 + m2: 0 | | 0 µ p λ = 1: ε (k)=(0, 1, i, 0)/√2, transverse polarizations, ± ± ± λ = 0: εµ(k)=( ~k , 0, 0, k )/m, longitudinal polarization. (10.10) 0 | | − 0 Complex Proca field: • 1 µν 2 µ = V † V + m V †V . (10.11) L −2 µν µ

,.Vµ and Vµ† obey the Proca eq ֒ → ~ ~ aλ(k) and aλ∗ (k) become independent amplitudes in Eq. (10.9). 122 CHAPTER 10. FREE VECTOR-BOSON FIELDS 10.3 Quantization of the elmg. ﬁeld

10.3.1 Preliminaries

Aµ = Aµ† hermitian ﬁeld operator with 1 = : F F µν : , F µν = ∂µAν ∂ν Aµ. (10.12) L −4 µν − ∂ .Canonical momentum variable: πµ = L = F µ0, i.e. π0 =0 ֒ → ∂(∂0Aµ) Contradiction to canonical commutator [A (t, ~x), π (t, ~y)] = 0 ! ⇒ 0 0 6 Origin of the problem: Aµ contains unphysical degrees of freedom because of gauge invariance. Possible solutions:

Fix the gauge of the operator Aµ by constraints. .No quantization of unphysical degrees of freedom ֒ • → Example: Radiation gauge A0 = Φ = 0, ~ A~ = 0. ∇ Disadvantage: Covariant gauge ∂A = 0 not compatible with [A (t, ~x), π (t, ~y)] = ig δ(~x ~y). µ ν µν − Impose gauge constraints on physical states (instead of ﬁeld operators). .Unphysical dergees of freedom are quantized, but can be decoupled in observables ֒ • → Corresponding Gupta–Bleuler procedure described in the following.

10.3.2 Gupta–Bleuler quantization: Starting assumptions:

1. Add covariant gauge-ﬁxing term to Lagrangian: 1 1 = : F F µν : :(∂A)2 : . (10.13) L −4 µν − 2ξ

gauge-ﬁxing term with arbitrary gauge parameter ξ > 0 | {z } 2. Demand constraint not as operator equation ∂A = 0, but as constraint on expectation values:

ψ ∂A ψ =! 0, ψ = Hilbert space of physical states. (10.14) h | | i ∀ | i ∈ Hphys 10.3. QUANTIZATION OF THE ELMG. FIELD 123

Quantization: Canonical momentum variable: • 1 πµ = F µ0 gµ0 (∂A). (10.15) − ξ

Canonical commutators: • [Aµ(t, ~x), πν(t, ~y)] = igµνδ(~x ~y), − [Aµ(t, ~x), Aν(t, ~y)] = 0, [πµ(t, ~x), πν(t, ~y)] = 0. (10.16)

For ξ = 1 (used in the following!): ⇒ Aµ(t, ~x), A˙ ν(t, ~y) = i gµν + gµ0gν0(ξ 1) δ(~x ~y) = igµν δ(~x ~y), − − − ξ=1 − − h i Aµ(t, ~x), Aν(t, ~y) = 0,

h µ ν i µ0 νk µk ν0 A˙ (t, ~x), A˙ (t, ~y) = i(ξ 1) g g + g g ∂k,xδ(~x ~y) = 0, (10.17) − − − ξ=1 h i i.e. Ak (k =1, 2, 3) behave like scalar ﬁelds, but A0 has “wrong” sign in commutator. EOM: • 1 Aµ 1 ∂µ(∂A) = Aµ = 0. (10.18) − − ξ ξ=1

Problem: Find Aµ(x) obeying Eqs. (10.14), (10.17), and (10.18). :Fourier ansatz ֒ → 3 µ ˜ ~ µ ikx ~ µ +ikx µ A (x) = dk aλ(k)ελ(k)e− + aλ† (k)ελ(k)∗e = A (x)†, (10.19) Z Xλ=0 µ with the “extended helicity basis” for k = k0(1,~e):

µ ε1,2(k) = (0,~e1,2), transversal d.o.f. where ~e 2 =1, ~e ~e =0, ~e ~e =0, ~e ~e = ~e, 1,2 · 1,2 1 · 2 1 × 2 µ µ µ so that ε (k)= ε1 (k) iε2 (k) /√2 = helicity states, ± ± µ ε3 (k) = (0,~e), longitudinal d.o.f. µ ~ ε0 (k) = (1, 0), scalar d.o.f. (10.20) 3 µ ν µν ελ(k) ελ′ (k)∗ = gλλ′ , ( gλλ′ ) ελ(k)ελ′ (k)∗ = g . (10.21) ⇒ · ′ − − λ,λX=0 124 CHAPTER 10. FREE VECTOR-BOSON FIELDS

Insertion of ansatz:

EOM (10.18) only demands k2 = 0, i.e. k = ~k . • 0 | | Commutators (10.17) lead to • 3 a (~k), a† ′ (~k′) = g ′ 2k (2π) δ(~k ~k′), λ λ − λλ 0 − h ~ ~ i ~ ~ aλ(k), aλ′ (k′) = aλ† (k), aλ† ′ (k′) = 0. (10.22) h i h i ( ) .a † as for scalar ﬁeld, but the roles of a and a† are interchanged ֒ → 1,2,3 0 0 Gauge condition (10.14), ψ ∂A ψ = 0, already results from demand ∂ A(+) ψ =! 0: • h | | i | i only e−ikx part

(+) ( ) (+) ( ) |{z} 0= ∂A ψ † = ψ ∂A − , ψ ∂A ψ = ψ ∂A + ∂A − ψ =0. | i h | h | | i h | | i : Condition on a ֒ → λ 3 (+) ikx 0 = ∂A ψ = dk˜ e− (k ε (k)) a (~k) ψ . | i · λ λ | i λ=0 Z k ε0 = k0 = k ε3, X · − · k ε1,2 = 0 | · {z } 0 = a (~k) a (~k) ψ . (10.23) ⇒ 0 − 3 | i h i Change of basis:

εµ (k) (1,~e) = εµ(k)+ εµ(k), εµ (k) (1, ~e) = εµ(k) εµ(k), L ≡ 0 3 N ≡ − 0 − 3 i.e. ε = 1 (ε + ε ), ε = 1 (ε ε ), ε2 = ε2 =0, ε ε =2. 0 2 L N 3 2 L − N L N L · N a (~k)εµ(k)+ a (~k)εµ(k)= a (~k)εµ (k)+ a (~k)εµ (k), ⇒ 0 0 3 3 L L N N with a = 1 (a + a ), a = 1 (a a ), L 2 0 3 N 2 0 − 3 ~ ~ ~ ~ [aL(k), aL† (k′)] = [aN (k), aN† (k′)]=0,

0 3 [a (~k), a† (~k′)] = [a (~k), a† (~k′)] = k (2π) δ(~k ~k′) etc. (10.24) L N N L − − Gauge condition (10.23): ⇒ a (~k) ψ =0. (10.25) N | i 10.3. QUANTIZATION OF THE ELMG. FIELD 125

Fock space: 1-particle states: • ~k,λ a† (~k) 0 , where a (~k) 0 =0, λ =0, 1, 2, 3. (10.26) | i ≡ λ | i λ | i Wave packets: f = dk˜ f (~k) ~k,λ . (10.27) | λi λ | i Z Orthogonality / normalization of states:

f f ′ ′ = dk˜ dk˜′ f ∗(~k)f ′ ′ (~k′) 0 a (~k)a† ′ (~k′) 0 h λ| λ i λ λ h | λ λ | i Z Z 3 ′ = g ′ 2k0(2π) δ(~k ~k ) − λλ − | {z } = g ′ dk˜ f ∗(~k)f ′ (~k), (no summation ) (10.28) − λλ λ λ λ Z P ! f 2 = f f = dk˜ f (~k) 2 < 0, indeﬁnite metric ֒ → || 0|| h 0| 0i − | 0 | Z f 2 > 0, λ =1, 2, 3. (10.29) || λ|| Classiﬁcation of states: • ψ ∂A(+) ψ =0 , H ≡ | i | | i n o ψ ψ generated by a† (~k) , HT ≡ | T i|| T i 1,2 n o (+) ′ ψ′ ψ′ generated by a† (~k) and ∂A ψ′ =0 . (10.30) H ≡ | i|| i 0,3 | i Properties: n o

– ′ , trivial. H ⊂ H – , since ∂A(+) ψ =0 ψ . HT ⊂ H | T i ∀ | T i ∈ HT – = T ′, completeness. H H ⊗ H ( ) ( ) Note: ψ = ψ ψ′ without ordering issues, since [a † , a †′ ]=0for λ = λ′. | i | T i⊗| i λ λ 6

Inspection of ψ′ : | i ψ′ = linear combination of | i a† ...a† 0 , λi =0, 3, or equivalently λi = L, N. (10.31) λn λ1 | i Gauge condition (10.25): ! 0 = a (~p) a† (~k ) ...a† (~k ) a† (~k ) ...a† (~k ) 0 N N 1 N nN L 1 L nL | i = a† (~k ) ...a† (~k ) a (~p) a† (~k ) ...a† (~k ) 0 . (10.32) N 1 N nN N L 1 L nL | i = 0for n > 0, see (10.24) 6 L | {z } ψ′ only generated by a† (~k)! ⇒ | i N 126 CHAPTER 10. FREE VECTOR-BOSON FIELDS

General form of ψ′ : | i

ψ′ = c 0 + N, 1 + N, 2 + ..., (10.33) | i 0 | i | i | i N, n = generated from n> 0 ops. a† . | i N N, n N, n′ =0, since [a , a† ]=0. h | i N N 2 2 2 ψ′ = ψ′ ψ′ = c 0 0 = c . ⇒ || || h | i | 0| h | i | 0|

Physical observables: General form:

3 1, λ =0, G = dkg˜ (~k) a† (~k)a (~k)η , η = λ λ λ λ −+1, λ =1, 2, 3. Z Xλ=0 ˜ ~ ~ ~ ˜ ~ ~ ~ = dkg(k) aλ† (k)aλ(k) + dkg(k) aλ† (k)aλ(k)ηλ . (10.34) Z λX=1,2 Z λX=0,3 ′ GT G ≡ ≡

Expectation value in| state ψ ={z ψ ψ′ }: | {z } | i | T i⊗| i

ψ G ψ = ψ G ψ ψ′ ψ′ + ψ ψ ψ′ G′ ψ′ , h | | i h T | T | T ih | i h T | T ih | | i 2 = c0 | | = dkg˜ (~k) ψ′ a†a a†a ψ′ h | 3 3 − 0 0| i | {z } Z | {z }

= 2 dkg˜ (~k) ψ′ a† a + a† a ψ′ = 0 − h | N L L N | i Z ψ G ψ ψ G ψ c 2 ψ G ψ h | | i = h T | T | T i| 0| = h T | T | T i. (10.35) ψ ψ ψ ψ c 2 ψ ψ h | i h T | T i| 0| h T | T i

ψ′ part of ψ irrelevant for observables. ⇒ | i | i Hilbert space of physical states: = / ′ . Hphys H H ∼ HT Example: 4-momentum

3 µ ˜ µ ~ ~ P = dkk ηλ aλ† (k)aλ(k). (10.36) Z Xλ=0 can be replaced by in T λ=1,2 H P | {z } Comment: holds for general ξ = 1 as well, but proof non-trivial. Hphys ∼ HT 6 10.4. PHOTON PROPAGATOR 127 10.4 Photon propagator

Deﬁnition:

µ ν iDµν(x, y) = 0 T Aµ(x)Aν(y) 0 , photon propagator. (10.37) F h | | i x y Explicit calculation: Insert Fourier representation (10.19) (for ξ = 1) and use canonical commutators • (10.22), or

derive and solve diﬀerential equation for propagator (shown for general ξ in the • following). Wave operator and EOM: 1 µν = gµν 1 ∂µ∂ν , EOM: µνA (x) = 0 (10.38) D − − ξ D ν 1 µν = gµν 1 gµ0gν0 ∂ ∂ D − − ξ t t µν a ∂t∂t ≡ | 1 {z } + 1 gµ0gνk + gµkgν0 ∂ ∂ + terms without ∂ . (10.39) ξ − t k t µν b ∂t∂ ≡ k k Application of µν to i|Dµν (x, y): {z } Dx F Note: Apply ∂t to θ-functions of T ordering as well (product rule!).

∂ 0 T F (x)G(y) 0 = 0 (∂ T )F (x)G(y) 0 + 0 T (∂ F (x))G(y) 0 x0 h | | i h | x0 | i h | x0 | i = (∂x0 θ(x0 y0)) 0 F (x)G(y) 0 − h | | i + (∂x0 θ(y0 x0)) 0 G(y)F (x) 0 − h | | i =|δ(x0 y0) 0{z[F (x),G(y)] 0} − h | | i = δ(x y ) 0 [F (x),G(y)] 0 + 0 T F˙ (x)G(y) 0 . (10.40) 0 − 0 h | | i h | | i

µν 0 T A (x)A (y) 0 ⇒ Dx h | ν ρ | i = 0 T ( µν A (x)) A (y) 0 + bµν 0 (∂ T )(∂kA (x))A (y) 0 h | Dx ν ρ | i k h | x0 ν ρ | i = 0, EOM +2aµν 0 (∂ T )(∂ A (x))A (y) 0 + aµν 0 (∂2 T )A (x)A (y) 0 | h {z| x0 } x0 ν ρ | i h | x0 ν ρ | i = bµν δ(x y ) 0 [∂kA (x), A (y)] 0 k 0 − 0 h | ν ρ | i = 0 + aµν 0 (∂ T )(∂ A (x))A (y) 0 + aµν ∂ 0 (∂ T )A (x)A (y) 0 | h | x0 x{z0 ν ρ | i} x0 h | x0 ν ρ | i 128 CHAPTER 10. FREE VECTOR-BOSON FIELDS

= aµν δ(x y ) 0 [A˙ (x), A (y)] 0 + aµν ∂ δ(x y ) 0 [A (x), A (y)] 0 0 − 0 h | ν ρ | i x0 0 − 0 h | ν ρ | i = i[gνρ+gν0gρ0(ξ 1)]δ(x y) − − = 0 | 1 {z } | {z } = gµν 1 gµ0gν0 i [g + g g (ξ 1)] δ(x y) − − ξ νρ ν0 ρ0 − − = iδµ δ(x y). (10.41) ρ − µν Identiﬁcation of DF (x, y) as Green function: 1 gµν 1 ∂µ∂ν D (x, y) = δµ δ(x y). (10.42) x − − ξ x x νρ ρ − Solution in momentum space:

4 d k ik(x y) D (x, y)= e− − D˜ (k). (10.43) νρ (2π)4 νρ Z 1 gµν k2 + 1 kµkν D˜ (k) = δµ ֒ → − − ξ νρ ρ 1 1 − g k k D˜ (k) = g k2 + 1 k k = − νρ + ν ρ (1 ξ). (10.44) νρ − νρ − ξ ν ρ k2 (k2)2 − Implementing causal behaviour via Feynman’s prescription yields:

4 µν µ ν µν d k ik(x y) ig ik k iD (x, y) = e− − − + (1 ξ) F (2π)4 k2 + iǫ (k2 + iǫ)2 − Z 4 µν d k ik(x y) ig = e− − − . (10.45) ξ=1 (2π)4 k2 + iǫ Z

Comment: Propagator does not exist for ξ , where the gauge-ﬁxing term in disappears µν →∞ L and the wave operator is singular (i.e. contains zero modes = gauge d.o.f.). D

Chapter 11

Interacting vector-boson ﬁelds

11.1 Electromagnetic interaction

Charged particles and elmg. gauge invariance:

Charged particles complex ﬁelds / non-hermitian ﬁeld operators Φ, Φ†, • → otherwise: particle antiparticle. ≡ Examples: Φ, Φ† = φ,φ† (scalar); ψ, ψ (Dirac fermion); Vµ,Vµ† (vector). Charged conservation EOM invariant under global elmg. gauge transformation: • → iqω iqω Φ Φ′ = e− Φ, Φ† Φ†′ = e Φ†, ω = const. (11.1) → → Lagrangian is invariant under global transformation: •

(Φ, ∂Φ,... ) = (Φ′, ∂Φ′,... ). (11.2) LΦ LΦ Noether current conserved elmg. current. →

129 130 CHAPTER 11. INTERACTING VECTOR-BOSON FIELDS

“Gauge principle”: Total Lagrangian should be invariant under local gauge transformations:

iqω(x) Φ Φ′ = e− Φ, etc., ω = ω(x) = arbitray function. (11.3) → :Φ terms cause problems ∂ ֒ → µ iqω(x) ∂ Φ ∂ Φ′ = ∂ e− Φ µ → µ µ iqω iqω = iq(∂µω)e− Φ + e− (∂µΦ) . (11.4) − ω-dependence does ω-dependence cancels not cancel in Φ because of global in general L invariance of | {z } | {z LΦ}

Idea: replace ∂µ by covariant derivative

Dµ = ∂µ + iqAµ(x) (11.5)

and transform the new ﬁeld Aµ(x) such that

! iqω(x) D Φ (D Φ)′ = D′ Φ′ = e− (D Φ). (11.6) µ → µ µ µ Explicitly:

iqω iqω D′ Φ′ = (∂ + iqA′ )(e− Φ) = e− ∂ iq(∂ ω) + iqA′ Φ. (11.7) µ µ µ µ − µ µ h ! i =iqAµ | {z } A′ = A + ∂ ω, elmg. gauge transformation (11.8) ⇒ µ µ µ .A (x) can be identiﬁed with photon ﬁeld ֒ → µ

Introduction of elmg. interaction by minimal substitution:

(Φ, ∂Φ,... ) = globally invariant LΦ ∂ D , addition of µ → µ LA,0 ↓ 1 = (Φ,DΦ,... ) F F µν = locally invariant (11.9) L LΦ − 4 µν 11.1. ELECTROMAGNETIC INTERACTION 131

Examples:

Scalar quantum electrodynamics: •

2 1 µν = (Dφ)†(Dφ) m φ†φ F F ,D = ∂ + iqA , (11.10) L − − 4 µν µ µ µ φ describes a scalar (spin-0) boson with charge q and mass m.

EOMs:

µ µ 2 (∂µ + iqAµ)(∂ + iqA )+ m φ = 0, KG eq. with elmg. interaction, (11.11)

µν ν ∂µF = j , Maxwell eq. (11.12)

Elmg. current:

∂ ∂φ ∂ ∂φ† jµ = L + L ∂(∂µφ) ∂ω ∂(∂µφ†) ∂ω µ µ 2 µ = iq (∂ φ†)φ φ†(∂ φ) 2q A φ†φ. (11.13) − − − Spinor quantum electrodynamics: • 1 = ψ (i /D m) ψ F F µν,D = ∂ + iqA , (11.14) L − − 4 µν µ µ µ 1 ψ describes a Dirac fermion (spin- 2 ) boson with charge q and mass m. EOMs:

(i/∂ q /A m) ψ = 0, Dirac eq. with elmg. interaction, (11.15) − − µν ν ∂µF = j , Maxwell eq. (11.16)

Elmg. current: (note fermionic signs !)

∂ ∂ψ ∂ψ ∂ jµ = L + L = qψγµψ. (11.17) −∂(∂µψ) ∂ω ∂ω ∂(∂µψ) 132 CHAPTER 11. INTERACTING VECTOR-BOSON FIELDS 11.2 Perturbation theory for spinor electrodynamics

Lagrangian and quantization: (covariant gauge) (A, ψ, ψ) = + + , (11.18) L Lψ,0 LA,0 Lint (ψ, ψ) = : ψ (i/∂ m ) ψ : , Dirac fermion of mass m and charge q = Q e, Lψ,0 − f f f 1 1 (A) = : F F µν : :(∂A)2 : , photon ﬁeld, LA,0 −4 µν − 2ξ (A, ψ, ψ) = Q e : ψ /Aψ := : j Aµ : , elmg. interaction. Lint − f − µ Hamiltonian: (A, ψ, ψ) = (A, ψ, ψ), since no derivative involved. (11.19) Hint −Lint Quantization of free ﬁelds as usual free propagators: → µ ν 0 T Aµ(x)Aν(y) 0 = iDµν (x, y) (11.20) h | | i free F x y

0 T ψ(x)ψ¯(y) 0 = iS (x, y) (11.21) h | | i free F x y

11.2.1 Expansion of the S-operator :(Apply Wick theorem as in Yukawa theory (see Sect.9.2 ֒ → S = T exp ( iQ e) d4x : ψ(x) /A(x) ψ(x): − f Z = 1 + ( iQ e) d4x : ψ(x) /A(x) ψ(x): + (e2) (11.22) − f O Z

x x1 x2 = 1 + +

+ x1 x2 + x1 x2 + x1 x2

+ + + x1 x2 x1 x2 x1 x2

+ + (e3) x1 x2 O 11.2. PERTURBATION THEORY FOR SPINOR ELECTRODYNAMICS 133

Feynman rules for graphical representation of the terms en: ∝

1. Draw all possible diagrams with n vertices

(any number of exernal lines, including disconnected diagrams).

2. Translate graphs into analytical expressions as follows:

External lines ˆ=non-contracted ﬁelds: •

Aµ(x)= x ψ(x)= x ψ¯(x)= (11.23) x Internal lines ˆ=contracted ﬁelds (=propagators): •

µ ν µ ν A (x1)A (x2) = x1 x2

ψ(x1)ψ(x2) = (11.24) x1 x2 Vertices ˆ=interaction terms: •

x µ iQ eγ = (11.25) − f µ

3. Order terms opposite to the fermion ﬂow indicated by the arrows.

4. For each closed fermion loop take Dirac trace and multiply by ( 1). − 5. Integrate the sum of all terms according to 1 d4x ... d4x : ... : . (11.26) n! 1 n Z 134 CHAPTER 11. INTERACTING VECTOR-BOSON FIELDS

11.2.2 Feynman rules for S-matrix elements Consider n m particle process: →

i = a† ...a† 0 , f = 0 aB ...aB , (11.27) | i A1 An | i h | h | m 1 where A1,...,Bm = photons or (anti)fermions A, f, f¯.

!Contributions to f S i only from terms a† ...a† aA ...aA in S ֒ → h | | i ∝ B1 Bm n 1 Procedure as in Yukawa model, new ingredients for photons:

External photons: contractions with operators Aµ(x) • µ ˜ ikx µ ~ A (x)aλ† (~p) 0 = dk e− ελ′ (k) aλ′ (k)aλ† (~p) + ... 0 (11.28) | i ′ | i λ =1,2 † † Z ′ ~ ′ ~ X h = aλ(~p)aλ (k)+[aλ (k), aλ(~p)] i

3 0 ~ | ={z (2π) 2p}δ(~p k)δλλ′ | {z −} ipx µ = e− ε (p) 0 + ... (11.29) λ | i :Contractions with incoming /outgoing photons ֒ →

µ ipx µ A (x)a† (~p) 0 = e− ε (p) 0 , λ | i λ | i µ ipx µ 0 a (~p)A (x) = 0 e ε (p)∗. (11.30) h | λ h | λ Internal photons: Fourier representation of propagator • 4 µν µ ν µ ν µν d k ik(x1 x2) ig ik k A (x )A (x ) = iD (x , x )= e− − − + (1 ξ) . 1 2 F 1 2 (2π)4 k2 + iǫ (k2 + iǫ)2 − Z (11.31)

Space–time integrals d4x at vertices imply momentum conservation, • i Z 4 loop integrals d pl remain open, fermionic signs as usual ... Z 11.2. PERTURBATION THEORY FOR SPINOR ELECTRODYNAMICS 135

3. Insert the explicit expressions (fermionic terms ordered opposite to arrows):

f uσ(p) f u¯σ(p) ←−−p −−→p f¯ vσ(p) f¯ v¯σ(p) −−→p ←−−p µ µ µ µ A ε (p) A ε (p)∗ ←−−p −−→p µν µ ν i µ ν ig ip p −2 + 2 2 (1 ξ) p /p mf + iǫ p p + iǫ (p + iǫ) − −

µ iQ eγ − f µ (11.32)

Example: + ( 1) − ×

6. The coherent sum of all diagrams yields i . Mfi

Straightforward generalization to more charged fermions f: Propagation of free ﬁelds completely independent, interaction: = Q e : ψ /Aψ : . Lint − f f f f Each fermion f hasP its own propagator and γff¯ vertex. ⇒ 136 CHAPTER 11. INTERACTING VECTOR-BOSON FIELDS

Comments: Gauge-parameter independence of S-matrix elements: • Momentum terms pµpν in photon propagator do not contribute to amplitudes. ∝ .(ξ-dependence completely cancels (proof non-trivial ֒ → Gauge freedom of external states: • Amplitudes with an incoming/outgoing photon of momentum p obey the following M (non-trivial) Ward identity:

µ ( ) µ = ε (p) ∗ T (p) p T (p)= 0, (11.33) M λ µ ⇒ µ where all external states other than the photon must be on shell. Physical (transverse) polarization vectors can be changed according to ⇒ εµ (p) εˆµ (p)= εµ (p)+ apµ with a = arbitrary. (11.34) 1,2 → 1,2 1,2 Comment: A Lorentz transformation in general leads to such replacements.

Convenient construction of transverse polarizations: • µ 2 2 µ µ Choose any gauge vector n = (1, ~n) with ~n = 1, n = 0, pµn = 0 and deﬁne ε1,2(p) such that 6 µ µ pµ ε1,2(p) = nµ ε1,2(p)= 0. (11.35) Comment: This is possible due to the gauge freedom (11.34).

Completeness relation: ⇒ µ ν ν µ µ ν µν p n + p n ε (p)ε (p)∗ = g + . (11.36) λ λ − p n λ=1,2 · X does not contribute to amplitudes because of Ward identity (11.33) | {z } .Convenient in photon spin summation of squared amplitudes ֒ →

Unphysical parts ψ′ = c0 0 + N, 1 + ... of photon states ψ = ψT ψ′ do • not inﬂuence transition| i amplitudes,| i | becausei S is derived from the| i exponential| i⊗| ofi the Hamiltonian H = H H′, so that S = S S′. Schematically: T ⊕ T ⊗ f S i f S i f ′ S′ i′ h | | i = h T | T | T i h | | i . (11.37) f i f i f i || ||·|| || || T ||·|| T || || ′||·|| ′|| can only be a phase factor, ′ ′ ′ ∗ because f S i = c0,f c0,i 0 0 , see Sect.| h 10.3.2| {z| i } h | i 11.3. IMPORTANT PROCESSES OF (SPINOR) QED 137 11.3 Important processes of (spinor) QED

11.3.1 Elastic ep scattering Process: e−(p, σ) + p(k, τ) e−(p′, σ′) + p(k′, τ ′) (11.38) → Diagram: p p′ e− e− µ γ p p Qp = Qe =1, − ′ − ν p + k = p′ + k′. p p k k′ Amplitude in Born approximation (tree level):

i = u¯ ′ (p′)( iQ e) γ u (p) u¯ ′ (k′)( iQ e) γ u (k) M σ − e µ σ τ − p ν τ h µν i h µ ν i ig i(p p′) (p p′) − + − − (1 ξ) × (p p )2 + iǫ ((p p )2 + iǫ)2 − − ′ − ′ iǫ irrelevant no contribution due to Dirac eqs.: ′ ′ at Born level u¯ ′ (p )(/p /p )uσ(p) = 0 σ − | 2 {z } | {z } iQeQpe µ ′ ′ = 2 u¯σ (p′) γµ uσ(p) u¯τ (k′) γ uτ (k) . (11.39) (p p′) − h i h i 1 2 = 2 ⇒ |M| 4 ′ ′ |M| σ,σX,τ,τ 2 2 4 Qe Qpe µ ν = Tr (/p′ + m )γ (/p + m )γ Tr (/k′ + m )γ (/k + m )γ 4(p p )4 { e µ e ν} { p p } − ′ 4 4e 2 µ ν ν µ 2 µν = p′ p + p′ p (pp′ m )g k′ k + k′ k (kk′ m )g (p p )4 { µ ν ν µ − − e µν}{ − − p } − ′ 2e4 = ... = t2 +2st + 2(s m2 m2)2 , (11.40) t2 − e − p n 2 2 o 2 where s =(p + k) , t =(p p′) , u =(p k′) . (11.41) − − 138 CHAPTER 11. INTERACTING VECTOR-BOSON FIELDS

Kinematics in proton rest frame: (xz-plane = scattering plane) pµ = (E , 0, 0,p ), p = E2 m2, e e e e − e µ 2 2 p′ = (Ee′ ,pe′ sin θ, 0,pe′ cospθ), pe′ = Ee′ me , µ − k = (mp, 0, 0, 0). p (11.42) s = m2 + m2 +2pk = m2 + m2 +2m E , λ(s, m2, m2)= ... =2m p . ⇒ e p e p p e e p p e q (11.43) Note: Ee′ = Ee′ (Ee, θ) with Ee = Ee′ (Ee, 0). Ee given by experiment, measured: Ee′ or θ. 2 2 Relation between Ee, Ee′ , θ from k′ = mp, e.g. derived via t: 2 2 t = 2me 2pp′ = 2(me EeEe′ + pepe′ cos θ) − 2 2 − 2 = (k k′) =2m 2kk′ =2m 2k(p + k p′)= 2k(p p′) − p − p − − − − = 2mp(Ee Ee′ ). (11.44) − 2 − me EeEe′ + pepe′ cos θ Ee Ee′ = − , can be solved for Ee′ (11.45) ⇒ − − mp or, e.g., expanded for mp 2 2 2 →∞ me Ee + pe cos θ 2 = − + (1/mp) − mp O p2(1 cos θ) = e − + .... (11.46) mp 2-particle phase space: 1 1 dΦ2 = dϕ1 dt 2 , derived as in Exercise 7.1. (11.47) 4(2π) 2 2 Z Z Z λ(s, me , mp) azimuthal angle integral 2π λ(x,y,zq)= x2 + y2 + z2 2xy 2xz 2yz | {z→ } − − − Cross section: | {z } 1 2 dσ = dΦ2 . (11.48) 2 2 |M| Z 2 λ(s, me , mp) Z

qflux factor Differential cross section in virtuality| Q2 {z t of the} photon: ≡ − 4 dσ 1 1 2e 2 2 2 2 2 = 2 2 2 t +2st + 2(s me mp) dQ 16π λ(s, me, mp) t − − 2 2 2 n 2 2 o πα 8m E 2(m + m +2mpEe) = p e e p +1 . (11.49) 2m2p2 Q4 − Q2 p e = Lorentz invariant, since dσ and Q2 are invariant, e2 e2 where α = = =1/137.0... = fine-structure constant. (11.50) 4π 4πε0~c 11.3. IMPORTANT PROCESSES OF (SPINOR) QED 139

Interesting kinematical limits:

Non-relativistic limit: p ,p′ m m • e e ≪ e ≪ p 2 Use approximation (11.46) of large mp for Q : θ Q2 = t = 2p2(1 cos θ) + (1/m ) =4p2 sin2 + ..., − e − O p e 2 dQ2 = 2p2 d cos θ + ... (11.51) − e ( ) ( ) E ′ m p ′ and neglect all terms of e e , e relative to the leading term. O m ∼ m m p p e :Diﬀerential cross section in angle θ ֒ → dσ πα2 8m2m2 πα2m2 = p e + ... = e + ..., Rutherford cross section. 2 4 4 4 θ d cos θ mp Q 2pe sin 2 (11.52)

Comment: The scattering is determined by the particle charges only, the spin does not play a role in the non-relativistic limit.

Relativistic electron: m p ,p′ m • e ≪ e e ≪ p Again use expansion (11.51) for Q2 and neglect all m terms (i.e. p E ). e e → e :Diﬀerential cross section in angle θ ֒ → dσ πα2 8m2E2 2m2 πα2 cos2 θ = p e p + ... = 2 + ..., Mott cross section. d cos θ m2 Q4 − Q2 2E2 sin4 θ p e 2 (11.53)

Comments:

– Electron helicity is conserved in the relativistic scattering process. 2 θ The factor cos 2 results from e− spin rotation.

– For Ee above 100MeV (mp 1 GeV) the ﬁnite extension of the proton becomes visible.∼ ∼ .Formfactor for proton charge distribution necessary ֒ → – For Q2 above m2 inelastic scattering becomes relevant (proton breaks up). ∼ p .Measured in terms of structure functions, described by the parton model ֒ → 140 CHAPTER 11. INTERACTING VECTOR-BOSON FIELDS

11.3.2 Other important processes in QED

Bremsstrahlung in potential scattering: e− + A e− + A + γ • →

Compton scattering: e− + γ e− + γ • →

+ Pair annihilation: e + e− γγ • → e+ e+

e− e− + Pair creation: e + e− ff¯ • → e+ f¯

e− f + + Bhabha scattering: e + e− e + e− • → e+ e+ e+ e+

e− e− e− e− γγ scattering: γγ γγ • →

+ 4 more diagrams Note: Higher-order processes of QED violate the superposition principle for elmg. ﬁelds ! etc. • Bibliography

[1] J. D. Bjorken and S. D. Drell, “Relativistic Quantum Mechanics”, McGraw-Hill (1964), German translation BI-Wissenschaftsverlag (1990)

[2] J. D. Bjorken and S. D. Drell, “Relativistic Quantum Fields”, McGraw-Hill (1965), Reprint: Dover (2012). German translation BI-Wissenschaftsverlag (1990)

[3] C. Itzykson and J. B. Zuber, “Quantum Field Theory,” New York, Usa: McGraw-hill (1980). Reprint: Dover (2006)

[4] M. Maggiore, “A Modern introduction to quantum ﬁeld theory,” Oxford University Press (2005)

[5] M. E. Peskin and D. V. Schroeder, “An Introduction to quantum ﬁeld theory,” Read- ing, USA: Addison-Wesley (1995) 842 p

[6] S. Weinberg, “The Quantum theory of ﬁelds. Vol. 1: Foundations,” Cambridge, UK: Univ. Pr. (1995) 609 p

[7] M. D. Schwartz, “Quantum Field Theory and the Standard Model,” Cambridge Uni- versity Press (2013), 859 p

[8] M. Srednicki, “Quantum ﬁeld theory,” Cambridge, UK: Univ. Pr. (2007) 641 p The parts “Spin zero” and “Spin one-half” are also available at http://arxiv.org/abs/hep-th/0409035 and http://arxiv.org/abs/hep-th/0409036

[9] H. Goldstein, C. Poole, J. Safko, “Classical Mechanics,” Addison Wesley (2002).

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