Draft for Internal Circulations: v1: Spring Semester, 2012, v2: Spring Semester, 2013 v3: Spring Semester, 2014, v4: Spring Semester, 2015 Lecture Notes on Quantum Field Theory
Yong Zhang1
School of Physics and Technology, Wuhan University, China
(Spring 2015)
Abstract These lectures notes are written for both third-year undergraduate students and first-year graduate students in the School of Physics and Technology, University Wuhan, China. They are mainly based on lecture notes of Sidney Coleman from Harvard and lecture notes of Michael Luke from Toronto and lecture notes of David Tong from Cambridge and the standard textbook of Peskin & Schroeder, so I do not claim any originality. These notes certainly have all kinds of typos or errors, so they will be updated from time to time. I do take the full responsibility for all kinds of typos or errors (besides errors in English writing), and please let me know of them.
The third version of these notes are typeset by a team including: Kun Zhang2, Graduate student (Id: 2013202020002), Participant in the Spring semester, 2012 and 2014 Yu-Jiang Bi3, Graduate student (Id: 2013202020004), Participant in the Spring semester, 2012 and 2014 The fourth version of these notes are typeset by Kun Zhang, Graduate student (Id: 2013202020002), Participant in the Spring semester, 2012 and 2014-2015
1yong [email protected] 2kun [email protected] [email protected]
1 Draft for Internal Circulations: v1: Spring Semester, 2012, v2: Spring Semester, 2013 v3: Spring Semester, 2014, v4: Spring Semester, 2015 Acknowledgements
I thank all participants in class including advanced undergraduate students, first-year graduate students and French students for their patience and persis- tence and for their various enlightening questions. I especially thank students who are willing to devote their precious time to the typewriting of these lecture notes in Latex. Main References to Lecture Notes
* [Luke] Michael Luke (Toronto): online lecture notes on QFT (Fall, 2012); the link to Luke’s homepage
* [Tong] David Tong (Cambridge): online lecture notes on QFT (October 2012); the link to Tong’s homepage
* [Coleman] Sidney Coleman (Harvard): online lecture notes; the link to Coleman’s lecture notes. the link to Coleman’s teaching videos;
* [PS] Michael E. Peskin and Daniel V. Schroeder: An Introduction to Quantum Field Theory, 1995 Westview Press;
* [MS] Franz Mandl and Graham Shaw: Quantum Field Theory (Second Edition), 2010 John Wiley & Sons, Ltd.
* [Zhou] Bang-Rong Zhou (Chinese Academy of Sciences): Quantum Field Theory (in Chinese), 2007 Higher Education Press; Main References to Homeworks
* Luke’s Problem Sets #1-6 or Tong’s Problem Sets #1-4. Research Projects
* See Yong Zhang’s English and Chinese homepages.
2 To our parents and our teachers!
To be the best researcher is to be the best person first of all: Respect and listen to our parents and our teachers always!
3 Quantum Field Theory (I) focuses on Feynman diagrams and basic concepts.
The aim of this course is to study the simulation of both quantum field theory and quantum gravity on quantum computer.
4 Contents
I Quantum Field Theory (I): Basic Concepts and Feynman Diagrams 11
1 Overview of Quantum Field Theory 12 1.1 Definition of quantum field theory ...... 12 1.2 Introduction to particle physics and Feynman diagrams ...... 13 1.2.1 Feynman diagrams in the pseudo-nucleon meson interaction . . . . . 13 1.2.2 Feynman diagrams in the Yukawa interaction ...... 18 1.2.3 Feynman diagrams in quantum electrodynamics ...... 20 1.3 The canonical quantization procedure ...... 22 1.4 Research projects in this course ...... 22
2 From Classical Mechanics to Quantum Mechanics 24 2.1 Classical particle mechanics ...... 24 2.1.1 Lagrangian formulation of classical particle mechanics ...... 24 2.1.2 Hamiltonian formulation of classical particle mechanics ...... 25 2.1.3 Noether’s theorem: symmetries and conservation laws ...... 25 2.1.4 Exactly solved problems ...... 27 2.1.5 Perturbation theory ...... 28 2.1.6 Scattering theory ...... 28 2.1.7 Why talk about classical particle mechanics in detail ...... 29 2.2 Advanced quantum mechanics ...... 29 2.2.1 The canonical quantization procedure ...... 29 2.2.2 Fundamental principles of quantum mechanics ...... 30 2.2.3 The Schr¨oedinger,Heisenberg and Dirac picture ...... 31 2.2.4 Non-relativistic quantum many-body mechanics ...... 33 2.2.5 Compatible observable ...... 34 2.2.6 The Heisenberg uncertainty principle ...... 34 2.2.7 Symmetries and conservation laws ...... 35 2.2.8 Exactly solved problems in quantum mechanics ...... 35 2.2.9 Simple harmonic oscillator ...... 36 2.2.10 Time-independent perturbation theory ...... 37 2.2.11 Time-dependent perturbation theory ...... 37 2.2.12 Angular momentum and spin ...... 39 2.2.13 Identical particles ...... 39 2.2.14 Scattering theory in quantum mechanics ...... 40
5 3 Classical Field Theory 41 3.1 Special relativity ...... 41 3.1.1 Lorentz transformation ...... 41 3.1.2 Four-vector calculation ...... 42 3.1.3 Causality ...... 43 3.1.4 Mass-energy relation ...... 44 3.1.5 Lorentz group ...... 45 3.1.6 Classification of particles ...... 46 3.2 Classical field theory ...... 46 3.2.1 Concepts of fields ...... 46 3.2.2 Electromagnetic fields ...... 47 3.2.3 Lagrangian, Hamiltonian and the action principle ...... 48 3.2.4 A simple string theory ...... 49 3.2.5 Real scalar field theory ...... 50 3.2.6 Complex scalar field theory ...... 51 3.2.7 Multi-component scalar field theory ...... 52 3.2.8 Electrodynamics ...... 53 3.2.9 General relativity ...... 54
4 Symmetries and Conservation Laws 56 4.1 Symmetries play a crucial role in field theories ...... 56 4.2 Noether’s theorem in field theory ...... 56 4.3 Space-time symmetries and conservation laws ...... 58 4.3.1 Space-time translation invariance and energy-momentum tensor . . . 58 4.3.2 Lorentz transformation invariance and angular-momentum tensor . . 60 4.4 Internal symmetries and conservation laws ...... 64 4.4.1 Definitions ...... 64 4.4.2 Noether’s theorem ...... 65 4.4.3 SO(2) invariant real scalar field theory ...... 65 4.4.4 U(1) invariant complex scalar field theory ...... 67 4.4.5 Non-Abelian internal symmetries ...... 68 4.5 Discrete symmetries ...... 69 4.5.1 Parity ...... 69 4.5.2 Time reversal ...... 70 4.5.3 Charge conjugation ...... 70
5 Constructing Quantum Scalar Field Theory 71 5.1 Quantum mechanics and special relativity ...... 71 5.1.1 Natural units ...... 71 5.1.2 Particle number unfixed at high energy (Special relativity) . . . . . 72 5.1.3 No position operator at short distance (Quantum mechanics) . . . . 73 5.1.4 Micro-Causality and algebra of local observables ...... 74 5.1.5 A naive relativistic single particle quantum mechanics ...... 75 5.2 Comparisons of quantum mechanics with quantum field theory ...... 76 5.3 Definition of Fock space ...... 76 5.4 Rotation invariant Fock space ...... 77 5.4.1 Fock space ...... 77 5.4.2 Occupation number representation (ONR) ...... 79 5.4.3 Hints from simple harmonic oscillator (SHO) ...... 80
6 5.4.4 The operator formalism of Fock space in a cubic box ...... 80 5.4.5 Drop the box normalization and take the continuum limit ...... 81 5.5 Lorentz invariant Fock space ...... 81 5.5.1 Lorentz group ...... 81 5.5.2 Definition of Lorentz invariant normalized states ...... 82 5.5.3 Lorentz invariant normalized state ...... 82 5.6 Constructing quantum scalar field theory ...... 84 5.6.1 Notation ...... 84 5.6.2 Constructing quantum scalar field ...... 85 5.7 Canonical quantization of classical field theory ...... 87 5.7.1 Classical field theory ...... 87 5.7.2 Canonical quantization ...... 89 5.7.3 Vacuum energy, Casimir force and cosmological constant problem . . 91 5.7.4 Normal ordered product ...... 93 5.8 Symmetries and conservation laws ...... 94 5.8.1 SO(2) invariant quantum real scalar field theory ...... 94 5.8.2 U(1) invariant quantum complex scalar field theory ...... 95 5.8.3 Discrete symmetries ...... 96 5.8.3.1 Parity ...... 96 5.8.3.2 Time reversal ...... 96 5.8.3.3 Charge conjugation ...... 97 5.8.3.4 The CPT theorem ...... 98 5.9 Micro-causality (Locality) ...... 98 5.9.1 Real scalar field theory ...... 98 5.9.2 Complex scalar field theory ...... 101 5.10 Remarks on canonical quantization ...... 101
6 Feynman Propagator, Wick’s Theorem and Dyson’s Formula 103 6.1 Retarded Green function ...... 103 6.1.1 Real scalar field theory with a classical source ...... 103 6.1.2 The retarded Green function ...... 104 6.1.3 Analytic formulation of D (x y)...... 105 R − 6.1.4 Canonical quantization of a real scalar field theory with a classical source ...... 107 6.2 Advanced Green function ...... 108 6.3 The Feynman Propagator D (x y)...... 109 F − 6.3.1 Definition of Feynman propagator with time-ordered product . . . . 109 6.3.2 Definition of Feynman propagator with contractions ...... 110 6.3.3 Definition with the Green function ...... 111 6.3.4 Definition with contour integrals ...... 111 6.3.5 Definition of Feynman propagator with iε 0+ prescription . . . . 113 → 6.4 Wick’s theorem ...... 113 6.4.1 The Time-ordered product ...... 113 6.4.2 Example for Wick’s theorem n = 3 and n =4...... 114 6.4.3 The Wick theorem ...... 115 6.4.4 Remarks on Wick’s theorem ...... 115 6.5 Dyson’s formula ...... 116 6.5.1 Interaction Picture (Dirac Picture) ...... 117 6.5.2 Unitary evolution operator ...... 118
7 6.5.3 Time-dependent perturbation theory ...... 118 6.5.4 Dyson’s formula ...... 119 6.5.5 Examples ...... 121
7 Feynman Diagrams and Feynman Rules to Compute Scattering Matrix, Cross Section and Decay Width 122 7.1 Scattering matrix (operator) ...... 122 7.1.1 Ideal model for scattering process ...... 122 7.1.2 Scattering operator S ...... 123 7.2 Calculation of two-“nucleon” scattering amplitude ...... 123 7.2.1 Model ...... 123 7.2.2 Two-“nucleon” scattering matrix ...... 124 7.2.3 Computing methods ...... 125 7.3 Feynman diagrams ...... 125 7.3.1 Main theorem ...... 125 7.3.2 Correspondence between algebra and diagrams ...... 125 7.3.3 Conventions of drawing external lines ...... 126 7.3.4 Conventions on diagrams vertices & internal lines ...... 128 7.3.5 Application ...... 128 7.4 Feynman rules in coordinate space-time (FRA)...... 129 7.4.1 Calculation hint ...... 129 7.4.2 Feynman rules in coordinate space-time (FRA)...... 131 7.5 Feynman rules in momentum space (FRB)...... 132 7.5.1 Calculation hint ...... 132 7.5.2 Feynman rules in momentum space ...... 132 7.6 Feynman rules C ...... 133 7.6.1 Calculation hint ...... 133 7.6.2 Feynman rules C (simplified FRB)...... 133 7.7 Application of FRC ...... 134 7.8 Remarks on Feynman propagator ...... 136 7.9 Cross sections & decay widths – measurable quantities in high energy physics137 7.9.1 QFT in a box with volume V = L3 ...... 137 7.9.2 Calculation of differential probability ...... 139 7.9.3 Cross section & decay width ...... 140 7.9.4 Calculation of Cross-sections and Decay-widths ...... 141
8 Return to Non-Relativistic Quantum Mechanics 144 8.1 The Schroedinger equation ...... 144
9 Dirac Fields: From Classical to Quantum 145 9.1 The Dirac equation and Dirac algebra ...... 145 9.1.1 The Dirac algebra ...... 145 9.1.2 Two widely used representations of the Dirac algebra ...... 146 9.1.3 The Lagrangian formulation ...... 147 9.1.4 Plane wave solutions of Dirac equation in Dirac representation . . . 147 9.2 Dirac equation and Clifford algebra ...... 149 9.2.1 The Clifford algebra ...... 149 9.2.2 Dirac equation with γµ matrices ...... 150 9.2.3 Plane wave solutions of Dirac equation with γµ matrices ...... 150
8 9.3 Lorentz transformation and parity of Dirac bispinor fields ...... 151 9.3.1 Classification of quantities out of Dirac bispinor fields under Lorentz transformation and parity ...... 151 9.3.2 Explicit formulation of D(Λ) ...... 153 9.4 Canonical quantization of free Dirac field theory ...... 155 9.4.1 Canonical quantization ...... 155 9.4.2 Fock space with the Fermi-Dirac statistics ...... 156 9.4.3 Symmetries and conservation laws: energy, momentum and charge . 157 9.5 Feynman propagator for a Dirac particle ...... 159 9.6 Feynman diagrams and Feynmans rules for fermion fields ...... 160 9.6.1 FDs & FRs for vertices ...... 161 9.6.2 Internal lines ...... 161 9.6.3 External lines ...... 161 9.6.4 Combinational factor (Symmetry factor) ...... 162 9.7 Special Feynman rules for fermions lines ...... 162 9.7.1 Mapping from matrix to number ...... 162 9.7.2 Feynman rules for a single fermion line from initial state to the final state ...... 162 9.7.3 Relative sign between FDs ...... 163 9.7.4 Minus sign from a loop of fermion lines ...... 163 9.8 Examples ...... 163 9.8.1 First order ...... 163 9.8.2 Second order ...... 164 9.8.3 Fourth order ...... 165 9.9 Spin-sums and cross section ...... 166 9.9.1 Spin-sums ...... 166 9.9.2 Examples ...... 168
10 Quantum Electrodynamics 169 10.1 How to quantize the Maxwell fields? ...... 169 10.2 Quantization of massive vector fields ...... 170 10.3 Plan: From massive electrodynamics to quantum electrodynamics ...... 174 10.4 Quantum Electrodynamics: local U(1) gauge invariance ...... 174 10.5 Remark on the covariant derivative ...... 175 10.6 Construction: From massive electrodynamics to quantum electrodynamics . 176 10.7 Quantum Electrodynamics: Feynman diagrams and Feynman rules . . . . . 180 10.8 Representative scattering processes in QED ...... 182 10.9 The calculation of the differential cross section of the Bhabha scattering in second order of coupling constant ...... 184 10.10The calculation of the differential cross section of the Compton scattering in second order of coupling constant ...... 188
11 Symmetries and Conservation Laws (II) 189 11.1 Translation group ...... 189 11.2 The Lorentz group and its representation theory ...... 189 11.2.1 Definition of the Lorentz group ...... 189 11.2.2 Representation theory of the Lorentz group ...... 189 11.2.3 Scalar, Weyl spinor, Dirac spinor and vector ...... 189 11.2.4 Parity ...... 189
9 11.2.5 The Lagrangian of the Weyl spinor ...... 189 11.2.6 The Lagragian of the Dirac spinor ...... 189 11.3 Transformation laws of scalar, Dirac spinor and vector fields under the Poincar´etransformation ...... 189 11.3.1 Scalar fields ...... 189 11.3.2 Dirac spinor fields ...... 189 11.3.3 Vector fields ...... 189
12 Research Projects 191 12.1 Basic concepts: fundamental problems ...... 191 12.2 Model: The Higgs physics ...... 191 12.3 Model: The Neutrino physics ...... 191 12.4 Mathematical Physics: Regularization and renormalization theories . . . . . 191 12.5 Theory: Symmetries, anomalies and breakings ...... 191 12.6 Model: The mechanism of quark confinement ...... 191 12.7 Mathematical Physics: The existence of the Yang-Mills theory ...... 191 12.8 Mathematical Physics: Integrable quantum field theories and the Yang– Baxter equation ...... 191 12.9 Interdisciplinary topics between quantum field theories and quantum infor- mation & computation ...... 192
II Renormalization and Symmetries 193
13 Notes on BPHZ Renormalization 194
III Path Integrals and Non-Abelian Gauge Field Theories 245
IV The Standard Model and Particle Physics 246
V Integrable Field Theories and Conformal Field Theories 247
VI Quantum Field Theories in Condense Matter Physics 248
VII General Relativity, Cosmology and Quantum Gravity 249
VIII Supersymmetries, Superstring and Supergravity 250
IX Quantum Field Theories on Quantum Computer 251
X Quantum Gravity on Quantum Computer 252
10 Part I
Quantum Field Theory (I): Basic Concepts and Feynman Diagrams
11 Lecture 1 Overview of Quantum Field Theory
1.1 Definition of quantum field theory
Def 1: QFT=Special Relativity + Quantum Mechanics. Note: QFT is a comprehensive subject for undergraduate student to revisit what they have learnt.
Quantum Mechanics
1 Special Relativity Preliminaries to QFT P Pq Electrodynamics @ @R Classical Mechanics
Special relativity PPq General relativity P 1 Pq Gravity Quantum gravity Quantum mechanics 1
Quantum field theory PPq Superstring theory Quantum gravity 1
Note: If quantum mechanics is changed, the entire modern physics has to be changed as well.
Def 2: QFT=Quantization of Classical Field Theory.
E.g. 1: Relativistic quantization of Electrodynamics: Quantum Electrodynamics.
E.g. 2: Non-Relativistic quantization of Electromagnetic Field: Quantum Optics.
Def 3: Quantum many-particle system with unfixed particle-number.
Particle # fixed Particle # un-fixed
⇓ ⇓ Non-relativistic QFT: Relativistic QFT: Condensed matter physics High energy physics
E.g. 1: In high energy physics, the particles can be created or annihilated, so that the particle number is unfixed, which is described by relativistic QFT.
12 E.g. 2: In condensed matter physics, particle number is a conserved quantity, which is described by non-relativistic QFT.
Def 4: Quantum field theory is the present language in which the laws of nature are written.
Note 1: Particle/wave duality equals particle/field duality. Particles are described by four dimensional vector (E, ~p), and waves are characterized by (ν, λ) or (ω, k). The particle wave duality allows the relations h E = hν = ~ω, ~p = = ~k, (1.1.1) | | λ where h = 2π~. Note 2: In QFT, each type of fundamental particle is a derived object of quantization of the associated field.
Photon EM field; ⇐⇒ electron Dirac field; ⇐⇒ God Particle(Higgs) Scalar field; ⇐⇒ Quark Quark field. ⇐⇒ Note 3: In QM, identical particles are indistinguishable, which is a kind of assump- tion. In QFT, particles of the same type are indistinguishable, because they are associated with the same quantum field.
1.2 Introduction to particle physics and Feynman diagrams
Classification of particles. mass, charge, spin, parity, lifetime. particle and antiparticle. fermion and boson. lepton and hadrons. hadrons: meson and baryon. baryons: nucleon and hyperons. particle creation and annihilation: quantum field theories. interactions: gravity; electromagnetic; weak; strong. standard model: unification of electromagnetic; weak; strong. string theory: unification of gravity; electromagnetic; weak; strong. high energy experiments. cross section and lifetime.
1.2.1 Feynman diagrams in the pseudo-nucleon meson interaction This course focuses on perturbative quantum field theory using Feynman diagrams. The key words of this course includes Feynman Diagrams, Feynman Rules and Feynman Inte- grals. To compute Feynman diagrams, one has to map a Feynman diagram to a Feynman integral with Feynman rules. The “Nucleon” (pseudo-nucleon) meson interaction as a toy model to understand Yukawa interaction and QED.
13 The model is characterized by the Lagrangian
= + , (1.2.1) L L0 Lint where is the free part included the free “nucleon” (ψ) and “meson” (φ), L0 L L
1 µ 2 (ψ) = ∂ ψ†∂ ψ m ψ†ψ; (1.2.2) L0 2 µ − 1 (φ) = ∂ φ∂µφ µ2φ2. (1.2.3) L0 2 µ −
And the interaction part has the formalism = gψ ψφ. The parameter g is named Lint − † as coupling constant. Note that we also have the Hamiltonian formalism = and H0 L0 = gψ ψφ. Hint †
Table 1.1: “Nucleon”-meson interaction “Nucleon” Anti-“Nucleon” Meson Pseudo-nucleon Anti-pseudo-nucleon “N” “N” φ something something π0 spin-0 spin-0 spin-0 mass “m” mass “m” mass “µ” charge +1 charge −1 charge 0
complex scalar fields ψ(x), ψ∗(x) real scalar φ∗(x) = φ(x)
E.g.1. Decay of a meson into “nucleon”-“anti-nucleon”.
φ(k) “N”(p) + “N”(q). (1.2.4) −→
p “N”
φ
k q “N” (1.2.5)
The decay amplitude is calculated as
S(φ “N” + “N”) = (2π)4δ4(k p q) i , (1.2.6) → − − × A where the delta function is required for energy-momentum conservation. In the way of computing the amplitude i , we can apply the time-dependent per- A turbation theory with Dyson’s series to derive the Feynman diagrams and the as- sociated Feynman rules. The amplitude i can be expanded as A i = i (1) + i (2) + + i (n) + , (1.2.7) A A A ··· A ···
14 where the label n stands for the number of constant ~ and unusually set as ~ = 1. In the first order i (1), it is related to the Feynman diagram (1.2.5). The Feynman A rules summarize as: the external lines represented for meson, “nucleon” and anti- “nucleon” contribute the factor 1. The interaction vertex gives rise to the factor ig. Therefore, the first order amplitude is − i (1) = 1 1 1 ( ig) = ig. (1.2.8) A × × × − −
E.g.2. “Nucleon”-“nucleon” scattering.
“N”(p ) + “N”(p ) “N”(p0 ) + “N”(p0 ). (1.2.9) 1 2 −→ 1 2
p1 p10
p2 p20
At the time t = , two-incoming “nucleons” are freely moving. After scattering, −∞ t = + , two particles are freely moving again. The grey region represents for the ∞ scattering interaction. The scattering matrix S has the formalism
4 S( p , p p0 , p0 ) = (2π) δ(p0 + p0 p p )i . (1.2.10) | 1 2i → | 1 2i 1 2 − 1 − 2 A In interaction picture, the initial and final state take the form
ψ(t = ) = p , p , ψ(t = + ) = p0 , p0 . (1.2.11) | −∞ i | 1 2i | ∞ i | 1 2i Then applying the time-dependent perturbation to derive the Dyson’s series
+ ∞ (n) UI = U (tf , ti), (1.2.12) n=0 X the scattering operation (matrix) can be expressed as
+ + ∞ ∞ S = U (+ , ) = S(n) = 1 + S(n), (1.2.13) I ∞ −∞ n=0 n=1 X X + ∞ (n) S( p , p p0 , p0 ) = p0 , p0 p , p + p0 , p0 S p , p . (1.2.14) | 1 2i → | 1 2i h 1 2| 1 2i h 1 2| | 1 2i n=1 X The expansion of the amplitude can be calculated i (0) = i (1) = 0 and A A 4 (2) (2) (2π) δ(p0 + p0 p p )i = p0 p0 S p p . (1.2.15) 1 2 − 1 − 2 A h 1 2| | 1 2i
With Wick’s theorem, Feynman diagrams and Feynman rules can be determined.
15 p1 p0 “N” 1 “N”
(2) p1 − p10 i = + (p10 p20 ) A ↔
“N” “N” p2 p20
The internal line stands for the Feynman propagator for meson, and contributes i the factor 2 2 . (p1 p ) µ − 10 − (2) i i = 1 1 ( ig) ( ig) 1 1 + (p0 p0 ) A × × − × (p p )2 µ2 × − × × 1 ↔ 2 1 − 10 − i i = ( ig)2 + . (1.2.16) − (p p )2 µ2 (p p )2 µ2 1 − 10 − 1 − 20 − E.g.3. Anti-“nucleon”-anti-“nucleon” scattering.
“N”(p) + “N”(q) “N”(p0) + “N”(q0). (1.2.17) −→
q q0 “N” “N”
q − q0 + (p q ) 0 ↔ 0
“N” “N” p p0
E.g.4. “Nucleon”-anti-“nucleon” scattering.
“N”(p) + “N”(q) “N”(p0) + “N”(q0). (1.2.18) −→
p p0 “N” ‘N” p p0 “N” “N” p + q p − p0 +
“N” “N” “N” “N” q q0 q q0
E.g.5. “Nucleon”-meson scattering.
“N”(p) + φ(q) “N”(p) + φ(q0). (1.2.19) −→
q q0 φ φ p + q + (q q0) ↔ “N” “N” p p0
16 E.g.6. Anti-“nucleon”-meson scattering.
“N”(p) + φ(q) “N”(p) + φ(q0). (1.2.20) −→
q q0 φ φ p + q + (q q0) ↔ “N” “N” p p0
E.g.7. Nucleon-anti-nucleon annihilation.
“N”(p) + “N”(q) φ(p0) + φ(q0). (1.2.21) −→
p p0 “N” φ
p − p0 + (p q ) 0 ↔ 0
“N” φ q q0
E.g.8. Meson-meson scattering.
φ(p) + φ(q) φ(p0) + φ(q0). (1.2.22) −→ p p0
k + p
k k + p p0 −
k q −
q q0
Remark: about two-particle scattering at order O(g2). 1). Virtual particle on internal line is meson. “N” + “N” “N” + “N”; (1.2.23) −→ “N” + “N” “N” + “N”; (1.2.24) −→ “N” + “N” “N” + “N”. (1.2.25) −→ 2). Virtual particle on internal line is nucleon. “N” + “N” φ + φ; (1.2.26) −→ “N” + φ “N” + φ; (1.2.27) −→ “N” + φ “N” + φ. (1.2.28) −→
17 1.2.2 Feynman diagrams in the Yukawa interaction “Nucleon-meson” interaction as the Yukawa interaction is the standard model of particle physics!
Table 1.2: The Yukawa interaction “Nucleon” Anti-“Nucleon” Meson “N” “N” φ proton anti-proton π0-meson 1 1 spin- 2 spin- 2 spin-0 mass “m” mass “m” mass “µ” m charge +1 charge −1 charge 0
Dirac field (spinor-field) ψ(x) real scalar φ∗(x) = φ(x)
E.g.1. The decay of a meson.
φ(p + q) N(p, r) + N(q, s). (1.2.29) −→ p, r N
φ
p + q q, s N
φ particle with four momentum p + q incoming. p + q
outgoing nucleon with four momentum p and spin r, r = 1 , 1 . p, r 2 − 2
outgoing anti-nucleon with four momentum q and spin s, s = 1 , 1 . q, s 2 − 2
The outline arrows represents for the momentum and the inline arrow characters the charge flow. E.g.2. Nucleon-nucleon scattering.
N(p, r) + N(q, s) N(p0, r0) + N(q0, s0). (1.2.30) −→ q, s q , s N 0 0 N
q − q0 + (q , s p , r ) 0 0 ↔ 0 0
N N p, r p0, r0
18 E.g.3. Anti-nucleon-anti-nucleon scattering.
N(p, r) + N(q, s) N(p0, r0) + N(q0, s0). (1.2.31) −→
q, s q0, s0 N N
q − q0 + (q , s p , r ) 0 0 ↔ 0 0
N N p, r p0, r0
E.g.4. Nucleon-anti-nucleon scattering.
N(p, r) + N(q, s) N(p0, r0) + N(q0, s0). (1.2.32) −→
p, r p0, r0 N N p, r p0, r0 N N
p + q p − p0 + (p, r ↔ q0, s0) N N q, s q0, s0 N N q, s q0, s0
E.g.5. Nucleon-meson scattering.
N(p, r) + φ(q) N(p0, r0) + φ(q0). (1.2.33) −→
q q0 φ φ p + q + (q q0) ↔ N N p, r p0, r0
E.g.6. Anti-nucleon-meson scattering.
N(p, r) + φ(q) N(p0, r0) + φ(q0). (1.2.34) −→
q q0 φ φ p + q + (q q0) ↔ N N p, r p0, r0
E.g.7. “Nucleon”-anti-“nucleon” annihilation.
N(p, r) + N(q, s) φ(p0) + φ(q0). (1.2.35) −→
19 p, r p0 N φ
p − p0 + (p q ) 0 ↔ 0
N φ q, s q0
E.g.8. Meson-meson scattering.
φ(p) + φ(q) φ(p0) + φ(q0). (1.2.36) −→
p p0
k + p
k k + p p0 −
k q −
q q0
1.2.3 Feynman diagrams in quantum electrodynamics
Table 1.3: Comparison between Yukawa interaction and QED
Matter Mediator (Interaction) ”Nucleon”-meson interaction ”Nucleon” (spinless, complex scalar) meson (spinless, real scalar) 1 Yukawa interaction Nucleon (spin- 2 , spinor ) meson (spinless, real scalar) 1 QED electron (spin- 2 , spinor) photon (spin-1, vector)
Example: typical Feynman diagrams in quantum electrodynamics.
a). The Moller scattering between two electrons.
e− + e− e− + e−. (1.2.37) −→
p, s p0, s0 e− e−
µ p − p 0 (p0, s0 q0, r0) ν − ↔
e− e− q, r q0, r0
20 The on-line arrow represents an electron and the out-line for momentum labeled by p, q, p0 and q0, which have the form (Ep, ~p). Parameters r, s, r0 and s0 represents for the spin degree of the electron. For example, r = 1 stands for spin up and r = 1 for 2 − 2 spin down. The wave line stands for a virtual photon in the scattering process. Such the diagram means that two-electron scattering is performed by an interchange of a virtual photon. Note that electron is a quanta of Dirac field and photon is a quanta of electrodynamics field. Such two Feynman diagrams are indistinguishable in physics because two outgoing electrons (p0, s0) and (q0, r0) are identical particles and we have the minus sign due to Fermi-Dirac statistics. Similarly, the scattering process between two positrons:
e+ + e+ e+ + e+. (1.2.38) −→
p, s p0, s0 e+ e+
µ p − p 0 (p0, s0 q0, r0) ν − ↔
e+ e+ q, r q0, r0 b). The Bhabha scattering between an electron and a positron.
+ + e− + e e− + e . (1.2.39) −→
p, s p0, s0 e− e− p, s p0, s0 e− e− p − p0 p + q (q, r ↔− p0, s0)
+ + + + e e e e q, r q0, r0 q, r q0, r0 c). The pair annihilation between an electron and a positron into two photons.
+ e− + e 2γ. (1.2.40) −→
ν p, s p0, ε1 e− γ ν p − p 0 + (p0, ε1 q0, ε2) µ ↔
e+ γ q, r µ q0, ε2 d). The Compton scattering between an electron and a photon.
e− + γ e− + γ. (1.2.41) −→
21 ν µ q, ε q0, ε γ 1 2 γ p + q ν µ + (q, ε q , ε ) 1 ↔ 0 2
e− e− p, s p0, s0
e). Two-Photon scattering.
γ + γ γ + γ. (1.2.42) −→
−→
1.3 The canonical quantization procedure
Canonical quantization is a procedure of deriving QM (quantum mechanics) (or QFT (quantum field theory)) from the Hamiltonian formulation of CPM (Classical particle mechanics)(or CFT (classical field theory)).
Hˆ Hˆ Quantization QM QFT
H H CPM CFT
−→ Continuum limit
Note 1. In classical mechanics, we both have Lagrangian formalism L(q, q˙) and Hamilto- nian formalism H(q, p) to characterize the equation of motion.
Note 2. Non-relativistic quantum mechanics prefers the Hamiltonian formulation, for ex- ample the Schr¨odingerequation is described by Hamiltonian.
Note 3. In relativistic QFT, the Hamiltonian formalism is associated with the canonical quantization and the Lagrangian formalism is associated with the path integral formalism.
Note 4. The two formalisms are essentially equivalent, but they are used in different cir- cumstances.
1.4 Research projects in this course
1. Conventional projects in quantum field theories Traditional research in (high energy physics) HEP:
22 Particle physics: the study of phenomenology in experiments; • Doing experiments (in CERN); • QFT: theoretical study in high energy physics. • 2. A millennium problem in quantum field theory An Open Problem in QFT : One of Seven Millennium Problems Wikipedia : Prove that Yang-Mills theory (Non- Abelian Gauge Field Theory) actually exists and has a unique ground state.
Note 1: The Yang-Mills theory was proposed by C.N. Yang and Mills in 1954, when Yang was 32 years old.
Note 2: The Yang-Mills theory is Non-abelian gauge field theory, which is the theoret- ical formulation of the Standard Model.
3. New projects in quantum field theory Nowadays quantum mechanics has been updated with quantum computation and in- formation.
Note 1: Quantum information and computation (QIC) represents a further develop- ment of quantum mechanics.
Note 2: QIC focuses on fundamental principles and logic of quantum mechanics.
Note 3: QIC is a new type of advanced quantum mechanics.
Note 4: QIC is the study of information processing tasks that can be accomplished using quantum mechanical systems (or using fundamentals of quantum mechan- ics).
Question: ?=Special Relativity+ Quantum Information and Computation.
23 Lecture 2 From Classical Mechanics to Quantum Mechanics
2.1 Classical particle mechanics
2.1.1 Lagrangian formulation of classical particle mechanics
In a system of n-particles, the state is characterized by the generalized coordinates q , q , , q 1 2 ··· n and their time derivationsq ˙ , q˙ , , q˙ . The Lagrangian is defined as 1 2 ··· n L(q q , q˙ q˙ ) = T V. (2.1.1) 1 ··· n 1 ··· n − And the action is defined as t2 S = dtL. (2.1.2) Zt1 Note: L does not have explicit time dependence,
∂L = 0, (2.1.3) ∂t which is associated with the conservation of energy. Action principle which has another names including Hamilton’s principle, the principle of least action and variational principle: for arbitrary variation q q (t) + δq (t) with a → a a fixed boundaries δqa(t1) = δqa(t2) = 0, we have stationary action, namely δS = 0.
t2 ∂L ∂L δS = dt δq + δq˙ (2.1.4) ∂q a ∂q˙ a Zt1 a a with ∂L d ∂L d ∂L δq˙ = δq δq , (2.1.5) ∂q˙ a dt ∂q˙ a − dt ∂q˙ a a a a we have t2 ∂L d ∂L ∂L t2 δS = dt δqa + δqa . (2.1.6) ∂qa − dt ∂q˙a ∂q˙a Zt1 t1
Due fixed boundary condition and action principle, we can derive ∂L d ∂L = ( ), (2.1.7) ∂qa dt ∂q˙a which is the equation of motion, also named as Euler-Lagrangian equation.
24 2.1.2 Hamiltonian formulation of classical particle mechanics Canonical momentum: ∂L pa = , a = 1, 2, , n. (2.1.8) ∂q˙a ··· Hamiltonian:
H(q q , p p ) = p q˙ L(q q , q˙ q˙ ). (2.1.9) 1 ··· n 1 ··· n a a − 1 ··· n 1 ··· n a X Note: Hamiltonian is not a function ofq ˙a, and it is a function of qa and pa. Proof.
∂L ∂L dH = p dq˙ +q ˙ dp dq dq˙ a a a a − ∂q a − ∂q˙ a a a a X ∂L ∂L = q˙ dp dq + p dq˙ dq˙ a a − ∂q a a a − ∂q˙ a a a a a X X ∂L = q˙ dp dq . (2.1.10) a a − ∂q a a a X
Hamilton’s equations: ∂H ∂H q˙a = , p˙a = . (2.1.11) ∂pa −∂qa Optional problem: Derive Hamilton’s equations. Optional problem: Derive Newtonian mechanics in both Lagrangian formulation and Hamiltonian formulation.
2.1.3 Noether’s theorem: symmetries and conservation laws We have two different variations:
δqa: variation for deriving EoM; • Dqa: variation for symmetries without specifying EoM. • Def 2.1.1. A symmetry is a transformation to keep physics (EoM) unchanged. A trans- formation Dqa is called symmetry iff dF DL = (2.1.12) dt for some F (qa, q˙a, t) with arbitrary qa(t) which may not satisfy the EoM. Remark: t2 DS = dtDL = F (t ) F (t ), (2.1.13) 2 − 1 Zt1 S0 = S + DS, (2.1.14)
δS0 = δS + δ(DS) = δS + δF (t ) δF (t ), (2.1.15) 2 − 1 because the boundary terms at t1 and t2 are fixed, therefore the EoM is unchanged.
25 Thm 2.1.3.1 (Noether’s Theorem (Particle mechanics)). For every continuous symmetry, there is a conserved quantity.
Proof.
1). On the one hand, without EoM, we have DL = dF/dt for qa,q ˙a. ∀ 2). On the another hand, with EoM and canonical momentum,
∂L ∂L DL = ( Dqa + Dq˙a) ∂qa ∂q˙a a X a a = (p ˙aDq + paDq˙ ) (2.1.16) a X d = ( p Dqa), dt a a X we can derive the conserved quantity denoted as Q
Q = p Dqa F, (2.1.17) a − a X d dQ ( p Dqa F ) = = 0. (2.1.18) dt a − dt a X
E.g.1. Space translation invariance // momentum conservation.
1 1 L = m q˙2 + m q˙2 V (q q ). (2.1.19) 2 1 1 2 2 2 − 1 − 2 The space translation with infinitesimal constant α is defined as
q1 q1 + α, Dq1 = α; → (2.1.20) q q + α, Dq = α. 2 → 2 2 And the Lagrangian is unchanged under the space translation, namely
1 2 1 2 L0 = mq˙ + mq˙ V (q q ) = L, (2.1.21) 2 1 2 2 − 1 − 2 thus DL = 0 F = 0. (2.1.22) ⇒ Therefore, the conserved quantity can be defined as
1 2 Q = p1Dq + p2Dq = α(p1 + p2), (2.1.23)
d dP (p Dq + p Dq ) = = 0, (2.1.24) dt 1 1 2 2 dt
where P is the total momentum which is conserved, i.e., P = p1 + p2.
26 E.g.2. Time translation invariance// energy conservation. The time translation with infinitesimal constant α is defined in the following way
t t + α, (2.1.25) → qa(t) qa(t + α), (2.1.26) → L(t) L(t + α), (2.1.27) → dqa Dqa = qa(t + α) qa = α + O(α2), (2.1.28) − dt thus dL DL = L(t + α) L(t) = α + O(α2) F = αL. (2.1.29) − dt ⇒ Therefore, the conserved quantity Q is given by
Q = p Dq + p Dq αL 1 1 2 2 − = α(p q˙ + p q˙ L) (2.1.30) 1 1 2 2 − = αH,
dQ dH = α = 0 = H = const, (2.1.31) dt dt ⇒ namely the Hamiltonian (energy) is conserved.
2.1.4 Exactly solved problems . The Kepler problem with the inverse-square law of force. 4 1 L = m(r ˙2 + r2θ˙2) V (r), (2.1.32) 2 − where V (r) is the potential energy denoted as α V (r) = (2.1.33) − r and α is a constant.
. The simple Harmonic oscillator. 4 1 1 L = mq˙2 kq2, (2.1.34) 2 − 2 from which the EoM can be derived as
mq¨ + kq = 0 (2.1.35)
or q¨ + w2q = 0, (2.1.36)
k where w = m . q 27 2.1.5 Perturbation theory In perturbation theory, the Hamiltonian can be written as
H(qa, pa, t) = H0(qa, pa, t) + Hint0 (qa, pa, t), (2.1.37) where H0 is the Hamiltonian that the EoM can be exactly solved and Hint0 is the small perturbation term. Therefore, the total Hamiltonian H can be solved in perturbation theory.
2.1.6 Scattering theory The procedures of describing scattering phenomena are the same whether the mechanics is classical or quantum. —Goldstein
The quantity jin, incident density (flux density), is defined as the incident particle number per unit time and per unit area, where the unit area is normal to the incident direction. For jout, it is the number of scattering particles per unit time. The total cross section σ describes the scattering process in the way
jinσ = jout, (2.1.38) and note that the total cross section has the dimension of area. dσ The differential cross section dΩ is defined as dσ dj j = out , (2.1.39) in dΩ dΩ
djout hence dΩ specifies the number of particles scattering into per solid angle where dΩ = sin θdθdϕ = 2π sin θdθ. (2.1.40)
The angle θ is the degree between the incident particles and the scattered particles, named as scattering angle. Usually in scattering process, the central force is symmetrical around the incidental axis, therefore the angle ϕ in solid angle can be integrated out as 2π.
E.g. in classical particle mechanics, in the Rutherford scattering experiment, due to the conservation of particle number
dσ dj = j 2πbdb = j 2π sin θdθ , (2.1.41) out in in dΩ | | where quantity b is defined as the perpendicular distance between the incidental particle and the center of force, also called as impact parameter. Absolute value is required for the positivity of particle number. In repulsive interaction: dσ bdb = ; (2.1.42) dΩ sin θdθ In attractive interaction: dσ bdb = . (2.1.43) dΩ −sin θdθ
28 2.1.7 Why talk about classical particle mechanics in detail Topics in the course of QFT:
1). Canonical quantization of QFT: Hamiltonian formulation.
2). Symmetries and conservation laws: Lagrangian formulation.
3). Perturbative QFT: Feynman diagrams.
4). Calculation of cross section in High energy scattering experiments.
2.2 Advanced quantum mechanics
Quantum mechanics is defined as non-relativistic quantum particle mechanics. Advanced quantum mechanics present contents between quantum mechanics for undergraduate stu- dents and quantum field theory for graduate students.
2.2.1 The canonical quantization procedure Heisenberg’s quantum mechanics is derived from the canonical quantization procedure of the Hamiltonian formulation of classical particle mechanics. The Possion brackets is defined as
∂f ∂g ∂f ∂g f, g , (2.2.1) { } ≡ ∂p ∂q − ∂q ∂p a a a a a X with which the EoM can be rewritten as ∂H q˙a(t) = H, qa = ; (2.2.2) { } ∂pa ∂H p˙a(t) = H, pa = . (2.2.3) { } −∂qa
Note that dA ∂A = + H,A , (2.2.4) dt ∂t { } where A is the observable other than H.
. Canonical quantization procedure. 4 Step 1: (q (t), p (t)) (ˆq (t), pˆ (t)). (2.2.5) a a −→ a a Replace classical variables (qa(t), pa(t)) with operator-valued function of time sat- isfying the commutation relations
[ˆqa(t), qˆb(t)] = 0 = [ˆpa(t), pˆa(t)], (2.2.6)
[ˆqa(t), pˆb(t)] = iδab = i~δab. (2.2.7)
29 Step 2: H(q, p) Hˆ (ˆq, pˆ). (2.2.8) −→ Quantum Hamiltonian Hˆ has the same form of classical Hamiltonian H except that it is a function ofq ˆa andp ˆb. Note: The canonical quantization suffers from the ordering ambiguity, for instance pˆ2qˆ =p ˆqˆpˆ =q ˆpˆ2, but they are the same in the classical physics limit, namely 6 6 ~ 0, such as → pˆ2qˆ = pˆqˆpˆ = pˆ 2 qˆ = p2q. (2.2.9) h i h i h i h i Step 3: Heisenberg’s equation of motion:
dqˆa(t) ∂H = i[H,ˆ qˆa(t)] = , (2.2.10) dt ∂pˆa
dpˆa(t) ∂H = i[H,ˆ pˆa(t)] = . (2.2.11) dt −∂qˆa
For the time dependent observable Aˆ(t):
dAˆ(t) ∂Aˆ = + i[H,ˆ Aˆ]. (2.2.12) dt ∂t
Step 4: Prove the Hamiltonian Hˆ is bounded from below, namely, that ground state exists and the Hilbert space can be constructed.
Note 1: Heisenberg’s matrix (operator) quantum mechanics was originally derived via canonical quantization procedure.
Note 2: Canonical quantization procedure makes the relationship between CPM and QM clear.
2.2.2 Fundamental principles of quantum mechanics . 4 ψ : State. ¨¨* | i Static part - Aˆ: Observable. HHj : Hilbert space. H . 4
Unitary evolution: ¨* ¨ ∂ Dynamic part ψ(t) = U(t) ψ(0) , i~ ∂t ψ(t) = H ψ(t) . HHj | i | i | i | i Non-unitary evolution: Quantum measurement, wave function collapse, ψ ψ . | i → | 0i The standard quantum mechanics: System: Hilbert space . • H Note: A quantum binary digit is a two-dimensional Hilbert space.
30 State: A closed system is described by a pure vector ψ obeying the linear • | i ∈ H superposition principle, namely
a ψ1 + b ψ2 , a, b C. (2.2.13) | i | i ∈ H ∈ Note: Quantum computer is powerful mainly due to the superposition principle which allows a kind of parallel computation on a single quantum computer.
Observable Oˆ. Hermitian operator: Oˆ = Oˆ. • † Unitary evolution of a state vector in a closed system: •
ψ(t) = U(t) ψ(0) ,U †(t)U(t) = Id. (2.2.14) | i | i E.g. A unitary evolution of a closed system is governed by the Schr¨odingerequation ∂ i~ ψ(t) = H ψ(t) (2.2.15) ∂t| i | i
with H† = H. Note: A quantum gate is a unitary transformation acting on quantum binary digits.
Non-unitary evolution of state vector, also named as quantum measurement, quan- • tum jump, quantum transition, wave function collapse and information loss. Note: Quantum measurement is not well defined in the viewpoint of an expert against QM, but is the main computing resource in quantum information & computation.
The composite system of A and B, i.e., and , is described by the tensor • HA HB product , namely ψ ψ . HA ⊗ HB | iA ⊗ | iB ∈ HA ⊗ HB Note: Quantum entanglement arises in the quantum composite system, which dis- tinguishes classical physics from quantum physics and it plays the crucial role in quantum information & computation.
2.2.3 The Schr¨oedinger,Heisenberg and Dirac picture In quantum mechanics, the probability or matrix element or probability amplitude is a realistic observed quantity, but quantum mechanics is directly described by the state vectors and observables. So that a picture defines a choice of state vectors and observables to preserve the probability amplitude.
. 4
: ψS(0) ψS(t) . QM: Shr¨odingerpicture XXz | i → | i OˆS(t) = OˆS(0).
. 4
: ψ (0) = ψ (t) . QFT: Heisenberg picture | H i | H i XXz Oˆ (0) Oˆ (t). H → H
31 Transition amplitude is unchanged in any picture,
ψ (t) Oˆ ψ (t) = ψ (t) Oˆ ψ (t) . (2.2.16) h S | S| S i h H | H | H i The Schr¨odinger picture • OˆS(t) = OˆS(t0); (2.2.17) ψ (t) = U(t, t ) ψ (t ) . (2.2.18) | S i 0 | S 0 i The Heisenberg picture • ψ (t) = ψ (t ) ; (2.2.19) | H i | H 0 i OˆH (t) = U †(t, t0)OˆH (t0)U(t, t0). (2.2.20)
Table 2.1: The Shr¨odinger,Heisenberg and Dirac picture
Schr¨odinger Heisenberg Dirac (Interaction)
time-dependent
time-dependent time-independent H = H + H0 state vector 0 int ∂ ∂ i~ ∂t |ψ(t)iS = H|ψ(t)iS |ψ(t)iH = |ψ(0)iH i~ ∂t |ψ(t)iI = HI|ψ(t)iI
iH0t iH0t HI(t) = e Hint0 e−
time-independent time-dependent time-dependent observable d d OS(t) = OS(0) i dt OH = [OH,H] i dt OI = [OI,H0] O (0) = O (0) = O (0) initial value S H I
|ψS(0)iS = |ψS(0)iH = |ψS(0)iI
probability amplitude Shψ(t)|OS|ψ(t)iS = Hhψ(t)|OH|ψ(t)iH = Ihψ(t)|OI|ψ(t)iI
Note 1: Non-relativistic quantum mechanics prefers the Schr¨odingerpicture.
Note 2: Heisenberg’s picture is good for relativistic QFT.
Note 3: Interaction picture is good for time-dependent perturbation theory. For example, perturbative QFT which is a collection of Feynman diagrams.
Note 4: In the case of time-independent Hamiltonian, we have the following equations in different pictures.
iH(t t0) ψ(t) S = e− − ψ(t0) S; (2.2.21) | i iHt | iHti OH(t) = e OH(0)e− ; (2.2.22) iH0t iH0t OI(t) = e OI(0)e− ; (2.2.23) iH0t ψ(t) = e− ψ(t) , (2.2.24) | iS | iI where the free Hamiltonian H0 can be exactly solved.
Prove d i ψ(t) = H (t) ψ(t) . (2.2.25) dt| iI I | iI
32 Proof. d d i ψ(t) = i eiH0t ψ(t) dt| iI dt | iS d = H eiH0t ψ(t) + eiH0ti ψ(t) − 0 | iS dt| iS iH0t = H ψ(t) + e (H + H0 ) ψ(t) − 0| iI 0 int | iS iH0t iH0t = H ψ(t) + e H ψ(t) + e H0 ψ(t) − 0| iI 0| iS int| iS iH0t iH0t = e H0 e− ψ(t) int | iI = H (t) ψ(t) . (2.2.26) I | iI
In the interaction picture, the equation of time evolution d i ψ(t) = H (t) ψ(t) (2.2.27) dt| iI I | iI can be solved in the Dyson’s series. Introduce the unitary operator UI(t, t0) satisfying
UI†UI = Id and UI(t0, t0) = 1. And the equation of time evolution can be rewritten as d i~ UI(t, t0) = HI(t)UI(t, t0). (2.2.28) dt Integrate out above equation, we can obtain i t U (t, t ) = U (t , t ) + ( ) dt H (t )U (t , t ), (2.2.29) I 0 I 0 0 − 1 I 1 I 1 0 ~ Zt0 and then we can perform the iteration method to obtain the relation
+ t t1 tn 1 ∞ i − U (t, t ) = 1 + ( )n dt dt dt H (t )H (t ) H (t ). (2.2.30) I 0 − 1 2 ··· n I 1 I 2 ··· I n n=1 ~ t0 t0 t0 X Z Z Z Note: In QFT, the Dyson’s series have another compact formulation which is used to derive Feynman diagrams.
2.2.4 Non-relativistic quantum many-body mechanics An N-particle system with Particle number = N, same mass m; • External potential: U(x); • Inter-particle potential: V (~x ~x ), 1 < j < k < N. • i − j The equation of motion has the form ∂ i~ ψ(t, ~x1, , ~xn) = Hψ(t, ~x1, , ~xn), (2.2.31) ∂t ··· ··· with the Hamiltonian H given by
N N j 1 ~2 − H = ( 2 + U(x )) + V (~x ~x ). (2.2.32) −2m∇j j j − k Xj=1 Xj=1 Xk=1
33 Note 1: Particle number is fixed, namely N is a conserved quantity, [H,N] = 0. Note 2: Position operators Xˆ , Xˆ , , Xˆ are well defined; 1 2 ··· n Note 3: Momentum operators Pˆ , Pˆ , , Pˆ are well defined and other observables are 1 2 ··· n independent coordinates ~x , ~x , , ~x . 1 2 ··· n However, quantum field theory tells another story, 1. Particle number is not-fixed; 2. Position operators Xˆ , Xˆ , , Xˆ do not exist; 1 2 ··· n 3. Observable Oˆ(t, ~x) depends on space-time.
Table 2.2: Comparison between Quantum Mechanics and Quantum Field Theory
Quantum Mechanics Quantum Field Theory Particle Number Fixed Un-fixed time parameter label/parameter label/parameter
Position operator Xˆi well-defined No Xˆi because particles can be destroyed
Position parameter ~xi well-defined ~xi well-defined as labels or parameters
Momentum operator Pˆi well-defined Pˆi well-defined Observable independent of coordinate ˆ,O(t) dependent on (t, ~x), Oˆ(t, ~x) Special reality No Yes Lorentz transformation No Yes
ψ(t, ~x1, ~x2, ··· , ~xn) wave function (first quantization) quantum field operator (second quantization)
2.2.5 Compatible observable Observables A and B are called compatible when [A, B] = AB BA = 0, (2.2.33) − and incompatible when [A, B] = 0. 6 Note: The concept of compatible observable is associated with micro-causality in QFT.
2.2.6 The Heisenberg uncertainty principle The Heisenberg’s uncertainty relation 1 (∆A)2 (∆B)2 [A, B] 2 , (2.2.34) h ih i ≥ 4 |h i| where ∆A = A A , (2.2.35) − h i (∆A)2 = A2 A 2. (2.2.36) h i h i − h i E.g. ∆x∆p ~ . When ∆x 0, ∆p + , which implies p + . Small distance x ≥ 4 → x → ∞ x → ∞ means high energy physics in which particles can be created and annihilated, so that the position of a particle becomes as non-sense.
Note: the uncertainty relation ∆x∆p ~ is meaningful in QM, but questionable in QFT. x ≥ 4
34 2.2.7 Symmetries and conservation laws Symmetry in quantum mechanics is a transformation D, iff [D,H] = 0. The conserved dQ charge is denoted as Q, which is a constant of the motion, namely dt = 0, then [Q, H] = 0.
E.g. D is a unitary transformation, and we can construct the transformation as
i D = 1 εQ + O(ε2). (2.2.37) − ~ The unitary constraint of the transformation D requires the generator Q as a Hermitian operator, namely Q† = Q.
Note: In the Schr¨odingerpicture, the equation of time evolution has the form
∂ i~ ψ(t) = H ψ(t) . (2.2.38) ∂t| i | i Because the symmetry transformation D preserves the EoM, we have
∂ i~ (D ψ(t) ) = H(D ψ(t) ). (2.2.39) ∂t | i | i
1 If D− exists and D is irreverent of time, then
∂ 1 i~ ψ(t) = D− HD ψ(t) = H ψ(t) , (2.2.40) ∂t| i | i | i which implies [D,H] = 0.
Note: In QM, symmetries and conservation laws can be helpful in some sense, for example, the degeneracies of quantum states due to symmetry, but in modern QFT, symmetries and conservation laws play the essential (key) roles which guide us to specify the formulation of the action.
2.2.8 Exactly solved problems in quantum mechanics Examples of exactly solved problems in quantum mechanics:
e.g.1: Free particles.
e.g.2: Many problems in one dimension.
e.g.3: Transmission-reflection problem.
e.g.4: Harmonic oscillator.
e.g.5: The central force problem.
e.g.6: The Hydrogen atom problem.
Note: In QFT, most problems can be solved in a perturbative approach using Feynman diagram.
35 2.2.9 Simple harmonic oscillator The Hamiltonian of the simple harmonic oscillator takes the form ˆ P~ 2 1 Hˆ = + mω2Xˆ 2, ω = 0. (2.2.41) 2m 2 6 where the position operator Xˆi and momentum operator Pˆj obey the commutative relation ˆ ˆ [Xi, Pj] = i~δij. (2.2.42) Introduce the Canonical transformation: P p = , q = √mωX, (2.2.43) √mω and rewrite the Hamiltonian as ω H = (p2 + q2), (2.2.44) 2 where the lower indices of X, P, q, p are neglected for simplicity or one can understand the harmonic oscillator as a one-dimensional harmonic oscillator. With the new defined raising and lowering operators denoted as q + ip q ip a = , a† = − , (2.2.45) √2 √2 with [a, a†] = 1, (2.2.46) the Hamiltonian can be rewritten as 1 H = ω(a†a + ). (2.2.47) 2 The number operator is defined as N = a†a, (2.2.48) thus 1 H = ω(N + ). (2.2.49) 2 Due to ψ a a ψ 0, define the ground state 0 as a 0 = 0, and the first excited h | † | i > | i | i state as 1 = a† 0 . | i | i 1 1 H 0 = ω(N + ) 0 = ω 0 = E 1 , (2.2.50) | i 2 | i 2 | i 0| i 1 3 H 1 = ω(N + ) 1 = ω 1 = E 1 . (2.2.51) | i 2 | i 2 | i 1| i The gap of eigenvalue is E E = ω = 0. (2.2.52) 1 − 0 6 The eigenstate n is denoted as | i 1 n n = (a†) 0 (2.2.53) | i √n! | i 1 with the eigenvalue En = (n + )~ω, namely H n = En n . 2 | i | i Thm 2.2.9.1. A simple harmonic oscillator has a unique ground state 0 and there is a | i gap between 0 and its first excited state. | i Note: Free quantum field theory (QFT without interaction) can be regarded as a collection of an infinite number of simple harmonic oscillator.
36 2.2.10 Time-independent perturbation theory The full Hamiltonian can be expanded as two parts
H = H0 + λHint0 , (2.2.54) where λ is a small real parameter in the range 0 < λ < 1 and Hint0 is timeless. For the (0) (0) free Hamiltonian H , it has the eigenstate n with the eigenvalue En , namely 0 | i H n(0) = E(0) n(0) . (2.2.55) 0| i n | i And for the full Hamiltonian H, we define
H n = E n . (2.2.56) | i n| i We can expand the eigenstate and eigenvalue of the full Hamiltonian in terms of the power of parameter λ
n = n(0) + λ n(1) + λ2 n(2) + ; (2.2.57) | i | i | i | i ··· E = E(0) + λ∆(1) + λ2∆(2) + . (2.2.58) n n n n ··· where the energy shift ∆n is defined as
∆ E E(0). (2.2.59) n ≡ n − n In the perturbative theory, the result formula of the eigenstate and eigenvalue of full Hamiltonian show as k0 H n(0) n = n(0) + λ k0 h | int0 | i + ; (2.2.60) | i | i | i (0) (0) ··· k=n En Ek X6 − (0) (0) 2 (0) (0) (0) 2 n Hint0 k E = E + λ n H0 n + λ h | | i + . (2.2.61) n n h | int| i (0) (0) ··· k=n En Ek X6 − (0) (0) Note: When En E is very small, the high energy physics arises naturally where QFT − k has to be considered.
2.2.11 Time-dependent perturbation theory The full Hamiltonian can be split as
H = H0 + Hint0 (t), (2.2.62) where Hint0 (t) is time-dependent.
In the interaction (Dirac) picture, the observable obeys the equation
dOI (t) 1 = [OI ,H0], (2.2.63) dt i~ and the state follows the Shr¨odinger-like equation d i~ ψI (t) = H0 (t) ψI (t) (2.2.64) dt| i I | i
37 iH0t iH0t with HI0 (t) = e Hint0 e− .
The state ψI (t) can be expanded in terms of the basis of free Hamiltonian, namely n | (0)i | i with H n = En n , 0| i | i ψ (t) = C (t) n , (2.2.65) | I i n | i n X where C (t) = n ψ (t) . n h | I i Note: (0) iEn t iH0t − ψ (t) = e− ψ (t) = C (t)e ~ n ; (2.2.66) | S i | I i n | i n X d iωnmt i~ Cn(t) = n H0 m e Cm(t) (2.2.67) dt h | int| i m X with (0) (0) En Em ωnm = − = ωmn. (2.2.68) ~ −
d Note: The equation of time evolution of state vector in interaction picture, i.e., i~ ψI (t) = dt | i H (t) ψ (t) , can be solved in Dyson’s series in time-dependent perturbative theory which I0 | I i is used to derive Feynman diagrams in time-dependent perturbative QFT.
Denote the initial state ψ (t ) as i and introduce the time evolution operator U (t, t ), | I 0 i | i I 0 ψ (t) = U (t, t ) i = n n U (t, t ) i , (2.2.69) | I i I 0 | i | ih | I 0 | i n X where i t i t t1 U (t, t ) = 1 H (t )dt + ( )2 dt dt H (t )H (t ) + . (2.2.70) I 0 − I 1 1 − 1 2 I 1 I 2 ··· ~ Zt0 ~ Zt0 Zt0 C (t) as the matrix element of time evolution operator, namely C (t) = n U (t, t ) i , n n h | I 0 | i can take the perturbation expansion:
C (t) = C(0) + C(1) + C(2) + , (2.2.71) n n n n ··· where
(0) Cn = δni; (2.2.72) i t C(1) = n H (t ) i dt n − h | I 1 | i 1 ~ Zt0 t i iω t1 = e ni n H0 i dt ; (2.2.73) − h | int| i 1 ~ Zt0 C(2) = . (2.2.74) n ··· The transition probability can be calculated as
2 P ( i n ) = C(1) + C(2) + (2.2.75) | i → | i n n ··· for n = i. 6
38 2.2.12 Angular momentum and spin
The angular momentum operator denoted as J~ = (J1,J2,J3) satisfies the commutative relation [Ji,Jj] = i~εijkJk, (2.2.76) which is the Lie algebra of SU(2).
The angular momentum composes the orbital angular momentum L~ = ~x ~p and • ~ ~ ~ ~ ~ × spin angular momentum S = 2~σ, namely J = L + S. The spinor representation of SU(2) group defines the spin angular momentum in • QM, such as j, m with m = j, j + 1, , j. | i − − ··· * Spin- 1 system, j = 1 (electron, position, proton, neutron): 1 , 1 stands for 2 2 | 2 2 i spin-up state; 1 , 1 stands for spin-down state. | 2 − 2 i * Spin-0 system: j = 0, 00 . | i * Spin-1 system: j = 1, 1, 1 , 1, 0 , 1, 1 . | − i | i | i Note: In QFT, the spin of a particle is defined as a spinor representation of the Lorentz group SO(1, 3) = SU(2) SU(2) arising from the special relativity. ∼ ⊗ 2.2.13 Identical particles Principle: Identical particles can not be distinguished in QM.
Note: Such the principle can be explained in QFT.
Bosons: The system of bosons are totally symmetrical under the exchange of any • pair P N bosons = + N bosons , (2.2.77) ij| i | i where Pij interchanges between arbitrary pair.
Fermions: The system of fermions are totally anti-symmetrical under the exchange • of any pair P N fermions = N fermions . (2.2.78) ij| i −| i
E.g.1: two-boson system: k k , k k , 1 ( k k + k k ). | 0i| 0i | 00i| 00i √2 | 0i| 00i | 00i| 0i E.g.2: two-fermion system: 1 ( k k k k ). √2 | 0i| 00i − | 00i| 0i Spin-statistics theorem: half-integer spin particles are fermions and integer spin particles are bosons.
Note 1: In QFT, the spin-statistics theorem can be verified.
Note 2: In canonical quantization, fermionic field theories are quantized with anti-commutator A, B = AB + BA and bosonic field theories are quantized with commutator { } [A, B] = AB BA. −
39 2.2.14 Scattering theory in quantum mechanics Consider the scattering problem in QM with central force interaction, with the incident flux denoted as ~jin and outgoing flux denoted as ~jout, the differential cross section dσ defines as ~j = dσ ~j . (2.2.79) | out| | in| The scattering flux describes the scattered particle in per solid angle, namely ~j = | out| ~j r2dΩ. Then the differential cross section takes the form | scatt| dσ ~j r2 = | scatt| , (2.2.80) dΩ ~j | in| dσ and the total cross section can be integrated out σ = dΩ dΩ.
After the scattering, the free particle takes the asymptoticalR wave function expressed as
ikr ik r ik r e e · e · + f(θ) , (2.2.81) → r where the function f(θ) is named as scattering amplitude and it can be proved that the square of absolute value of f(θ) equals to the differential cross section, namely
dσ = f(θ) 2. (2.2.82) dΩ | |
Consider the symmetry in scattering.
Two-spinless-boson scattering: • dσ = f(θ) + f(π θ) 2. (2.2.83) dΩ | − |
Two-spinless-fermion scattering: • dσ = f(θ) f(π θ) 2. (2.2.84) dΩ | − − |
Two-electron unpolarized beam scattering: • dσ 1 3 = f(θ) + f(π θ) 2 + f(θ) f(π θ) 2. (2.2.85) dΩ 4| − | 4| − − |
1 The factor 4 contributes from the spin-singlet state which is spin-antisymmetric, therefore f(θ) + f(π θ) is associated with a space -symmetric function. And |3 − | the factor 4 contributes from spin-triplet states which are spin-symmetric, therefore f(θ) f(π θ) is associated with the space-anti-symmetric function. | − − |
40 Lecture 3 Classical Field Theory
About Lecture II:
[Luke] P.P. 40-59; • [Tong] P.P. 15-19; • [Coleman] P.P. 40-69; • [PS] P.P. 17-19; • [Zhou] P.P. 25-33. •
3.1 Special relativity
3.1.1 Lorentz transformation
Setup 1: Two inertial frames O and O0,
O :(ct, x, y, z) O0(ct0, x0, y0, z0). (3.1.1) →
Setup 2: Lorentz transformation (LT):
ct ct0 µ x Λ ν x0 , (3.1.2) y −→ y0 z z 0 where µ and ν are index labels, µ, ν = 0, 1, 2, 3.
Assumptions 1: Light speed in vacuum is constant, namely c = c ; |o |o0 Assumptions 2: Equation of motion is invariant under LT in two frames,
= 0 Λ = 0. (3.1.3) ··· −→ · · · Lorentz invariance:
2 2 2 2 2 2 2 2 2 ( s) = ct x y z = ct0 x0 y0 z0 . (3.1.4) 4 − − − − − −
Eg. 1: Lorentz boost along x-axis, β = v , γ = 1 , Lorentz boost along x-axis: c √1 β2 −
41 t t0 6 6 v -x - x - x0
γ βγ 0 0 − βγ γ 0 0 Λµ = − . (3.1.5) ν 0 0 1 0 0 0 0 1 Eg. 2: Lorentz rotation around z-axis,
1 0 0 0 0 cos θ sin θ 0 Λµ = − . (3.1.6) ν 0 sin θ cos θ 0 0 0 0 1 Physics in special relativity:
Eg. 1: Time dilation;
Eg. 2: Length contraction;
Eg. 3: Causality = Locality = unchanged time-ordering.
3.1.2 Four-vector calculation Three-vectors: ~r = (x, y, z); ~r ~r = x2 + y2 + z2; (3.1.7) · ~p = (p , p , p ); ~p ~p = p2 + p2 + p2; (3.1.8) x y z · x y z ~ = (∂ , ∂ , ∂ ); ~ ~ = ∂2 + ∂2 + ∂2. (3.1.9) ∇ x y z ∇ · ∇ x y z Contra-variant four vectors:
Space-time vector: • xµ = (t, ~r) = (x0, x1, x2, x3) = (t, x, y, z); (3.1.10)
Energy-momentum vector: • pµ = (E, ~pc); (3.1.11)
Four-derivative: • ∂ ∂ ∂ ∂ ∂ ∂ = ( , , , ). (3.1.12) µ ≡ ∂xµ c∂t ∂x ∂y ∂z Covariant four vectors:
Space-time vector: • x = (t, ~r) = (t, x, y, z); (3.1.13) µ − − − − Energy-momentum vector: • p = (E, ~pc); (3.1.14) µ −
42 Four-derivative: • ∂ ∂ ∂ ∂ ∂ ∂µ = ( , , , ). (3.1.15) ≡ ∂xµ c∂t −∂x −∂y −∂z
ν xµ = ηµνx = (t, ~r) = (t, x, y, z) − − − − (3.1.16) = (x0, x1, x2, x3).
Minkowski space-time metric denoted as ηµν represented for the flat space-time without gravity, expressed as 1 0 0 0 0 1 0 0 ηµν = − , (3.1.17) 0 0 1 0 − 0 0 0 1 − with its inverse ηνσ satisfying
νσ σ ηµνη = δµ (unit matrix). (3.1.18)
Contravariant vector and covariant vector is transformed via Minkowski metric:
ν µ µν xµ = ηµνx , x = η xν; (3.1.19) ν µ µν pµ = ηµνp , p = η pν; (3.1.20) ν µ µν ∂µ = ηµν∂ , ∂ = η ∂ν. (3.1.21)
Inner product is invariant quantity under LT:
x2 = xµx = η xµxν = ct2 x2 y2 z2. (3.1.22) µ µν − − − p2 = pµp = p2 ~p2c2 = E2 ~p2c2 = m c4, (3.1.23) µ 0 − − 0 where quantity m0 is denoted as mass at rest frame, therefore we have the mass-energy relation 2 2 4 2 2 E = m0c + ~p c , (3.1.24) and the mass deficient violation relation
∆E = ∆mc2. (3.1.25)
3.1.3 Causality
Causality = Locality = unchanged time-ordering. The space-time can be divided into three regions: s2 > 0, time-like region; 4 s2 = 0, light-cone; 4 s2 < 0, space-like region; 4 where s2 = c2( t)2 x2 y2 z2. (3.1.26) 4 4 − 4 − 4 − 4
43 t 4s2 > 0
4s2 < 0
x
4s2 = 0
Thm 1: Time ordering is unchanged in time-like region;
Thm 2: Time ordering can be violated in space-like region.
Example: Mother was born at the event point O1 and the child is born at O2, where t2 > t1. In the space-like region, after the Lorentz boost, the time ordering can be changed, namely t2 < t1, shown as t t t2: Child born
O2 Λ O1 x O1 x O2
t1: Mother born
t2 > t1 t2 < t1
Note: The causality means that no instantaneous interaction in space-like region.
3.1.4 Mass-energy relation Mass-energy relation: 2 2 E = mc = γm0c . (3.1.27) Optional problem: derive ∆E = ∆mc2.
Note 1: Energy and matter can be converted to each other. e.g. Atom bomb; Nuclear power station.
Note 2: At v c, particle number is not fixed. ≈ e.g. 1. p (proton) + p p + p + π0 (meson). (3.1.28) −→ e.g. 2. p + p p + p + p +p ¯ (anti-proton). (3.1.29) −→ Fermilab, Stanford. e.g. 3. p +p ¯ + + Higgs particle. (3.1.30) −→ · · · ··· LHC, CERN.
44 3.1.5 Lorentz group
µ µ µ ν Definition: Lorentz transformation matrix Λ = Λ ν defined by x0 = Λ νx satisfying µ ν x0 xµ0 = x xν, namely
µ ν α β α β µ ν ηµνΛ αΛ βx x = ηαβx x , ηµνΛ αΛ β = ηαβ. (3.1.31)
Note: In the infinitesimal transformation
ν ν ν 2 Λ δ = δ δ + ω δ + o(ω ), (3.1.32)
ν where ω δ is small quantity. And we can check the following relations
1 ν ν ν 2 (Λ− ) = δ ω + o(ω ); (3.1.33) δ δ − δ ω = ω . (3.1.34) µν − νµ Proof. η (δν + ων )(δµ + ωµ ) = η = ω = ω . (3.1.35) µν δ δ ρ ρ ρδ ⇒ µν − νµ
E.g.1. Lorentz boost along x-direction with y z fixed, in the infinitesimal formalism, we − have ω = ω , ω0 = ω1 . (3.1.36) 01 − 10 1 0 vx γ βγ 0 0 0 c 0 0 − vx v − βγ γ 0 0 small x 0 0 0 vx Λµ = − c δµ + − c + o(( )2). ν 0 0 1 0 −−−−−→ ν 0 0 0 0 c 0 0 0 1 0 0 0 0 (3.1.37) Note that the infinitesimal term in above Lorentz transformation stands for a sym- metric matrix.
E.g.2. Lorentz rotation around z-axis, in the infinitesimal formalism, we have
ω = ω , ωi = ωj (3.1.38) ij − ji j − i with i, j = 1, 2, 3.
1 0 0 0 0 0 0 0 0 cos θ sin θ 0 θ small 0 0 θ 0 Λµ = − δµ + − +o(θ2). (3.1.39) ν 0 sin θ cos θ 0 −−−−→ ν 0 θ 0 0 0 0 0 1 0 0 0 0 Note that the infinitesimal term in above Lorentz transformation stands for an anti-symmetric matrix.
µ Thm 3.1.5.1. The set of all Lorentz transformations Λ = Λ ν is a group called the Lorentz group denoted as SO(1, 3).
Proof. See group theory.
45 Note 1. If det(Λ) = 1, the Lorentz group is called proper Lorentz group denoted as SO+(1, 3). The determinant of the Lorentz transformation is important because of the Jacobian determinant which specifies the transformation of integral mea- sure under Lorentz transformation, namely dt0dx0dy0dz0 = det(Λ)dtdxdydz. Note that dx4 is a Lorentz transformation invariant.
0 Note 2. If Λ 0 1 and det(Λ) = 1, the Lorentz group is called proper orthochronous ≥ 0 Lorentz group denoted as SO↑ (1, 3). Λ 1 means time ordering or time arrow + 0 ≥ is unchanged under Lorentz transformation.
Note 3. Real physics prefers the proper and orthochronous Lorentz group which is called the Lorentz group in the course.
Note 4. SO(1, 3) = SU(2) SU(2). The angular momentum state j, m where j is integer ∼ ⊗ | i or half-integer forms a representation of the SU(2) group, so the representation of the Lorentz group is denoted as j , m j , m or j j , m m . | 1 1i ⊗ | 2 2i | 1 2 1 2i
3.1.6 Classification of particles
At high energy physics, particles including electrons, photons, etc, are classified as repre- sentations of the Lorenz group, labeled by (j1, j2), where j1, j2 are integers or half integers denoting angular momentum.
Table 3.1: Classification of fundamental particles representation Particles Fields Spin /Statistics LG representation Messon or Higgs Scalar, φ(x) Spinless /Boson Invariant, (0, 0)
Photon Vector, Aµ(x) Spin 1 /Boson Covariant, (1, 0) or (0, 1) Electron Spinor, ψ(x) Spin 1 /Fermion Spinor transformation, ( 1 , 0) (0, 1 ) 2 2 ⊕ 2
3.2 Classical field theory
Before the special relativity, the concept of field is a kind of mathematical object, but afterwards, it becomes a real physical object.
3.2.1 Concepts of fields
Definition. 1. Field is a physical quantity depends on every space-time point. E.g. E~ (t, ~x) = E~~x(t), B~ (t, ~x) = B~~x(t), where ~x is not dynamical variable thus becomes as label.
Definition. 2. A field is an object with infinity number of degrees of freedom, because t R+, ~x R3. ∈ ∈ Definition. 3. A field represents an interaction between particles, which can not propagate instantaneously. And interaction between particles is mediated by fields or associated quanta.
46 e− γ e− e− e− Photon e− e−
Photon: quanta of E.M. fields No instantaneous interaction
Two-electron interacts by the interchange of virtual photon.
Definition. 4. The dynamics of fields at space-time point completely depends on physics around the point, so mathematics of local fields prefer partial differential equation, such as Maxwell equations.
Definition. 5. Particle-Wave duality. Particle-Field duality.
3.2.2 Electromagnetic fields Note: Be careful of notations, because of SI unit, Gauss unit and natural unit. In natural unit, we set c = ε0 = µ0 = 1.
Maxwell equations in vacuum:
∂E~ E~ = 0; = B~ ; (3.2.1) ∇ · ∂t ∇ × ∂B~ B~ = 0; = E.~ ∇ · ∂t −∇ × Let us prove electrodynamics is a Lorentz covariant theory. Namely, we show Maxwell equations have a Lorentz covariant form.
In Gauge potential, with A~ denoted the magnetic potential and ϕ denoted the electric potential, the four-dimensional potential is defined as
Aµ(t, ~x) = (ϕ, A~). (3.2.2) with ∂A~ E~ = ϕ ; B~ = A.~ (3.2.3) −∇ − ∂t ∇ × Define the field strength: F µν = ∂µAν ∂νAν. (3.2.4) − Note that F 0i = Ei; F ij = εijkB (3.2.5) − − k with i, j = 1, 2, 3. Therefore F µν has the matrix expression
0 E E E − x − y − z Ex 0 Bz By F µν = − . (3.2.6) E B 0 B y z − x E B B 0 z − y x ab Fµν = ηµaηνbF . (3.2.7)
47 The Maxwell equations • ∂B~ B~ = 0, = E,~ (3.2.8) ∇ · ∂t −∇ × are equivalent with the Bianchi identity
∂µFνλ + ∂νFλµ + ∂λFµν = 0. (3.2.9)
The Maxwell equations • ∂E~ E~ = 0, = B,~ (3.2.10) ∇ · ∂t ∇ × have other form shown as µν ∂µF = 0. (3.2.11)
3.2.3 Lagrangian, Hamiltonian and the action principle The field is denoted as φ (t, ~x), a = 1, 2, ,N. • a ··· The state is described by φ (t, ~x) and ∂ φ (t, ~x). • a µ a The Lagrangian density is the function of φ (t, ~x) and ∂ φ (t, ~x), namely (φa, ∂ φ ). • a µ a L µ a The action is defined as S = d4x (φa, ∂ φ ). • L µ a Note: d4x, and S are both LorentzR invariant scalars. L Under infinitesimal variation φ φ + δφ with fixed boundary δφ (t ) = δφ (t ) = 0, a → a a a 1 a 2 the action principle states δS = 0, which is equivalent with the equation of motion.
∂ ∂ 0 = δS = d4x( L δφ + L δ(∂ φ )). (3.2.12) ∂φ a ∂(∂ φ ) µ a Z a µ a I II Because of | {z } | {z } ∂ II = L ∂µ(δφa) ∂(∂µφa) (3.2.13) ∂ ∂ =∂µ( L δφa) ∂µ( L )δφa, ∂(∂µφa) − ∂(∂µφa) so ∂ ∂ ∂ 0 = δS = d4x L ∂ ( L ) δφ + ∂ ( L δφ ) . (3.2.14) ∂φ − µ ∂(∂ φ ) a µ ∂(∂ φ ) a Z a µ a µ a 4 ∂ The 4-total derivative term d x∂ ( L δφ ) means the surface term after integration µ ∂(∂µφa) a which is supposed to vanish under boundary condition. And because δφa is arbitrary R variation, so ∂ ∂ La = ∂µ( L a ), (3.2.15) ∂φ ∂(∂µφ ) which is called equation of motion or Euler-Lagrangian equation.
Define the conjugate momentum as ∂ ∂ Πa = L , φ˙a = φa(t, ~x). (3.2.16) ∂φ˙a ∂t
48 The Hamiltonian density has the formalism = Π φ˙ . (3.2.17) H a a − L And Hamiltonian has the expression
H = d3~x . (3.2.18) H Z 3.2.4 A simple string theory Parameters L, ρ and T denote the string length, mass density and the string tension respec- tively. The boundary points are fixed, namely 0 x L. The transverse displacement ≤ ≤ quantity is denoted as φ(t, ~x). The corresponding Lagrangian density is 1 1 = ρφ˙2 T φ2, (3.2.19) L 2 − 2 x ˙ ∂ where φ = ∂t φ and φx = ∂xφ. The equation of motion can be calculated in the way ∂ ∂ ∂ L = 0, L = ρφ,˙ L = T φx, (3.2.20) ∂φ ∂φ˙ ∂φx − T ρφ¨ T φ = 0, φ¨ ( )φ = 0. (3.2.21) − xx − ρ xx Define the canonical momentum ∂L Π = = ρφ.˙ (3.2.22) ∂φ˙ The Hamiltonian density is 1 1 = Πφ˙ = ρφ˙2 + T φ2 0. (3.2.23) H − L 2 2 x ≥ Thm 3.2.4.1. Such the string φ(t, x) is equivalent to an infinite set of decoupled harmonic oscillators. Proof. Assume + 2 ∞ nπx φ(t, x) = sin q (t). (3.2.24) a a n r n=1 X + 2 ∞ nπx φ˙ = sin q˙ (t); (3.2.25) a a n r n=1 X + 2 ∞ nπ nπx φ = cos q (t). (3.2.26) x a a a n r n=1 X (3.2.27) So + a ∞ 1 1 nπ L = dx = ρq˙2 T ( )2q2 . (3.2.28) L 2 n − 2 a n 0 n=1 Z X The equation of motion can be derived in the way
∂L ∂L nπ 2 = ρq˙n, = T ( ) qn, (3.2.29) ∂q˙n ∂qn − a 2 q¨n + wnqn = 0 (3.2.30) T nπ with wn = ρ a . q 49 3.2.5 Real scalar field theory Real scalar field (Klein-Gorden field) φ(t, ~x) is described by the Lagrangian density 1 = (∂ φ∂µφ µ2φ2) L 2 µ − 1 = (φ˙2 ( φ)2 µ2φ2) (3.2.31) 2 − ∇ − with µ2 > 0. The corresponding canonical momentum is ∂ Π = L = φ,˙ (3.2.32) ∂φ˙ and the Hamiltonian density is 1 = (φ˙2 + ( φ)2 + µ2φ2) 0. (3.2.33) H 2 ∇ ≥ Equation of motion can be found in the way ∂ ∂ L = µ2φ, L = ∂µφ, (3.2.34) ∂φ − ∂(∂µφ) 2 ( + µ )φ = 0, (3.2.35) which is called the Klein-Gordon equation associated with the real scalar field.
Plain-wave solution of Klein-Gordon equation has the formalism 4 d k 2 2 ik x φ(t, ~x) = δ(k µ )e− · φ(k). (3.2.36) (2π)3/2 − Z 4 3 The integration is in four dimension, namely d k = dk0d ~k. k2 µ2 = k2 w2, w = ~k2 + µ2, (3.2.37) − 0 − ~k ~k where w~k is associated with the frequency. Substitute the solution into the differential equation, we can find the term 2 ik x 2 2 ik x ( + µ )e− · = ( k + µ )e− · , (3.2.38) − and with delta function, we have δ(k2 µ2)(k2 µ2) = 0. Note that − − 2 2 1 1 0 δ(k µ ) = δ(k0 + w~k) + δ(k w~k), (3.2.39) − 2w~k 2w~k − which implies the negative energy.
With interaction, the Lagrangian has the expression = + , (3.2.40) L L0 Lint where = V (φ). And the equation of motion becomes as L − ∂ ( + µ2)φ = V (φ). (3.2.41) −∂φ λ 4 E.g. V (φ) = 4! φ with λ as the coupling constant, λ ( + µ2)φ = φ3. (3.2.42) −3! Note: Up to now, no analytic solution to the equation of motion with interaction was been found.
50 3.2.6 Complex scalar field theory Two real scalar fields φ and φ are described by the Lagrangian • 1 2 = (φ ) + (φ ) V (φ , φ ), (3.2.43) L L0 1 L0 2 − 1 2 where the free part of the Lagrangian has the formalism 1 1 (φ ) = ∂ φ ∂µφ m2φ2, (3.2.44) L0 i 2 µ i i − 2 i And for instance the interaction part has the expression
2 2 V (φ1, φ2) = g(φ1 + φ2), (3.2.45)
where g is the coupling constant.
Two-component real scalar field vector φ~ is defined as • φ φ~ = 1 , φ~T = φ φ . (3.2.46) φ 1 2 2 The Lagrangian can be rewritten as 1 1 = ∂ φ~T ∂µφ~ m2φ~T φ~ V (φ~T , φ~). (3.2.47) L 2 µ − 2 −
Complex scalar fields are defined as • 1 ψ(x) = (φ1 + iφ2), (3.2.48) √2 1 ψ†(x) = (φ1 iφ2). (3.2.49) √2 − which give the Lagrangian as
µ 2 = ∂ ψ†∂ ψ m ψ†ψ V (ψ†, ψ). (3.2.50) L µ − − The equations of motion are expressed as
∂V ( + m2)ψ = ; (3.2.51) − ∂ψ 2 ∂V ( + m )ψ† = . (3.2.52) −∂ψ† Canonical momentums show as ∂ ∂ Πψ = L = ψ˙†;Π = L = ψ,˙ (3.2.53) ˙ ψ† ˙ ∂ψ ∂ψ† which give rise the Hamiltonian as
= Πψψ˙ + Π ψ˙† (3.2.54) H ψ† − L 2 = ψ˙†ψ˙ + ψ† ψ + m ψ†ψ + V (ψ†, ψ) 0. (3.2.55) ∇ ∇ ≥
51 Note. “Nucleon”-meson theory, which is also named as the scalar Yukawa theory. The lagrangian density shows as = + gψ†ψφ. (3.2.56) L Lφ Lψ − ψ: complex scalar field.
“nucleon”=pseudo-nucleon = spinless with charge.
Note. proton and anti-proton are nucleons with spin 1/2.
φ: real scalar fields. meson = spinless without charge.
g: coupling constant. [ψ] = [φ] = [g] = 1 in mass dimension.
The equations of motion are
2 ( + µ )φ = gψ†ψ; − (3.2.57) 2 ( ( + m )ψ = gψ†φ. −
3.2.7 Multi-component scalar field theory Define the N-component real scalar field vector as •
φ1 φ ~ 2 ~T φ = . ; φ = φ1 φ2 φN . (3.2.58) . ··· φ N
N N 1 = (∂ φ ∂µφ µ2φ φ ) V ( φ2) L 2 µ a a − a a − a a=1 a=1 X X 1 = (∂ φ~T ∂µφ~ m2φ~T φ~) V (φ~T φ~). (3.2.59) 2 µ − −
Define the N-component complex scalar field vector as •
ψ1 ψ2 ψ~ = . ; ψ~† = ψ† ψ† ψ† . (3.2.60) . 1 2 ··· N ψ N
N N 1 µ 2 = (∂ ψ†∂ ψ m ψ†ψ ) V ( ψ†ψ ) L 2 µ a a − a a − a a a=1 a=1 X X 1 µ 2 = (∂ ψ~†∂ ψ~ m ψ~†ψ~) V (ψ~†ψ~). (3.2.61) 2 µ − −
52 3.2.8 Electrodynamics
µ With the four-dimension potential A = (ϕ, A~), the electromagnetic tensor Fµν is defined as F = ∂ A ∂ A . Under the gauge transformation A A + ∂ J µ, F is gauge µν µ ν − ν µ µ → µ µ µν invariance. For redundant degrees of freedom of electromagnetic field, we have two kinds of gauge fixing shown as
Lorentz Gauge: ∂µAµ = 0; (3.2.62) Coloumb Gauge: A~ = 0. (3.2.63) ∇ · In vacuum, the electromagnetic field is governed by the Lagrangian 1 = F F µν L −4 µν 1 = (∂ A ∂ A )F µν −4 µ ν − ν µ 1 = ∂ A F µν. (3.2.64) −2 µ ν The equation of motion is found in the way
∂ ∂ L = F µν, L = 0, (3.2.65) ∂(∂µAν) − ∂A˙ ν
µν ∂µF = 0. (3.2.66) The time components of the canonical momenta are
∂ Πµ = L . (3.2.67) ∂(∂0Aµ)
0 i ∂ i Note that Π = 0 and Π = L = E . The Hamiltonian density can be derived as ∂(∂0Ai) 1 = ΠµA˙ = (E~ 2 + B~ 2) + ϕ E.~ (3.2.68) H µ − L 2 ∇ ·
Note: 1 1 = F F µν = (E~ 2 B~ 2); (3.2.69) L −4 µν 2 − 1 H = d3x = d3x(E~ 2 + B~ 2) 0; (3.2.70) H 2 ≥ Z Z P~ = d3xE~ B.~ (3.2.71) × Z
With current, the Electric magnetic field is governed by the Lagrangian 1 = F F µν J µA , (3.2.72) L −4 µν − µ where J µ is four-dimension current J µ = (ρ,~j). The equation of motion can be derived as
µν ν ∂µF = J . (3.2.73)
53 The equation of motion can be reformulated as
∂ (∂µAν ∂νAµ) = J ν; (3.2.74) µ − Aν ∂ν(∂ Aµ) = J ν. (3.2.75) − µ In Lorentz gauge, we have the equation
ν ν A = J , (3.2.76) namely ∂2 ∂2 ϕ 2ϕ = ρ; A~ 2A~ = ~j. (3.2.77) ∂t2 − ∇ ∂t2 − ∇ Note that the equation is Lorentz covariant.
Note 1: The retarded solution of the equation of motion shows as
v ρ (t c ), ~x0 3 ϕ(t, ~r) = − d ~x0. (3.2.78) 4π~r Z Note 2: To solve the equation of motion in electromagnetic field theory, the Green function usually is applied. In quantum field theory, the Feynman propagator is also Green function.
Note 3: The current conservation can be derived from the equation of motion, namely ∂ρ ∂ F µν = J ν = ∂ J µ = 0, + ~j = 0 (3.2.79) µ ⇒ µ ∂t ∇ · by the relation µν ν ∂ν∂µF = ∂νJ = 0. (3.2.80) And the conserved quantity Q can be defined as
Q d3xρ (3.2.81) ≡ Z dQ due to dt = 0. Note 4: Conservation law in field theory is described by the local current conservation.
Note 5: Photon is massless. If photon is massive, then it is described by the Proca La- grangian 1 µ2 = F F µν J Aµ + A Aµ. (3.2.82) L −4 µν − µ 2 µ
3.2.9 General relativity Physical quantities defined in general relativity:
Metric field: g (x); • µν Christoffel connection: Γi (g (x)); • jk µν Curvature tensor: Rα (Γi ); • µβγ jk
54 Ricci tensor: R = gαβR ; • µν αµβν Scalar curvature: R = gµνR . • µν Action in vacuum takes the formula
S = d4x√ gR/16πG (3.2.83) − Z with g = det(gµν) and G as the gravity constant. And the equation of motion can be derived 1 R g = 0. (3.2.84) µν − 2 µν Note 1: With matter, the equation of motion becomes as 1 R g = 8πGT , (3.2.85) µν − 2 µν µν
where Tµν is the energy-momentum tensor. Note 2: Quantum gravity is the theory of quantization of the general relativity, which is an open probelm in 21st century.
55 Lecture 4 Symmetries and Conservation Laws
About Lecture II:
[Luke] P.P. 40-59; • [Tong] P.P. 15-19; • [Coleman] P.P. 40-69; • [PS] P.P. 17-19; • [Zhou] P.P. 25-33. • 4.1 Symmetries play a crucial role in field theories
Lagrangian density in real scalar field with interaction takes the form
1 1 λ = ∂ φ∂µφ µ2φ2 φ4, (4.1.1) L 2 µ − 2 − 4! and the equation of motion is λ ( + µ2)ϕ = ϕ3. (4.1.2) −3! Up to now, no analytic solution has been found for this equation. Without solutions of equation of motion, one can perform following ways to understand the physics
Way-out 1: Perturbative approach (Feynman diagram);
Way-out 2: Symmetries and conservation laws (without solving equation of motion);
Way-out 3: Numerical simulation (Lattice QCD);
Way-out 4: . ······ Note: Symmetries in quantum field theory are almost able to determine the formulation of the action, e.g., the Yang-Mills theory.
4.2 Noether’s theorem in field theory
We have two different variations:
δqa, δφ(x): variation for deriving equation of motion; • Dqa, Dφ(x): variation for symmetries without specifying equation of motion. •
56 The fields are denoted as φa(t, ~x). The state is determined by φa(x) and first derivation ∂ φ (x), namely by coordinate and momentum. The Lagrangian takes the form = µ a L (φ , ∂ φ ) without explicit dependence on x. And the action can be obtained S = L a µ a d4x (φ , ∂ φ ). L a µ a RDef 4.2.1 (Symmetry). A transformation in field theory is called a symmetry iff
D = ∂ F µ (4.2.1) L µ µ for some F (φa, ∂µφa), which must be hold for arbitrary φa(x), but not necessarily satis- fying equation of motion.
Remark:
DS = d4xD = d4x∂ F µ L µ Z Z t2 = d3xF 0 + dt F~ dS,~ (4.2.2) · Z t1 Z I
S0 = S + DS (4.2.3) where the time boundary is fixed and spatial boundary vanishes, thus two terms do not have contribution in deriving equation of motion, namely δS0 = δS.
Thm 4.2.0.1 (Noether’s theorem). For every continuous symmetry, there is a conserved current.
Proof.
1). From the definition of symmetry: D = ∂ F µ. L µ 2). From Hamilton’s principle:
∂ ∂ D = ( L Dφ + L D(∂ φ )) L ∂φ a ∂(∂ φ ) µ a a a µ a X ∂ ∂ = (∂ ( L )Dφ + L ∂ (Dφ )) µ ∂(∂ φ ) a ∂(∂ φ ) µ a (4.2.4) a µ a µ a X ∂ = ∂ ( L Dφ ), µ ∂(∂ φ ) a a µ a X therefore ∂ = ∂ ( L Dφ F µ) = 0. (4.2.5) ⇒ µ ∂(∂ φ ) a − a µ a X Define the conserved current as ∂ J µ = L Dφ F µ, (4.2.6) ∂(∂ φ ) a − a µ a X and it turns out µ ∂µJ = 0. (4.2.7)
57 E.g. In Electromagnetic theory, the four-dimension current is defined as J µ = (ρ,~j), where µ ρ is charge density and ~j is current. The current conservation ∂µJ = 0 gives rise to the relation ∂ρ + ~j = 0. (4.2.8) ∂t ∇ · We can define the conserved quantity Q as
Q = d3xρ (4.2.9) Z due to dQ ∂ρ = d3x = d3x( ~j) = ~j dS~ = 0. (4.2.10) dt ∂t −∇ · − · ZV ZV IV
4.3 Space-time symmetries and conservation laws
4.3.1 Space-time translation invariance and energy-momentum tensor
Space-time translation takes the form
T x x0 = T x = x ε, (4.3.1) −−→ − where ε is an infinitesimal constant. And the physical system described by the field is unchanged, namely
UT φ (x) φ0 (x0) = φ (x), (4.3.2) a −−−→ a a
0(x0) = (x), (4.3.3) L L where x and x0 respectively denote the representations of a space-time point in two coor- dinate systems. In active transformation, we have
+ n n 1 ∞ ε d φ0(x) = U φ(x)U † = φ(T − x) = φ(x + ε) = φ(x). (4.3.4) T T n! dxn n=0 X And the variations of Dφ and D can be calculated as a L
Dφ (x) = φ0 (x) φ (x) a a − a = φ (x + ε) φ (x) (4.3.5) a − a µ 2 = ε ∂µφa(x) + o(ε ),
D = 0(x) (x) L L − L = (x + ε) (x) (4.3.6) L − L = εµ∂ + o(ε2). µL According to the definition 4.2.1 of symmetry, the function F µ of space-time transforma- tion takes the form F µ = εµ . (4.3.7) L
58 From Noether’s theorem 4.2.0.1, the conserved current can be defined
∂ J µ = L Dφ F µ ∂(∂ φ ) a − a µ a X ∂ = L (εν∂ φ ) εµ ∂(∂ φ ) ν a − L a µ a (4.3.8) X ∂ = ε L ∂νφ ηµν ν ∂(∂ φ ) a − L " a µ a # Xµ ν = εν(T ) ,
T µν = (T µ)ν, (4.3.9)
∂ T µν = L ∂νφ ηµν , (4.3.10) ∂(∂ φ ) a − L a µ a X where T µν is named as energy momentum tensor. The conservation law expresses as
∂ J µ = 0 = ∂ T µν = 0. (4.3.11) µ ⇒ µ Note: T µν is crucial in general relativity.
Table 4.1: Energy momentum tensor T µν
µ ν ∂µ(T ) = 0 εν Charge density and current ε = (1, 0, 0, 0) T 00: energy density (T 0)ν : charge density T µ0: energy term T 0j: momentum density ε = (0, ~x) T i0: energy current (T i)ν : current T µj: momentum term T ij: momentum current
E.g. In scalar field theory 1 1 = ∂ φ∂µφ µ2φ2, (4.3.12) L 2 µ − 2 we have
T 00 = (T 0)0 ∂ ˙ = L φa (4.3.13) ˙ − L a ∂φa X = , H where is energy density, and H
H = d3xT 00 = d3x , (4.3.14) H Z Z which implies the energy is conserved quantity.
59 T 0i = (T 0)i ∂ = L ∂iφ 0 ˙ a − a ∂φa X (4.3.15) i = Πa∂ φa a X = pi, where pi is momentum density, and
P i = d3xT 0i = d3xpi, (4.3.16) Z Z which implies the momentum is conserved quantity. Notice the conceptual difference between canonical momentum and conserved momentum.
4.3.2 Lorentz transformation invariance and angular-momentum tensor µ Def 4.3.1. The constant Lorentz transformation is denoted as Λ = Λ ν satisfying
µ ν ηρσ = ηµνΛ ρΛ σ, (4.3.17)
where ηρσ is Minkowski flat space-time metric defined in (3.1.17). Lemma 4.3.2.1. The infinitesimal Lorentz transformation takes the form
2 Λµν = ηµν + ωµν + o(ω ) (4.3.18) with infinitesimal constant ωµν satisfying
ω = ω . (4.3.19) µν − νµ Details and examples see previous subsection 3.1.5.
Note 1. 1 2 Λ− = η ω + o(ω ). (4.3.20) µν µν − µν Note 2.
ν 2 (Λx)µ = xµ + ωµνx + o(ω ); (4.3.21) 1 ν 2 Λ− x = x ω x + o(ω ). (4.3.22) µ µ − µν Definition of the scalar field theory under the active Lorentz transformation
• Λ x x0 = Λx; (4.3.23) −−→
• UΛ φ(x) φ0(x0) = φ(x); (4.3.24) −−−→ UΛ (x) 0(x0) = (x). (4.3.25) L −−−→L L
60 • 1 φ0(x) = φ(Λ− x) = UΛφ(x)UΛ† ; (4.3.26) 1 0(x) = (Λ− x) = U (x)U † . (4.3.27) L L ΛL Λ The set Λ forms the Lorentz group, and the set U forms the representation of { } { Λ} Lorentz group.
Note: The scalar field φ(x), Lagrangian density (φ, ∂ φ), integral measure d4x and nat- L µ urally the action S = d4x are both scalars under Lorentz transformation. L R Remark: The same physics
Frame 1 Frame 2 x x0
Apply the Noether’s theorem to derive the conserved current.
Dφ(x) = φ0(x) φ(x) 1− = φ(Λ− x) φ(x) − σ 2 = φ(xρ ωρσx + o(ω )) φ(x) −σ ρ − = ωρσx ∂ φ(x) − σ ρ = ωσρx ∂ φ(x). (4.3.28)
Note that ω = ω . ρσ − σρ
D = 0(x) (x) L L 1− L = (Λ− x) (x) L − L = ω xσ∂ρ σρ L = ∂ρ(ω xσ ). (4.3.29) σρ L From = ∂ F µ, the quantity F can be found as L µ F µ = ηµρω xσ , (4.3.30) σρ L and the associated conserved current can be defined ∂ J µ = L Dφ F µ ∂(∂ φ ) a − a µ a X ∂ = L ω xσ∂ρφ ηµρω xσ ∂(∂ φ ) σρ a − σρ L a µ a X ∂ = ω L xσ∂ρφ ηµρxσ σρ ∂(∂ φ ) a − L ( a µ a ) X ∂ = ω xσ L ∂ρφ ηµρ σρ ∂(∂ φ ) a − L ( a µ a !) X = ω xσT µρ . (4.3.31) σρ { }
61 Note that xσT µρ is an anti-symmetric tensor, therefore we can symmetrize the term xσT µρ to eliminate the redundant zero term. 1 1 xσT µρ = (xσT µρ + xρT µσ) + (xσT µρ xρT µσ) (4.3.32) 2 2 − symmetry anti-symmetry Thus the conserved current can| be written{z as } | {z } 1 J µ = ω (xσT µρ xρT µσ). (4.3.33) 2 σρ − Define the angular-momentum tensor as
(M µ)σρ = M µσρ = xσT µρ xρT µσ (4.3.34) − satisfying the conserved relation µσρ ∂µM = 0. (4.3.35)
Note 1. µ: Current and charge index σ, ρ: Lorentz transformation index M µσρ: σ = 0, ρ = i: Lorentz boost index σ, ρ = i, j: Lorentz rotation index
E.g. when µ = 0, σ, ρ = i, j,
M 0ij = xiT 0j xjT 0i. (4.3.36) − Recall that T 0i is momentum density. Then M 0ij can be viewed as angular momentum density, namely
J ij = d3xM 0ij Z = d3x(xiT 0j xjT 0i) − Z (4.3.37)
Note 2. M µσρ = xσT µρ xρT µσ (4.3.38) − which represents the orbital angular-momentum for spinless particle without spin angular-momentum.
E.g. 1: Angular-momentum for Lorentz boost σ = 0, ρ = i. The current is denoted as M µ0i, then the conserved charge can be calculated in the way J 0i = d3xM 00i. R J 0i = d3xM 00i Z = d3x(x0T 0i xiT 00) − Z = tP i d3xxiT 00, (4.3.39) − Z 62 where T 0i is the momentum density, i.e., P i = d3xT 0i, and T 00 is the mass- energy density. The conserved relation d J 0i = 0 gives rise to dt R d (tP i d3xxiT 00) = 0, (4.3.40) dt − Z dP i d P i + t = d3xxiT 00, (4.3.41) dt dt Z dP i where the momentum is conserved quantity, i.e., dt = 0. Therefore
d P i = d3xxiT 00. (4.3.42) dt Z E.g. T 00 = ρ, where ρ is the mass density, which gives rise to
d P i = constant = d3xρxi (4.3.43) dt Z suggesting that the center of mass moves steadily.
E.g. 2: Point-particle in General Relativity. Particle trajectory is denoted as
ξµ(τ) = (ξ(τ), ξ~(τ)), (4.3.44)
where τ is proper time, and space-time point is denoted as xµ = (t, ~x). Einstein equation gives rise to R T (4.3.45) µν ∝ µν where dξµ(τ) dξν(τ) T µν(t, ~r) = m dτδ(4)(xµ ξµ(τ)) , (4.3.46) 0 − dτ · dτ Z
dξν T 0ν(t, ~r) = m δ(3)(~r ξ~(t)) , (4.3.47) 0 − dτ
T 0i = P iδ(3)(~r ξ~(t)). (4.3.48) − Momentum: P µ = d3xT 0µ(t, ~r), (4.3.49) Z Angular-momentum:
J ij = d3x(xiT 0j xjT 0i) − (4.3.50) Z = ξiP j ξjP i. −
63 Field (system) Space-time (frame) Active transformation changed fixed Passive transformation fixed changed
4.4 Internal symmetries and conservation laws
4.4.1 Definitions ∆. Active and passive transformation. Active transformation: Λ 1 φ(x) φ0(x) = φ(Λ− x); (4.4.1) −−→ Passive transformation: Λ φ(x) φ0(x) = φ(Λx). (4.4.2) −−→ E.g. Rotation in active transformation.
R 1 φ(x) φ0(x) = φ(R− x). (4.4.3) −−→ y y
~ey
R ~ex x x active transformation
R 1 φ(~e ) φ0(~e ) = φ(~e ) = φ(R− ~e ). (4.4.4) x −−→ y x y The field is characterized as the circle in blue. In active transformation, the frame or coordinate is unchanged and the system or field is rotated. E.g. Rotation in passive transformation.
R φ(x) φ0(x) = φ(Rx). (4.4.5) −−→ y x
R ~ex x y −~ey passive transformation
R φ(~e ) φ0( ~e ) = φ(~e ) = φ (R( ~e )) . (4.4.6) x −−→ − y x − y In passive transformation, the system or field is unchanged and the frame or coordinate is rotated.
∆. Classification of symmetries. ∆. Space-time symmetry and internal symmetry. . ∆. Comparison between space-time symmetry and internal symmetry.
64 Space-time symmetry Internal symmetry Space-time transformation U(1) symmetry for Continuous symmetry Lorentz transformation charge conservation Parity Discrete symmetry Charge conjugation Time reversal
Field Space-time Space-time symmetry changed changed Internal symmetry changed fixed
Space-time symmetry Internal symmetry Space-time Changed Fixed Field Changed Changed Example: Space-time translation, U(1) symmetries,
symmetry Lorentz transformation U(1) ∼= SO(2) µν Example: ∂µT = 0 (Electric)Charge µρσ ∂ρ conservation law ∂µM = 0 + ~j = 0 ∂t ∇ ·
4.4.2 Noether’s theorem Note: Noether’s theorem only applies to continuous symmetry.
Noether’s theorem for internal and continuous symmetry.
x x, (4.4.7) → φ(x) φ0(x), (4.4.8) → 0, (4.4.9) L → L which gives rise to Dφ(x) = φ0(x) φ(x), (4.4.10) − µ D = 0 = ∂ F = 0, (4.4.11) L L − L µ so that F µ is a constant and can be set as zero. So the conserved current is defined as
µ ∂ J = L a Dφa, (4.4.12) ∂(∂µφ )
3 0 3 ∂ a 3 a Q = d xJ = d x ( L Dφ ) = d x (ΠaDφ ). (4.4.13) ∂φ˙ Z Z a Z
4.4.3 SO(2) invariant real scalar field theory The Lagrangian takes the form
= (φ ) + (φ ) V (φ , φ ), (4.4.14) L L0 1 L0 2 − 1 2
65 2 where φ1 and φ2 are real scalar fields with same m . The free part of the Lagrangian shows as 1 1 (φ ) = ∂ φ ∂µφ m2φ2 (4.4.15) L0 i 2 µ i i − 2 i and the interaction part has the formalism
2 2 V (φ1, φ2) = g(φ1 + φ2), (4.4.16) where g is the coupling constant.
Thm 4.4.3.1. Lagrangian (4.4.14) is SO(2) rotation invariance. L φ φ 10 = R 1 (4.4.17) φ φ 20 2 Proof. SO(2) stands for 2 dimensional special orthogonal group. R SO(2) has the ∈ matrix representation shown as
cos λ sin λ R = (4.4.18) sin λ cos λ − T T where λ is constant and R R = RR = 112. With the vector φ~ defined as φ φ~ = 1 , (4.4.19) φ 2 the Lagrangian can be rewritten as 1 1 = ∂ φ~T ∂µφ~ m2φ~T φ~ gφ~T φ.~ (4.4.20) L 2 µ − 2 −
T T T Define the after rotation field as φ~0 = Rφ~ and φ~0 = φ~ R , therefore
T T T T φ~0 φ~0 = φ~ R Rφ~ = φ~ φ~; (4.4.21) T µ T T µ T µ ∂µφ~0 ∂ φ~0 = ∂µφ~ R R∂ φ~ = ∂µφ~ ∂ φ.~ (4.4.22) Thus (φ~0, ∂ φ~0) = (φ,~ ∂ φ~). (4.4.23) L µ L µ
Apply the Noether’s theorm.
With D = 0 = 0, (4.4.24) L L − L from D = ∂ F µ, we can find F µ is constant. Rewrite the rotation matrix R into L µ infinitesimal formalism
cos λ sin λ R = sin λ cos λ − 0 λ = 11 + + o(λ2). (4.4.25) 2 λ 0 − 66 Therefore
2 φ10 = φ1 + λφ2 + o(λ ); (4.4.26) 2 φ0 = φ λφ + o(λ ), (4.4.27) 2 2 − 1 which give rise to Dφ = φ0 φ = λφ ; (4.4.28) 1 1 − 1 2 Dφ = φ0 φ = λφ . (4.4.29) 2 2 − 2 − 1 The conserved current can be calculated as
µ ∂ ∂ J = L Dφ1 + L Dφ2 ∂(∂µφ1) ∂(∂µφ2) = λ(∂µφ φ ∂µφ φ ) (4.4.30) 1 2 − 2 1 with the associated conserved charge θ
θ = d3~xJ 0 = d3~x(φ˙ φ φ φ˙ ). (4.4.31) 1 2 − 1 2 Z Z In classical theory, we do not know the exact meaning of the conserved charge θ.
4.4.4 U(1) invariant complex scalar field theory Introduce the new complex field ψ(x) 1 ψ(x) = (φ1 + iφ2); (4.4.32) √2 1 ψ†(x) = (φ1 iφ2), (4.4.33) √2 − which give the Lagrangian (4.4.14) as
µ 2 = ∂ ψ†∂ ψ m ψ†ψ V (ψ†, ψ). (4.4.34) L0 µ − − with V (ψ†ψ) = gψ†ψ. The SO(2) rotation of the field doublet (φ1, φ2) becomes the U(1) transformation of the complex field ψ(x), namely 1 ψ0(x) = (φ0 + iφ0 ) √2 1 2 1 = (cos λφ1 + sin λφ2 + i( sin λφ1 + cos λφ2)) √2 − 1 iλ iλ = (e− φ1 + ie− φ2) √2 iλ = e− ψ(x); (4.4.35) 1 ψ0†(x) = (φ0 iφ0 ) √2 1 − 2 1 = (cos λφ1 + sin λφ2 i( sin λφ1 + cos λφ2)) √2 − − 1 iλ iλ = (e φ1 ie φ2) √2 − iλ = e ψ†(x). (4.4.36) (4.4.37)
67 U(1) invariance of Lagrangian with eiλ U(1) , i.e., L { ∈ } iλ iλ ψ0†(x)ψ0(x) = ψ†(x)e e− ψ(x) = ψ†(x)ψ(x). (4.4.38)
The conserved current can be calculated as
µ ∂ ∂ J = L Dψ + L Dψ† ∂(∂µψ) ∂(∂µψ†) µ µ = iλ((∂ ψ)ψ† (∂ ψ†)ψ) (4.4.39) − with the charge density 0 J = i(ψψ˙ † ψ˙†ψ). (4.4.40) −
4.4.5 Non-Abelian internal symmetries The standard model of particle physics includes non-Abelian gauge field theories.
Note:
Lie group Product Example
Abelian group R1R2 = R2R1 SO(2) ∼= U(1)
Non-Abelian group R1R2 = R1R2 SO(N) with N 3, U(N) with N 2 6 ≥ ≥
E.g. 1. SO(N) with N 3 invariant quantum field theory. ≥
φ1 φ ~ 2 ~T φ = . , φ = φ1, φ2, φN . (4.4.41) . ··· φ N 1 1 = ∂ φ~T ∂µφ~ m2φ~T φ~ V (φ~T , φ~). (4.4.42) L 2 µ − 2 − The after rotation field is denoted as φ~ = Rφ~ and φ~ T = φ~T RT , where R SO(N) 0 0 ∈ with RT R = RRT = 11. Note that R R = R R in general. 1 2 6 2 1 The Lagrangian (4.4.42) is invariant under transformation R due to L T T T T φ~0 φ~0 = φ~ R Rφ~ = φ~ φ~; (4.4.43)
T µ T T µ T µ ∂µφ~0 ∂ φ~0 = ∂µφ~ R R∂ φ~ = ∂µφ~ ∂ φ.~ (4.4.44)
In the case N = 3, apply the Noether’s theorem, we can find the conserved currents denoted as
J µ12 = (∂µφ )φ (∂µφ )φ ; (4.4.45) 1 2 − 2 1 J µ13 = (∂µφ )φ (∂µφ )φ ; (4.4.46) 1 3 − 3 1 J µ23 = (∂µφ )φ (∂µφ )φ . (4.4.47) 2 3 − 3 2
68 E.g. 2. U(N) with N 2 invariant quantum field theory. ≥
ψ1 ψ2 ψ~ = . , ψ~† = ψ†, ψ†, ψ† . (4.4.48) . 1 2 ··· N ψ N 1 µ 1 2 = ∂ ψ~†∂ ψ~ m ψ~†ψ~ V (ψ~†, ψ~). (4.4.49) L 2 µ − 2 − The after rotation field is denoted as ψ~ = Uψ~ and ψ~ = ψ~ U , where U U(N) 0 0† † † ∈ with U U = UU = 11. Note that U U = U U in general. † † 1 2 6 2 1 The Lagrangian (4.4.49) is invariant under transformation U due to L
ψ~0†ψ~0 = ψ~†U †Uψ~ = ψ~†ψ~; (4.4.50)
µ µ µ ∂µψ~0†∂ ψ~0 = ∂µψ~†U †U∂ ψ~ = ∂µψ~†∂ ψ.~ (4.4.51)
4.5 Discrete symmetries
Noether’s theorem is only defined for continuous symmetries. For discrete symmetry, no conservation law can be defined. However, discrete symmetries play important roles in quantum field theories.
4.5.1 Parity A parity transformation (space reflection or mirror reflection) is defined as
t t, ~x ~x. (4.5.1) → → − In four dimensional vector, it can be characterized as xµ x . → µ The scalar field is classified by
P φ(t, ~x) φ0(t, ~x) = φ(t, ~x), (4.5.2) −→ ± − where “ + ” sign represents for scalar and “ ” represents for pseudo-scalar. − E.g. 1. For N-component real scalar field, see previous subsection 3.2.7, it transformed under the parity operation in the way
P φ~(t, ~x) φ~0 = Rφ~(t, ~x), (4.5.3) −→ − where R satisfies some constraints, for example RT R = RRT = 11.
E.g. 2. In the case of electromagnetic field Aµ(x) in subsection 3.2.2, the parity transfor- mation works as P µ A (t, ~x) A0 (t, ~x) = A (t, ~x). (4.5.4) µ −→ µ −
69 4.5.2 Time reversal The time reversal transformation is defined as
t t, ~x ~x. (4.5.5) → − → In four dimensional vector, we have xµ x . → − µ E.g. 1. Real scalar field theory:
T φ(t, ~x) φ0(t, ~x) = φ( t, ~x). (4.5.6) −→ − Pseudo scalar field theory:
T φ(t, ~x) φ0(t, ~x) = φ( t, ~x). (4.5.7) −→ − − E.g. 2. Electromagnetic field:
T µ A (t, ~x) A0 (t, ~x) = A ( t, ~x). (4.5.8) µ −→ µ − Note: Parity and time reversal transformations are Lorentz transformation with
det(P ) = det(T ) = 1, (4.5.9) − namely P,T do not belong to the SO↑ (1, 3) group instead of P,T SO(1, 3), details refer + ∈ to Lorentz group in subsection 3.1.5.
4.5.3 Charge conjugation Charge conjugation is a special operation defined in quantum field theory, and arises from a transformation between particles and anti-particles.
For real scalar field φ(x), charge conjugation acts as • C φ(x) φ0(x) = φ(x), (4.5.10) −→ which implies the anti-particle of meson is itself, namely π0 C π0. −→ For the real complex field ψ(x), charge conjugation gives rise to • C ψ(x) ψ0(x) = ψ†(x); (4.5.11) −→ C ψ†(x) ψ0†(x) = ψ(x). (4.5.12) −→ Note 1. Local quantum field theory (describing standard model) is invariant under the joint action of C, P and T , which is called the CPT theorem.
Note 2. Under charge conjugation, the electromagnetic field (describing the photon) is transformed as C A (x) A0 (x) = A (x) (4.5.13) µ −→ µ − µ with the “ ” sign due to the CPT theorem. −
70 Lecture 5 Constructing Quantum Scalar Field Theory
About Lecture III:
[Luke] P.P. 20-39; • [Tong] P.P. 7-37; • [Coleman] P.P. 17-40; • [PS] P.P. 15-29; • [Zhou] P.P. 50-70. •
5.1 Quantum mechanics and special relativity
5.1.1 Natural units SI (International system of units). • Meter(m) Length → Primary units: Second(s) Time → Kilogram(kg) Mess →
ML2 E = T 2 = [Joule]; 1eV = 1.602 10 19J; × − 1GeV = 1.602 10 10J = 109eV ; × − V = L , c = 2.998 108m s 1; T × · − h 34 25 ~ = = 1.054 10− J s = 6.577 10− GeV s. 2π × · × ·
2 Fine structure constant: α = 1 e = 1 . 4πε0 ~c 137.04 Natural units (NU) • Set the basic constants as ~ = 1, c = 1, which gives rise to
1 25 ~ = 1 [time] = 6.577 10− sec; ↔ GeV ∼ × (5.1.1) 1 16 ~c = 1 [length] = 1.973 10− m = 197fm. ↔ GeV ∼ ×
71 We can choose the dimension of energy or mass as the basic dimension, [E] = GeV = [mass], to perform the dimension analysis, and the dimension of quantity X is defined as
[X] = massd = [md] = [Ed] = d. (5.1.2) h i E.g. [c] = [~] = [k] = 0, [e] = 0; (5.1.3) 1 [time] = [length] = = 1. (5.1.4) GeV − Optional Problem: What’s the Plank mass and its unit?
5.1.2 Particle number unfixed at high energy (Special relativity) In special relativity, see previous subsection 3.1, we have mass-energy relation
2 2 E = mc = γm0c . (5.1.5)
Optional problem: derive ∆E = ∆mc2.
Note 1. The mass-energy relation (5.1.5) tells energy and matter can be converted to each other such as atom bomb and nuclear power station.
Note 2. At v c, particle number is not fixed. ≈ E.g. 1. p (proton) + p p + p + π0 (meson). (5.1.6) −→ E.g. 2. p + p p + p + p +p ¯ (anti-proton). (5.1.7) −→ Fermilab, Stanford. E.g. 3. p +p ¯ + + Higgs particle. (5.1.8) −→ · · · ··· LHC, CERN.
Optional problem: Calculate the threshold energy EL of the incident particle at which all particles after collision are at rest in the CoM (center-of-mass frame).
E.g. 1.
Proton Target proton Before collision (EL, ~pLc) (mp, 0) at the lab frame
p After collision p at the CoM frame π0
72 2 4 2 4 (m 0 + 2m ) c 2m c π p − p EL = 2 2mpc (0.134 + 0.938 2)2 2 0.9382 (5.1.9) = × − × 2 0.938 × = 1.22GeV,
2 1 2 EL = γmpc = mpc , (5.1.10) 1 (v /c)2 − L v pm c2 L = 1 ( p )2 0.639. (5.1.11) c s − EL ≈
E.g. 2. p + p p + p + p +p, ¯ (5.1.12) −→ (E + m c2) (~p c + 0)2 = (4m c2)2 0, (5.1.13) L p − L p − 2 EL = 7mpc , (5.1.14) v 1 L = 1 ( )2 0.989. (5.1.15) c r − 7 ≈ E.g. 3. E 7T eV 7000m c2, (5.1.16) L ≈ ≈ p v 1 L = 1 ( )2 0.999 9. (5.1.17) c r − 7000 ≈ ···
5.1.3 No position operator at short distance (Quantum mechanics) A particle of mass m can not be localized in a region smaller than its Compton wavelength λc, which is defined as ~ ~c λ = = . (5.1.18) c mc mc2 For example, we have a particle in a container with the reflection wall
L λ c
The Heisenberg uncertainty relation x p ~ gives rise to 4 · 4 ≥ 2 ~ x L, p . (5.1.19) 4 ∼ 4 ∼ 2L On the other hand, the mass-energy relation E = m2c4 + p2c2 suggests
p~c E p c = . (5.1.20) 4 ≈ 4 · 2L
73 When E 2mc2 = mc2 + mc2, (5.1.21) 4 ≥ namely ~ λc L = , (5.1.22) ≤ 4mc 4 particle-antiparticle pair can be created.
L λ /4 c E.g. Trap an electron in a container, the typical length scale L is the Bohr radii, 2 ~ 8 L = 0.529 10− cm. (5.1.23) mc2 ≈ × The Compton wavelength of electron can be calculated as
~c 197fm 11 λe = 2 = 4 10− cm. (5.1.24) mec 0.511 ≈ × Therefore we can see L λ , so it is safe enough to apply non-relativistic quantum e mechanics in condensed matter physics.
5.1.4 Micro-Causality and algebra of local observables
Two observables OˆA(t1, ~x1) and OˆB(t2, ~x2) should be commutative at space-like separated region, namely [Oˆ (t , ~x ), Oˆ (t , ~x )] = 0, (x x )2 < 0, (5.1.25) A 1 1 B 2 2 1 − 2 because special relativity requires that particles in space-like regions can not have in- stantaneous interference with each other. In other words, interaction between particles is propagated via fields.
e− e−
Interaction in space-like region via fields
Note: A commutative commutator between two operators OˆA and OˆB means that they have common eigenstates, so quantum measurements by OˆA and OˆB are not affected with each other, i.e.,
OˆA OˆB ψ - ψ0 - ψ0 . | i | i | i
For example, Alice and Bob are space-like separated, (x x )2 < 0 with the observables A − B ˆ A ˆ B OA = σZ , OB = σX , (5.1.26) which are the Pauli matrices, so OˆA and OˆB are space-time independent operators and they have global action in the entire space-time. Experiments between Alice and Bob can be perform in the following steps.
74 Step 1. Alice has the spin-up eigenstate satisfying σA = . | ↑i Z | ↑i | ↑i B Step 2. Bob performs the measurement σX which gives rise to 1 ( ). (5.1.27) | ↑i −→ √2 | ↑i ± | ↓i
Step 3. Alice performs the measurement and then immediately realizes what Bob has done on her spin due to [σA, σB ] = 0, (5.1.28) Z X 6 which violates causality.
5.1.5 A naive relativistic single particle quantum mechanics Inside the light-cone (time-like region) is the physical region and particles can not travel outside the light-cone. In non-relativistic quantum mechanics, however, particles can travel faster than the speed of light c and can be found outside the light-cone.
E.g. The spinless, free moving particle is described by the Hamiltonian
P~ 2 H = , (5.1.29) 2m where the position operator and momentum operator satisfy the commutative relation
[Xi,Pj] = i~δij, i, j = 1, 2, 3. (5.1.30)
The equation of motion follows the Schr¨odingerequation expressed as
∂ i~ ψ(t) = H ψ(t) . (5.1.31) ∂t| i | i Recall that we have
3 1 i~k ~x 1 = d ~k ~k ~k , ~k ~x = e− · . (5.1.32) | ih | h | i (2π)3/2 Z Suppose at t = 0, the particle at ~x = 0 with ψ(0) = ~x = 0 . After time t, the state | i | i evolves as ψ(t) , and we can compute the probability amplitude of finding this particle | i at ~x,
iHt ~x ψ(t) = ~x e− ψ(0) h | i h | | i 3 iHt = d ~k ~x e− ~k ~k ~x = 0 h | | ih | i Z 3~ (5.1.33) d k i~k2/2mt i~k ~x = e− e · (2π)3 Z m 2 = ( )3/2eim~x /2t, 2πit so the probability amplitude from point (t = 0, ~x = 0) to arbitrary place (t, ~x) like the region outside the light-cone is non-zero, which violates causality.
75 A Naive single particle relativistic quantum mechanics can be constructed as
H = P~ 2 + m2,H ~k = ~k2 + m2 ~k . (5.1.34) | i | i q q The probability amplitude can be calculated as
3~ d k i√~k2+m2t i~k ~x ~x ψ(t) = e− e · (5.1.35) h | i (2π)3 Z mr e− ∞ (z m)r 2 2 = i 2 zdze− − sinh z m t. (5.1.36) 2π r m − Z p Optional problem: Derive relation (5.1.36) from (5.1.35).
Note 1. Outside the light-cone, r = ~x > t, we can find | | ~x ψ(t) = 0, (5.1.37) h | i 6 which still violates causality.
Note 2. As r 1/m, e mr/r 0, the probability amplitude becomes as − → ~x ψ(t) = 0, (5.1.38) h | i therefore at the length scale greater than the Compton wavelength, the probability amplitude outside the light cone is almost zero, due to no relativistic corrections at that scale.
5.2 Comparisons of quantum mechanics with quantum field theory
Quantum Mechanics Quantum Field Theory Particle Number Fixed Un-fixed Position operator Defined Not defined, labels/Parameters Momentum operator Defined Defined Observable Hermitian, space independent Hermitian, space dependent Causality No Yes Time Label/Parameter Label/Parameter
5.3 Definition of Fock space
Def 5.3.1. Fock space is the Hilbert space for any number of particle denoted by
+ ∞ = (n) = (0) (1) (n) (5.3.1) F ⊕H H ⊕ H ⊕ · · · ⊕ H ⊕ · · · n=0 X with (n) as the n-particle Hilbert space. H Note 1. Fock space is the state space of quantum field theory.
76 Note 2. More details refer to “Second Quantization” in Advanced Quantum Mechanics.
Note 3.
Construction of QFT
Direct construction Canonical quantization (Second quantization) of classical filed theory
Rotation invariant Lorentz invariant Fock space Fock space
5.4 Rotation invariant Fock space
5.4.1 Fock space We consider the free moving spinless particles like meson and Higgs particle, which are characterized by the scalars and obey the Boson statistics. Hamiltonian is denoted as Hˆ ˆ and the momentum as P~ . In following lecture notes, the hat notation may be omitted for convenience.
The Hilbert space (0) denotes the no-particle space, called the vacuum. The vac- • H uum state with the notation vac , instead of 0 in Luke’s notes, satisfies the nor- | i | i malization condition vac vac = 1 (5.4.1) h | i with no observed energy and no observed momentum, namely
H vac = 0, P~ vac = 0. (5.4.2) | i | i (1) denotes the one-particle space. State has the notation ~k with the normalized •H | i relation ~k k~ = δ(3)(~k k~ ). (5.4.3) h | 0i − 0 Observables H and P~ give rise to the relations
H ~k = ω ~k , P~ ~k = ~k ~k , (5.4.4) | i ~k| i | i | i ~ 2 2 where ω~k = k + m represents for the energy. p (2) describes the two-particle space with the state ~k ,~k of two identical or indis- •H | 1 2i tinguishable particles satisfying the Boson statistics
~k ,~k = ~k ,~k . (5.4.5) | 1 2i | 2 1i The normalization relation is somehow complicated due to identical particles.
(3) (3) (3) (3) ~k ,~k ~k0 ,~k0 = δ (k~ k~ )δ (k~ k~ ) + δ (k~ k~ )δ (k~ k~ ). (5.4.6) h 1 2| 1 2i 1 − 10 2 − 20 1 − 20 2 − 10
77 Observables H and P~ give rise to the relations
P~ ~k ,~k = (~k + ~k ) ~k ,~k , (5.4.7) | 1 2i 1 2 | 1 2i ~ ~ ~ ~ H k1, k2 = (ω~ + ω~ ) k1, k2 . (5.4.8) | i k1 k2 | i The completeness relation in Fock space shows as
∞ 1 11= vac vac + d3~k d3~k d3~k ~k ~k ~k ~k ~k ~k . (5.4.9) | ih | n! 1 2 ··· n| 1 2 ··· nih 1 2 ··· n| n=1 X Z Note 1. (n) may be not Hilbert space because of completeness relation requirement. H 1 Note 2. The factor n! is from the Boson-Einstein statistics. Note 3. The n-particle wave-function
ψ (t) = d3~k d3~k d3~k ~k ~k ~k ~k ~k ~k ψ (t) , (5.4.10) | n i 1 2 ··· n | 1 2 ··· nih 1 2 ··· n| n i Z is complicated due to 3n integral variables.
Thm 5.4.1.1. Such the above construction of the Fock space is invariant under spatial rotation.
Proof. The rotation operation denoted as R SO(3) gives rise to the expression ∈ ~k R ~kR = R~k (5.4.11) −−→ T with R R = 113 3. And the rotation operator acting on the state is denoted as O(R) gives × rise to the expression O(R) ~k ~kR = O(R) ~k = R~k (5.4.12) | i −−−−→| i | i | i with O†(R)O(R) = O(R)O†(R) = 11.
The normalized relation is rotation invariant, i.e.,
R R (3) (3) R R ~k ~k0 = ~k ~k0 = δ (~k ~k0) = δ (~k ~k0 ), (5.4.13) h | i h | i − − The integral measure is rotation invariant
d3~k = d3~kR = d3~k det(R) (5.4.14) Z Z Z due to Jacobian determinant relation. The completeness relation is also rotation invari- ance, i.e., d3~k ~k ~k = d3~kR ~kR ~kR . (5.4.15) | ih | | ih | Z Z We can check the relation d3~k ~k ~k = 1 (5.4.16) h | i Z to see the inner product and the integral measure are both indeed rotation invariant.
78 5.4.2 Occupation number representation (ONR)
Constructing the occupation number representation in the following steps.
Step 1. Confine the system in a periodic cubical box with length L.
L
System L
L
Step 2. Due to translation invariance, the allowed values of ~k are discrete with the for- malism ~ 2π 2π 2π k = ( nx, ny, nz), nx, ny, nz Z. (5.4.17) L L L ∈
ky 2π/L
2π/L
k O x
Step 3. n(~k) denotes the particle number with momentum ~k,
n( ) = n(~k ), n(~k ), . (5.4.18) | · i | 1 2 · · · i
Step 4. N(~k) denotes the occupation number operator with momentum ~k satisfying
N(~k1) n( ) = n(~k1) n( ) , | · i | · i (5.4.19) N(~k ) n( ) = n(~k ) n( ) . 2 | · i 2 | · i
Step 5. Hamiltonian and momentum operators have the formalism
~ ~ ~ ~ H = ω~kN(k), P = kN(k). (5.4.20) X~k X~k
Note: n( ) is still in the state formalism. | · i
79 5.4.3 Hints from simple harmonic oscillator (SHO) Statement: Occupation number representation of Fock space
one to one correspondence Representation of infinite number of independent Harmonic oscillators
Note: In simple harmonic oscillator, “N” denotes the excited energy level; in occupation number representation, “N” denotes the occupation number of particles.
5.4.4 The operator formalism of Fock space in a cubic box
Define the creation operator a† and annihilation operator a , satisfying ~k ~k
(3) a~k, a~† = δ~ ~ , (5.4.21) k0 k,k0 h i
a~k, a~k = a~† , a~† = 0. (5.4.22) 0 k k0 In vacuum state, the annihilation operator willh givei rise to
a vac = 0, for ~k. (5.4.23) ~k| i ∀ And the one-particle state can be created in the way
~k = a† vac . (5.4.24) | i ~k| i Similarly, the two-particle state is given by ~ ~ k1, k2 = a~† a~† vac . (5.4.25) | i k1 k2 | i Hamiltonian and momentum can be rewritten as respectively
H = ω a† a , ~k ~k ~k X~k P~ = ~ka† a . (5.4.26) ~k ~k X~k
Define the particle number operator as N(~k) = a† a and N = a† a , then the Hamil- ~k ~k ~k ~k ~k tonian and momentum have the expression P ~ H = ω~kN(k), X~k P~ = ~kN(~k). (5.4.27) X~k With the commutative relation of the creation and annihilation operator, we can check the following relations ~ ~ k k0 = vac a~ka~† vac = vac [a~k, a~† ] vac h | i h | k0 | i h | k0 | i (3) (5.4.28) = vac vac δ~k,~k = δ~ ~ , h | i 0 k,k0
80 H ~k = Ha† vac = H, a† vac | i ~k| i ~k | i h i (5.4.29) = ω~k a~† a~k , a~† vac 0 k0 0 k | i ~ Xk0 = ω a† vac = ω ~k . ~k ~k| i ~k| i
5.4.5 Drop the box normalization and take the continuum limit Drop the box normalization and replace it with the infinity continuum limit, namely L . → ∞ L
L
L −→ ∞
The commutative relation of creation and annihilation operator becomes as
(3) ~ ~ a~k, a~† = δ (k k0). (5.4.30) k0 − h i And the normalized relation takes the form
(3) ~k ~k0 = δ (~k ~k0). (5.4.31) h | i − The summation in Hamiltonian and momentum turns into integration, i.e.,
3 H = d ~k ω a† a , (5.4.32) ~k ~k ~k Z 3 P = d ~k ~k a† a . (5.4.33) ~k ~k Z Note 1. Any observables in free quantum field theory can be written in terms of creation and annihilation operators.
Note 2. a and a† are operator-valued distributions, i.e., only integrations with a , a† are ~k ~k ~k ~k meaningful in mathematics.
Note 3. Why we have a† and a in quantum field theory? Because they represent particle ~k ~k creation and annihilation in experiments. Note 4. Quantum free field theory is a collection of independent harmonic oscillators.
5.5 Lorentz invariant Fock space
5.5.1 Lorentz group A Lorentz transformation is generated by both Lorentz boosts and rotations. The Lorentz group is a set of all Lorentz transformations. The proper Lorentz group is the Lorentz group with det(Λ) = 1; the proper orthochronous Lorentz group is the Lorentz group with 0 det(Λ) = 1 and Λ 0 > 0. Usually, the proper orthochronous Lorentz group is called the Lorentz group for simplicity. More details refer to previous Subsection 3.1.5.
81 QFT Harmonic oscillators
a~k annihilation operator lowering operator a† creation operator rasing operator ~k N~k particle number excitation energy level 1 H0 = 2 ω vacuum energy zero-point energy Lowest energy state vac ground state 0 | i | i
5.5.2 Definition of Lorentz invariant normalized states The Lorentz transformation is denoted as Λ
k Λ kΛ = Λk, (5.5.1) −−→ with µ µ ν 0 1 2 3 k0 = Λ νk , k = (k , k , k , k ), (5.5.2) and the corresponding unitary transformation is denoted as U(Λ)
U(Λ) Λ k k0 = U(Λ) k = k , (5.5.3) | i −−−−→| i | i | i with U †U = UU † = 11.
Def 5.5.1. The Lorentz invariant one-particle state is defined as
k (2π)3/2 2ω ~k , (5.5.4) | i ≡ ~k| i q ~ 2 2 where ω~k = k + m . p
Note 1. The normalization relation for state k shows as | i 3 3 k k0 = (2π) 2ω δ (~k ~k0). (5.5.5) h | i ~k − And the completeness relation becomes as
d3~k 1 d3~k ~k ~k = k k . (5.5.6) | ih | (2π)3 2ω | ih | Z Z ~k
3/2 Note 2. The factor 2ω~k is the compensation factor from Lorentz boost and (2π) is due to the Feynman rules for the Feynman diagrams. p
5.5.3 Lorentz invariant normalized state
Thm 5.5.3.1. k k = (2π)32ω δ3(~k ~k ) is Lorentz invariant. h | 0i ~k − 0 Proof.
82 3~ Step 1. Verify the integral measure d k is Lorentz transformation invariant. 2ω~k The four dimensional integralR measure d4k = dk0dk1dk2dk3 is Lorentz invariant due to d4kΛ = det(Λ)d4k = d4k (5.5.7) where det(Λ) is the Jacobian determinant. The k2 = m2 is called on-shell mass condition, therefore we have δ(k2 m2) = δ(k2 ω2) − 0 − ~k 1 0 0 (5.5.8) = (δ(k + ω~k) + δ(k ω~k)). 2ω~k − which is Lorentz invariant. The step function, θ(k0) = 1, for k0 > 0. (5.5.9) is also Lorentz invariant.
k0 δ(k2 − m2)θ(k0)
~k
So we have the integral
d3~k d4kδ(k2 m2)θ(k0) = , (5.5.10) − 2ω~ k k0=ω Z Z ~k
which is Lorentz transformation invariant. Step 2. Verify the completeness relation is Lorentz transformation invariant.
3 U(Λ)1U †(Λ) = U(Λ) d ~k ~k ~k U †(Λ) = 1. (5.5.11) | ih | Z With the relation k = (2π)3/2 2ω ~k , (5.5.12) | i ~k| i we can find q d3~k U(Λ) 2ω ~k ~k U †(Λ) 2ω ~k| ih | Z ~k d3~kΛ = U(Λ)2ω ~k ~k U †(Λ) 2ω ~k| ih | Z ~kΛ 1 d3~kΛ = kΛ kΛ (2π)3 2ω | ih | Z ~kΛ 1 d3~k = k k (2π)3 2ω | ih | Z ~k = 1. (5.5.13)
83 Step 3. Check the normalization condition is Lorentz transformation invariant.
3 3 k k0 = (2π) 2ω δ (~k ~k0). (5.5.14) h | i ~k − 1 d3~k 3 k k0 = 1 , (5.5.15) (2π) 2ω~k h | i Z 3 2 1 | {z } |{z} where 1 and 2 are Lorentz| transformation{z } invariant, so 3 is Lorentz transfor-
mation invariant, namely 1 + 2 3 . →
5.6 Constructing quantum scalar field theory
5.6.1 Notation Rotation invariant one-particle state is given by
~k = a† vac (5.6.1) | i ~k| i with (3) ~ ~ a~k, a~† = δ (k k0), (5.6.2) k0 − h i a~k, a~k = a~† , a~† = 0. (5.6.3) 0 k k0 Hamiltonian and momentum can be reformulated h asi
1 3 H = d ~kω (a† a + a a† ); (5.6.4) 2 ~k ~k ~k ~k ~k Z 1 3 P~ = d ~k~k(a† a + a a† ). (5.6.5) 2 ~k ~k ~k ~k Z Then Lorentz invariant state can be constructed via
k = (2π)3/2 2ω ~k | i ~k| i 3/2q = (2π) 2ω a† vac ~k ~k| i q = β† vac , (5.6.6) k| i where 3/2 βk = (2π) 2ω~ka~k, (5.6.7) 3 q (3) ~ ~ [βk, β† ] = (2π) 2ω~ δ (k k0). (5.6.8) k0 k −
Note: Other formalism of creation and annihilation operator can be introduced.
a(~k) α(~k) = , (5.6.9) 3/2 (2π) 2ω~k
1 p(3) [α , α† ] = δ (~k ~k0). (5.6.10) ~k ~k 3 0 (2π) 2ω~k −
84 5.6.2 Constructing quantum scalar field
3 ik x ik x φˆ(t, ~x) = d ~k α(~k)e− · + α†(~k)e · ; (5.6.11) Z 3~ ˆ d k ik x ik x φ(t, ~x) = 3 β(k)e− · + β†(k)e · ; (5.6.12) (2π) 2ω~ Z k d3~k φˆ(t, ~x) = a(k)e ik x + a (k)eik x , (5.6.13) 3/2 − · † · (2π) 2ω~ Z k p where φˆ = φˆ and k x = ω t ~k ~x. The corresponding canonical momentum is † · ~k − · 3~ ˙ d k ω~k ik x ik x Π(ˆ t, ~x) = φˆ(t, ~x) = i a(k)e− · a†(k)e · . (5.6.14) (2π)3/2 − 2 ! − Z r
α(~k) prefers to the fourier expansion of φˆ(t, ~x); • β(k) is associated with Lorentz invariant state, i.e., k = β (k) vac ; • | i † | i a(~k) is closely related to the creation or annihilation operator in Harmonic oscillator. •
The equation of motion named as Klein-Gordon equation shows as ¨ φˆ = 2φˆ µ2φˆ = ( + µ2)φˆ(t, ~x) = 0. (5.6.15) ∇ − ⇒ Lemma 5.6.2.1. The commutative relation
φˆ(t, ~x), Π(ˆ t, ~y) = iδ(3)(~x ~y) (5.6.16) − h i is equivalent with ~ ~ 1 (3) ~ ~ α(k), α†(k0) = 3 δ (k k0). (5.6.17) (2π) 2ω~ − h i k Proof. For simplicity, the hat notation above operator is omitted in following proof. Choose the Fourier transform solution of the field operator and canonical momentum shown as
3 ik x ik x φ(t, ~x) = d ~k α(~k)e− · + α†(~k)e · ; (5.6.18) Z 3 ik x ik x Π(t, ~x) = φ˙(t, ~x) = d ~k ( iω )(α(~k)e− · + α†(~k)e · ) . (5.6.19) − ~k Z Set the time parameter t as 0, namely
3 i~k ~x i~k ~x φ(~x) = φ(t = 0, ~x) = d ~k α(~k)e · + α†(~k)e− · ; Z 3 i~k ~x = d ~k(α(~k) + α†( ~k))e · ; (5.6.20) − Z 3 i~k ~x Π(~x) = d ~k( iω )(α(~k) α†( ~k))e · . (5.6.21) − ~k − − Z 85 Perform the inverse of the Fourier transform, we find the equations
3 d ~x i~k ~x α(~k) + α†( ~k) = φ(~x)e− · ; (5.6.22) − (2π)3 Z 3 d ~x i~k ~x iω (α(~k) α†( ~k)) = Π(~x)e− · , (5.6.23) − ~k − − (2π)3 Z (5.6.24) and the solutions show as
3 1 d ~x i i~k ~x α(~k) = φ(~x) + Π(~x) e− · ; (5.6.25) 2 (2π)3 ω Z ~k 3 ~ 1 d ~x i i~k ~x α†(k) = 3 φ(~x) Π(~x) e · . (5.6.26) 2 (2π) − ω~ Z k (5.6.27)
The commutative relation can be calculated as
3 3 1 d ~xd ~y i i ~ ~ ~ ~ ik ~x+ik0 ~y α(k), α†(k) = 6 φ(~x) + Π(~x), φ(~y) Π(~y) e− · · 4 (2π) ω~ − ω~ h i Z k k0 1 d3~xd3~y i i i~k ~x+i~k0 ~y i~k ~x+i~k0 ~y = 6 φ(~x), Π(~y) e− · · + Π(~x), φ(~y) e− · · 4 (2π) −ω~ ω~ Z k0 k 1 d3~xd3~y i (3) i~k ~x+i~k0 ~y = 6 iδ (~x ~y) e− · · 4 (2π) −ω~ − Z k0 i (3) i~k ~x+i~k ~y i~k ~x+i~k ~y + iδ (~x ~y) e− · 0· e− · 0· ω~ − − k 1 3 1 i(~k ~k ) ~x 1 i(~k ~k ) ~x = d ~x e− − 0 · + e− − 0 · 4 ω~ ω~ Z k0 k 1 1 1 (3) ~ ~ 1 (3) ~ ~ = 3 δ (k k0) + δ (k k0) 4 (2π) ω~ − ω~ − k0 k 1 (3) ~ ~ = 3 δ (k k0). (5.6.28) (2π) 2ω~k −
Thm 5.6.2.1. The Heisenberg equation of motion
dΠ(ˆ t, ~x) = i H,ˆ Π(ˆ t, ~x) (5.6.29) dt h i is equivalent with the Klein-Gordon equation
2 ( + µ )φˆ(t, ~x) = 0. (5.6.30)
Proof. For simplicity, the hat notation above operator is omitted in following proof.
LHS = Π(˙ t, ~x) = φ¨(t, ~x). (5.6.31)
86 i RHS = d3~y Π2(t, ~y), Π(t, ~x) + φ2(t, ~y), Π(t, ~x) + µ2 φ2(t, ~y), Π(t, ~x) 2 ∇y Z = 0 + i d3~y φ(t, ~y)[ φ(t, ~y), Π(t, ~x)] + iµ2 d3~yφ(t, ~y)[φ(t, ~y), Π(t, ~x)] ∇y ∇y Z Z = i d3~y φ(t, ~y) (iδ(3)(~x ~y)) + iµ2 d3~yφ(t, ~y)iδ(3)(~x ~y) ∇y ∇y − − Z Z = i d3~y ( φ(t, ~y)iδ(3)(~x ~y)) 2φ(t, ~y)iδ(3)(~x ~y) µ2φ(t, ~x) ∇y ∇y − − ∇y − − Z = 2φ(t, ~x) µ2φ(t, ~x). (5.6.32) ∇ − Therefore we have the Klein-Gordon equation
φ¨(t, ~x) 2φ(t, ~x) + µ2φ(t, ~x) = 0. (5.6.33) − ∇
Why does relativistic quantum field theory prefer the Heisenberg picture?
(1). The particle creation or annihilation can be easily described in the operator formalism.
(2). The equations of motion or the Heisenberg equations are Lorentz covariant, for ex- amples, the Klein-Gordon equation, the Maxwell equations and the Dirac equation.
(3). The classical-quantum correspondence is obvious. The expectation value of the Heisenberg equations at ~ 0 is the Hamilton’s equations. → 5.7 Canonical quantization of classical field theory
Canonical quantization is a procedure of deriving QM (quantum mechanics) (or QFT (quantum field theory)) from the Hamiltonian formulation of CPM (Classical particle mechanics)(or CFT (classical field theory)).
Hˆ Hˆ Quantization QM QFT
H H CPM CFT
−→ Continuum limit
5.7.1 Classical field theory What is a field? Field is a quantity defined on every space-time point. See previous Section 3.2.
The field is denoted as φ (t, ~x), a = 1, 2, ,N. • a ··· The state is described by φ (t, ~x) and ∂ φ (t, ~x). • a µ a
87 The Lagrangian density is the function of φ (t, ~x) and ∂ φ (t, ~x), namely (φa, ∂ φ ). • a µ a L µ a The action is defined as S = d4x (φa, ∂ φ ). • L µ a R The equation of motion or Euler-Lagrangian equation. • ∂ ∂ La = ∂µ( L a ), (5.7.1) ∂φ ∂(∂µφ )
Note: d4x, and S are both Lorentz invariant scalars. L Define the conjugate momentum as
∂ ∂ Πa = L , φ˙a = φa(t, ~x). (5.7.2) ∂φ˙a ∂t
The Hamiltonian density has the formalism
= Π φ˙ . (5.7.3) H a a − L And Hamiltonian has the expression
H = d3~x . (5.7.4) H Z
E.g. Real scalar field (Klein-Gorden field) φ(t, ~x) is described by the Lagrangian density
1 = (∂ φ∂µφ µ2φ2) L 2 µ − 1 = (φ˙2 ( φ)2 µ2φ2) (5.7.5) 2 − ∇ − with µ2 > 0. The corresponding canonical momentum is
∂ Π = L = φ,˙ (5.7.6) ∂φ˙ and the Hamiltonian density is
1 = (φ˙2 + ( φ)2 + µ2φ2) 0. (5.7.7) H 2 ∇ ≥
Equation of motion can be found in the way
∂ ∂ L = µ2φ, L = ∂µφ, (5.7.8) ∂φ − ∂(∂µφ)
2 ( + µ )φ = 0, (5.7.9) which is called the Klein-Gordon equation associated with the real scalar field.
88 5.7.2 Canonical quantization The canonical quantization procedure proceeds in following steps. Step 1. Replace the field and canonical momentum with operators φ (t, ~x), Π (t, ~y) a b → φˆa, Πˆ b, satisfying the equal time commutation relations,
φˆa(t, ~x), φˆb(t, ~y) = 0 = Πˆ a(t, ~x), Πˆ b(t, ~y) , (5.7.10) h i h i φˆ (t, ~x), Πˆ (t, ~y) = iδ δ(3)(~x ~y). (5.7.11) a b ab − h i Step 2. Rewrite the Hamiltonian as operator, i.e.,
Hˆ = d3~x (φˆ , Πˆ ). (5.7.12) H a a Z Step 3. Equation of motion in Heisenberg picture can be derived as
∂φˆ a = i H,ˆ φˆ , (5.7.13) ∂t a ∂Πˆ h i a = i H,ˆ Πˆ . (5.7.14) ∂t a h i ˙ In real scalar field theory, canonical momentum operator takes the form Π(ˆ t, ~x) = φˆ(t, ~x), and Hamiltonian operator has the formalism 1 Hˆ = d3~x Πˆ 2(t, ~x) + ( φˆ(t, ~x))2 + µ2φˆ2(t, ~x) . (5.7.15) 2 ∇ Z h i The equation of motion named as Klein-Gordon equation shows as ¨ φˆ = 2φˆ µ2φˆ = ( + µ2)φˆ(t, ~x) = 0. (5.7.16) ∇ − ⇒ which has the plane-wave solution
d3~k φˆ(t, ~x) = a(k)e ik x + a (k)eik x , (5.7.17) 3/2 − · † · (2π) 2ω~ Z k where φˆ = φˆ and k x = ω ~k ~x. Thep corresponding canonical momentum is † · ~k − · 3~ ˙ d k ω~k ik x ik x Π(ˆ t, ~x) = φˆ(t, ~x) = i a(k)e− · a†(k)e · . (5.7.18) (2π)3/2 − 2 ! − Z r
Lemma 5.7.2.1. The Hamiltonian operator formalism
1 ˙ Hˆ = d3~x(φˆ2 + ( φˆ)2 + µ2φˆ2) (5.7.19) 2 ∇ Z is equivalent with the expression
1 3 Hˆ = d ~kω (a(~k)a†(~k) + a†(~k)a(~k)). (5.7.20) 2 ~k Z
89 Proof. For simplicity, the hat notation above operator is omitted in following proof.
3~ d k 1 ik x ik x φ(t, ~x) = (a(~k)e− · + a†(~k)e · ). (5.7.21) (2π)3/2 2ω Z ~k 3~ p d k ω~k ik x ik x Π(t, ~x) = φ˙(t, ~x) = i (a(~k)e− · a†(~k)e · ). (5.7.22) (2π)3/2 − 2 − Z r ! Hamiltonian can be divided into two parts, namely H = I+II, where 1 1 I = d3~xΠ2 = d3xφ˙2(t, ~x) 2 2 Z Z 3 3 1 d ~k d ~k0 ω~ ω~ = d3~x ( i k )( i k0 ) 2 (2π)3/2 (2π)3/2 − 2 − 2 Z Z Z r r ik x ik x ik x ik x a(~k)e− · a†(~k)e · a(~k0)e− 0· a†(~k0)e 0· . (5.7.23) × − − Doing the calculations separately, i.e., I = 1 + 2 + 3 + 4 . Note that we have the assistant relations shown as ~ ~ ik x ik0 x i(ω~ +ω~ )t i(k+k0) ~x e− · − · = e− k k0 e · , (5.7.24) 3 d ~x i(~k+~k ) ~x (3) e− 0 · = δ (~k + ~k0), (5.7.25) (2π)3 Z d3~xδ(3)(x) = 1, (5.7.26) Z therefore 3 1 d ~x 3 3 2 √ω~kω~k ik x ik x 1 = d ~k d ~k0( i) 0 a(~k)a(~k0)e− · − 0· 2 (2π)3 − 2 Z Z Z 1 √ω~ ω~ 3~ 3~ (3) ~ ~ 2 k k0 ~ ~ i(ω~ +ω~ )t = d k d k0δ (k + k0)( i) a(k)a(k0)e− k k0 2 − 2 Z Z 1 3 1 2iω t = d ~k ω a(~k)a( ~k) e− ~k ; (5.7.27) 2 −2 ~k − Z
3 1 d ~x 3 3 2 √ω~kω~k ik x+ik x 2 = d ~k d ~k0( i) 0 a(~k)( a†(~k0))e− · 0· 2 (2π)3 − 2 − Z Z Z 1 3 1 = d ~k ω a(~k)a†(~k) ; (5.7.28) 2 2 ~k Z
3 1 d ~x 3 3 2 √ω~kω~k ik x ik x 3 = d ~k d ~k0( i) 0 ( a†(~k))a(~k0)e · − 0· 2 (2π)3 − 2 − Z Z Z 1 3 1 = d ~k ω a†(~k)a(~k) ; (5.7.29) 2 2 ~k Z
3 1 d ~x 3 3 2 √ω~kω~k ik x+ik x 4 = d ~k d ~k0( i) 0 ( a†(~k))( a†(~k0))e · 0· 2 (2π)3 − 2 − − Z Z Z 1 3 1 2iω t = d ~k ω a†(~k)a†( ~k) e ~k ; (5.7.30) 2 −2 ~k − Z 90 I = 1 + 2 + 3 + 4
1 3 1 2 2iω t 2 2iω t = d ~k ( ω a(~k)a( ~k)e ~k ω a (~k)a ( ~k)e ~k ~k − ~k † † 2 2ω~ − − − − Z k +ω2a(~k)a (~k) + ω2a (~k)a(~k) . (5.7.31) ~k † ~k † The part II can be calculated as
1 II = d3~x( φ φ + µ2φ2) 2 ∇ ∇ Z 1 3 1 2 2 = d ~k (~k + µ ) a(~k)a†(~k) + a†(~k)a(~k) 2 2ω~ Z k 2iω t 2iω t +a(~k)a( ~k)e− ~k + a†(~k)a†( ~k)e ~k . (5.7.32) − − Finally, we have
H = I + II 1 3 1 2 2 2 = d ~k a(~k)a†(~k)(ω + ~k + µ ) + a†(~k)a(~k) 2 2ω ~k Z ~k 2 2 2 2iω t 2 2 2 (ω + ~k + µ ) + a(~k)a( ~k)e− ~k ( ω + ~k + µ ) × ~k − − ~k 2iω t 2 2 2 +a†(~k)a†( ~k)e ~k ( ω + ~k + µ ) − − ~k 1 3 = d ~k ω a(~k)a†(~k) + a†(~k)a(~k) , (5.7.33) 2 ~k Z where (3) a(~k), a†(~k0) = δ (~k ~k0). (5.7.34) − h i
Lemma 5.7.2.2. The momentum operator formalism
˙ Pˆ = d3~xφˆ φˆ (5.7.35) ∇ Z is equivalent with the expression
1 3 Pˆ = d ~k~k a(~k)a†(~k) + a†(~k)a(~k) . (5.7.36) 2 Z 5.7.3 Vacuum energy, Casimir force and cosmological constant problem With the Hamiltonian (5.7.20) denoted as
1 3 H = d ~kω (a(~k)a†(~k) + a†(~k)a(~k)) 2 ~k Z 1 3 (3) = d kω (2a†(~k)a(~k) + δ (0)) 2 ~k Z 3 1 3 (3) = d kω a†(~k)a(~k) + d kω δ (0), ~k 2 ~k Z Z (5.7.37)
91 (3) where a(~k), a†(~k) = δ (0), the vacuum energy can be calculated as h i E vac H vac vac ≡ h | | i 1 = d3k ~k2 + m2δ(3)(0). (5.7.38) 2 Z q The delta function has the expansion formalism shown as 3 (3) d x i~k ~x δ (~k) = e · , (5.7.39) (2π)3 Z therefore d3x V δ(3)(0) = = . (5.7.40) (2π)3 (2π)3 Z And the vacuum energy can be rewritten as 1 V E = ( d3k ~k2 + m2) . (5.7.41) vac 2 × (2π)3 Z q Define the vacuum energy density as E 1 d3k ρ = vac = ~k2 + m2. (5.7.42) vac V 2 (2π)3 Z q When V + , the vacuum energy becomes infinity, i.e., E + , which is • → ∞ vac → ∞ named as infra-red divergence. When V is fixed, the energy density is limitless, namely ρ + , which is called • vac → ∞ ultra-violet divergence.
Note 1. In the box normalization with periodic boundary condition, we have the Hamil- tonian 1 H = ~ω~ (a† a~ + a~ a† ) 2 k ~k k k ~k X~k 1 = ~ω~ (a† a~ + ), (5.7.43) k ~k k 2 X~k 1 where 2 ~ω~k denotes the zero point energy. In quantum mechanics, we only have one simple harmonic oscillator, however, in quantum field theory, there are infinity number of oscillators, which contributes to the infinity large of vacuum energy. Drop the box normalization and replace it with the infinity continuum limit, i.e., d3~k replace the summation ~k with integral (2π)3 , we still have the divergence vacuum energy. P R Note 2. Although the vacuum energy is infinity large, the vacuum can be disturbed for something, for example, the Casimir force, more details refer to the Wikipedia Casimir force. Note 3. The observed vacuum energy of the universe is almost zero, so there is a contra- diction between theoretical calculation and experiment, + = ρthero ρexper 0. (5.7.44) ∞ vac ≫ vac ≈ See Wikipedia cosmological constant problem.
92 5.7.4 Normal ordered product Remark 1. The vacuum energy is the absolute energy, and we are only interested in energy difference.
Remark 2. Without gravity, the vacuum energy Evac can not be detected.
Def 5.7.1. Normal ordered product is introduced to remove the vacuum energy by hand
: a†(~k)a(~k) :=: a(~k)a†(~k) := a†(~k)a(~k). (5.7.45)
1). Inside normal ordering, : :, bosonic operators are commutative. ······ 2). In the result, creation operators are always on LHS, annihilation operators are on RHS.
3). Normal ordering is defined for free field operator
1 3 : H : = d kω : a† a + a a† : 2 ~k ~k ~k ~k ~k Z 3 = d kω a† a , (5.7.46) ~k ~k ~k Z E = vac : H : vac = 0. (5.7.47) vac h | | i The field operator has the plane-wave solution (5.6.11) shown as
d3~k φ(t, ~x) = a(k)e ik x + a (k)eik x , (5.7.48) 3/2 − · † · (2π) 2ω~ Z k ik x p ik x where the wave function e− · is associated with positive frequency and e · is for negative frequency. The field operator can be divided into two parts
(+) ( ) φ(t, ~x) = φ (t, ~x) + φ − (t, ~x) (5.7.49) with
3~ (+) d k ik x φ (t, ~x) = a(~k)e− · , (5.7.50) (2π)3/2 2ω Z ~k 3~ ( ) d kp ik x φ − (t, ~x) = a†(~k)e · , (5.7.51) (2π)3/2 2ω Z ~k (+) p ( ) where φ (t, ~x) is related to the positive energy with annihilation operator and φ − (t, ~x) is related to the negative energy with creation operator.
: φ(x1)φ(x2): (+) ( ) (+) ( ) = : φ (x1) + φ − (x1) φ (x2) + φ − (x2) : (5.7.52) (+) (+) ( ) (+) ( ) (+) ( ) ( ) = φ (x1)φ (x2) + φ − (x1)φ (x2) + φ − (x2)φ (x1) + φ − (x1)φ − (x2).
Note that without normal ordering, field operator φ(x1)φ(x2) is supposed to have the term (+) ( ) φ (x1)φ − (x2).
93 5.8 Symmetries and conservation laws
5.8.1 SO(2) invariant quantum real scalar field theory
Two real scalar fields φ1 and φ2 are described by the Lagrangian
= (φ ) + (φ ) V (φ , φ ), (5.8.1) L L0 1 L0 2 − 1 2 where the free part of the Lagrangian has the formalism 1 1 (φ ) = ∂ φ ∂µφ m2φ2, (5.8.2) L0 i 2 µ i i − 2 i And for instance the interaction part has the expression
2 2 V (φ1, φ2) = g(φ1 + φ2), (5.8.3) where g is the coupling constant. More details refer to the previous Subsection 3.2.6.
Consider a free field theory without the interacting part V (φ1, φ2) and apply the canonical quantization procedure to obtain the plane-wave solutions of the field operators φi, i = 1, 2,
3~ d k ik x ik x φ = a (~k)e + a†(~k)e , (5.8.4) i 3/2 i − · i · (2π) 2ω~ Z k h i where p (3) a (~k), a†(~k0) = δ δ (~k ~k0). (5.8.5) i j ij − The vacuum is defined by h i a (~k) vac = 0, (5.8.6) i | i and one-particle states are given by
~k, 1 = a†(~k) vac , (5.8.7) | i 1 | i
~k, 2 = a†(~k) vac . (5.8.8) | i 2 | i
The conserved charge Q from SO(2) invariance takes the form
3 3 Q = d ~x φ˙ φ φ˙ φ = i d ~k a†(~k)a (~k) a†(~k)a (~k) , (5.8.9) 1 2 − 2 1 1 2 − 2 1 Z Z which has no explicit physical interpretation. With new creation and annihilation opera- tors defined by ~ ~ ~ ~ a1(k) + ia2(k) a1†(k) ia2†(k) b~ = , b† = − , (5.8.10) k √2 ~k √2 ~ ~ ~ ~ a1(k) ia2(k) a1†(k) + ia2†(k) c~ = − , c† = , (5.8.11) k √2 ~k √2 the conserved charge Q has a new form
3 Q = d ~k b† b c† c ~k ~k − ~k ~k Z = (+1)N+ ( 1)N . (5.8.12) b − c
94 We have b-type of particles and c-type of particles respectively denoted by b vac = c vac = 0, (5.8.13) ~k| i ~k| i ~k, b = b† vac , ~k, c = c† vac . (5.8.14) | i ~k| i | i ~k| i A b-type particle has charge +1, and a c-type particle has charge 1, namley − Q, b† = (+1)b† , Q, c† = ( 1)c† ; (5.8.15) ~k ~k ~k − ~k h i h i Q ~k, b = +1 ~k, b ,Q ~k, c = 1 ~k, c . (5.8.16) | i | i | i − | i Note that the c-type particle is the anti-particle of b-type particle.
5.8.2 U(1) invariant quantum complex scalar field theory Introduce new fields which are complex scalar fields denoted by 1 ψ(x) = (φ1 + iφ2), (5.8.17) √2 1 ψ†(x) = (φ1 iφ2). (5.8.18) √2 − which give the free part of the Lagrangian
µ 2 = ∂ ψ†∂ ψ m ψ†ψ. (5.8.19) L0 µ − Canonical momentums show as ∂ ∂ Πψ = L = ψ˙†;Π = L = ψ.˙ (5.8.20) ˙ ψ† ˙ ∂ψ ∂ψ†
Applying the canonical quantization procedure for above complex scalar field theory, sug- gesting that the field operator and canonical momentum obey the commutative relation [ψ(t, ~x), Π (t, ~y)] = iδ(3)(~x ~y), (5.8.21) ψ − the field operator has the formalism 3~ d k ik x ik x ψ(x) = (b e− · + c† e · ), (5.8.22) (2π)3/2 2ω ~k ~k Z ~k 3~ d kp ik x ik x ψ†(x) = (b† e · + c e− · ), (5.8.23) (2π)3/2 2ω ~k ~k Z ~k where the operators b~k and c~k satisfy the followingp commutative relations,
b~ , b~ = 0 = c~ , c~ ; (5.8.24) k k0 k k0 (3) ~ ~ b~k, b~† = δ (k k0) = c~k, c~† . (5.8.25) k0 − k0 In U(1) invariance, the chargeh densityi given by h i
0 J = i(ψψ˙ † ψ˙†ψ) (5.8.26) − gives rise to the conserved charge
3 0 3 Q = d xJ = d ~k b† b c† c . (5.8.27) ~k ~k − ~k ~k Z Z
95 Note 1. The complex field operator can be defined in terms of two real field operators, i.e., 1 ψ(x) = (φ1 + iφ2) , (5.8.28) √2 1 ψ†(x) = (φ1 iφ2) . (5.8.29) √2 −
Note 2. SO(2) group is isomorphic to U(1) group.
5.8.3 Discrete symmetries 5.8.3.1 Parity A parity transformation is characterized as
x P xµ. (5.8.30) µ −→ Define the unitary operator UP as
U ~k P U ~k = ~k . (5.8.31) | i −−→ P | i | − i The scalar field is classified by
φ0(t, ~x) = U φ(t, ~x)U † = φ(t, ~x), (5.8.32) P P ± − where “ + ” sign represents for scalar and “ ” represents for pseudo-scalar. − In terms of the creation and annihilation operator formalism of field operator,
3~ d k iω t+i~k ~x iω t i~k ~x φ(t, ~x) = a(k)e ~k + a (k)e ~k , (5.8.33) 3/2 − · † − · (2π) 2ω~ Z k under the parity action, we canp see the creation and annihilation operator is transformed in the way a a~k ~k UP U † = − . (5.8.34) a† P a† ~k ! ± ~k ! −
5.8.3.2 Time reversal The time reversal is defined as x T xµ. (5.8.35) µ −→− Under the time reversal action, the field operator is transformed as
φ0(t, ~x) = U 0 φ(t, ~x)U 0 † = φ( t, ~x), (5.8.36) T T ± − where the operator UT0 is an anti-unitary operator and can be constructed as the product of a unitary operator UT and a complex conjugation K, namely UT0 = UT K. As an anti-linear and anti-unitary operator, K satisfies the following relations
KiK† = i; (5.8.37) − Kα Kβ = α β ∗; (5.8.38) h | i h | i K(λ α + λ β ) = λ∗ α + λ∗ β . (5.8.39) 1| i 2| i 1| i 2| i
96 Under the time reversal action, the canonical momentum as the time derivative of field operator, i.e., Πφ = φ˙, is transformed as
φ˙0(t, ~x) = U 0 φ˙(t, ~x)U 0 † = ( 1)φ˙( t, ~x). (5.8.40) T T ± − − From the commutative relation
φ(t, ~x), φ˙(t, ~y) = iδ(3)(~x ~y), (5.8.41) − h i and the anti-unitary operator U K relation U Ki(U K) = i, we can see that the time T T T † − reversal operator has to be anti-unitary.
5.8.3.3 Charge conjugation Charge conjugation interchanges the particle with its antiparticle defined by
UC C ψ(x) ψ (x) = U ψ(x)U † = ψ†(x); (5.8.42) −−→ C C UC C ψ†(x) ψ† (x) = U ψ†(x)U † = ψ(x), (5.8.43) −−→ C C (5.8.44) which is equivalent with b~ c~ U k U † = k . (5.8.45) C c C b ~k ~k In pseudo-nucleon theory, under the charge conjugation, the “nucleon” state “N”(k) is | i transformed as UC “N”(k) = UC b† vac = c† vac = “N”(k) . (5.8.46) | i ~k| i ~k| i | i Note that the vacuum state is unchanged under UP , UT and UC action.
Thm 5.8.3.1.
iH(t t ) iH(t t ) χ(t ) e− f − i ψ(t ) = χ(t ) e− f − i ψ(t ) . (5.8.47) h f | | i i h f | | i i Proof.
iH(t t ) iH(t t ) χ(t ) e− f − i ψ(t ) = χ(t ) U † U e− f − i U † U ψ(t ) h f | | i i h f | C C C C | i i iH(t t ) = χ(t ) e− f − i ψ(t ) (5.8.48) h f | | i i due to the Hamiltonian is invariant under charge conjugation, namely UC HUC† = H.
In the “nucleon” scattering, we have the Feynman diagram
p1 p0 “N” 1 “N”
(2) p1 − p10 i = + (p10 p20 ) A ↔
“N” “N” p2 p20
97 to compute the scattering amplitude
i i i (2) = ( ig)2 + . (5.8.49) A − (p p )2 µ2 (p p )2 µ2 1 − 10 − 1 − 20 − For anti-“nucleon” scattering, we have
p1 p10 “N” “N”
(2) q − q0 i = + (p10 p20 ) A0 ↔
“N” “N” p2 p20
(2) 2 i i i 0 = ( ig) + . (5.8.50) A − (p p )2 µ2 (p p )2 µ2 1 − 10 − 1 − 20 − According to Theorem 5.8.3.1, without specific calculations, we hvae
“N”(p0 ), “N”(p0 ) S “N”(p ), “N”(p ) = “N”(p0 ), “N”(p0 ) S “N”(p ), “N”(p ) , h 1 2 | | 1 2 i h 1 2 | | 1 2 i (5.8.51) namely i (2) = i (2). A A0 Note: Symmetry is crucial in quantum field theory because it has nothing to do with Feynman diagrams or perturbative expansion.
5.8.3.4 The CPT theorem 5.9 Micro-causality (Locality)
5.9.1 Real scalar field theory Thm 5.9.1.1. In free scalar field theory, the micro-causality relation [φ(t, ~x), φ(t, ~y)] = 0 is ensured by equal-time commutation
[φ(x), φ(y)] = 0, for (x y)2 < 0 (5.9.1) − assumed in canonical quantization.
Statement 5.9.1.1. φ(t = 0, ~x) vac is explained as the creation of a particle at position | i ~x.
Proof. The field has the expansion solution (5.6.11)
3~ d k ik x ik x φ(t, ~x) = (β(k)e− · + β†(k)e · ), (5.9.2) (2π)32ω Z ~k where
k = β†(k) vac , (5.9.3) | i | i p k = (2π)32ω δ(3)(~p ~k). (5.9.4) h | i ~k −
98 And at time 0, acting on the vacuum, the field gives rise to
3~ d k ik x φ(0, ~x) vac = e− · k . (5.9.5) | i (2π)32ω | i Z ~k In the momentum representation, we have
3~ d k i~k ~x p φ(0, ~x) vac = e− · p k h | | i (2π)32ω h | i Z ~k i~p ~x = e− · . (5.9.6)
Recall in non-relativistic quantum mechanics, we have
Xˆ ~x = ~x ~x , (5.9.7) | i | i Pˆ ~p = ~p ~p . (5.9.8) | i | i 1 i~p ~x ~p ~x = e− · . (5.9.9) h | i (2π)3/2
Hence φ(t = 0, ~x) vac can be viewed as a particle at position ~x. | i
Statement 5.9.1.2. D(x y) = vac φ(x)φ(y) vac represents the amplitude of creating − h | | i a particle at y and annihilating it at x, namely the propagation of the particle from y to x. x D(x y) = − y
Proof.
(+) ( ) D(x y) = vac φ (x)φ − (y) vac − h | (+) ( ) | i = vac [φ (x), φ − (y)] vac h | | i d3~kd3~k 0 ~ ~ ik x+ik0 y = 3 [a(k), a†(k0)]e− · · (2π) 2ω~ 2ω~ Z k k0 3~ d kp ik (x y) = e− · − (2π)32ω Z ~k = 0. (5.9.10) 6
Note: In space-like region (x y)2 < 0, D(x y) = 0 or D(y x) = 0 suggests that the − − 6 − 6 causality is violated.
Lemma 5.9.1.1. D(x y) is a Lorentz invariant scalar. − Proof. Note that in relation (5.9.10), we have obtained that
3~ d k ik (x y) D(x y) = e− · − , (5.9.11) − (2π)32ω Z ~k 99 d3~k where 3 is a Lorentz invariant measure and the term in the exponent function is (2π) 2ω~k a LorentzR invariant scalar product. Therefore, we have 3~ d k iΛk Λ(x y) Λ(D(x y)) = e− · − − (2π)32ω Z ~k 3~ d k ik (x y) = e− · − (2π)32ω Z ~k = D(x y). (5.9.12) −
Lemma 5.9.1.2. D(x y) = D(y x) . − |tx=ty − |tx=ty Proof.
(D(x y) D(y x)) − − − tx=ty 3 d k i~k (~x ~y) i~k (~x ~y) = (e · − e− · − ) (2π)32ω − Z ~k~ = 0. (5.9.13)
Note:[φ(x), φ(y)]tx=ty = 0.
Lemma 5.9.1.3. In special relativity, for space-like separated events (x y)2 < 0, there − always exists Λ giving rise to tx = ty. Proof. t t
x y Λ y x
tx > ty tx = ty
Lemma 5.9.1.4. (D(x y) D(y x))(x y)2<0 = 0. − − − − Proof.
Λ (D(x y) D(y x))(x y)2<0 (D(Λ(x y)) D(Λ(y x)))tx=ty = 0. (5.9.14) − − − − −−→ − − −
100 Proof. Proof for the Theorem 5.9.1.1.
+ + [φ(x), φ(y)](x y)2<0 = φ (x), φ−(y) + φ−(x), φ (y) (x y)2<0 − − = (D(x y) D(y x))(x y)2<0 − − − − Λ (D(Λ(x y)) D(Λ(y x))) −−→ − − − tx=ty = 0. (5.9.15)
N ote 1. Micro-causality is violated in the sense of D(x y) = 0, but such violation can not − 6 be observed in the sense of quantum measurement due to [φ(x), φ(y)](x y)2<0 = 0 (compatible observable). −
N ote 2. Micro-causality predicts the existence of anti-particle for every particle.
x y D(x y) = = D(y x) = − − y x Particle Anti-particle (eg. Electron) (eg. Positron)
5.9.2 Complex scalar field theory Theorem: The micro-causality in free scalar field theory,
[ψ(x), ψ(y)] = 0, for (x y)2 < 0. (5.9.16) − is ensured by the equal time commutator [ψ(t, ~x), ψ(t, ~y)] = 0 in the canonical quantization procedure.
5.10 Remarks on canonical quantization
Advantages: i. Particle interpretation; (Hamiltonian has a discrete spectrum of particle.)
ii. Particle creation and annihilation;
iii. Micro-Causality and anti-particle;
iv. Classical-quantum correspondence,
~ 0 Heisenberg equations of motion → Hamilton’s equations. −−−−→ Disadvantages:
i. No explicit Lorentz-invariance in Hamiltonian formulation, e.g. equal-time commuta- tor depending on time.
ii. Infinity large vacuum energy.
iii. D(x y) = 0 implies the violation of micro-causality. − 6
101 Table 5.1: Comparison of canonical quantization with path integral quantization
Canonical quantization Path integral quantization Hamiltonian formalism Lagrangian formalism Formalism Operator-valued function Ordinary-valued function Particle interpretation Explicit Not explicit Classical-quantum Hamilton’s equations Action/Action principle correspondence Heisenberg equations ⇐⇒ Lorentz covariant Not explicit Explicit Vacuum energy + + ∞ ∞
102 Lecture 6 Feynman Propagator, Wick’s Theorem and Dyson’s Formula
About Lecture IV:
[Luke] P.P. 65-80; • [Tong] P.P. 38-41 & 47-48; • [Coleman] P.P. 70-82; • [PS] P.P. 29-33 & 77-90; • [Zhou] P.P. 70-72 & 127-146. • 6.1 Retarded Green function
6.1.1 Real scalar field theory with a classical source Def 6.1.1. In free scalar theory, Lagrangian with quadratic terms of fields has the form 1 1 = ∂ φ∂µφ m2φ2. (6.1.1) L0 2 µ − 2 More details refer to previous Section 3.2.5 and 5.7.1. Def 6.1.2. With interaction, Lagrangian can have non-quadratic terms of fields such as λ = , = φ4, (6.1.2) L L0 − Lint Lint 4! where λ is coupling constant. In the model of real scalar field theory with a classical source, the Lagrangian is given by = + ρ(x)φ(x), (6.1.3) L L0 where ρ(x) is a real number function which can be turned on and off, namely
ρ(x)t>tf = ρ(x)t ρ(x) ρ(x)t ti tf t 103 Note: ρ(x) is not a field operator, just an ordinary function. The corresponding canonical momentum and Hamiltonian can be found as ∂ Πφ = L = φ,˙ (6.1.5) ∂φ˙ 1 1 1 = Π φ˙ = φ˙2 + ( φ)2 + m2φ2 ρ(x)φ(x). (6.1.6) H φ − L 2 2 ∇ 2 − The equation of motion can be derived in the way ∂ 2 L = m φ + ρ(x), ∂µ( L ) = φ, (6.1.7) ∂φ − ∂(∂µφ) 2 ( + m )φ = ρ(x). (6.1.8) To understand the meaning of the source term ρ(x), as an example, see electrodynamics 4π 2ϕ = ρ (6.1.9) ∇ − ε where ϕ is electric potential determined by charge-density distribution ρ. 6.1.2 The retarded Green function With the equation of motion 2 ( + m )φ = ρ(x), (6.1.10) the boundary condition is set as φ(x) t= = φ0(x), (6.1.11) | −∞ where φ (x) is free field since t , i.e., there is no interaction. 0 → −∞ To solve the equation of motion, we introduce the Green function, namely φ(x) = φ (x) + i d4yD (x y)ρ(y), (6.1.12) 0 R − Z where D (x y) is named as retarded Green function satisfying R − ( + m2)D (x y) = iδ(4)(x y) (6.1.13) R − − − with D (x y) = 0, x0 < y0. R − Note: The term “Retarded” means that the future is determined by the past. As the diagram shown as t I I: forward light-cone, time-like region II II x II: outside light-cone, space-like region III: backward light-cone, time-like region III 104 in which the origin denotes y and the forward light-cone denotes x, we have = 0 Domain II, III. D (x y) (6.1.14) R − = 0 Domain I, x0 > y0. 6 6.1.3 Analytic formulation of D (x y) R − Def 6.1.3. The retarded Green function can be defined as D (x y) = θ(x0 y0)[φ(x), φ(y)] R − − = θ(x0 y0) vac [φ(x), φ(y)] vac − h | | i x0>y0 ===== D(x y) D(y x) − − − 3 ik (x y) ik (x y) x0>y0 d ~k e e ===== − · − · − , (6.1.15) (2π)3 2w − 2w Z ~k ~k ! where d3~k e ik (x y) D(x y) = − · − . (6.1.16) − (2π)3 2w Z ~k Thm 6.1.3.1. With complex analysis, the retarded Green function has the formula 4 0 0 d k i ik (x y) D (x y) = θ(x y ) e− · − R − − (2π)4 k2 m2 ZCR − 3~ 0 ik0(x0 y0) 0 0 d k dk ie− − i~k (~x ~y) = θ(x y ) e · − (6.1.17), − (2π)3 2π (k0 w )(k0 + w ) Z "ICR − ~k ~k # in which the contour CR is given by Im(k0) w 0 −w~k ~k Re(k ) CR Proof. Since the retarded Green function satisfies the equation, ( + m2)D (x y) = iδ(4)(x y), (6.1.18) R − − − we have 4 d k i ik (x y) D (x y) = e · − . (6.1.19) R − (2π)4 k2 m2 Z − But the above integral is not well defined due to two poles, namley 1 1 1 2 2 = 2 2 = 0 0 . (6.1.20) k m k w (k w~ )(k + w~ ) − 0 − ~k − k k Besides, the retarded condition, i.e., D (x y) = 0, x0 < y0 (6.1.21) R − has to be satisfied. In the following, we construct two contour integrals and then apply the residue theorem. 105 In the region x0 < y0, we choose the contour C on the upper half plane • R0 Im(k0) Im(k0) No contribution CR0 =⇒ w 0 w 0 −w~k ~k Re(k ) −w~k ~k Re(k ) to compute 0 dk i ik0(x0 y0) IA = 0 0 e− − . (6.1.22) C 2π (k w~ )(k + w~ ) I R0 − k k The reason we choose the upper half plane is 0 i(iImk0)(x0 y0) Imk + Imk0(x0 y0) e− − → ∞ e − 0. (6.1.23) −−−−−−−−→ → Since no pole terms in CR0 , with the residue theorem, we have 0 dk i ik0(x0 y0) 0 0 IA = 0 0 e− − = 0 (x < y ), (6.1.24) C 2π (k w~ )(k + w~ ) I R0 − k k namley D (x y) = 0. (6.1.25) R − In the region x0 > y0, we choose the contour C on the lower half plane • R Im(k0) Im(k0) =⇒ 0 0 −w~ w~k Re(k ) Re(k ) k w −w~k ~k CR Integral vanishing on boundary to compute 0 dk i ik0(x0 y0) I = e− − , (6.1.26) R 2π (k0 w )(k0 + w ) ICR − ~k ~k because 0 i(iImk0)(x0 y0) Imk Imk0(x0 y0) e− − →−∞ e − 0. (6.1.27) −−−−−−−−→ → 106 Apply the residue theorem to compute the contour integral IR ik0(x0 y0) ik0(x0 y0) i e− − i e − IR = ( 2πi) + ( 2πi) 2π − 2w~ 2π − 2w~ k k0=w k k0= w ~k − − ~k 0 0 0 0 0 0 e ik (x y ) e ik (x y ) = − − − , (6.1.28) 2w~ − 2w~ k k0=w k k0= w ~k ~k − therefore 0 0 0 0 0 0 3~ iw~ (x y ) iw~ (x y ) x >y d k e− k − e k − i~k (~x ~y) D (x y) ===== e · − R − (2π)3 2w − 2w Z ~k ~k ! 3 ik (x y) ik (x y) x0>y0 d ~k e e ===== − · − · − (2π)3 2w − 2w Z ~k ~k ! = θ(x0 y0)[φ(x), φ(y)] − = θ(x0 y0) vac [φ(x), φ(y)] vac . (6.1.29) − h | | i 6.1.4 Canonical quantization of a real scalar field theory with a classical source With the analytical form of the retarded Green function, we have 3~ 4 d k 1 0 0 ik (x y) ik (x y) φ(x) = φ0(x) + i d y 3 θ(x y ) e− · − e · − ρ(y). (6.1.30) (2π) 2w~ − − Z Z k As x0 + , we have ρ(x0) = 0 and θ(x0 y0) = 1, so that → ∞ − 3 x0 + d ~k → ∞ 4 ik (x y) ik (x y) φ(x) ======φ0(x) + i d y 3 e− · − e · − ρ(y). (6.1.31) (2π) 2w~ − Z Z k Define the Fourier transform function 4 ik y ρ˜(k) = d ye · ρ(y), ρ˜( k) =ρ ˜∗(k), (6.1.32) − Z where k2 = m2 andρ ˜(k) is a known and fixed function. We have 3 x0 + d ~k → ∞ ik (x y) ik (x y) φ(x) ======φ0(x) + i 3 e− · − ρ˜(k) e · − ρ˜( k) (2π) 2w~ − − Z k 0 3~ x + d k ik x ik x ======→ ∞ b e− · + b† e · , (6.1.33) 3/2 ~k ~k (2π) 2w~ Z k where p i b = a + ρ˜(k), (6.1.34) ~k ~k 3/2 (2π) 2w~k satisfying the commutative relation p (3) ~ ~ [b~k, b~† ] = δ (k k0). (6.1.35) k0 − 107 As x0 + , the Hamiltonian takes the form → ∞ 3 3 : H := d ~x : := d ~kw b† b , (6.1.36) H ~k ~k ~k Z Z and the vacuum energy can be calculated as E = vac : H : vac vac h | | i d3~k = w ρ˜(k) 2 ~k (2π)32w | | Z ~k w dN(~k) (6.1.37) ≡ ~k Z with N dN(~k). dN(~k) is the expectation value of the number of particles with ≡ momentum ~k and N is the expectation value of the total number of particles. R 6.2 Advanced Green function Def 6.2.1. The advanced Green function is defined in the way ( + m2)D (x0 y0) = iδ4(x y); x A − − − (6.2.1) D (x y) = 0, x0 > y0. A − In terms of the advanced Green function, the past is determined by the future, see the diagrammatic interpretation, t I I: forward light-cone: time-like region II II x II: outside light-cone: space-like region III: backward light-cone: time-like region III where the origin denotes y and the backward-light cone denotes x, namley = 0 Domain I, II. D (x y) (6.2.2) A − = 0 Domain III, x0 < y0. 6 The analytical formulation of the advanced Green function D (x y) has the form A − D (x y) = θ(y0 x0)[φ(x), φ(y)], (6.2.3) A − − which can be defined with the contour integral 4 d k i ik (x y) D (x y) = e− · − (6.2.4) A − (2π)4 k2 m2 ZCA − with the contour CA given by 108 Im(k0) CA w −w~k ~k Re(k0) 6.3 The Feynman Propagator D (x y) F − Remark: Various approaches to defining the Feynman propagator: 1. Time-ordered product; 2. Wick’s theorem; 3. Green function; 4. Contour integrals; 5. iε 0+. → 6.3.1 Definition of Feynman propagator with time-ordered product Def 6.3.1. Feynman propagator can be defined in terms of the time-ordered product D (x y) = vac T φ(x)φ(y) vac , (6.3.1) F − h | | i where “T” stands for “time-ordering”, specifically φ(x)φ(y) x0 > y0; T φ(x)φ(y) = (6.3.2) φ(y)φ(x) x0 < y0. Note 1. φ(x)φ(y) are scalar fields, i.e., spinless particles, and they obey Bosonic statistics, namley inside time ordering, we have T φ(x)φ(y) = T φ(y)φ(x). (6.3.3) Note 2. In x0 = y0, ~x = ~y space-like region, we define 6 θ(x0 y0) + θ(y0 x0) = 1, (6.3.4) − − DF (x y) = vac φ(x)φ(y) vac . (6.3.5) − |tx=ty h | | i Note 3. As x0 = y0, ~x = ~y, Feynman propagator is ill-defined, D (x y) = D (0) = vac φ(x)φ(y) vac F − F h | | i d3~k 1 = (2π)3 2w Z ~k + . (6.3.6) → ∞ 109 Note 4. D (x y) = θ(x0 y0) vac φ(x)φ(y) vac + θ(y0 x0) vac φ(y)φ(x) vac F − − h | | i − h | | i = θ(x0 y0)D(x y) + θ(y0 x0)D(y x). (6.3.7) − − − − x x0 > y0: Amplitude of propagation of particle from y to x. y y x0 < y0: Amplitude of propagation of particle from x to y. x Note 5. D (x y) is defined for free theories, not for interaction theories. F − 6.3.2 Definition of Feynman propagator with contractions Thm 6.3.2.1. The Feynman propagator can be defined in terms of the contraction, namely D (x y) = φ(x)φ(y) F − = T φ(x)φ(y) : φ(x)φ(y): . (6.3.8) − Note: Definition of Feynman propagator with contraction is the simplest example for Wick’s theorem. See the following Section 6.4. Proof. Field operator can be divided into two parts (5.7.49) + φ(x) = φ (x) + φ−(x) (6.3.9) where φ+(x) is positive frequency part associated with annihilation operator and φ+(x) is negative frequency part associated with creation operator. With normal ordering, operator φ(x)φ(y) gives rise to + + : φ(x)φ(y) : = : (φ (x) + φ−(x))(φ (y) + φ−(y)) : + + + + = φ (x)φ (y) + φ−(x)φ−(y) + φ−(x)φ (y) + φ−(y)φ (x).(6.3.10) For time ordering: When x0 > y0, • T φ(x)φ(y) = φ(x)φ(y) + + = (φ (x) + φ−(x))(φ (y) + φ−(y)) (6.3.11) + + + + = φ (x)φ (y) + φ−(x)φ−(y) + φ−(x)φ (y) + φ (x)φ−(y). So we have 0 0 x >y + + T φ(x)φ(y) : φ(x)φ(y): ===== φ (x)φ−(y) φ−(y)φ (x) − + − = φ (x), φ−(y) + = vac φ (x), φ−(y) vac h | | i = vac φ(x)φ(y) vac h | | i = θ(x0 y0)D(x y). (6.3.12) − − 110 When x0 < y0, similarly, we have • T φ(x)φ(y) : φ(x)φ(y) := θ(y0 x0)D(y x). (6.3.13) − − − From above all, we have the conclusion relations T φ(x)φ(y) =: φ(x)φ(y) : +D (x y) (6.3.14) F − and D (x y) = vac T φ(x)φ(y) vac . (6.3.15) F − h | | i 6.3.3 Definition with the Green function Def 6.3.2. The Feynman propagator D (x y) is the Green function satisfying F − ( + m2)D (x y) = iδ(4)(x y) (6.3.16) x F − − − with D (x y) = θ(x0 y0)D(x y) + θ(y0 x0)D(y x). (6.3.17) F − − − − − Note that the Feynman propagator is neither the retarded Green function given by D (x y) = θ(x0 y0)D(x y) (6.3.18) R − − − nor the advanced Green function given by D (x y) = θ(y0 x0)D(y x), (6.3.19) A − − − instead of the superposition of the retarded Green function and advanced Green function, namley D (x y) = D (x y) + D (x y). (6.3.20) F − R − A − 6.3.4 Definition with contour integrals Thm 6.3.4.1. The Feynman propagator can be defined as the complex integral shown as 4 d k i ik (x y) D (x y) = e− · − (6.3.21) F − (2π)4 k2 m2 ZC1,C2 − where x0 > y0, performing clockwise contour C on the lower half-plane. 1 (6.3.22) x0 < y0, performing anti-clockwise contour C on the upper half-plane. 2 Proof. In the condition x0 > y0, we perform the contour integral with the clockwise contour • C1, 111 Im(k0) Im(k0) −w~k =⇒ 0 0 w~k Re(k ) Re(k ) w~k C1 because 0 i(iIm(k0))(x0 y0) Im(k0)(x0 y0) Im(k ) : e− − = e − 0. (6.3.23) → −∞ → With the residue theorem, we have 0 0 0 0 0 0 ik (x y ) iω~ (x y ) dk ie− − ( 2πi) i iω (x0 y0) e− k − = − e− ~k − = , (6.3.24) 2π (k0 + ω )(k0 ω ) 2ω 2π 2ω IC1 ~k − ~k ~k ~k 0 where the pole is k = w~k, therefore 3 ik (x y) x0>y0 d ~k e D (x y) ===== − · − = θ(x0 y0)D(x y). (6.3.25) F − (2π)3 2ω − − Z ~k In the condition y0 > x0, we perform the anti-clockwise C contour integral, • 2 Im(k0) Im(k0) C2 −w~k =⇒ 0 0 w~k Re(k ) Re(k ) −w~k because 0 i(iIm(k0))(x0 y0) Im(k0)(x0 y0) Im(k ) + : e− − = e − 0. (6.3.26) → ∞ → We apply the residue theorem to the above contour integral 0 0 0 0 0 0 ik (x y ) iω~ (x y ) dk ie− − (2πi) i iω (x0 y0) e k − = e ~k − = (6.3.27) 2π (k0 + ω )(k0 ω ) 2ω 2π 2ω IC2 ~k − ~k − ~k ~k with the pole at k0 = w , thus − ~k 3 ik (x y) y0>x0 d ~k e D (x y) ===== · − = θ(y0 x0)D(y x). (6.3.28) F − (2π)3 2ω − − Z ~k 112 6.3.5 Definition of Feynman propagator with iε 0+ prescription → Thm 6.3.5.1. The Feynman propagator has the iε 0+ prescription → 4 d k i ik (x y) D (x y) = e− · − , (6.3.29) F − (2π)4 k2 m2 + iε Z − where ε 0+. → Proof. Because ε is an infinitesimal quantity, we have [k0 (ω iε)][k0 + (ω iε)] − ~k − ~k − = (k0)2 (ω iε)2 − ~k − = (k0)2 ω2 + iε, (6.3.30) − ~k therefore 1 1 1 = . (6.3.31) k2 m2 + iε k0 (ω iε) · k0 + (ω iε) − − ~k − ~k − And the distribution of two poles is shown in the following Im(k0) −w~k + iε 0 w~k − iε Re(k ) 6.4 Wick’s theorem 6.4.1 The Time-ordered product Def 6.4.1. The time-ordered product is defined in the way t > t > > t φ(x )φ(x ) φ(x ); 1 2 ··· n 1 2 ··· n t2 > t1 > > tn φ(x2)φ(x1) φ(xn); T φ(x1)φ(x2) φ(xn) = . ··· . ··· (6.4.1) ··· . . tn > tn 1 > > t1 φ(xn)φ(xn 1) φ(x1). − ··· − ··· Operators of the earliest time on the rightmost; • Operators of next to the earliest time on the next to the rightmost; • Operators of next to the latest time on the next to the leftmost; • Operators of the latest time on the leftmost. • 113 6.4.2 Example for Wick’s theorem n = 3 and n = 4 For simplicity, change the notation as φi = φ(xi). E.g. 1. When n = 2, we have T φ1φ2 = φ1φ2 + : φ1φ2 : . (6.4.2) E.g. 2. When n = 3, we have T (φ1φ2φ3) =: φ1φ2φ3 : + φ1φ2φ3 + φ1φ2φ3 + φ1φ2φ3 . (6.4.3) Check the above relation in the following way. Suppose we have t3 = min(t1, t2, t3), (6.4.4) therefore T (φ1φ2φ3) = T (φ1φ2)φ3 + = T (φ1φ2)φ3 + φ3−T (φ1φ2) + [T (φ1φ2), φ3−] . (6.4.5) I II Doing the calculation separately.| {z } | {z } + I = T (φ1φ2)φ3 + φ3−T (φ1φ2) + + = : φ1φ2 : φ3 + φ1φ2 φ3 + φ3− : φ1φ2 : + φ1φ2 φ3− + = φ1φ2 φ3+ : φ1φ2φ3 : + : φ3−φ1φ2 : = φ1φ2 φ3+ : φ1φ2φ3 :; (6.4.6) II = [T φ1φ2, φ3−] = : φ1φ2 :, φ3− + + + + = [φ1 φ2 + φ1−φ 2− + φ1−φ2 + φ2−φ1 , φ3−] + + + + = [φ1 φ2 , φ3−] + [φ1−φ2 , φ3−] + [φ2−φ1 , φ3−] + + + + + + = [φ1 , φ3−]φ2 + [φ2 , φ3−]φ1 + [φ2 , φ3−]φ1− + [φ1 , φ3−]φ2− = φ1φ3 φ2 + φ2φ3 φ1. (6.4.7) Therefore, I + II =: φ1φ2φ3 : + φ1φ2φ3 + φ1φ2φ3 + φ1φ2φ3 . (6.4.8) E.g. 3. When n = 4, we have T (φ1φ2φ3φ4) =: φ1φ2φ3φ4 : + φ1φ2 : φ3φ4 : + : φ1φ2 : φ3φ4 + φ1φ4 : φ2φ3 : + : φ1φ4 : φ2φ3 + φ1φ3 : φ2φ4 : + : φ1φ3 : φ2φ4 + φ1φ2 φ3φ4 + φ1φ3 φ2φ4 + φ1φ4 φ2φ3 . (6.4.9) 2 Note that there is 6 one-contraction terms, i.e., C4 = 6, and 3 two-contraction 1 2 2 terms, i.e., 2 C4 C2 = 3. 114 6.4.3 The Wick theorem Thm 6.4.3.1 (Wick’s theorem). Time-ordered product T (φ(x )φ(x ) φ(x )) can be ex- 1 2 ··· n pressed as a summation of products of Feynman propagator (or contractions) and normal- ordered products, namely T (φ φ φ ) = : φ φ φ : no-contraction 1 2 ··· n 1 2 ··· n + φ1φ2 : φ3 φn : only one contraction 2 ··· +Cn terms with one contraction ) + φ1φ2 φ3φ4 : φ5 φn : two contractions 1 2 2 ··· + 2 CnCn 2 terms with two contractions ) − + ··· +all possible terms with [n/2] 1 contractions − + φ1φ2 φ3φ4 φn 1φn ··· − n is even with n/2 contractions + ) ··· + φ1φ2 φ3φ4 φn 2φn 1 φn ··· − − n is odd with (n 1)/2 contractions + ) − ··· (6.4.10) Proof. The proof is done by induction. 1. As n = 1, we have T φ(x) =: φ(x) : which is trivial. 2. As n = 2, we have the Feynman propagator defined with contraction as the example of Wick’s theorem T (φ1φ2) =: φ1φ2 : + φ1φ2 . (6.4.11) Proof see Theorem 6.3.2.1. 3. Assume Wick’s theorem is solid for n 1, then denote − T (φ1φ2 φn 1) = W (φ1φ2 φn 1). (6.4.12) ··· − ··· − Suppose t = min(t , t , , t ), then we have n 1 2 ··· n T (φ1φ2 φn) = W (φ1φ2 φn 1)φn − ··· ··· + = T (φ1φ2 φn 1)(φn + φn−) − ··· + = T (φ1φ2 φn 1)φn + φn−T (φ1φ2 φn 1) ··· − ··· − +[T (φ1φ2 φn 1), φn−], (6.4.13) ··· − + where the term T (φ1φ2 φn 1)φn + φn−T (φ1φ2 φn 1) are all possible contractions ··· − ··· − without φn and the term [T (φ1φ2 φn 1), φn−] is all possible contractions with φn. ··· − 6.4.4 Remarks on Wick’s theorem Remark 1. From Wick’ theorem, we can immediately see vac T (φ φ φ ) vac = 0 (6.4.14) h | 1 2 3 | i due to no place for all contraction. 115 Remark 2. From Wick’s theorem, we have the diagrammatical representation 1 2 1 2 1 2 vac T (φ φ φ φ ) vac = + + h | 1 2 3 4 | i 3 4 3 4 3 4 (6.4.15) where the connected line stands for contraction φiφj = i j (6.4.16) Remark 3. For normal ordering, : φ1φ2 : is an operator. For the contraction φ3φ4, it is just a number. : φ1φ2φ3φ4 : =: φ1φ2 : φ3φ4 . (6.4.17) Remark 4. ( ) ( ) : φ φ φ :, φ − =: φ φ φ , φ − : . (6.4.18) 1 2 ··· n n+1 1 2 ··· n n+1 h i h i Remark 5. For different fields which are commutative, such as [φ(x1), ψ(x2)] = 0, (6.4.19) the contraction is 0, namely φ(x1)ψ(x2) = 0. (6.4.20) Remark 6. With normal ordering, the annihilation and creation operators have the relation : aa† + a†a :=: a†a + a†a := 2a†a; (6.4.21) : aa† a†a =: aa† aa† := 0. (6.4.22) − − 6.5 Dyson’s formula With interaction, the Hamiltonian can be divided into two parts H = H0 + Hint (6.5.1) where H is the total Hamiltonian and H0 is the free particle’s Hamiltonian and Hint is the interaction part. Fact 1. Equation of motion of interacting fields can not be exactly solvable in most cases. Fact 2. Equation of motion of free fields can be exactly solved and the creation and anni- hilation operators can be well-defined. As a sort of way out for interaction theory, we can choose a suitable framework in which field operators have the evolution of free operators. 116 6.5.1 Interaction Picture (Dirac Picture) A picture defines a choice of operators and state vectors to preserve matrix elements or probability amplitude. For example, the Shr¨odingerpicture defines as the evolution of state vector with fixed operators, namely O (t) = O (0), S S (6.5.2) i d ψ(t) = H ψ(t) . dt | iS | iS In interaction picture or Dirac picture, the operator and state vector are denoted as OI (t) and ψ(t) respectively, and they have the following constraints. | iI Constraint 1. ψ(t) O ψ(t) = ψ(t) O (t) ψ(t) . (6.5.3) Sh | S| iS I h | I | iI Constraint 2. OS(0) = OI (0), (6.5.4) ψ(0) = ψ(0) . (6.5.5) | iS | iI In interaction picture, the operator is described in the way iH0t iH0t OI (t) = e OSe− , (6.5.6) and the time evolution of operator is governed by the free Hamiltonian, i.e., d i O (t) = [O (t),H ] . (6.5.7) dt I I 0 In terms of state vector, we have ψ(t) = eiH0t ψ(t) . (6.5.8) | iI | iS Thm 6.5.1.1. The time evolution equation of state vector in interaction picture takes the form d i ψ(t) = H (t) ψ(t) (6.5.9) dt| iI I | iI iH0t iH0t with HI (t) = e Hinte− . Proof see Section 2.2.3. Remark 1. When Hint = 0, the state vector in interaction picture evolves as the Shr¨odinger equation, namely ψ(t) = ψ(t) . | iI | iH Remark 2. More details about comparison between Shr¨odingerpicture, Heisenberg picture and interaction picture see previous Section 2.2.3. 117 6.5.2 Unitary evolution operator In the interaction picture, we are supposed to solve the time evolution of equation shown as d i ψ(t) = H ψ(t) , (6.5.10) dt| iI I | i iH0t iH0t where HI (t) = e Hinte− . With the new-defined unitary evolution operator UI (t, t0), we have the solution ψ(t) = U (t, t ) ψ(t ) , (6.5.11) | iI I 0 | 0 iI where i d U (t, t ) = H (t)U (t, t ), dt I 0 I I 0 (6.5.12) U (t , t ) = 1. I 0 0 Note: The scattering operator (S-matrix) is defined as. S = U (+ , ). (6.5.13) I ∞ −∞ 6.5.3 Time-dependent perturbation theory Applying the recursive solution, we get t U (t, t ) = 1 i dt H (t )U (t , t ). (6.5.14) I 0 − 1 I 1 I 1 0 Zt0 In the first-order approximation, we have the solution t U (1)(t, t ) = 1 i dt H (t ). (6.5.15) I 0 − 1 I 1 Zt0 In terms of the second-order approximation, t U (2) = 1 i dt H (t )U (1)(t , t ) I − 1 I 1 I 1 0 Zt0 t t t1 = 1 + ( i) dt H (t ) + ( i)2 dt dt H (t )H (t ). (6.5.16) − 1 I 1 − 1 2 I 1 I 2 Zt0 Zt0 Zt0 Then for n-order approximation, we obtain n t t1 tn 1 U (n)(t, t ) = 1 + ( i)n dt dt − dt H (t )H (t ) H (t ). (6.5.17) I 0 − 1 2 ··· n I 1 I 2 ··· I n n=1 t0 t0 t0 X Z Z Z As n + , we can see t t , corresponding with the diagram shown below. → ∞ n → 0 t0 tn tn 1 ··· t2 t1 t − (6.5.18) Therefore, we have U(t , t ) = 1. And when N + , we have the exact solution. n 0 → ∞ + ∞ Tf t1 tn 1 U (T ,T ) = 1 + ( i)n dt dt − dt H (t )H (t ) H (t ). (6.5.19) I f i − 1 2 ··· n I 1 I 2 ··· I n n=1 Ti Ti Ti X Z Z Z 118 6.5.4 Dyson’s formula The term with n = 2 in the exact solution (6.5.19) has the expression Tf t1 U (2)(T ,T ) = ( i)2 dt dt H (t )H (t ) I f i − 1 2 I 1 I 2 ZTi ZTi T t t >t f 1 =====(1 2 i)2 dt dt T (H (t )H (t )) − 1 2 I 1 I 2 ZTi ZTi (6.5.20) Tf t1 Tf t2 2 1 = ( i) dt1 dt2 + dt2 dt1 T (HI (t1)HI (t2)). − 2! Ti Ti Ti Ti Z Z Z Z (2) (2) I I 1 2 | {z } | {z } 1 (2) (2) where 2! is normalization factor and the integration domains I1 and I2 are shown as respectively t2 t2 Tf Tf (2) (2) I1 I2 Ti Ti Ti Tf t1 Ti Tf t1 (2) (2) (2) We can get the two integration domains into one union, namley I = I1 I2 . t2 S Tf I2 Ti Ti Tf t1 (2) Then UI (Tf ,Ti) can be rewritten as ( i)2 Tf Tf U (2)(T ,T ) = − dt dt T (H (t )H (t )), (6.5.21) I f i 2! 1 2 I 1 I 2 ZTi ZTi where the time ordering is naturally introduced here. Remarks: 1). Inside the time ordering, operators are commutative and become classical, namely T (HI (t1)HI (t2)) = T (HI (t2)HI (t1)). (6.5.22) 119 2). T (H (t )H (t )) is a symmetric function under permutation t t . I 1 I 2 1 ↔ 2 3). T (H (t )H (t ) H (t )) is a symmetric function under permutation (t , t , , t ), I 1 I 2 ··· I n 1 2 ··· n and note that #SN = N!. Thm 6.5.4.1 (Dyson’s formula). The Dyson’s formula gives rise to the solution of unitary operator as + ∞ ( i)n Tf Tf Tf U (T ,T ) = 1 + − dt dt dt T (H (t )H (t ) H (t )), I f i n! 1 2 ··· n I 1 I 2 ··· I N n=1 Ti Ti Ti X Z Z Z Tf = T exp i dtH (t) , (6.5.23) − I ZTi where “T ” denotes time ordering. Proof. We present the proof order by order. 1). As n = 0, we have the trivial relation (0) UI (Tf ,Ti) = 1. (6.5.24) 2). As n = 1, we have Tf U (1)(T ,T ) = i dt H (t ) (6.5.25) I f i − 1 I 1 ZTi where no time-ordering is involved. 3). As n = 2, see the above relation (6.5.21). 4). For n-th order, we have Tf t1 tn 1 U (n)(T ,T ) = ( i)n dt dt − dt H (t )H (t ) H (t ) I f i − 1 2 ··· n I 1 I 2 ··· I n ZTi ZTi ZTi Tf t1 tn 1 t1>t2> >tn − ======(··· i)n dt dt dt T (H (t )H (t ) H (t )) − 1 2 ··· n I 1 I 2 ··· I n ZTi ZTi ZTi (N) I1 ( i)n | {z } = − + + + n! ··· ··· ··· ··· ZZ Z ZZ Z ZZ Z (n) (n) (n) I1 I2 In! T (H|(t ){zH (t }) | H ({zt )) } | {z } × I 1 I 2 ··· I n ( i)n Tf Tf Tf = − dt dt dt T (H (t )H (t ) H (t )). n! 1 2 ··· n I 1 I 2 ··· I n ZTi ZTi ZTi I(n) (6.5.26) | {z } Anther simple proof for Dyson’s formula is shown below. 120 Proof. From Dyson’s formula t U (t, t ) = T exp i dt H (t ) , (6.5.27) I 0 − 1 I 1 Zt0 we can check the equation of time evolution directly d d t i U (t, t ) = T i exp i dt H (t ) dt I 0 dt − 1 I 1 Zt0 t = TH (t) exp i dt H (t ) I − 1 I 1 Zt0 t = H (t)T exp i dt H (t ) I − 1 I 1 Zt0 = HI (t)UI (t, t0). (6.5.28) Note that the time-ordering has to do with parameter t1 and is irrelevant with t. 6.5.5 Examples E.g.1. When HI (t) is independent of time, i.e., HI (t) = HI , we have the original expo- nential function of time evolution operator shown as t U (t, t ) = T exp i H dt I 0 − I 1 Zt0 = T exp( i(t t )H ) − − 0 I = exp( i(t t )H ). (6.5.29) − − 0 I E.g.2. When Hamiltonians at different time are commutative, HI (t1)HI (t2) = HI (t2)HI (t1), (6.5.30) the time ordering in Dyson’s formula becomes unnecessary, so t U (t, t ) = exp i dt0H (t0) . (6.5.31) I 0 − I Zt0 The above commutative relation is essentially different from the micro-causality which means in space-like region we have [HI (t, ~x),HI (t, ~y)] = 0 (6.5.32) with H (t, ~x) = d3~x (t, ~x) and H (t, ~y) = d3~y (t, ~y). I H I H R R 121 Lecture 7 Feynman Diagrams and Feynman Rules to Compute Scattering Matrix, Cross Section and Decay Width About Lecture V: [Luke] P.P. 65-105; • [Tong] P.P. 48-75; • [Coleman] P.P. 70-139; • [PS] P.P. 70-115; • [Zhou] P.P. 127-170. • 7.1 Scattering matrix (operator) 7.1.1 Ideal model for scattering process With the interaction, the Hamiltonian can have the form H = H0 + f(t)Hint where f(t) is a “turning-on-and-off” function, vanishing in the far past and far future. See the diagram below. f(t) 0 0 4 T 4 t T + , + , 4 0. → ∞ 4 → ∞ T → Note 1. Before and after scattering, only eigenstates of H0 are allowed. Note 2. There is one-to-one correspondences between asymptotic eigenstates of H and eigenstates of H0. 122 i S-matrix f | i | i Scattering area . . T t = Black box t = + f( −∞) = 0 f(t) = 1 f(+ )∞ = 0 −∞ ∞ 7.1.2 Scattering operator S With the interaction, H = H0 + Hint, (7.1.1) in interaction picture, we have the time evolution equation d i ψ(t) = H (t) ψ(t) . (7.1.2) dt| iI I | iI In the far past (without subscription I), initial state i = ψ( ) is an eigenstate of H . | i | −∞ i 0 In the far future, we have ψ(+ ) = U(+ , ) ψ( ) | ∞ ∞ −∞ | −∞ i (7.1.3) = S ψ( ) = S i , | −∞ i | i where S = U(+ , ) is defined as the scattering operator matrix (S-matrix). ∞ −∞ The probability of finding final state f , eigenstate of H in t = + , is given by | i 0 ∞ f ψ( ) = f S i = S (7.1.4) h | −∞ i h | | i fi where Sfi is the S-matrix element. 7.2 Calculation of two-“nucleon” scattering amplitude 7.2.1 Model The toy model named as “Nucleon”-meson theory, which is also named as the scalar Yukawa theory, takes the Lagrangian density = + gψ†ψφ. (7.2.1) L Lφ Lψ − ψ is the complex scalar field. “nucleon” is the pseudo-nucleon due to spinless with • charge. Proton and anti-proton are nucleons with spin 1/2. φ is the real scalar field with spinless and without charge. • g is the coupling constant with [ψ] = [φ] = [g] = 1 in mass dimension. • The equations of motion show as 2 + µ φ = gψ†ψ; − (7.2.2) 2 ( + m ψ = gψ†φ. − 123 In the Dirac (interaction) picture, the free complex field operator has the solution 3~ d k ik x ik x ψ(x) = (β e− · + γ†e · ). (7.2.3) (2π)32ω k k Z ~k with 3/2√ βk = (2π) 2ωkb~k, (7.2.4) 3/2 γk = (2π) 2ω~kc~k. (7.2.5) And the free real field operator has the expressionq 3~ d k ik x ik x φ(x) = (a e− · + a† e · ). (7.2.6) (2π)3/2 2ω ~k ~k Z ~k p 7.2.2 Two-“nucleon” scattering matrix In the scattering process “N”(p ) + “N”(p ) “N”(p0 ) + “N”(p0 ), (7.2.7) 1 2 −→ 1 2 the rough scattering process can be depicted as p1 p10 p2 p20 t where pi stands for the four-dimension momentum assignment. The inline arrow denotes the charge of “nucleon”. The time arrow is read from left hand side to the right hand side. The two-particle incoming state is the initial state i = p , p = β†(p )β†(p ) vac . (7.2.8) | i | 1 2i 1 2 | i The two-particle outgoing state is the final state f = p0 , p0 = β†(p0 )β†(p0 ) vac . (7.2.9) | i | 1 2i 1 2 | i Note: Assume that the final state differs with the initial state, i.e., f i = 0. h | i With the Dyson’s formula, the scattering operator can be calculated as + ∞ 4 S = T exp( ig d x ψ†(x)ψ(x)φ(x)) . (7.2.10) − Z−∞ The calculation task is to obtain f S i . With the expansion formalism of scattering h | | i operator, in the order of coupling constant g, we have g(0) = 1, f S(0) i . h | | i g(1) = g, f S(1) i . h | | i g(i) = gi, f S(i) i . h | | i 124 7.2.3 Computing methods Three approaches to compute the scattering amplitude: 1. Application of Wick’s theorem (rigorous and complicate). 2. Feynma diagrams and Feynman rules (intuitive and guesswork). 3. Deriving 2 from 1. 7.3 Feynman diagrams 7.3.1 Main theorem The algebraic terms of f S 1 i are one to one correspondence with Feynman diagrams, h | − | i namely f S 1 i = (2π)4δ(4)(P P )i (7.3.1) h | − | i I − F A = Sum of all connected, amputated Feynman diagrams with PI incoming and PF outgoing (7.3.2) where PI is the total initial momentum and PF is the total final momentum PI = pi,PF = pf . (7.3.3) Xi Xf Remark 1. Connected diagrams have no disjoint sub-diagrams. Remark 2. External lines are amputated, because of no contribution. = 7.3.2 Correspondence between algebra and diagrams Table 7.1: Feynaman diagram components correspondent to the algebraic expression Feynman diagrams f S 1 i h | − | i External lines i , f | i | i Vertices Hint Internal lines Contractions (Feynman propagator) Remark: With different Feynman rules, the Feynman diagrams set the one-to-one corre- spondent to the S-matrix elements. 125 A Feynman FR integrals Feynman B diagrams ⇐⇒ FR ⇐⇒ C S-matrix FR element Mapping rules where different Feynaman rules suggest FRA Feynman rules in coordinate space-time; • FRB Feynman rules in momentum space; • FRC simplified Feynman rules in moment space. • 7.3.3 Conventions of drawing external lines As the scattering diagram shown below, • p1 p10 |ii |fi p2 p20 −∞ +∞ t the initial state is on the left hand side and right hand side is for finial state. The middle part is for interaction. The time arrow is from the left to the right. Solid lines are for “nucleons”. • ψ or ψ† Dotted lines are for mesons. φ One line represents for one particle. • Each line has directed momentum. • p1 p10 k Arrow on the solid lines stands for the charge of “nucleon”. No arrow is on dotted • lines because of no charge of meson. 126 Arrow Initial Final (charge) (incoming) (outgoing) Q = 1, anti-nucleon Q = +1, nucleon Outgoing arrow − “N” “N” Q = +1, nucleon Q = 1, anti-nucleon Incoming arrow − “N” “N” No arrow Q = 0, meson Q=0, meson E.g. 1. incoming outcoming Q = 1 “N” “N” Q = 1 Q = −1 “N” “N” Q = −1 outcoming incoming “N” + “N” “N” + “N” −→ Q + Q = 0 = Q + Q E.g. 2. N φ “ ” “N” φ “N” + “N” −→ 0 = Q + Q E.g. 3. p1 p0 1 p1 p10 “N” “N” φ φ “N” “N” φ φ p2 p0 2 p2 p20 “N” + “N” “N” + “N” φ + φ φ + φ −→ −→ Q + Q = 2 = Q + Q 0 + 0 = 0 + 0 − The charge arrow and momentum arrow have the rough rules listed as Same Opposite Charge arrow +1 1 − Momentum “N” “N” 127 7.3.4 Conventions on diagrams vertices & internal lines Vertex • ψ φ 4 Vertex: ig d xψ†(x)ψ(x)φ(x) x − ψ† R Internal lines (contraction) • ψ : D (x1 x2) = ψ(x1)ψ†(x2) . x1 x2 F − (7.3.4) φ : D (x1 x2) = φ(x1)φ(x2) . x1 x2 F − (7.3.5) The solid lines do not specify online arrow, which depends on the external lines. 7.3.5 Application In the scattering process “N”(p ) + “N”(p ) “N”(p0 ) + “N”(p0 ), (7.3.6) 1 2 −→ 1 2 the scattering amplitude can be calculated order by order. The zeroth order is 0, namely • f S(0) i = f 1 i = 0. (7.3.7) h | | i h | | i The first order is 0, because • p1 p10 No enough internal lines to f S(1) i = = h | | i form a connected diagram = 0 p 2 p20 (7.3.8) In the second order, we can have the Feynman diagram shown as • p1 p10 “N” “N” x1 f S(2) i = + (p0 p0 ) h | | i 1 ←→ 2 x2 “N” “N” p 2 p20 (7.3.9) 128 7.4 Feynman rules in coordinate space-time (FRA) 7.4.1 Calculation hint Example. “N” + “N” “N” + “N”. −→ p1 p10 x1 (2) 1 f S i = S [ +(p10 p20 )] h | | i x2 ↔ p 2 p20 (7.4.1) (2) 4 4 ip1 x1 ip2 x2 ip x1 ip x2 f S i =( ig) d x ( ig) d x e− · e− · e 10 · e 20 · φ(x )φ(x ) h | | i − 1 − 2 1 2 Z Z + (p0 p0 ) 1 ↔ 2 2 4 4 i(p1 p ) x1 i(p2 p ) x2 =( ig) d x d x e− − 10 · e− − 20 · + (p0 p0 ) φ(x )φ(x ) . − 1 2 1 ↔ 2 1 2 Z (7.4.2) Note: 1 = 1 2!. S 2! × Algebraic proof of this example. Proof. 3 i = p , p = (2π) 2w 2w b†(~p )b†(~p ) vac ; (7.4.3) | i | 1 2i ~p1 ~p2 1 2 | i 3p p f = p0 , p0 = (2π) 2w~p 2w~p b†(~p0 )b†(~p0 ) vac . (7.4.4) | i | 1 2i 10 20 1 2 | i q+ q ∞ 4 S = T exp i d xgψ†(x)ψ(x)φ(x) . (7.4.5) − Z−∞ Compute f S i . h | | i a. Zero-th order of g. f S(0) i = f i = 0 (7.4.6) h | | i h | i which is from experiment setup. b. First order of g. (1) 4 S = ig d xψ†(x)ψ(x)φ(x). (7.4.7) − Z (1) 4 f S i = ig d x f ψ†(x)ψ(x)φ(x) i = 0. (7.4.8) h | | i − h | | i Z Analysis i = β†β† vac ; f = vac ββ. (7.4.9) | i | i h | h | 129 Thus we need ββ to cancel β†β† and β†β† to cancel ββ. β†β†ββ are needed. ψ β + γ†; ψ† β† + γ. (7.4.10) ∼ ∼ ψ†ψ β†β + γβ + β†γ† + γγ†. (7.4.11) ∼ (1) f S i f β†β i = 0. (7.4.12) h | | i ∼ h | | i c. Second order of g. 2 (2) ( ig) 4 4 f S i = − d x d x f T (ψ†(x )ψ(x )φ(x )ψ†(x )ψ(x )φ(x )) i h | | i 2! 1 2h | 1 1 1 2 2 2 | i 2 Z (7.4.13) ( ig) 4 4 = − d x d x f : ψ†(x )ψ(x )ψ†(x )ψ(x ): i φ(x )φ(x ) . 2! 1 2h | 1 1 2 2 | i 1 2 Z Trick 1. Apply the Wick’s theorem. Trick 2. Insert vac vac to replace normal ordering. | ih | 2 (2) ( ig) 4 4 f S i = − d x d x f ψ†(x )ψ(x ) vac vac ψ†(x )ψ(x ) i h | | i 2! 1 2h | 1 1 | ih | 2 2 | i Z φ(x )φ(x ) . × 1 2 (7.4.14) Trick 3. Contraction between fields and states. vac ψ(x )ψ(x ) i h | 1 2 | i = vac ψ(x )ψ(x ) p , p h | 1 2 | 1 2i = vac ψ(x ) p vac ψ(x ) p + vac ψ(x ) p vac ψ(x ) p (7.4.15) h | 1 | 1ih | 2 | 2i h | 1 | 2ih | 2 | 1i ψ(x ) p ψ(x ) p +(p p ). ≡ 1 | 1i 2 | 2i 1 ↔ 2 e.g. 3 ik x d ~k e− · ψ(x ) p = vac β(k)β†(p ) vac . (7.4.16) 1 | 1i (2π)3 2w h | 1 | i Z ~k ip1 x1 ip2 x2 vac ψ(x )ψ(x ) i = e− · − · + (p p ). (7.4.17) h | 1 2 | i 1 ↔ 2 f ψ†(x1)ψ†(x2) vac = p10 ψ†(x1) vac p20 ψ†(x2) vac + (p10 p20 ) h | | i h | | ih | | i ↔ (7.4.18) ip x1+ip x2 =e 10 · 20 · + (p0 p0 ). 1 ↔ 2 Trick 4. Symmetry under interchange x x . 1 ↔ 2 For the interchange x x , we have 1 ↔ 2 4 4 4 4 d x1d x2 = d x2d x1; (7.4.19) φ(x1)φ(x2) = φ(x2)φ(x1) . (7.4.20) 130 Thus ( ig)2 f S(2) i = − d4x d4x φ(x )φ(x ) h | | i 2! 1 2 1 2 Z ip1 x1 ip2 x2 ip x1+ip x2 (e− · − · + (x x ))(e 10 · 20 · + (p0 p0 )) × 1 ↔ 2 1 ↔ 2 =( ig)2 d4x d4x φ(x )φ(x ) − 1 2 1 2 Z ip1 x1 ip2 x2 ip x1+ip x2 e− · − · (e 10 · 20 · + (p0 p0 )) × 1 ↔ 2 (7.4.21) 7.4.2 Feynman rules in coordinate space-time (FRA) 1. Draw all possible connected diagrams. 2. For each external line p x ip x ψ(x) p vac ψ(x) p = e− · incoming −→ | i ≡ h | | i (7.4.22) p0 x ip x p0 ψ(x) p0 ψ(x) vac = e 0· outgoing −→ h | ≡ h | | i (7.4.23) 3. For internal line ψ ( ig) d4x x −→ − R † ψ (7.4.24) 4. For internal line φ(x )φ(x ) x1 x2 −→ 1 2 (7.4.25) ψ(x )ψ(x ) x1 x2 −→ 1 2 (7.4.26) 5. Divided by symmetric factor 1 1 = m (7.4.27) S n! × where 1/n! is from Taylor expansion of S matrix, and m is the number of same Feynman diagrams which is due to permutation of vertexes or Bose statistics. 131 7.5 Feynman rules in momentum space (FRB) 7.5.1 Calculation hint 4 d k i ik (x1 x2) φ(x )φ(x ) = e · − . (7.5.1) 2 1 (2π)4 k2 m2 + iε Z − 4 (2) 2 4 4 d k i ik (x1 x2) f S i =( ig) d x d x e · − h | | i − 1 2 (2π)4 k2 m2 + iε Z Z − ip1 x1 ip2 x2 ip x1+ip x2 e− · − · (e 10 · 20 · + (p0 p0 )) × 1 ↔ 2 4 2 4 4 d k i =( ig) d x1d x2 − (2π)4 k2 m2 + iε (7.5.2) Z Z − i(p +k p1) x1 i(p k p2) x2 (e 10 − · e 20 − − · + (p0 p0 )) × 1 ↔ 2 d4k i =( ig)2 d4x d4x − 1 2 (2π)4 k2 m2 + iε Z Z − 4 (4) 4 (4) ((2π) δ (p0 + k p )(2π) δ (p0 p k) + (p0 p0 )). × 1 − 1 2 − 2 − 1 ↔ 2 p1 p10 (2) f S i = k +(p0 p0 ) h | | i 1 ↔ 2 p 2 p20 (7.5.3) 7.5.2 Feynman rules in momentum space 1. Draw all possible connected diagrams. 2. For each external line, assign the factor 1. Note: External lines have no contribution, so they are amputated. 3. For each vertex, assign the factor ( ig)(2π)4δ(4)(p + k p ). − 10 − 1 p1 p10 4 (4) 0 ( ig)(2π) δ (p + k p1) −→ − 1 − k (7.5.4) 4. For internal line k d4k i (2π)4 k2 m2+iε −→ − (7.5.5) R 5. Symmetry factor. 132 7.6 Feynman rules C 7.6.1 Calculation hint Integrate over internal momentum k. (2) 4 (4) 2 i i f S i =(2π) δ (p + p p0 p0 )( ig) + h | | i 1 2 − 1 − 2 − (p p )2 µ2 (p p )2 µ2 1 − 10 − 2 − 20 − + (p0 p0 ). 1 ↔ 2 (7.6.1) p1 p10 (2) p − p f S i = 1 10 +(p0 p0 ) h | | i 1 ↔ 2 p 2 p20 (7.6.2) 7.6.2 Feynman rules C (simplified FRB) 1 Draw all possible connected Feynman diagrams with momentum assign. 4 1 Impose energy momentum at each vertex. 4 0 1 Write down the factor for the total energy-momentum conservation. 4 00 (2π)4δ(4)(P P ) F − I 2 For each external line, assign “1”. 4 3 For each vertex, assign “ ig”. 4 − 4 For momentum determined internal line 4 0 p1 p − 1 i 2 2 (p1 p ) µ −→ − 10 − (7.6.3) 4 For momentum in-determined internal line 4 0 k d4k i (2π)4 k2 µ2+iε −→ − (7.6.4) R 5 Symmetry factor. 4 133 7.7 Application of FRC e.g.1. Decay of a meson into “nucleon”-“anti-nucleon”. φ “N” + “N”. (7.7.1) −→ To compute “N”“N” S φ . h | | i p1 “N” (1) ig S : −→ − p p 2 “N” (7.7.2) In center of mass frame, ~p = 0 and ~p1 + ~p2 = 0. “N”“N” S(1) φ = (2π)4δ(4)(p + p p)( ig). (7.7.3) h | | i 1 2 − − Remark 1: Feynman diagrams. 1. Diagrammatic representation tool of S-matrix element. 2. (Feynman) Diagrammatic movement of particles in space time. (physics) Remark 2: Feynman rules. 1. FRA: in coordinate space-time. 2. FRB: integrate out vertex. 3. FRC : integrate out internal momentum. e.g.2. φ(p ) + φ(p ) φ(p0 ) + φ(p0 ). (7.7.4) 1 2 −→ 1 2 To compute φφ S φφ , at least fourth order is non-zero. h | | i φφ S(4) φφ = 0. (7.7.5) h | | i 6 0 p1 p1 k + p1 0 k k + p1 p − 1 k p2 − p 0 2 p2 (7.7.6) 134 d4k i i i (4) =( ig)4 A − (2π)4 (k2 m2 + iε) ((k2 + p )2 m2 + iε) Z − 1 − (7.7.7) i i × ((k + p p )2 m2 + iε) ((k p )2 m2 + iε) 1 − 10 − − 2 − Remark: 1 1 = m. (7.7.8) S n! × In this example, n = 4 and m = 4!, which is due to the permutations of four vertexes x1, x2, x3 and x4. x1 x2 x3 x4 e.g.3. “N”(p ) + “N”(p ) “N”(p0 ) + “N”(p0 ). (7.7.9) 1 2 −→ 1 2 In second order O(g2). p1 p10 “N” “N” p1 p10 “N” “N” p + p f S(2) i = p1 − p10 + 1 2 h | | i “N” “N” “N” “N” p2 p0 p 2 2 p20 (7.7.10) i i i (2) = ( ig)2 + . (7.7.11) A − (p p )2 µ2 (p + p )2 µ2 1 − 10 − 1 2 − The symmetry factor S is 1 S = 2! = 1. (7.7.12) 2! × e.g.4. “N”(p ) + “N”(p ) φ(p0 ) + φ(p0 ). (7.7.13) 1 2 −→ 1 2 p1 p10 “N” φ (2) p − p f S i = 1 10 + (p0 p0 ) h | | i 1 ↔ 2 “N” φ p 2 p20 (7.7.14) i i i (2) = ( ig)2 + . (7.7.15) A − (p p )2 m2 (p p )2 m2 1 − 10 − 1 − 20 − 135 e.g.5. “N”(p ) + φ(p ) “N”(p0 ) + φ(p0 ). (7.7.16) 1 2 −→ 1 2 p1 p20 “N” φ p1 p20 “N” φ p + p f S(2) i = p1 − p20 + 1 2 h | | i φ “N” φ “N” p2 p p 10 2 p10 (7.7.17) i i i (2) = ( ig)2 + . (7.7.18) A − (p p )2 m2 (p + p )2 m2 1 − 20 − 1 2 − Note: no iε term in the result, because we do not have integrals. Remark: about two-particle scattering at order O(g2). 1). Virtual particle on internal line is meson. “N” + “N” “N” + “N”; (7.7.19) −→ “N” + “N” “N” + “N”; (7.7.20) −→ “N” + “N” “N” + “N”. (7.7.21) −→ 2). Virtual particle on internal line is nucleon. “N” + “N” φ + φ; (7.7.22) −→ “N” + φ N + φ; (7.7.23) −→ “N” + φ “N” + φ. (7.7.24) −→ Note: the charge conservation due to U(1) invariance favours the above process, but forbids “N” + “N” φ + φ. (7.7.25) −→ 7.8 Remarks on Feynman propagator 1). Internal lines k2 = m2, due to < kµ < + . 6 −∞ ∞ The Feynman propagator represents the propagation of a virtual particle with k2 = m2. 6 x D (x y) = F − y 2). Lorentz transformation D (Λ(x y)); F − D (x y) is invariant under space-time translation D (x + a y a); F − F − − space-time reflection D (y x). F − 136 Hence Feynman integrals also have such symmetries. Note: Hamiltonian formulation of QFT is not explicitly Lorentz invariant, but Feyn- man rules with Feynman diagrams define an explicitly Lorentz invariant perturbative QFT. 7.9 Cross sections & decay widths – measurable quantities in high energy physics Question: δ(4)(P P ). Afi ∝ F − I Scattering amplitude δ(4)(P P ) = δ = 1; • ∝ F − I ⇒ Scattering probability (δ(4)(P P ))2 =R δ2 = . • ∝ F − I ⇒ ∞ Answer 1. Take the box normalization. R Answer 2. Take the wave-packet. 7.9.1 QFT in a box with volume V = L3 L L L Periodic boundary condition: 2π 2π 2π p = n , p = n , p = n . (7.9.1) 1 L 1 2 L 2 3 L 3 Lemma 1. V = d3~p. (7.9.2) (2π)3 X~p Z Proof. (2π)3 p p p = n n n , (7.9.3) 4 14 24 3 L3 4 14 24 3 (2π)3 = dp dp dp = . (7.9.4) ⇒ 1 2 3 L3 Z X~p Lemma 2. V (3) lim δ~p ,~p = δ (~pi ~pj), (7.9.5) V + (2π)3 i j − → ∞ where δ~pi,~pj is Kronecker delta function 1, ~p = ~p δ = i j . (7.9.6) ~pi,~pj 0, ~p = ~p i 6 j 137 Proof. δ~pi,~pj f(~pj) = f(~pi), (7.9.7) X~pj (2π)3 d3~pδ(3)(~p ~p )f(~p ) = f(~p ) = δ(3)(~p ~p )f(~p ), (7.9.8) i − j j i V i − j j Z X~pj (2π)3 = δ = δ(3)(~p ~p ). (7.9.9) ⇒ ~pi,~pj V i − j Lemma 3. 2 (4) VT (4) lim δ (p p0) = δ (p p0), (7.9.10) V,T + VT − (2π)4 − → ∞ h i where V is space box, and T is time box. L T 2 V : T : L T 2 L − space box time box Proof. V (3) δ = δ (~p ~p0). (7.9.11) (2π)3 ~p,~p0 − Square it 2 2 V (3) δ = δ (~p ~p0) , (7.9.12) (2π)6 ~p,~p0 − h i due to δ~p,~p0 δ~p,~p0 = δ~p,~p0 . (7.9.13) Replace 7.9.5 in the former formula, 2 V (3) (3) δ (~p ~p0) = δ (~p ~p0) . (7.9.14) (2π)3 − − h i Similarly T 2 δ(p p0 ) = δ(p p0 ) . (7.9.15) (2π) 0 − 0 0 − 0 2 VT (4) (4) = δ (p p0) = δ (p p0) . (7.9.16) ⇒ (2π)4 − − h i Box-normalization a , a† = δ , (7.9.17) ~pi ~pj ~pi,~pj h i a , a = 0 = a† , a† . (7.9.18) ~pi ~pj ~pi ~pj Fock space h i ~p = a† vac , (7.9.19) | i ~p| i 138 ~p,~q = a†a† vac . (7.9.20) | i ~p ~q| i Field operators 1 ip x ip x φ(t, ~x) = (a~pe− · + a†e · ). (7.9.21) 2V p0 ~p X~p Note: p 1 ip x φ(t, ~x) vac = e · ~p . (7.9.22) | i 2V p0 | i X~p p Remark 1. V + [φ(x), φ(y)] ======→ ∞ D(x y) D(y x). (7.9.23) − − − Remark 2. √V a(~p) = a . (7.9.24) (2π)3/2 ~p (3) a(~p), a†(~q) = δ (~p ~q). (7.9.25) − h i Feynman rules in a box. ~p = a† vac . (7.9.26) | i ~p| i 1 ip x ip x φ(t, ~x) = (a~pe− · + a†e · ). (7.9.27) 2V p0 ~p X~p External lines add a factor 1 pand other lines (vertexes) are unchanged. √2V p0 7.9.2 Calculation of differential probability Reformulation of S-matrix in a box. VT 4 (4) 1 1 f S 1 i = i fi (2π) δVT (PF PI ) . (7.9.28) h | − | i A − 2wf V √2wiV Yf Yi p 2 VT 2 8 (4) 2 1 1 f S 1 i = fi (2π) δVT (PF PI ) . (7.9.29) |h | − | i| |A | | − | 2wf V 2wiV Yf Yi Summation over momentum of outgoing particle. Lemma. V S = S d3~p . (7.9.30) × × (2π)3 f X~pf Z S is symmetry factor due to statistics. 1 S = (7.9.31) i mi! if there are mi identical particles in theQ final state. Define the new symbol wVT , 3 VT 2 8 (4) 2 1 d ~pf wVT = fi (2π) δVT (PF PI ) 3 S. (7.9.32) |A | | − | 2wiV (2π) 2wf × Yi Yf 139 3 d ~pf where 3 is Lorentz measure. (2π) 2wf With Lemma 3, 3 VT 2 4 (4) 1 d ~pf wVT = fi (2π) δ (PF PI ) VT 3 S |A | − × × 2wiV (2π) 2wf × Yi Yf (7.9.33) VT 2 1 = fi VT D |A | × × 2wiV × Yi where 3 4 (4) d ~pf D = (2π) δ (PF PI ) 3 S. (7.9.34) − (2π) 2wf × Yf 7.9.3 Cross section & decay width Differential cross section differential probability 1 dσ ≡ unit time × unit flux (7.9.35) w 1 = VT . T × F 1). Only with one incident particle, i.e. particle decay. . . ~v ~v F = | |. (7.9.36) 1 V 2). With two incident particles. v1 . . v2 ~v ~v F = | 1 − 2|. (7.9.37) 2 V e.g. 1. Differential decay width of a rest particle, E = M. Change the notation, dσ dΓ. → wVT 1 2 dΓ = = fi V D T 2EV |A | × × (7.9.38) 1 = 2 D. 2M |Afi| × 140 e.g. 2. 1 + 2 10 + 20 + + n0. (7.9.39) −→ ··· w 1 dσ = VT T × F2 (7.9.40) 2 1 = |Afi| D. 4E E × ~v ~v × 1 2 | 1 − 2| 7.9.4 Calculation of Cross-sections and Decay-widths For two-particle final states in CoM 1 + 2 + 10 + 20. (7.9.41) · · · → P = p + p + = P = p0 + p0 . (7.9.42) I 1 2 ··· F 1 2 In CoM ~p0 + ~p0 = 0 = ~p0 = ~p0 . (7.9.43) 1 2 ⇒ | 1| | 2| E = E + E + = E0 + E0 (7.9.44) T 1 2 ··· 1 2 and ET is a fixed number. Then: (4) (3) δ (p0 + p0 P ) = δ (~p0 + ~p0 )δ(E0 + E0 E ). (7.9.45) 1 2 − I 1 2 1 2 − T Lemma: 1 ~p10 0 1 D(1 + 2 + 10 + 20) = | |2 dΩ10 . (7.9.46) · · · → S 16π ET where ~p is the solution of E = E + E and dΩ is the solid angle, i.e. dΩ | 10 |0 T 10 20 10 10 ≡ d cos θ10 dϕ10 . Proof. 3 3 1 d ~p10 d ~p20 4 (4) D = (2π) δ (p0 + p0 P ) S (2π)32E (2π)32E 1 2 − I Z 10 Z 20 3 1 d ~p10 = δ(E0 + E0 E ) (7.9.47) S (2π)22E 2E 1 2 − T Z 10 20 2 1 ~p10 d ~p10 dΩ10 = | | | | δ(E0 + E0 E ). S (2π)22E E 1 2 − T Z 10 20 Since 2 2 2 2 δ(E0 + E0 E ) = δ( ~p + m + ~p + m E ) 1 2 − T | 10 | | 20 | − T = δ(qf( ~p0 )) q | 1| (7.9.48) 1 = δ( ~p0 ~p0 ), f ( ~p ) | 1|0 − | 1| 0 | 10 | ∂(E10 + E20 ET ) ∂E10 ∂E20 f 0( ~p0 ) = − = + , (7.9.49) | 1| ∂( ~p ) ∂( ~p ) ∂( ~p ) | 10 | | 10 | | 20 | 2 2 2 2 2 2 E0 = ~p0 + m ,E0 = ~p0 + m , (7.9.50) 1 | 1| 2 | 2| 141 ~p10 ~p20 1 1 ~p10 ET f 0( ~p10 ) = | | + | | = ~p10 ( + ) = | | . (7.9.51) | | E10 E20 | | E10 E20 E10 E20 Then 2 1 ~p10 d ~p10 dΩ10 E10 E20 1 1 ~p10 0 D = | | | | δ( ~p0 ~p0 ) = | | dΩ0 . (7.9.52) S 4π24E E × ~p E | 1| − | 1|0 S × 16π2 E 1 Z 10 20 | 10 | T T e.g.1 Decay width φ “N” + “N”. → 0 p1 “N” φ p 0 p2 “N” (1) = ig. (7.9.53) Afi − In CoM, φ particle is at rest, i.e. ~p = 0 = ~p10 + ~p20 . 2 2 2 2 (µ, 0) = ( ~p + m , ~p0 ) + ( ~p + m , ~p0 ), (7.9.54) | 10 | 1 | 20 | 2 q q E = µ = 2 ~p 2 + m2, (7.9.55) T | 10 | q 2 µ 2 ~p10 0 = m . (7.9.56) | | r 4 − 1 Γ(1)(φ “N” + “N”) = (1) 2D → |CoM 2µ |Afi | Z 1 2 ~p10 0 = ig | | dΩ0 2µ | − | × 16π2E 1 Z T 2 2 (7.9.57) g µ 2 = m dΩ0 32π2µ 4 − 1 r Z 2 2 g µ 2 = 2 m . 8πµ r 4 − Where S = 1, because “N”, “N” are not identical particles. Note: Particle lifetime is τ = 1 1 , where g is coupling constant. g is weaker, τ T ∝ g2 is longer. e.g.2 Cross-section for 1 + 2 1 + 2 in CoM. → 0 0 1 1 1 ~p dΩ 1 (dσ) = 2 | 10 |0 10 , (7.9.58) CoM 4E E |Afi| ~v ~v × 16π2 × E × S 1 2 | 1 − 2| T 142 dσ 2 ~p 1 ( ) = |Afi| | 10 |0 , (7.9.59) dΩ CoM 64π2 E E ~v ~v E × S 10 1 2| 1 − 2| T dσ 2 ~p 1 ( ) = |Afi| | 10 |0 . (7.9.60) d cos θ CoM 32π E E ~v ~v E × S 10 1 2| 1 − 2| T ~p1 ~p1 ~p2 ~p1 ~v1 = , ~v2 = , (7.9.61) ≡ γm E1 ≡ γm −E1 1 1 ET ~v1 ~v2 = ~p1 0( + ) = ~p1 0 , (7.9.62) | − | | | E1 E2 | | E1E2 where E + E = E = E + E , and ~p is the solution of E + E E = 0. 1 2 T 10 10 | 1|0 1 2 − T Then we have 1 1 1 = ~v1 ~v2 ET E1E2 ET E1E2 × ~p1 + ~p2 | − | E1 E2 1 1 = (7.9.63) E1+E 2 ET E1E2 × ~p1 0 | | E1E2 1 = , E2 ~p T | 1|0 dσ 1 ~p 1 = |Afi| | 10 |0 . (7.9.64) dΩ 64π2 × E2 × ~p × S 10 CoM T | 1|0 In the special case “N” + “N” “N” + “N”, (7.9.65) −→ we have m = m = m = m , and ~p = ~p , 1 2 10 20 | 1|0 | 10 |0 dσ 1 1 = = |Afi| . (7.9.66) ⇒ dΩ 64π2 × E2 × S 10 CoM T 1 = 1 for “N”“N” “N”“N”, 1 = 1 for “N”“N” “N”“N”. S 2! → S 1 → 143 Lecture 8 Return to Non-Relativistic Quantum Mechanics 8.1 The Schroedinger equation 144 Lecture 9 Dirac Fields: From Classical to Quantum About Lecture VI: [Luke] P.P. 125-167; • [Tong] P.P. 81-123; • [Coleman] P.P. 221-337; • [PS] P.P. 35-76 & 115-123; • [Zhou] P.P. 34-45 & 81-104. • 9.1 The Dirac equation and Dirac algebra Dirac fields describe dynamics of massive spin-1/2 particles like electrons, protons, and so on. In non-relativistic quantum mechanics, Pauli realized that an electron is described by a two-component column vector ψ = ∗ (9.1.1) ∗ with Pauli matrices 112 and σx, σy, σz. In relativistic quantum field theory, Dirac realized that we had to introduce a 4-component column vector, called the Dirac field, ∗ ψ = ∗ (9.1.2) ∗ ∗ to describe the motion of electron with Dirac matrices β, ~α = (α , α , α ) which are 4 4 1 2 3 × matrices. 9.1.1 The Dirac algebra The principles of writing the Dirac equation give rise to the Dirac algebra generated by Dirac matrices β, ~α = (α1, α2, α3). 1). The Lorentz invariance of the Dirac equation ∂ (i + i~α ~ βm)ψ(t, ~x) = 0, (9.1.3) ∂t · ∇ − implies that ~α, β are constant matrices and that both the time derivative and spatial derivatives are first-order derivatives. 145 2). Hamiltonian is a Hermitian operator. With the Dirac equation ∂ i ψ(t, ~x) = ( i~α ~ + βm)ψ(t, ~x), (9.1.4) ∂t − · ∇ and the canonical quantization in quantum mechanics ∂ i H, i~ ~p, (9.1.5) ∂t ↔ − ∇ ↔ the Hamiltonian has the form H = i~α ~ + βm = ~α ~p + βm. (9.1.6) − · ∇ · That Hamiltonian is a Hermitian operator suggests that H† = H ~α† = ~α, β† = β. (9.1.7) ⇐⇒ 3). The mass-energy relation E2 = m2 + ~p2 leads to the following calculation, ∂ i ψ = Hψ = Eψ = (~α ~p + βm)ψ Eψ = (~α ~p + βm)ψ (9.1.8) ∂t · ⇒ · (~α ~p + βm)(~α ~p + βm)ψ = E2ψ, (9.1.9) · · 2 2 2 αipiαjpj + αipiβm + βmαjpj + m β = E , (9.1.10) α α + α α i j j i p p + mp (α β + βα ) + m2β2 = E2 = δ p p + m2. (9.1.11) 2 i j i i i ij i j which derives the algebraic relations of the Dirac algebra, αiαj + αjαi = 2δij; αiβ + βαi = 0; (9.1.12) 2 β = 112. 2 2 Note: αi = 11, β = 11and tr(α ) = tr(β2α ) = tr(βα β) = tr(α ) = 0 = tr(α ) = tr(β) = 0, (9.1.13) i i i − i ⇒ i so αi and β are Hermitian traceless square identity matrices. 9.1.2 Two widely used representations of the Dirac algebra 1). The Weyl representation. ~σ 0 0 11 ~α = , β = 2 . (9.1.14) W 0 ~σ W 11 0 − 2 Note: The Weyl representation presents a suitable description on physics of Neutrino. 2). The Dirac representation (Standard representation). 0 ~σ 11 0 ~α = , β = 2 . (9.1.15) D ~σ 0 D 0 11 − 2 146 With the unitary matrix, 1 11 11 H = 2 2 , (9.1.16) √2 112 112 − these two representations are related to each other via the similarity transformation ~αD = H~αW H†, βD = HβW H†. (9.1.17) Thm. The Dirac algebra generated by ~α , β is invariant under any unitary transfor- { i } mation U, ~α0 = U ~αU †, β0 = UβU † (9.1.18) satisfying the same algebraic constraints defining the Dirac algebra. 9.1.3 The Lagrangian formulation The Lagrangian density is ∂ = ψ†(i + i~α ~ βm)ψ (9.1.19) L ∂t · ∇ − and its component formalism is ∂ = ψ†(i δ + i(~α) ~ β m)ψ (9.1.20) L a ∂t ab ab∇ − ab b where ψa† is the component of a 1 4 matrix, ψ is the component of a 4 1 matrix and × a × (~α) , β is in 4 4 matrix, and ψ is called bispinor. ab ab × EoM is derived from the Euler-Lagrangian equation ∂ L = 0, (9.1.21) ∂ψa† which gives rise to ∂ = (i + i~α ~ βm)ψ = 0. (9.1.22) ⇒ ∂t · ∇ − 9.1.4 Plane wave solutions of Dirac equation in Dirac representation The Dirac equation has the form in the momentum representation ∂ i ψ = (~α ~p + β m)ψ, (9.1.23) ∂t D · D with p0 = ~p 2 + m2. Substitute the positive-frequency solution | | p (+) ip x ψ (x) = u~pe− · , (9.1.24) into the Dirac equation, we have ∂ i ψ(+)(x) = p0ψ(+)(x) = (~α ~p + mβ )ψ(+)(x), (9.1.25) ∂t D · D which is (p0 ~α ~p)u = mβ u . (9.1.26) − D · ~p D ~p 147 Substitute the negative-frequency solution ( ) ip x ψ − (x) = v~pe · , (9.1.27) into the Dirac equation, we have ∂ ( ) 0 ( ) ( ) i ψ − (x) = p ψ − (x) = (~α ~p + mβ )ψ − (x), (9.1.28) ∂t − D · D which is (p0 + ~α ~p)v = mβ v . (9.1.29) D · ~p − D ~p In the rest frame, i.e. ~p = 0, p0 = m, 0 p u~p = mβDu~p; βDu~p = u~p; 0 = (9.1.30) p v = mβDv . ⇒ βDv = v , ~p − ~p ~p − ~p which have the following linearly independent solutions spanning the four dimensional vector space, 1 0 (1) 0 (2) 1 u = √2m , u = √2m , (9.1.31) ~p=0 0 ~p=0 0 0 0 0 0 (1) 0 (2) 0 v = √2m , v = √2m . (9.1.32) ~p=0 1 ~p=0 0 0 1 Note: √2m is the normalization factor which is very meaningful in deriving Feynman rules of Dirac fields or in calculating the cross section. In the frame of p = p = 0, p = 0, E = p0, x y z 6 (E (α ) p )u = mβ u ; − D 3 z ~p D ~p (9.1.33) (E + (α ) p )v = mβ v , D 3 z ~p − D ~p which have the following linearly independent solutions spanning the four dimensional vector space, √E + m 0 0 √E + m u(1) = , u(2) = , (9.1.34) pz √E m pz 0 − 0 √E m − − √E m 0 − 0 √E m v(1) = , v(2) = − − . (9.1.35) pz √E + m pz 0 0 √E + m Note: pz = 0, E = m, the above solutions return to the solutions in the rest frame. Thm. (r) (s) (r) (s) u~p=0† βDu~p=0 = u~p †βDu~p = 2mδrs; (9.1.36) (r) (s) (r) (s) v †β v = v †β v = 2mδ , (9.1.37) ~p=0 D ~p=0 ~p D ~p − rs which are Lorentz invariant scalars. 148 9.2 Dirac equation and Clifford algebra The Clifford algebra is isomorphism to the Dirac algebra, and with the Clifford algebra, the Dirac equation has an explicit Lorentz covariant formalism which is widely used in relativistic quantum field theory. 9.2.1 The Clifford algebra Def. Define γµ matrix as γ0 = β, γi = βαi, i = 1, 2, 3, (9.2.1) satisfying 1 1 γµ, γν = 2gµν = 2 − , (9.2.2) { } 1 − 1 − ν γµ = gµνγ , (9.2.3) and the algebra generated by the γµ matrices is called the Clifford algebra. The γµ matrices have the following interesting properties: 1). 0 0 i i (γ )† = γ , (γ )† = γ . (9.2.4) − 2). (γ0)2 = 11 , (γi)2 = 11 . (9.2.5) 4 − 4 3). µ ν (γ )† = gµνγ = γµ. (9.2.6) 4). The Dirac adjoint of the γµ matrices are defined as µ 0 µ 0 µ µ γ γ (γ )†γ , γ = γ . (9.2.7) ≡ µν i µ ν 5). The rank-2 anti-symmetric tensor, σ = 2 [γ , γ ], have σ00 = 0, σ0i = iαi, σij = iαiαj. (9.2.8) − 1 µ ν α β 6). The fifth Gamma matrix γ5 given by γ5 = 4! εµναβγ γ γ γ with εµναβ is the totally anti-symmetric tensor, has 5 5 2 γ = γ , (γ ) = 11 , γ = γ† = γ . (9.2.9) 5 4 5 5 − 5 Note 1: γ = iγ0γ1γ2γ3 = iα1α2α3. 5 − Note 2: The γ5 has the respective forms in the Dirac representation and the Weyl representation, 0 11 11 0 (γ ) = 2 , (γ ) = 2 . (9.2.10) 5 D 11 0 5 W 0 11 2 − 2 149 9.2.2 Dirac equation with γµ matrices In the first place, we reformulate the Lagrangian with γµ matrices, ∂ 0 0 ∂ = ψ†(i + i~α ~ βm)ψ = (ψ†γ )γ (i + i~α ~ βm)ψ. (9.2.11) L ∂t · ∇ − ∂t · ∇ − 0 Secondly, with the Dirac adjoint ψ = ψ†γ , we have ∂ = ψγ0i + ψi~γ ~ mψψ. (9.2.12) L ∂t · ∇ − Thirdly, with ∂ ∂ γµ = (γ0, γi), ∂ = ( , ), (9.2.13) µ ∂t ∂xi we have = ψ(iγµ∂ m)ψ = ψ ((iγµ) ∂ mδ )ψ . (9.2.14) L µ − a ab µ − ab b The Euler-Lagrangian equation, EoM, is the Dirac equation ∂ µ L = 0 = (iγ ∂µ m)ψ = 0. (9.2.15) ∂ψ ⇒ − Notation: With the Dirac slash a/ γµa , b/ γµb satisfying a/b/+b/a/ = 2a b, we define ≡ µ ≡ µ · ∂/ γµ∂ , to reformulate the Dirac equation as ≡ µ (i∂/ m)ψ = 0 (9.2.16) − which is obviously covariant under Lorentz transformations. 9.2.3 Plane wave solutions of Dirac equation with γµ matrices Substitute the positive-frequency and negative-frequency solutions (+) (r) ip x ψ (x) = u e− · ; ~p r = 1, 2 (9.2.17) ( ) (r) ip x ( ψ − (x) = v~p e · into the Dirac equation, we have (r) (r) (p/ m)u~p = 0; (p/ + m)v~p = 0; (r−) (r) (9.2.18) ( u (p/ m) = 0. ( v (p/ + m) = 0. ~p − ~p µ (r) (r) 0 with p/ = γ pµ and u~p = u~p †γ . (r) (s) The set of u~p , r = 1, 2 and v~p , s = 1, 2 forms a kind of the orthonormal basis in the four-dimensional Hilbert space: Thm. 1. The completeness relation. 2 (r) (r) r=1 u~p u~p = p/ + m; 2 (s) (s) (9.2.19) ( v v = p/ m. Ps=1 ~p ~p − Thm. 2. The orthonormality condition.P (r) (s) (r) (s) u~p u~p = 2mδrs = v~p v~p ; (r) (s) − (9.2.20) ( u~p v~p = 0. 150 9.3 Lorentz transformation and parity of Dirac bispinor field- s µ A Lorentz transformation Λ ν given by µ µ µ ν Λ: x x0 = Λ x (9.3.1) → ν 2 2 T satisfies x = x0 or Λ ηΛ = η with η denoting the Minkowski metric. The Dirac bispinor field ψ(x) transforms under the Lorentz transformation Λ 1 ψ(x) ψ0(x) = D(Λ)ψ(Λ− x), (9.3.2) −−→ the associated component formalism, 1 ψa0 (x) = D(Λ)abψ(λ− x)b. (9.3.3) where the D(Λ) is called the bispinor representation of Lorentz group. The Hermitian conjugation of the Dirac bispinor field ψ(x) tranforms Λ 1 ψ†(x) ψ†(Λ− x)D†(Λ); (9.3.4) −−→ and its Dirac adjoint Λ 1 0 1 ψ(x) ψ†(Λ− x)D†(Λ)γ = ψ†(Λ− x)D(Λ); (9.3.5) −−→ 0 0 with D(Λ) = γ D†(Λ)γ . The Parity means µ µ P :(t, ~x) x0 = (t, ~x) = P x , (9.3.6) → − which gives rise to ∂ ∂ ∂ ∂ ( , ) ( , ). (9.3.7) ∂t ∂x −→ ∂t −∂x The Dirac bispinor field ψ(x), its Hermitian conjugation, and its Dirac adjoint trans- form respectively under parity ψ(x) P γ0ψ(P x); (9.3.8) −−→ and P 0 ψ†(x) ψ†(P x)γ ; (9.3.9) −−→ and ψ(x) P ψ(P x)γ0. (9.3.10) −−→ Note that Gamma matrices γµ are unchanged under Lorentz transformation and parity. 9.3.1 Classification of quantities out of Dirac bispinor fields under Lorentz transformation and parity 1). Scalar field quantity is invariant under Lorentz transformation (LT) and parity. Example: ψ(x)ψ(x) is a scalar field quantity. About Lorentz transformation of ψ(x)ψ(x): Λ 1 1 ψ(x)ψ(x) ψ0(x)ψ0(x) =ψ(Λ− x)D(Λ)D(Λ)ψ(Λ− x) −−→ (9.3.11) 1 1 =ψ(Λ− x)ψ(Λ− x) 151 if and only if D(Λ)D(Λ) = 114. (9.3.12) About parity of ψ(x)ψ(x): P 0 0 ψ(x)ψ(x) ψ0(x)ψ0(x) =ψ(P x)γ γ ψ(P x) −−→ (9.3.13) =ψ(P x)ψ(P x) which does not impose any constraints on the Dirac bispinor representation D(Λ). 2). Vector field quantity is covariant under LT and invariant under parity. Note: parity of vector V µ: (V 0, V~ ) P (V 0, V~ ), (9.3.14) −−→ − V µ P V P V µ. (9.3.15) −−→ µ −−→ Example: ψ(x)γµψ(x) is a vector field quantity: µ Λ µ 1 µ 1 ψ(x)γ ψ(x) ψ0(x)γ ψ0(x) = ψ(Λ− x)D(Λ)γ D(Λ)ψ(Λ− x) −−→ (9.3.16) µ 1 ν 1 =Λ νψ(Λ− x)γ ψ(Λ− x), which gives rise to the constraint equation on the Dirac bispinor representation D(Λ): µ µ ν D(Λ)γ D(Λ) = Λ νγ . (9.3.17) About parity of ψ(x)γµψ(x), µ P µ 0 µ 0 ψ(x)γ ψ(x) ψ0(x)γ ψ0(x) = ψ(P x)γ γ γ ψ(P x) −−→ (9.3.18) =ψ(P x)γµψ(P x). 3). Tensor field quantity. When the constraint equations 9.3.12 and 9.3.17 are satisfied, we have µ ν Λ µ ν 1 α β 1 ψ(x)γ γ ψ(x) Λ Λ ψ(Λ− x)γ γ ψ(Λ− x). (9.3.19) −−→ α β Example for antisymmetric tensor field quantity: i T µν = ψ(x)σµνψ(x), σµν = [γµ, γν]. (9.3.20) 2 4). Pseudo-scalar quantity is invariant under LT and “ ” under parity. − Example: ψ(x)γ5ψ(x) is a pseudo-scalar field quantity. 5 Λ 1 5 1 ψ(x)γ ψ(x) ψ(Λ− x)D(Λ)γ D(Λ)ψ(Λ− x) −−→ (9.3.21) 1 5 1 =ψ(Λ− x)γ ψ(Λ− x), which imposes the constraint equation on the Dirac bispinor representation D(Λ): D(Λ)γ5D(Λ) = γ5 (9.3.22) which can be derived from (9.3.17) due to det(Λ) = 1. About parity of ψ(x)γ5ψ(x), we have ψ(x)γ5ψ(x) P ψ(P x)γ0γ5γ0ψ(P x) −−→ (9.3.23) = ψ(P x)γ5ψ(P x). − 152 5). Pseudo-vector, called axial vector, is covariant under LT and “ ” under parity. − Note: parity of axial vector: Aµ P A . (9.3.24) −−→ − µ Example ψ(x)γµγ5ψ(x) is a pseudo-vector field quantity. µ 5 Λ µ 1 ν 5 1 ψ(x)γ γ ψ(x) Λ ψ(Λ− x)γ γ ψ(Λ− x), (9.3.25) −−→ ν when the constraint conditions (9.3.12) and (9.3.17) are satisfied. About parity of ψ(x)γ5ψ(x), we have ψ(x)γµγ5ψ(x) P ψ(P x)γ0γµγ5γ0ψ(P x) −−→ (9.3.26) = ψ(P x)γ γ5ψ(P x). − µ In summary, we derive two constraint equations: (9.3.12) and (9.3.17) for the Dirac bispinor representation: D(Λ)D(Λ) = 114; (9.3.27) µ µ ν D(Λ)γ D(Λ) = Λ νγ ; (9.3.28) which are to be solved in the following subsection. 9.3.2 Explicit formulation of D(Λ) The Lorentz transformation Λ satisfying ΛT ηΛ = η, (9.3.29) has a solution give by i Λ = exp( ω J ρσ), (9.3.30) −2 ρσ with ω = ω , where the orbital angular-momentum tensor J ρσ has the form ρσ − σρ (J ρσ) = i(δρ δσ δρ δσ ) (9.3.31) αβ α β − β α satisfying the algebra generated by the generators of the Lorentz group. The Dirac spinor representation D(Λ) defined as D(Λ) 1 ψ (x) ψ0 (x) = D(Λ) ψ (Λ− x). (9.3.32) a −−−→ a ab b has the form i D(Λ) = exp( ω Sρσ), (9.3.33) −2 ρσ where the spin angular-momentum tensor Sρσ is given by i 1 Sρσ = [γρ, γσ] = σρσ (9.3.34) 4 2 with σ0i = iαi, σij = iαiαj (9.3.35) − 153 in the Dirac algebra, which satisfy the algebra generated by the generators of the Lorentz group. As an exercise, we prove the lemma to be used soon. Lemma 1. µ ρσ ρσ µ ν [γ ,S ] = [J ] νγ . (9.3.36) Also, we have another lemma: Lemma 2. For the Lorentz boost: D†(Λ) = D(Λ); 1 For the Lorentz rotation: D†(Λ) = D− (Λ). Proof. To prove Lemma 2, we perform the following calculation step by step. Step 1. In the Weyl representation, the spin angular-momentum tensor Sρσ has the form i σi 0 S0i = , (9.3.37) 2 0 σi − 1 σk 0 Sij = ε Lk,L = . (9.3.38) ijk k 2 0 σk where σi is the Pauli matrix. Step 2. Consider the Lorentz boost in the i-th direction, and denote φ~ = (ω0i) = (φ, 0, 0) (9.3.39) where φ is a real number, so the Dirac spinor representation D(Λ) has the form 1 D(Λ) = exp( iφ S0i) = exp( φ~ ~α) − i 2 · φ~ ~σ (9.3.40) exp · 0 = 2 . φ~ ~σ 0 exp −2· ! Step 3. Consider the Lorentz rotation around the k-th direction, namely, take k ωij = εijkθ , (9.3.41) k with the antisymmetric rank-three tensor εijk and real numbers θ , so the Dirac spinor representation D(Λ) has the form i D(Λ) = exp( ω ε L ) = exp( iθ~ L~ ) −2 ij ijk k − · ~ (9.3.42) exp iθ ~σ o = − 2· . iθ~ ~σ 0 exp − 2· ! Note: D(Λ) is a 4 4 matrix with two 2 2 block matrices. × × Now, we prove that the Dirac spinor representation (9.3.33) satisfies the two constraint equations: (9.3.12) and (9.3.17), derived in the last subsection. 1 Thm. 1. D(Λ) = D− (Λ). 154 Proof. 0 0 D(Λ) = γ D†(Λ)γ . (9.3.43) For the Lorentz boost, 0 0 0 0 D(Λ) = γ D†(Λ)γ = γ D(Λ)γ 1 1 = γ0 exp( φ~ ~α)γ0 = exp( φ~ ~α) (9.3.44) 2 · −2 · 1 = D− (Λ) where the anticommutator γ0, αi = 0 has been used since γ0 = β. { } For the Lorentz rotation, 0 0 0 1 0 D(Λ) = γ D†(Λ)γ = γ D− (Λ)γ = γ0 exp(iθ~ L~ )γ0 = exp(iθ~ L~ ) (9.3.45) · · 1 = D− (Λ), 0 where the commutator [αiαj, γ ] = 0 has been exploited. µ µ ν Thm. 2. D(Λ)γ D(Λ) = Λ νγ . Proof. In first order of ωρσ, we have i i i (1 + ω Sρσ)γµ(1 ω Sρσ) = (1 ω J ρσ)µ γν (9.3.46) 2 ρσ − 2 ρσ − 2 ρσ ν which gives rise to the Lemma 1. In higher order expansion of ωρσ, we do similar calcula- tion. 9.4 Canonical quantization of free Dirac field theory Spin-statistics theorem (Fierz, 1939; Pauli, 1940; Schwinger 1950): Particles with integral spin (spin 0, spin 1, ) are quantized as Bosons with commu- ··· tators; particles with half integral spin (spin 1/2, spin 3/2, ) are quantized as fermions ··· with anti-commutators. The Dirac field theory describes the dynamics of spin-1/2 fermions, so the associated canonical quantization makes use of anti-commutators. 9.4.1 Canonical quantization The free Dirac field ψ(t, ~x), with four component-fields ψa(t, ~x), a = 1, 2, 3, 4, has the Lagrangian µ = ψ(iγ ∂µ m)ψ L − (9.4.1) i = iψ†ψ˙ + iψ γ ∂ ψ mψ ψ , a a a ab i b − a a so that the conjugate momentum of the component-field ψa(t, ~x) is given by ∂ Πψa L = iψa†. (9.4.2) ≡ ∂ψ˙a In accordance with the Spin-statistics theorem, we have the equal-time anti-commutators given by 155 (I). The equal-time anti-commutators of Dirac component-fields. (3) ψ (t, ~x), ψ†(t, ~y) = δ δ (~x ~y), (9.4.3) { a b } ab − ψ (t, ~x), ψ (t, ~y) = 0 = ψ†(t, ~x), ψ†(t, ~y) . (9.4.4) { a b } { a b } (II). The equal-time anti-commutators of creation and annihilation operators. The plane-wave solution of the free Dirac equation has the form 2 3 d ~p 1 (r) (r) ip x (r) (r) ip x ψ(t, ~x) = (b u e− · + c †v e · ), (9.4.5) (2π)3/2 2w ~p ~p ~p ~p r=1 ~p X Z with its Dirac conjugation p 2 3 d ~p 1 (r) (r) ip x (r) (r) ip x ψ(t, ~x) = (b †u e · + c v e− · ), (9.4.6) (2π)3/2 2w ~p ~p ~p ~p r=1 ~p X Z (r) (r) p (r) (r) (r) where b~p , c~p are annihilation operators and b~p †, c~p †v~p are creating operators, r = 1 for the spin-up polarization and r = 2 for the spin-down polarization. The equal-time anti-commutators of creation and annihilation operators have the form (r) (s) (3) (r) (s) b , b † = δrsδ (~p ~p0) = c , c † , (9.4.7) { ~p ~p0 } − { ~p ~p0 } (r) (s) (r) (s) b , b = 0 = b †, b † , (9.4.8) { ~p ~p0 } { ~p ~p0 } (r) (s) (r) (s) c , c = 0 = c †, c † . (9.4.9) { ~p ~p0 } { ~p ~p0 } Theorem. Two types of anti-commutative relations (I) and (II) are equivalent. 9.4.2 Fock space with the Fermi-Dirac statistics Let us construct the Fock space of fermions defined as particles obeying the Fermi-Dirac statistics. 1). Vacuum state vac (no-particle state) defined as | i b(r) vac = 0 = c(s) vac , (9.4.10) ~p | i ~p | i (r) where the b~p operator is about the annihilation of particle (such as electron) with spin (s) r, and the c~p operator is about the annihilation of anti-particle (such as positron) with spin s, and the label r = s = 1 represents spin up and the label r = s = 2 represents spin down. 2). One-particle state. Rotation invariant one-particle state defined as (r) ~p,r = b † vac , (9.4.11) | i ~p | i and Lorentz invariant one-particle state defined as p, r = (2π)3/2 2w ~p,r . (9.4.12) | i ~p| i The normalization condition of one-particle statep can be calculated (s) (r) (3) ~p0, s ~p,r = vac b , b † vac = δrsδ (~p ~p0). (9.4.13) h | i h |{ ~p0 ~p }| i − Similarly, one-antiparticle state can be defined too. 156 3). Multi-particle state. For example, a two-particle state defined as (r1) (r2) ~p1, r1; ~p2, r2 = b †b † vac | i ~p1 ~p2 | i (r2) (r1) = b †b † vac (9.4.14) − ~p2 ~p1 | i = ~p , r ; ~p , r −| 2 2 1 1i which gives rise to ~p,r; ~p,r = 0, interpreted as The Pauli exclusion principle: T- | i wo identical fermions are forbidden in the same quantum state. For a high school student, this principle is just a statement in universal physics and chemistry; for an undergraduate student major in theoretical physics, it is just an assumption in non- relativistic quantum mechanics; for a graduate student in theoretical physics, it is a derived theorem in relativistic quantum field theory. Hence, quantum field theory indeed represents a kind of progress in the human being’s knowledge of nature. 9.4.3 Symmetries and conservation laws: energy, momentum and charge Energy, momentum and charge in the free Dirac theory are physical quantities because they are conserved quantities associated with symmetries. 1). Energy-momentum tensor T µν due to the invariance of the action under the space- time translation transformation µ δψ(x) = a ∂µψ(x) (9.4.15) has the form ∂ T µν L ∂νψ = ψiγµ∂νψ. (9.4.16) ≡ ∂(∂µψ) As the upper index µ = 0, T 0ν is the charge density of the form 0ν ν T = iψ†∂ ψ (9.4.17) in which the case of ν = 0 denotes the energy density (the Hamiltonian density) 00 T = = iψ†ψ˙; (9.4.18) H and the other case of ν = i denotes the momentum density 0i T = iψ†∂ ψ. (9.4.19) − i 2). Total energy and total momentum. The energy density T 00 is the Hamiltonian density, = Π ψ˙ H ψ − L µ = iψ†ψ˙ ψ(iγ ∂ m)ψ (9.4.20) − µ − = iψ†ψ,˙ and the total energy is the total Hamiltonian 3 3 H = d ~x = d ~xiψ†ψ˙ H Z Z 2 3 (r) (r) (r) (r) = d ~pE (b †b c c †) ~p ~p ~p − ~p ~p (9.4.21) r=1 X Z 2 3 (r) (r) (r) (r) 3 (3) = d ~pE~p(b~p †b~p + c~p †c~p ) + 2 d ~pE~pδ (0), r=1 X Z Z 157 where the infinity can be removed by the normal ordering defined as 2 3 (r) (r) (r) (r) : H := d ~pE~p(b~p †b~p + c~p †c~p ). (9.4.22) r=1 X Z Note: Inside the normal ordering of fermion operators (also in the time ordering of fermion operators), fermion operators are anti-commutative of the form : c(r)c(r)† := : c(r)† c(r) := c(r)† c(r). (9.4.23) ~p ~p − ~p ~p − ~p ~p The total momentum is calculated as follows with the help of the normal ordering of fermion operators, i 0i 3 ∂ : P : = ~x : T := d ~x : ψ†( i ψ): − ∂xi Z Z 2 (9.4.24) 3 i (r) (r) (r) (r) = d ~pp (b~p †b~p + c~p †c~p ). r=1 X Z 3). Charge due to the U(1) invariance. The U(1) transformation given by δψ(x) = iαψ(x); (9.4.25) − δψ†(x) = iαψ(x) (9.4.26) gives rise to the conserved current ∂ jµ L δψ = ψγµψ, (9.4.27) ≡ ∂(∂µψ) with the charge density from the µ = 0 case 0 0 j = ψγ ψ = ψ†ψ, (9.4.28) and the total charge 3 Q = d ~xψ†ψ Z 2 (9.4.29) 3 (r) (r) (r) (r) = d ~p (b †b c †c ). ~p ~p − ~p ~p r=1 X Z A Dirac particle has +1 charge due to the commutative relation (r) (r) [Q, b~p †] = (+1)b~p †, (9.4.30) and a Dirac anti-particle has 1 charge due to the commutative relation − (r) (r) [Q, c †] = ( 1)c †, (9.4.31) ~p − ~p both of which can be used to test the charge number of a given multi-particle state in the Fock space. 158 9.5 Feynman propagator for a Dirac particle Inside time-ordering and normal ordering, exchange of two fermion operators introduces a minus sign. e.g. 1. T (ψ ψ ) = ψ ψ ; (9.5.1) 1 2 − 2 1 : ψ ψ := : ψ ψ : . (9.5.2) 1 2 − 2 1 e.g. 2. T (ψ ψ ψ ) =: ψ ψ ψ : + ψ ψ ψ + ψ ψ ψ ψ ψ ψ . (9.5.3) 1 2 3 1 2 3 1 2 3 1 2 3 − 1 3 2 e.g. 3. : ψ ψ ψ := : ψ ψ ψ := ψ ψ ψ = ψ ψ ψ . (9.5.4) 1 2 3 − 1 3 2 − 2 1 3 − 1 3 2 e.g. 4. : ψ ψ ψ ψ :=: ψ ψ ψ ψ :( 1)2. (9.5.5) 1 2 3 4 1 4 2 3 − S ψ (x)ψ (y) = T ψ (x)ψ (y) : ψ (x)ψ (y):, (9.5.6) αβ ≡ α β α β − α β S = vac T ψ (x)ψ (y) vac . (9.5.7) αβ h | α β | i When x0 > y0, ψ (x)ψ (y) = vac ψ (x)ψ (y) vac α β h | α β | i ( ) = vac ψ(+)(x)ψ − (y) vac h | α β | i 3 3 2 d ~pd ~p0 1 (r) (s) ip x+ip y (r) (s) = u u e− · 0· vac b b † vac (2π)3 2E 2E ~p,α ~p0,β h | ~p ~p0 | i ~p ~p0 r,s=1 Z X 3 p 2 p d ~p (r) (r) ip (x y) = u u e− · − (9.5.8) (2π)32E ~p,α ~p,β Z ~p r=1 3 X d ~p ip (x y) = (p/ + m) e− · − (2π)32E αβ Z ~p 3 µ d ~p ip (x y) = (iγ ∂ + m) e− · − µ αβ (2π)32E Z ~p = (iγµ∂ + m) θ(x0 y0) vac φ(x)φ(y) vac , µ αβ − h | | i where the Dirac field has been connected with real scalar field theory, and we have µ ψα(x)ψβ(y) = (iγ ∂µ + m)αβ φ(x)φ(y) 4 µ d p i ip (x y) = (iγ ∂ + m) e− · − µ αβ (2π)4 p2 m2 + iε (9.5.9) 4 Z − d p i(p/ + m)αβ ip (x y) = e− · − . (2π)4 p2 m2 + iε Z − Note: p2 = p/p/, p2 m2 = (p/ + m)(p/ m). (9.5.10) − − 159 4 d p i ip (x y) ψ(x)ψ(y) = e− · − , (9.5.11) (2π)4 p/ m + iε Z − ψ(x)ψ(y) = 0 = ψ(x)ψ(y), (9.5.12) ψ(x)φ(y) = 0, (9.5.13) d4p ie ip (x y) φ(x)φ(y) = − · − . (9.5.14) (2π)4 p2 m2 + iε Z − 9.6 Feynman diagrams and Feynmans rules for fermion field- s Nucleon-meson theory: The Lagrangian is 1 1 = ψ(iγµ∂ m)ψ + (∂ φ)2 µ2φ2 gψΓψφ, (9.6.1) L µ − 2 µ − 2 − where Γab = δab is called Yukawa interacton, Γab = (iγ5)ab is called Pseudo Yukawa interaction. The interaction Hamiltonian is = gψ¯ Γ ψ φ (9.6.2) Hint a ab b where g is coupling constant and φ(x) is pseudo scalar field satisfied φ(P x) = φ(x). (9.6.3) − Dyson’s formula: S = U(+ , ) = T exp( i d4x ) ∞ −∞ − Hint + Z (9.6.4) ∞ ( i)n = 1 + − d4x d4x d4x T ( (x ) (x ) x )). n! 1 2 ··· n Hint 1 Hint 2 ···H( n n=1 X Z Vertex: interaction; ¨¨* Feynman diagrams - Internal line: Feynman propagator; HHj External line: incoming or outgoing particles. FRA: in coordinate space time; ¨¨* Feynman rules - FRB: integrate out vertices; HHj FRC: integrate out possible internal momentum. 160 9.6.1 FDs & FRs for vertices = igψ¯Γψφ. (9.6.5) Hint igΓ igΓ − − The arrow on the line denotes the charge flow. In FRC, assign the factor igΓ to each vertex. − 9.6.2 Internal lines p i DF = p2 µ2+iε − p i(p/+m) DF = p2 m2+iε − p i( p/+m) DF = p2 −m2+iε − 9.6.3 External lines In FRC, assign factor “1” to incoming or outgoing meson fields. Incoming fermions line: ~p,s A (s) ip x FR : u~p e− · ; x FRB&FRC : u(s). (4 1) ~p × ψ(x) p, s vac ψ(s)(x) p, s | i ≡ h | | i 2 d3~p 1 0 (r) (r) ip0 x 3 (s) = vac b u e− · (2π) 2ω~pb † vac h | (2π)3 2ω ~p0 ~p0 ~p | i Z ~p0 r=1 (9.6.6) X p d3~p 1 0 p (r) ip0 x 3 (3) = u e− · (2π) 2ω~pδrsδ (~p ~p0) (2π)3 2ω ~p0 − Z ~p0 (s) ip x p = u~p e− · p. Incoming anti-fermion line: ~k,s FRC : v(s). (1 4) x ~k × Outgoing fermion line: ~p,s FRC : u(s). (1 4) x ~p × 161 Outgoing anti-fermion line: ~k,s FRC : v(s). (4 1) x ~k × 9.6.4 Combinational factor (Symmetry factor) 1 1 = n! = 1 (9.6.7) S n! × 1 1 where n! is permutations of vertices, and n! comes from Dyson’s formula. Once S = 1, positions of vertices in FDs are fixed. 9.7 Special Feynman rules for fermions lines 9.7.1 Mapping from matrix to number Scattering amplitude is a number. Feynman propagator S 4 4 F × Vertex igΓ 4 4 − ab × u&v 4 1 External line × u&v 1 4 × (1 4)(4 4)(4 4) (4 4)(4 1) × × × ··· × × row matrixu ¯&¯v iΓ SF column matrix u&v 9.7.2 Feynman rules for a single fermion line from initial state to the final state ~q1 ~q ~q ~p,r 2 3 ~p ~q1 − ~p ~q1 ~q2 − − ~p0,s (s) (r) u¯ ( igΓ)SF (~p ~q1 ~q2)( igΓ)SF (~p ~q1)( igΓ)u ~p0 − − − − − − ~p 1 4 4 4 4 4 4 4 4 4 4 4 4 1 × × × × × × × Remark 1: The arrow on the fermion line must flow consistently through the diagram, due to charge conservation. Remark 2: The order of matrix multiplication is given by starting at the end of an arrow and working back to the start. 162 9.7.3 Relative sign between FDs e.g. 1. 1 3 1 3 + (1 3) ↔ 2 4 2 4 FDs with same fermion lines but two boson lines exchanged, then they differ by a “+” sign due to Bose statistics. e.g. 2. 1 3 1 4 (3 − 4) ↔ 2 4 2 3 FDs width two fermion lines exchanged, then they differ by a relative sign “ ”. − 9.7.4 Minus sign from a loop of fermion lines ~p φ igΓ igΓ φ − − ~q ~q ~p − : ψ(x )( igΓ)ψ(x )φ(x )ψ(x )( igΓ)ψ(x )φ(x ): 1 − 1 1 2 − 2 2 = ( igΓ) ψ(x )ψ(x )( igΓ) ψ(x )ψ(x ): φ(x )φ(x ): (9.7.1) − − 1 2 − 2 1 1 2 = ( igΓ)S (x x )( igΓ)S (x x ): φ(x )φ(x ): − − F 1 − 2 − F 2 − 1 1 2 =Tr(( igΓ)S (x x )( igΓ)S (x x )) : φ(x )φ(x ): . − F 1 − 2 − F 2 − 1 1 2 Remark: Loop of fermion line is the trace with an additional“ ” sign − Tr(( igΓ)S (~p)( igΓ)S (~q ~p)). (9.7.2) − − F − F − 9.8 Examples 9.8.1 First order φ N + N. (9.8.1) −→ 163 ~p,r (r) N u 1 4 ~p × φ (s) N v 4 1 ~p,s −~p × − (1) (r) (s) i fi = u~p ( igΓ)v ~p. (9.8.2) A − − 9.8.2 Second order 1). N + φ N + φ. (9.8.3) −→ ~q ~q0 φ φ + (q q0) ↔ (r) (r0) u~p N N u~p0 ~p,r ~p0, r0 (2) (r0) (r) i = u ( igΓ)(SF (~p + ~q) + SF (~p ~q0))( igΓ)u . (9.8.4) Afi ~p0 − − − ~p 2). N + φ N + φ. (9.8.5) −→ ~q ~q0 φ φ ( + (q q0)) − ↔ (r) (r0) v~p N N v~p0 ~p,r ~p0, r0 (2) (r) (r0) i = v ( igΓ)(SF (~p + ~q) + SF (~p ~q0))( igΓ)v . (9.8.6) Afi − ~p − − − ~p0 n o There is a minus overall sign, due to the exchange of fermion operators. Remark: overall sign has no contribution to cross section. : ψ(x )Γψ(x )φ(x )ψ(x )Γψ(x )φ(x ): N(p, r) 1 1 1 2 2 2 | i (9.8.7) =( 1)3 :Γ ψ(x )ψ(x )Γψ(x )φ(x )φ(x ): ψ(x ) N(p, r) − 1 2 2 1 2 1 | i 3). N + N N + N. (9.8.8) −→ 164 ~q, s ~q , s N 0 0 N ( ~q − ~q0 (~q , s ~p , r )) − − 0 0 ↔ 0 0 N N ~p,r ~p0, r0 i (2) = ( ig)2 Afi − − (s0) (s) (r) (r0) (r0) (s) (r) (s0) (u Γu )(v Γv )DF (~q ~q0) (u Γu )(v Γv )DF (~q ~p0) . × ~q0 ~q ~p ~p0 − − ~p0 ~q ~p ~q0 − n (9.8.9)o : ψ(x )Γψ(x )φ(x )ψ(x )Γψ(x )φ(x ): N(p , r ),N(p , r ) 1 1 1 2 2 2 | 1 1 2 2 i (9.8.10) = : ψ(x )Γ φ(x )ψ(x )Γφ(x ): ψ(x )ψ(x ) N(p , r ),N(p , r ) . − 1 1 2 2 1 2 | 1 1 2 2 i 4). N + N N + N. (9.8.11) −→ ~p,r ~p0, r0 N N ~p,r ~p0, r0 N N ~p + ~q ~p − ~p0 (~p,r ↔− ~q0, s0) N N ~q, s ~q0, s0 N N ~q, s ~q0, s0 i (2) =( ig)2 Afi − (s) (r) (r0) (s0) (s) (s0) (r) (r0) (v Γu )(u Γv )DF (~p + ~q) (v Γv )(u Γu )DF (~p ~p0) . × ~q ~p ~p0 ~q0 − ~q ~q0 ~p ~p0 − n (9.8.12)o 9.8.3 Fourth order φ + φ φ + φ. (9.8.13) −→ ~p ~p0 φ φ ~k + ~p ~k ~k + ~p ~p0 − ~k ~q − φ φ ~q ~q0 165 4 (4) 4 d k i = ( ig) Tr(S (~k)ΓS (~k + ~p)ΓS (~k ~q)ΓS (~k + ~p ~p0)Γ). (9.8.14) Afi − − (2π)4 F F F − F − Z 9.9 Spin-sums and cross section 1 + 2 10 + 20. (9.9.1) −→ In CoM frame ~p0 ~p θ ~q = ~p − ~q0 dσ 1 ~p = | 0| 2 (9.9.2) dΩ 64π2(E + E )2 × ~p |Afi| 10 CoM 1 2 | | where dΩ10 = d cos θdϕ. (9.9.3) 9.9.1 Spin-sums It is not easy to measure the spin of particles in high energy experiments. Unpolarized incoming fermion particles and averaging over all incoming polariza- • tions. All polarized outgoing particles sum over all final polarizations. • 2 dσ 1 ~p 1 = | 0| 2 . (9.9.4) dΩ 64π2(E + E )2 × ~p 2 |Afi| 10 CoM 1 2 r,s=1 | | X Lemma. 2 1 2 X = u (~p0)Gu (~p) , (9.9.5) 2 s r r,s=1 X 1 X = Tr p/0 + m)G(p/ + m)G , (9.9.6) 2 0 0 where G is product of gamma matrices and G = γ G†eγ . Proof. e 2 1 X = u (~p0)Gu (~p) u (~p0)Gu (~p) † , (9.9.7) 2 s r s r r,s=1 X 166 0 † us(~p0)Gur(~p) † = us†(~p0)γ Gur(~p) 0 (9.9.8) = ur†(~p)G†(γ )†us(~p0) = ur(~p)Gus(~p0) 0 0 0 0 with G = γ G†γ and (γ )† = γ . e 2 e 1 X = u (~p0)Gu (~p)u (~p)Gu (~p0), (9.9.9) 2 s r r s r,s=1 X e 2 1 s r r s X = u (~p0)G u (~p)u (~p)G u (~p0) 2 α αβ β γ γδ δ r,s=1 X (9.9.10) 2 e 1 s s r r = u (~p0)u (~p0) G u (~p)u (~p) G . 2 δ α αβ β γ γδ r,s=1 X e Noted that r, s are spin index, and α, β, γ, δ are matrix component. Because of completeness relation 2 s s uδ(~p0)uα(~p0) = (p/0 + m)δα, (9.9.11) s=1 X 2 r r uβ(~p)uγ(~p) = (p/ + m)βγ, (9.9.12) r=1 X thus we have 1 X = (p/0 + m)δαGαβ(p/ + m)βγGγδ 2 (9.9.13) 1 = Tr (p/0 + m)G(p/ + m)G e. 2 e Corollary. 2 1 2 1 v (~p0)Gu (~p) = Tr p/0 m)G(p/ + m)G ; (9.9.14) 2 s r 2 − r,s=1 X e 2 1 2 1 u (~p0)Gv (~p) = Tr p/0 + m)G(p/ m)G ; (9.9.15) 2 s r 2 − r,s=1 X e 2 1 2 1 v (~p0)Gv (~p) = Tr p/0 m)G(p/ m)G . (9.9.16) 2 s r 2 − − r,s=1 X e 167 9.9.2 Examples See Luke’s notes P.P. 166-167. (2) 2 (s) (p/ + q/ + m) (p/ q/0 + m) (r) i = ig u γ5 + − γ5u Afi ~p0 (p + q)2 m2 + iε (p q )2 m2 + iε ~p − − 0 − (9.9.17) 2 (s) µ (r) = ig u γ qu F (p, p0, q) ~p0 ~p where 1 1 F (p, p0, q) = + . (9.9.18) 2p q + µ2 + iε 2p q µ2 + iε · 0 · − 2 2 1 (2) 1 4 2 µ ν i = g F (p, p0, q) q q Tr (p/0 + m)γ (p/ + m)γ 2 Afi 2 µ ν r,s=1 (9.9.19) X 4 2 2 2 2 = 2g F (p, p0, q) 2(p0 q)(p q) p p0µ + m µ . · · − · 168 Lecture 10 Quantum Electrodynamics About Lecture VII: [Luke] P.P. 168-193; • [Tong] P.P. 124-140; • [Coleman] empty; • [PS] P.P. 123-174; • [Zhou] P.P. 141-181. • [MS] P.P. 99-160. • This semester, we have spent much time in both technique details and solutions to homeworks, so that we only have a little time in Quantum Electrodynamics (QED), which is the key topic of this course. Fortunately, we have prepared everything for perturbative QED in sense of Feynman diagrams. Recall that we have discussed the pseudo-Yukawa theory (describing the interaction between pseudo-nucleons (complex scalar particles) and real scalar particles) and the Yukawa theory (describing the interaction between nucleons (spin-1/2 particles) and real scalar particles), both of which present a good guidance for perturbative QED with the interaction between electrons (spin-1/2 particles) and photons (real vector particles). 10.1 How to quantize the Maxwell fields? Physical quantities of electrodynamics in vacuum is the electric field denoted as E~ and magnetic field denoted as B~ , where the electromagnetic wave is transverse wave satisfying the relation ~k E~ = ~k B~ = 0. (10.1.1) · · So the electromagnetic wave only has two physical polarizations denoted as ε(1) and ε(2) respectively, see the diagram shown below ε(1) ε(2) ~k But the Lorentz covariant electrodynamics prefers the gauge potential description Aµ, µ = 0, 1, 2, 3. Such that it has four independent polarizations 169 (1). ε(0) the time direction; (2). ε(1), ε(2); (3). the propagation direction ε(3), where ε(0) and ε(3) are unphysical and must be removed, which is a difficult problem. 2 µ Way out 1: Introduce small mass term like mγAµA for the quantization of massive vector field theory and then m2 0. γ → Way out 2: Introduce a gauge fixing term, like the Coulomb gauge. Especially, the elec- trodynamics with the Lorentz gauge is for the Gupta-Bleuler quantization. Way out 3: Introduce ghost fields for the path integral quantization. 10.2 Quantization of massive vector fields Here we focus on the quantization of massive vector fields, which is Luke’s chosen approach. The Proca action describes the massive spin-1 particle, where the Lagrangian density is given by 1 1 = F F µν + m2 A Aµ. (10.2.1) L −4 µν 2 γ µ The Proca equation, the equation of motion can be derived as µν 2 ν ∂µF + mγA = 0, (10.2.2) where the Lorentz gauge µ ∂µA = 0 (10.2.3) is easily derived with m2 = 0. Then its derived relation k Aµ = 0 suggests that A only γ 6 µ µ has three independent polarizations. And the equation of motion has the plane wave solutions 3 3~ d k (r) (r) ik x (r) (r) ik x Aµ(x) = a e− · + a† ∗e · . (10.2.4) (2π)3/2 2ω ~k µ ~k µ r=1 ~k X Z and we have the constraint equationsp 2 2 k = mγ; µ (r) (10.2.5) ( k εµ = 0, r = 1, 2, 3. Since k2 = m2 > 0, we know that k is a time-like vector. The constraint k ε(r) = 0 γ · suggests that the polarization vector ε(r) is orthogonal to the momentum vector k and is a space-like vector. For instance, we choose ε(1), ε(2) and ε(3) to survive, which satisfy The normalization and orthogonality relation. • ε(r)ε(s)µ∗ = ε(r)ε(s)∗ = δ . (10.2.6) µ − rs 170 The completeness relation. • 3 kµkν ε(r)ε(r)∗ = η + . (10.2.7) µ ν − µν m2 r=1 γ X E.g. 1. In the rest frame, we have k = (mγ, 0, 0, 0); (10.2.8) ε(1) = (0, 1, 0, 0), ε(2) = (0, 0, 1, 0), ε(3) = (0, 0, 0, 1). (10.2.9) E.g. 2. In the frame moving along the z axis, we have 2 2 k = ( k3 + mγ, 0, 0, k3); (10.2.10) q (1) (2) (3) 1 2 2 ε = (0, 1, i, 0), ε = (0, 1, i, 0), ε = 2 (k3, 0, 0, k3 + mγ). (10.2.11) − mγ q Note: ε(3) will be required to be decoupled from the theory when m2 0. γ → Proof. Proof for the completeness relation (10.2.7). Firstly, we can introduce the Dirac notation to describe the momentum vector and polarization vector, ε(r) = r = ε(r) µ , (10.2.12) | i µ | i µ X ε(s)† = s = ν ε(s)∗, (10.2.13) h | h | ν ν X (r)∗ (s)µ where we have ν µ = η . The orthogonal relation (10.2.6) εµ ε = δ is equivalent h | i µν − rs with r s = δ . (10.2.14) h | i − rs The momentum vector k can be rewritten as k , which is orthogonal to the polarization µ (r) | i vector k εµ = 0, namely r k = 0. (10.2.15) h | i From the relations (10.2.14) and (10.2.15), we have 3 r r s = s ; (10.2.16) | ih | i −| i r=1 X 3 r r k = 0. (10.2.17) | ih | i r=1 X We assume 3 (r) (r)∗ εµ εν = Aηµν + Bkµkν. (10.2.18) r=1 X Therefore, we have 3 ε(r)ε(r)∗ε(s)ν = Aε(s)µ + Bk (k ε(s)). (10.2.19) µ ν µ · r=1 X 171 Comparing above relation with the relation (10.2.16), we can find Aε(s)µ = ε(s)µ, (10.2.20) − thus A = 1. Similarly, from − 3 (r) (r)∗ ν ν 2 εµ εν k = Aηµνk + Bkµk = 0, (10.2.21) r=1 X comparing it with the relation (10.2.17), we know k + Bk m2 = 0, (10.2.22) − µ µ γ 1 then B = 2 . So that we have the completeness relation mγ 3 kµkν ε(r)ε(r)∗ = η + . (10.2.23) µ ν − µν m2 r=1 γ X Back to the quantization of the massive vector fields, from the Lagrangian density (10.2.1), the canonical momentum can be calculated as ∂ Π0 = L = 0; (10.2.24) ∂(∂0A0) ∂ Πi = L = F 0i = Ei. (10.2.25) ∂(∂0Ai) − Through the canonical quantization procedure, we impose the equal-time commutator relations [A (t, ~x),E (t, ~y)] = iδ δ(3)(~x ~y); (10.2.26) i j ij − [Ai(t, ~x),Aj(t, ~y)] = 0 = [Ei(t, ~x),Ej(t, ~y)] , (10.2.27) (r) which give rise to the commutation relations of the creation operator a† and the anni- ~k (r) hilation operator a~k shown as (r) (s) rs (3) ~ ~ a~k , a~† = δ δ (k k0); (10.2.28) k0 − (r) (s) (r) (s) a~ , a~ = 0 = a† , a~ . (10.2.29) k k ~k k0 h i The Hamiltonian with the normal ordering can be written as 3 3 (r) (r) : H := d ~kω a† a . (10.2.30) ~k ~k ~k r=1 Z X 172 The Feynman propagator for the massive vector fields can be calculated as A (x)A (y) = vac TA (x)A (y) vac µ ν h | µ ν | i x0>y0 ===== vac A (x)A (y) vac h | µ ν | i (+) ( ) = vac A (x)A − (y) vac h | µ ν | i 3~ 3 d k ik (x y) (r) (r) = e− · − ε (k)ε ∗(k) (2π)32ω µ ν k r=1 Z X 3~ d k ik (x y) kµkν = e− · − η + (2π)32ω − µν m2 Z k γ (10.2.31) y y 3~ ∂µ∂µ d k ik (x y) = η e− · − − µν − m2 (2π)32ω γ Z k ∂y∂y = η µ µ φ(x)φ(y) − µν − m2 γ y y 4 ∂µ∂µ d k i ik (x y) = η e− · − − µν − m2 (2π)4 k2 m2 + iε γ Z − γ 4 d k kµkν i ik (x y) = η + e− · − . (2π)4 − µν m2 k2 m2 + iε Z γ − γ Feynman diagrams and Feynman rules for the massive vector field theories show below. Internal line (the massive photon propagator). • k i kµkν µ ν k2 m−2 +iε(ηµν m2 ) − γ − γ Incoming massive photon. • A (x) k, r = vac A (x) k, r µ | i h | µ | i 3 3 d ~k (s) = vac ε(s)(k )e ik0 xa k, r 3/2 µ 0 − · ~ h | (2π) 2ω k0 | i s=1 ~k X Z 0 (r) ik x = εµ (k)e− · . p (10.2.32) k, µ (r) εµ (k) Outgoing massive photon. • k, r A (x) = k, r A (x) vac h | ν h | ν | i (r)∗ ik x = εν (k)e · . (10.2.33) k, ν (r) εν ∗(k) 173 10.3 Plan: From massive electrodynamics to quantum elec- trodynamics The plan of deriving the quantum electrodynamics from the massive electrodynamics shows below. m2 0 Massive photon γ → Massless photon (the Proca theory) forbidden (with two polarizations) +electron +electron m2 0 Massive electrodynamics γ → Quantum electrodynamics with three polarizations allowable (with two polarizations) Note 1. The limitation m2 0 is forbidden in the massive vector field theories, due to γ → the denominator term 1/m2 + . γ → ∞ Note 2. Why massive electrodynamics allows to the limitation m2 0? Because we have γ → the U(1) gauge invariance in quantum electrodynamics. Note 3. As 1/m2 + is allowed, we can decouple ε(3) in calculation and directly apply γ → ∞ 2 ε(r)ε(r)∗ = η . (10.3.1) µ ν − µν r=1 X Note 4. In practice, we apply the spin (polarization) summation technique so that the exact form of ε(r) is not needed. 10.4 Quantum Electrodynamics: local U(1) gauge invari- ance The Lagrangian density of the Dirac field is given by = ψ(iγµ∂ m)ψ. (10.4.1) LDirac µ − The global (rigid) U(1) transformation gives rise to iλ ψ(x) e− ψ(x); (10.4.2) → ψ(x) eiλψ(x). (10.4.3) → From the global U(1) symmetry, the conserved current takes the form ∂ J µ = L δψ = ψγµψ, (10.4.4) ∂(∂µψ) µ by which the conservation relation ∂µJ = 0 gives rise to the relation µ ∂µ ψγ ψ = 0. (10.4.5) 174 In the case of local U(1) transformation, we have ieλ(x) (I): ψ(x) e− ψ(x); (10.4.6) → ψ(x) eieλ(x)ψ(x), (10.4.7) → where λ(x) is a scalar function of x. Such the local U(1) transformation gives rise to ψ(x)ψ(x) ψ(x)ψ(x); (10.4.8) → ieλ(x) ∂ ψ(x) ∂ e− ψ(x) µ → µ ieλ(x) ieλ(x) = ie∂ λ(x)e− ψ(x) + e− ∂ ψ(x). (10.4.9) − µ µ Therefore, we have ψ(x)iγµ∂ ψ(x) ψ(x)iγµ∂ ψ(x) + eψ(x)γµψ(x)∂ λ(x). (10.4.10) µ → µ µ In Electrodynamics given by the Lagrangian density 1 = F F µν, (10.4.11) LEM −4 µν the local U(1) transformation gives rise to the transformations (II :) A (x) A (x) + ∂ λ(x); µ → µ µ . (10.4.12) LEM → LEM 1 With the minimal coupling between the spin- 2 particle and photon, we have the interaction Lagrangian density = J µA = eψγµψA , (10.4.13) Lint − µ − µ which is transformed as eψγµψ∂ λ(x) (10.4.14) Lint → Lint − µ under the local U(1) transformation. Note the above additional term can be cancelled by another term (10.4.10). Therefore, the Lagrangian density of quantum electrodynamics given by = + + (10.4.15) LQED LDirac LEM Lint is invariant under the local U(1) gauge symmetry. In quantum electrodynamics, we have the local U(1) gauge invariance. In the Yang- Mills theory, which is the stone of the standard model of particle physics, we have the local abelian gauge invariance, such as the local SU(2) gauge invariance. 10.5 Remark on the covariant derivative With the covariant derivative Dµ D ∂ + ieA (10.5.1) µ ≡ µ µ 175 the Lagrangian density of quantum electrodynamics 1 = F F µν + ψ (iγµ(∂ + ieA ) m) ψ (10.5.2) LQED −4 µν µ µ − can be rewritten as 1 = F F µν + ψiγµD ψ mψψ. (10.5.3) LQED −4 µν µ − The covariant derivative of the Dirac field given by Dµψ(x) = (∂µ + ieAµ(x)) ψ(x), (10.5.4) transforms under the local U(1) transformation, ieλ(x) ieλ(x) D ψ(x) ie∂ λ(x)e− ψ(x) + e− ∂ ψ(x) µ → − µ µ ieλ(x) ieλ(x) +ieAµe− ψ(x) + ie∂µλ(x)e− ψ(x) ieλ(x) = e− Dµψ(x), (10.5.5) which informs ψiγµD ψ ψiγµD ψ. (10.5.6) µ → µ The equation of motion of the Dirac field has the form µ iγ Dµψ = mψ (10.5.7) which is iγµ∂ ψ mψ = J µψ = eψγµψA (x). (10.5.8) µ − µ 10.6 Construction: From massive electrodynamics to quan- tum electrodynamics In massive quantum electrodynamics, the Lagrangian density is given by 1 mγ = + m2 A Aµ. (10.6.1) LQED LQED 2 γ µ When m2 = 0, we do not have local U(1) gauge invariance because γ 6 A (x) A (x) + ∂ λ(x), µ → µ µ A (x)Aµ(x) A (x)Aµ(x) + 2∂ λ(x)Aµ(x) + λ2 . (10.6.2) µ → µ µ O 2 When mγ = 0, we have the local U(1) gauge invariance. Thm 10.6.0.1. When m2 0, namely mγ , γ → LQED → LQED (1). The second term in the massive photon propagator k k 1 iη + i µ ν (10.6.3) − µν m2 k2 m2 + iε γ − γ has no contribution before m2 0. γ → 176 (2). The second term in the completeness relation 3 kµkν ε(r)ε(r)∗ = η + . (10.6.4) µ ν − µν m2 r=1 γ X has no contribution before m2 0. γ → (3) 2 (3). The third polarization vector εµ has no contribution when m 0. γ → (1). Example to check the statement (1) of Theorem 10.6.0.1. The Bhabha scattering + + e−(p , r ) + e (p , r ) e−(p0 , r0 ) + e (p0 , r0 ), (10.6.5) 1 1 2 2 −→ 1 1 2 2 in the second order of the coupling constant, has the Feynman diagram shown as p1, r1 p10 , r10 e− e− p1 + p2 µ ν e+ e+ p2, r2 p20 , r20 In the massive quantum electrodynamics, the amplitude can be obtained from the above Feynman diagram. iη k k 1 (r ) i (2) = ( ie)2 v(r2)γµu(r1) − µν + i µ ν u(r0)γνv 20 ~p2 ~p1 2 2 2 2 2 ~p0 ~p0 A − k mγ + iε mγ k mγ + iε 1 2 − − iη (r ) = ( ie)2 v(r2)γµu(r1) − µν u(r0)γνv 20 ~p2 ~p1 2 2 ~p0 ~p0 − k mγ + iε 1 2 − k k 1 (r ) +( ie)2 v(r2)γµu(r1) i µ ν u(r0)γνv 20 , (10.6.6) ~p2 ~p1 2 2 2 ~p0 ~p0 − mγ k mγ + iε 1 2 − where k = p1 + p2. Lemma 10.6.0.1. v(r2)γµu(r1)k = 0. (10.6.7) ~p2 ~p1 µ Due to Lemma 10.6.0.1, we know that the second term of (10.6.6) vanishes. First way of proving the Lemma 10.6.0.1 shows below. Proof. v(r2)γµu(r1)k = v(r2)k/u(r1) ~p2 ~p1 µ ~p2 ~p1 = v(r2)(p/ + p/ )u(r1) ~p2 1 2 ~p1 = v(r2)p/ u(r1) + v(r2) p/ u(r1) ~p2 2 ~p1 ~p2 1 ~p1 = mv(r2)u(r1) + mv(r2)u(r1) − ~p2 ~p1 ~p2 ~p1 = 0. (10.6.8) 177 Second way of proving the Lemma 10.6.0.1 shows below. µ µ µ Proof. Due to the global U(1) invariance, we have ∂µJ = 0 with J = ψγ ψ. µ + µ + 0 = vac ∂µJ e e− = ∂µ vac J e e− h | | i h | |µ +i = ∂ vac ψγ ψ e e− µh | | i µ + = ∂µ ψγ ψ e e− | i! (r2) µ (r1) i(p1+p2) x = ∂ v γ u e− · µ ~p2 ~p1 (r2) µ (r1) i(p1+p2) x = iv γ u (p1 + p2)µe− · − ~p2 ~p1 (r2) µ (r1) ik x = iv γ u kµe− · − ~p2 ~p1 = 0. (10.6.9) kµkν Note: The second term 2 in the massive photon propagator has no contribution mγ mainly due to the global U(1) invariance. (2). Example to check the statement (2) of Theorem 10.6.0.1. The Compton scattering + e−(p, s) + γ(k, r) e−(p0, s0) + e γ(k0, r0), (10.6.10) −→ in the second order, has the Feynman diagram shown as k, r k0, r0 γ γ p + k ν µ + (k, r k , r ) ↔ 0 0 e− e− p, s p0, s0 In terms of the massive quantum electrodynamics, the amplitude read from Feynman diagram has the formalism (2) µν (r ) (r) i = M ε 0 ∗(k0)ε (k) (10.6.11) A µ ν with γµ (p/ + k/ + m) γν γν (p/ k/ + m) γµ M µν = ( ie)2( i)u(s0) + 0 − u(s). (10.6.12) − − ~p0 (p + k)2 m2 (p k)2 m2 ~p − 0 − − Therefore, the differential cross section with spin sum is proportion to dσ 2 i (2) dΩ ∝ A 3 3 (r) µν (r0)∗ (r) αβ∗ (r0) ∗ = M εµ (k0)εν (k) M εα (k0)εβ (k) r=1 X rX0=1 3 3 (r) µν αβ∗ (r) ∗ (r0)∗ (r0) = M M εν (k)εβ (k) εµ (k0)εα (k0) r=1 X rX0=1 kνkβ kµ0 kα0 = M µνM αβ∗ η + η + . (10.6.13) − νβ m2 − µα m2 γ γ 178 Lemma 10.6.0.2. µν M kµ0 = 0. (10.6.14) kµ0 kα0 Due to Lemma 10.6.0.2, we know that the term 2 in (10.6.13) has no contribution. mγ µν kν kβ Similarly, because M kν = 0, the term 2 in (10.6.13) also vanishes. mγ Proof of Lemma 10.6.0.2 shows below. Proof. ν ν µν 2 (s0) k/0 (p/ + k/ + m) γ γ (p/0 k/ + m) k/0 (s) M kµ0 = ( ie) ( i)u + − u − − ~p0 (p + k )2 m2 (p k)2 m2 ~p 0 0 0 2 − − − = ( ie) ( i)( 1 + 2 ) . (10.6.15) − − Doing the calculations separately. (s0) ν (s) u~p k/0 (p/0 + k/0 + m) γ u~p 1 = 0 (p + k )2 m2 0 0 − (s0) 2 ν (s) u ( p/k/0 + 2p0 k0 + k0 + mk/0)γ u = ~p0 − · ~p p 2 + k 2 + 2p k m2 0 0 0 · 0 − (s0) 2 ν (s) u ( mk/0 + 2p0 k0 + mγ + mk/0)γ u = ~p0 − · ~p 2p k + m2 0 · 0 γ = u(s0)γνu(s); (10.6.16) ~p0 ~p (s0) ν (s) u~p γ (p/0 k/ + m) k/0u~p 2 = 0 − (p k)2 m2 0 − − (s0) ν (s) u γ (p/0 k/0 + m) k/0u = ~p0 − ~p (p k )2 m2 − 0 − (s0) ν 2 (s) u γ k/0p/ + 2p k0 k0 + mk/0 u = ~p0 − · − ~p 2p k − · 0 (s0) ν 2 (s) u γ mk/0 + 2p k0 + mk/0 mγ u = ~p0 − · − ~p m2 2p k γ − · 0 = u(s0)γνu(s). (10.6.17) − ~p0 ~p Then we have µν 2 2 (s0) ν ν (s) M kµ0 = ( ie) ( i)( 1 + 2 ) = ( ie) ( i)u (γ γ )u = 0. (10.6.18) − − − − ~p0 − ~p Thm 10.6.0.2 (The Ward identity). The Ward identity µ k Mµ = 0 (10.6.19) (r) is for an arbitrary quantum electrodynamics process with an external photon εµ (k) due to the local U(1) gauge invariance. 179 (3). Example to check the statement (3) of Theorem 10.6.0.1. In the frame moving along the z axis, we set 2 2 k = ( k3 + mγ, 0, 0, k3); (10.6.20) q (1) (2) (3) 1 2 2 ε = (0, 1, i, 0), ε = (0, 1, i, 0), ε = 2 (k3, 0, 0, k3 + mγ). (10.6.21) − mγ q With the Ward identity (10.6.19), we have kµM = 0 = k2 + m2 M + k M = 0 µ ⇒ 3 γ 0 3 3 q 1 2 − 2 k3 mγ = M0 = − M3 = 1 + 2 M3. (10.6.22) ⇒ 2 2 − k3 k3 + mγ ! q (3) The third polarization vector εµ of the massive photon with has the contribution 2 2 k + mγ (3) µ k3 3 εµ M = M0 + M3 mγ q mγ 1 1 2 − 2 2 2 k3 mγ mγ = 1 + 2 + 1 + 2 M3 mγ − k3 ! k3 ! 2 4 2 4 k3 1 mγ mγ 1 mγ mγ = 1 2 + 4 + 1 + 2 + 4 M3 mγ − − 2 k3 O k3 !! 2 k3 O k3 !!! 3 mγ mγ = + 3 M3. (10.6.23) k3 O k3 !! (3) µ As m 0, we have εµ M = 0. γ → 10.7 Quantum Electrodynamics: Feynman diagrams and Feynman rules The Lagrangian of Quantum Electrodynamics (QED) is 1 = F F µν + ψ(iγµ∂ m)ψ + , (10.7.1) L −4 µν µ − LI in which the interaction Lagrangian density is = eψγµψA , and the vector field A LI − µ µ represents electromagnetic fields and the bispinor field ψ represent the Dirac field. Note that the photon as quanta of electromagnetic fields Aµ has the following properties, 1). scalar-like; massless; 2). with vector index; polarization up or down. 180 The interaction Hamiltonian density is = eψγµψA with coupling constant e, and HI µ the scattering matrix via the Dyson’s formula is S(+ , ) = T exp( i d4x). (10.7.2) ∞ −∞ − HI Z Feynman rules for Feynman diagrams including vector fields in QED: 1). Interaction vertex. Assign the factor ieγµ at each vertex. − ψ µ Aµ ieγµ − ψ 2). Internal line. Feynman propagator of vector fields. A (x)A (y) = vac TA (x)A (y) vac . (10.7.3) µ ν h | µ ν | i k iηµν µ ν − k2+iε 3). External lines. (r) Incoming photon with the polarization vector εµ (k). k, µ (r) ik x A k, r = ε (k)e− · . (10.7.4) µ| i µ (r) Outgoing photon with the polarization vector εν ∗(k). k, ν (r) ik x k, r A (x) = ε ∗(k)e · . (10.7.5) h | ν ν The polarization sum is the completeness relation of the polarization vectors, 2 (r) (r) ε ε ∗ = η (10.7.6) µ ν − µν r=1 X where r is the polarization parameter and µ, ν are the space-time indices. 181 10.8 Representative scattering processes in QED The following Feynman diagrams are new things from the viewpoint of QED, but are indeed the closest relatives of our old friends which we have already studied in detail in both the pseudo-Yukawa theory and the Yukawa theory. a). The Moller scattering between electrons. 3. e− + e− e− + e−. (10.8.1) −→ 2.N + N N + N. (10.8.2) −→ 1. “N” + “N” “N” + “N”. (10.8.3) −→ p, s p0, s0 e− e− µ p − p 0 (p0, s0 q0, r0) ν − ↔ e− e− q, r q0, r0 (2) 2 (s0) µ (s) iηµν (r0) ν (r) 2 i fi = ( ie) u~p γ u~p − 2 u~q γ u~q ( ie) (r0 s0, p0 q0) A − 0 (p p0) 0 − − ↔ ↔ − (s0) µ (s) (r0) (r) u γ u u γµu 2 ~p0 ~p ~q0 ~q = ( ie) ( i) (r0 s0, p0 q0) . − − (p p )2 − ↔ ↔ − 0 (10.8.4) Similarly, we can compute the scattering process between positrons: 3. e+ + e+ e+ + e+. (10.8.5) −→ 2. N + N N + N. (10.8.6) −→ 1. “N” + “N” “N” + “N”. (10.8.7) −→ b). The Bhabha scattering between an electron and a positron. + + 3. e− + e e− + e . (10.8.8) −→ 2.N + N N + N. (10.8.9) −→ 1. “N” + “N” “N” + “N”. (10.8.10) −→ p, s p0, s0 e− e− p, s p0, s0 e− e− p − p0 p + q (q, r ↔− p0, s0) + + + + e e e e q, r q0, r0 q, r q0, r0 182 (s0) µ (s) (r) (r0) (r) µ (s) (s0) (r0) u~p γ u~p v~q γµv~q v~q γ u~p u~p γµv~q i (2) = ( ie)2( i) 0 0 0 0 Afi − − − (p p )2 − (p+ q)2 − 0 (10.8.11) where the minus sign in front of ( ie)2 denotes the overall minus sign. − Note that the following notes are about the lecture given on 6/17, 2014. c). The pair annihilation between an electron and a positron into two photons. + 3. e− + e 2γ. (10.8.12) −→ 2.N + N 2φ. (10.8.13) −→ 1. “N” + “N” 2φ. (10.8.14) −→ ν p, s p0, ε1 e− γ ν p − p 0 + (p0, ε1 q0, ε2) µ ↔ e+ γ q, r µ q0, ε2 (2) 2 (r) γµ(p/ p/0 + m)γν γν(p/ q/0 + m)γµ (s) µ ν i = ( i)( ie) v − + − u ε ∗ε ∗. (10.8.15) A − − ~q (p p )2 m2 (p q )2 m2 ~p 2 1 − 0 − − 0 − d). The Compton scattering between an electron and a photon. 3. e− + γ e− + γ. (10.8.16) −→ 2.N + φ N + φ. (10.8.17) −→ 1. “N” + φ “N” + φ. (10.8.18) −→ (10.8.19) ν µ q, ε q0, ε γ 1 2 γ p + q ν µ + (q, ε q , ε ) 1 ↔ 0 2 e− e− p, s p0, s0 (2) 2 (s0) γµ(p/ + q/ + m)γν γν(p/ q/0 + m)γµ (s) µ ν ∗ i = ( ie) ( i)u~ 2 2 + − 2 2 u~p ε2 ε1 (10.8.20) A − − − p0 (p + q) m (p q ) m − − 0 − where the minus sign in front of ( ie)2 denotes the overall minus sign. − e). Two-Photon scattering. 2. γ + γ γ + γ. (10.8.21) −→ 1. φ + φ φ + φ. (10.8.22) −→ 183 −→ d4k 1 i (4) + (10.8.23) A ∝ (2π)4 k4 −→ ∞ Z in which the integral becomes the infinity, so only regularized and renormalized Feynman integrals make sense in quantum field theory. 10.9 The calculation of the differential cross section of the Bhabha scattering in second order of coupling constant The Bhabha scattering has the form of reaction, + + e (~p , r ) + e−(~p , r ) e (~p0 , s ) + e−(~p0 , s ), (10.9.1) 1 1 2 2 −→ 1 1 2 2 which gives rise to two Feynman diagrams in second order of coupling constant p2, r2 p20 , s2 e− e− p2, r2 p20 , s2 e− e− p p p1 − p10 1 + 2 − + + + + e e e e p1, r1 p10 , s1 p1, r1 p10 , s1 The associated Feynman integral has the form i (2) = i + i . (10.9.2) A Aa Ab with 2 1 α = ie u (~p0 )γ u (~p ) v (~p )γ v (~p0 ) ; (10.9.3) Aa − s2 2 α r2 2 (p p )2 r1 1 s1 1 1 − 10 2 1 α b = ie us2 (~p20 )γαvs1 (~p10 ) 2 [vr1 (~p1)γ ur2 (~p2)] . (10.9.4) A (p1 + p2) In the center-of-mass frame, the kinematics of the two-particle collision given by 0 0 e+ p1 = (E, ~p ) e+ θ e− p1 = (E, ~p) p2 = (E, ~p) − e p0 = (E, ~p0) − 2 − 184 2 p1 p10 = E ~p ~p0 cos θ; (10.9.5) · 2 − | | p1 p20 = E + ~p ~p0 cos θ; (10.9.6) · | | p p = E2 + ~p 2 ; (10.9.7) 1 · 2 | | 2 2 p0 p0 = E + ~p0 , (10.9.8) 1 · 2 and 2 2 2 ET = E1 + E2 = 2E,ET = 4E = (p1 + p2) , (10.9.9) and ~p0 = ~p m + = m , | | ⇐⇒ e e− 2 2 2 p1 p10 = E ~p ~p0 = E ~p cos θ, · − · − | | 2 2 2 θ (p p0 ) = (~p ~p0) = 4 ~p sin , (10.9.10) 1 − 1 − − − | | 2 the differential cross section of the Bhabha scattering with polarized particles in second order of coupling constant has the form dσ 1 ~p = 2 | 0| dΩ 64π2E2 |Afi| ~p 10 CoM T | | (10.9.11) 1 = 2 . 256E2π2 |Afi| In high energy particle physics, we usually do not measure the spin polarizations of incoming or outgoing particles, so we perform the spin average on incoming particles and the spin sum on outgoing particles, namely, 1 1 1 = (10.9.12) 2 2 4 r r s s r ,r ,s ,s X1 X2 X1 X2 1 X2 1 2 which gives rise to the following the scattering amplitude 2 (2) 1 2 i = i + i = (X + X + X + X∗ ), (10.9.13) Afi 4 | Aa Ab| a b ab ab r1r2s1s2 X with 1 X = 2 ; (10.9.14) a 4 |Aa| r ,r ,s ,s 1 X2 1 2 1 X = 2 ; (10.9.15) b 4 |Ab| r ,r ,s ,s 1 X2 1 2 1 X = ∗; (10.9.16) ab 4 AaAb r ,r ,s ,s 1 X2 1 2 1 X∗ = ∗ . (10.9.17) ab 4 AaAb r ,r ,s ,s 1 X2 1 2 For example, let us calculate Xb in detail: 2 1 α b = ie us2 (~p20 )γαvs1 (~p10 ) 2 [vr1 (~p1)γ ur2 (~p2)] . (10.9.18) A (p1 + p2) 185 and its complex conjugation 2 1 β † b∗ = b = ie vs1 (~p10 )γβus2 (~p20 ) 2 ur2 (~p2)γ vr1 (~p1) , (10.9.19) A A − (p1 + p2) h i with e e 0 0 β 0 β 0 β γ γ γ† γ = γ , γ γ (γ )†γ = γ . (10.9.20) β ≡ β β ≡ has the form e e 2 1 β b∗ = ie vs1 (~p10 )γβus2 (~p20 ) 2 ur2 (~p2)γ vr1 (~p1) (10.9.21) A − (p1 + p2) h i so that the Xb has the form as the product of traces of gamma matrices 4 1 e αβ Xb = 4 AαβB , (10.9.22) 4 (p1 + p2) where the spin sum is performed to derive the following trace formulaes: Aαβ = us2 (~p20 )γαvs1 (~p10 ) vs1 (~p10 )γβus2 (~p20 ) s1,s2 (10.9.23) X = Tr (p/0 + m)γ (p/0 m)γ , 2 α 1 − β and Bαβ = Tr (p/ m)γα(p/ + m)γβ . (10.9.24) 1 − 2 With the trace properties of gamma matrices, the Aαβ is expressed as A = Tr (p/0 γ + mγ )(p/0 γ mγ ) αβ 2 α α 1 β − β 2 = Tr p/0 γ p/0 γ mp/0 γ γ + mγ p/0 γ m γ γ 2 α 1 β − 2 α β α 1 β − α β 2 = Tr p/0 γ p/0 γ m Tr(γ γ ) (10.9.25) 2 α 1 β − α β µ ν 2 = p 0p 0(2g 2g 2η 2g + 2g 2g ) 4m g 2 1 µα νβ − µν αβ µβ αν − αβ 2 = 4 p0 p0 + p0 p0 (m + p0 p0 )g . 1α 2β 2α 1β − 1 · 2 αβ and similarly, the Bαβ has the form Bαβ = 4 pαpβ + pαpβ (m2 + p p )gαβ . (10.9.26) 1 2 2 1 − 1 · 2 With the kinematics of the two-particle scattering in the center-of-mass frame, the Xb is calculated as e4 Xb = 4 16 (p1 p10 )(p2 p20 ) + (p10 p2)(p1 p20 ) 4(p1 + p2) × × · · · · 2 (m + p p )p0 p0 + (p p0 )(p0 p ) + (p p0 )(p p0 ) − 1 · 2 1 · 2 1 · 2 1 · 2 2 · 2 1 · 1 (10.9.27) 2 2 2 (m + p p )p0 p0 p p (m + p0 p0 ) (m + p0 p0 )p p − 1 · 2 2 · 1 − 1 · 2 1 · 2 − 1 · 2 1 · 2 2 2 +4(m + p0 p0 )(p p + m ) , 1 · 2 1 · 2 and it can simplified as e4 Xb = 4 32 (p1 p10 )(p2 p20 ) 4(p1 + p2) × × · · (10.9.28) 2 2 4 + (p p0 )(p p0 ) + (p0 p0 )m + m p p + 2m . 1 · 2 2 · 1 1 · 2 1 · 2 186 which has a further simplified form 4 e 2 2 2 2 Xb = (E ~p ~p0 cos θ) + (E + ~p ~p0 cos θ) 2E4 − | | | | (10.9.29) 2 2 2 2 2 2 4 + m (E + ~p0 ) + m (E + ~p ) + 2m . | | In the approximation E m , E ~p = ~p , the X can be eventually simplified as e ≈ | 0| | | b 4 e 4 2 2 2 2 2 X = E + ~p ~p0 cos θ + (m E ) b E4 | | O (10.9.30) m2 =e4 1 + cos2 θ + ( ) O E2 which gives rise to the part of the differential cross section, 2 dσ 1 4 2 m 2 2 e 1 + cos θ + ( 2 ) dΩ0 ∝ 256π E O E 1 Xb (10.9.31) 1 m2 = α2 1 + cos2 θ + ( ) , 16E2 O E2 with the fine structure constant α = e2/4π or α = 1/137. Similarly, we can calculate Xa, 4 1 e αβ X = A0 B0 a 4 (p p )4 αβ 1 10 r ,r ,s ,s − 1 X2 1 2 1 e4 θ m2 = 128E4 1 + cos4 + ( ) 4 (p p )4 × × 2 O E2 1 − 10 (10.9.32) 1 e4 θ m2 = 128E4 1 + cos4 + ( ) 4 16E4 sin4 θ × × 2 O E2 2 2e4 θ m2 = 1 + cos4 + ( ) . sin4 θ 2 O E2 2 and the associated part of the differential cross section has the form 4 θ dσ 1 2 1 + cos 2 2 α 4 θ . (10.9.33) dΩ0 ∝ 8E sin 1 Xa 2 Furthermore, we can calculate Xab 4 2 2e 4 θ m Xab = − cos + ( ) , (10.9.34) sin2 θ 2 O E2 2 2 which happens to be a real number so X = X in sense of the approximation of ( m ). ab ab∗ O E2 As a conclusion, the differential cross section of the Bhabha scattering in second order of coupling constant has the form 2 4 θ 2 4 θ 2 dσ α 1 + cos 2 1 + cos θ 2 cos 2 m = 2 4 θ + 2 θ + ( 2 ) (10.9.35) dΩ0 8E sin 2 − sin O E 1 2 2 ! which has been verified in experiments, see the reference book [MS]. 187 10.10 The calculation of the differential cross section of the Compton scattering in second order of coupling con- stant 188 Lecture 11 Symmetries and Conservation Laws (II) 11.1 Translation group 11.2 The Lorentz group and its representation theory 11.2.1 Definition of the Lorentz group 11.2.2 Representation theory of the Lorentz group 11.2.3 Scalar, Weyl spinor, Dirac spinor and vector 11.2.4 Parity 11.2.5 The Lagrangian of the Weyl spinor 11.2.6 The Lagragian of the Dirac spinor 11.3 Transformation laws of scalar, Dirac spinor and vector fields under the Poincar´etransformation 11.3.1 Scalar fields 11.3.2 Dirac spinor fields 11.3.3 Vector fields Under the Lorentz transformation, the space and vector field Aµ(x) are transformed as Λ x x0 = Λx; (11.3.1) −−→ UΛ ν 1 A0 (x) A0 (x) = Λ A (Λ− x), (11.3.2) µ −−→ µ µ ν where Lorentz group representation of spin-1 field V (Λ) takes the form i Λ = exp ωρσS . (11.3.3) −2 ρσ The spin-1 angular momentum tensor Sρσ has the expression (S ) ν = i η η ν η η ν . (11.3.4) ρσ µ ρµ σ − σµ ρ With the orbital momentum L = i(x ∂ x ∂ ), (11.3.5) ρσ ρ σ − σ ρ 189 the total momentum tensor has the formalism ν ν ν (Mρσ)µ = (Sρσ)µ + ηµ Lρσ. (11.3.6) Calculate out the variation of transformation DA (x) = A0 (x) A (x) µ µ − µ ν 1 = Λ A (Λ− x) A (x) µ ν − µ i = ωρσ (M ) ν A (x). (11.3.7) −2 ρσ µ ν 190 Lecture 12 Research Projects 12.1 Basic concepts: fundamental problems Review articles. S-matrix. Vacuum energy. The cosmological constant problem. The Casimir force. And so on. 12.2 Model: The Higgs physics Review articles. 12.3 Model: The Neutrino physics Review articles. 12.4 Mathematical Physics: Regularization and renormal- ization theories Review articles. The BPHZ renormalization. The algebraic renormalization. The Connes- Kreimer Hopf algebra. 12.5 Theory: Symmetries, anomalies and breakings Review articles. The algebraic renormalization. The scaling transformation. The confor- mal transformation. 12.6 Model: The mechanism of quark confinement Review articles. Open problem. 12.7 Mathematical Physics: The existence of the Yang-Mills theory Review articles. Open problem. 12.8 Mathematical Physics: Integrable quantum field theo- ries and the Yang–Baxter equation Review articles. 191 12.9 Interdisciplinary topics between quantum field theo- ries and quantum information & computation Review articles. 192 Part II Renormalization and Symmetries 193 Lecture 13 Notes on BPHZ Renormalization A .pdf file is attached for the time being. Later on, it will be replaced by a .tex file. 194 Draft for Internal Circulations: v1: August 19, 2003, Universit¨atLeipzig, Germany. v2: September 4, 2004, Institute of Theoretical Physics, Chinese Academy of Sciences. v3: Almost unchanged from v2, April 16, 2015. Notes on BPHZ Renormalization1 2 Yong Zhang3 School of Physics and Technology, Wuhan University (April 2015) Abstract This article gives an introduction to the BPHZ renormalization in Zimmerman- n’s sense. The historical development of the BPHZ renormalization is firstly reviewed, and then as an example, a massive scalar model defined by the BPHZ renormalization is given. Afterwards, the Zimmermann‘s forest formula, the quantum action principle, the Zimmermann identities, the Callan–Symanzik equation, the algebraic renormalization procedure, charge constructions vi- a the LSZ reduction procedure, and the Connes–Kreimer Hopf algebra are presented. 1A part of my thesis for a PhD. degree (October, 2003) in Universit¨atLeipzig, Germany, under the supervision of Prof. Dr. Klaus Sibold, who is an expert on the algebraic renormalization procedure. 2These notes are uploaded online for who are interested in theoretical research of quantum field theory. 3yong [email protected]. 1 Contents 1 Notations 3 2 Introduction 6 2.1 The history of the BPHZ renormalization ...... 7 2.2 The massive ϕ4 model via the BPHZ renormalization ...... 8 3 The Zimmermann’s forest formula 11 3.1 A simple introduction ...... 11 3.2 A general calculation procedure ...... 13 3.3 Examples in two-loop ...... 15 4 The quantum action principle 19 4.1 The equation of motion ...... 20 4.2 The renormalized version in higher orders ...... 21 5 The Callan–Symanzik equation 24 5.1 The Zimmermann identities ...... 25 5.2 Deriving the Callan–Symanzik equation ...... 28 5.3 The different representations of the Callan–Symanzik equation ...... 31 6 The algebraic renormalization procedure 33 6.1 The construction of the Slavnov–Taylor identity ...... 34 6.2 The consistency condition to all orders ...... 36 7 Current constructions and charge constructions 37 7.1 The conformal algebra ...... 37 7.2 Current constructions and charge constructions ...... 38 8 The Connes–Kreimer Hopf algebra 41 8.1 The axioms of the Hopf algebra ...... 41 8.2 An example: the Connes–Kreimer Hopf algebra ...... 43 9 Acknowledgements 47 2 1 Notations Following general conventions, we would like to choose the natural units: ~ = 1, c = 1. But in the algebraic renormalization procedure, it is both reasonable and prudent to keep ~ in all formulas. On the one hand, in quantum field theories with spontaneously broken symmetries such as the standard model, an expansion in loop numbers is a better choice than in coupling constants. On the other hand, the quantum action principle represented by the 1PI Green functions is a suitable formulation when it is applied to the problem of “renormalization and symmetries”. Furthermore, the loop number in 1PI diagrams is right the power counting of ~, as suggests that keeping ~ in formulas makes much clear the induction proof based on the loop expansions. Moreover, in the sense of the dimensional analysis, keeping ~ is a good way of checking calculation if right or not. The metric ηµν in four dimensional flat space-time is chosen as (η ) = diag(1, 1, 1, 1). (1) µν − − − The Feynman propagator in the article is given by 4 d k i k(x1 x2) ∆F (x1 x2) = i~ e− − Dk, (2) − (2π)4 ∫ where ϵ is a positive parameter and the symbol Dk is denoted by 1 D = . (3) k k2 m2 + iϵ − Z [J], Z [J], and Γ[ϕ] • C The generating functional of the general Green functions, Z [J], is expanded as the power series of a given external source J(x), n ∞ 1 i Z [J] = d4x d4x d4x n! 1 2 ··· n n=0 ~ ∑ ( ) ∫ ∫ ∫ J(x )J(x ) J(x ) 0 T ϕ(x )ϕ(x ) ϕ(x ) 0 , (4) 1 2 ··· n ⟨ | 1 2 ··· n | ⟩ where a n-point general Green function is denoted by G (x , x x ) = 0 T ϕ(x )ϕ(x ) ϕ(x ) 0 . (5) n 1 2 ··· n ⟨ | 1 2 ··· n | ⟩ The generating functional of the connected Green functions, ZC [J], is defined by ZC [J] := i~ ln Z [J], (6) − which can also be expanded as n 1 ∞ 1 i − Z [J] = d4x d4x d4x C n! 1 2 ··· n n=1 ~ ∑ ( ) ∫ ∫ ∫ J(x )J(x ) J(x ) 0 T ϕ(x )ϕ(x ) ϕ(x ) 0 c, (7) 1 2 ··· n ⟨ | 1 2 ··· n | ⟩ where a n-point connected Green function is denoted by G (x , x x )c = 0 T ϕ(x )ϕ(x ) ϕ(x ) 0 c. (8) n 1 2 ··· n ⟨ | 1 2 ··· n | ⟩ 3 The generating functional of the 1PI(one particle irreducible) Green functions, Γ[ϕ], also called the vertex functional, is defined by the Legendre transformation, Γ[ϕ] := Z [J] d4x J(x) ϕ(x), (9) C − ∫ which defines ϕ(x) and J(x) by δZ [J] δΓ ϕ(x) = C ,J(x) = . (10) δJ(x) −δϕ(x) The generating functional of the 1PI(one particle irreducible) Green functions, Γ[ϕ], can also be expanded as a power series of the classical field ϕ(x) by ∞ 1 Γ[ϕ] = d4x d4x d4x n! 1 2 ··· n n=2 ∑ ∫ ∫ ∫ ϕ(x )ϕ(x ) ϕ(x ) 0 T ϕ(x )ϕ(x ) ϕ(x ) 0 1PI . (11) 1 2 ··· n ⟨ | 1 2 ··· n | ⟩ where a n-point 1PI Green function is denoted by Γ (x , x x ) = 0 T ϕ(x )ϕ(x ) ϕ(x ) 0 1PI . (12) n 1 2 ··· n ⟨ | 1 2 ··· n | ⟩ Z [J, ρ], Z [J, ρ], and Γ[ϕ, ρ] • C The generating functional of the general Green functions Z [J, ρ] with insertions of composite operators QA(x) is defined by i Z [J, ρ] := 0 T exp d4x J(x)ϕ(x) + ρA(x)Q (x) 0 (13) ⟨ | A | ⟩ ~ ∫ where the symbol A denotes quantum numbers or space-time indices. The generating functional of the connected Green functions, ZC [J, ρ], with insertions of composite operators QA(x), is defined by ZC [J, ρ] := i~ ln Z [J, ρ]. (14) − The generating functional of the 1PI(one particle irreducible) Green functions, Γ[ϕ, ρ] with insertions of composite operators QA(x), is defined by the transformation, Γ[ϕ, ρ] := Z [J, ρ] d4x J(x) ϕ(x), (15) C − ∫ which also defines ϕ(x) and J(x) by δZ [J, ρ] δΓ[ϕ, ρ] ϕ(x) = C ,J(x) = . (16) δJ(x) − δϕ(x) Insertions of composite operators Q (x) • A An insertions of composite operators QA(x) into the general Green function are defined by Q (y ) Q (y ) Q (y ) G (x , x , x ) A1 1 · A2 2 ··· Am m · n 1 2 ··· n := 0 TQ (y )Q (y ) Q (y )ϕ(x )ϕ(x ) ϕ(x ) 0 ⟨ | A1 1 A2 2 ··· Am m 1 2 ··· n | ⟩ ( i~)n+mδn+m = − Z [J, ρ] J=ρ=0. (17) δρA1 (y )δρA2 (y ) δρAm (y )δJ(x )δJ(x ) δJ(x ) | 1 2 ··· m 1 2 ··· n 4 An insertion of composite operators QA(x) into the connected Green function are defined by c QA1 (y1) QA2 (y2) QAm (ym) Gn(x1, x2, xn) · ··· · ··· c := 0 TQA1 (y1)QA2 (y2) QAm (ym)ϕ(x1)ϕ(x2) ϕ(xn) 0 ⟨ | ··· n+m 1 n+m ··· | ⟩ ( i~) − δ = − ZC [J, ρ] J=ρ=0. (18) δρA1 (y )δρA2 (y ) δρAm (y )δJ(x )δJ(x ) δJ(x ) | 1 2 ··· m 1 2 ··· n An insertions of composite operators QA(x) into the 1PI Green function are defined by QA1 (y1) QA2 (y2) QAm (ym) Γn(x1, x2, xn) · ··· · ··· 1PI := 0 TQA1 (y1)QA2 (y2) QAm (ym)ϕ(x1)ϕ(x2) ϕ(xn) 0 ⟨ | ··· m 1 n+m ··· | ⟩ ( i~) − δ = − Γ[ϕ, ρ] J=ρ=0. (19) δρA1 (y )δρA2 (y ) δρAm (y )δϕ(x )δϕ(x ) ϕ(x ) | 1 2 ··· m 1 2 ··· n In the simplest case, a insertion of one composite operator, Q (x) Γ can be expanded A · as QA(x) Γ = QA(x) + (~ QA(x)), (20) · O where the second term denotes non-local contributions from loop F eynman diagrams. Relating insertions of quantum composite operators to variations of external fields is a very pragmatic approach, which brings much convenience to deal with quantum corrections to classical composite operators and hence makes possible the algebraic renormalization principle: For example, it is a necessary way to construct the linearized Slavnov–Taylor identity operator in a non-abelian gauge field theory. Quantum action principle • With insertions of composite operators into the 1PI Green function, the quantum action principle can be represented by δΓ[ϕ, ρ] = ∆ (x) Γ[ϕ, ρ], δρA(x) A · δΓ[ϕ, ρ] = ∆a(x) Γ[ϕ, ρ], δϕa(x) · ∂Γ[ϕ, ρ] = ∆ Γ[ϕ, ρ], ∂λ λ · δΓ[ϕ, ρ] a ϕb(x) = ∆b (x) Γ[ϕ, ρ], δϕa(x) · δΓ[ϕ, ρ] δΓ[ϕ, ρ] a A = ∆A(x) Γ[ϕ, ρ], (21) δρ (x) δϕa(x) · where the symbol ϕa denotes the field component. In the BPHZ renormalization, a quantum composite operator represented by a corre- sponding normal product, the quantum action principle is denoted by δΓ[ϕ, ρ] δΓ = [ eff ] Γ[ϕ, ρ], δρA(x) δρA(x) · 5 δΓ[ϕ, ρ] δΓ = [ ren ] Γ[ϕ, ρ], δϕa(x) δϕa(x) · ∂Γ[ϕ, ρ] ∂Γ = [ ren ] Γ[ϕ, ρ], ∂λ ∂λ · δΓ[ϕ, ρ] δΓren ϕb(x) = [ϕb(x) ] Γ[ϕ, ρ]. (22) δϕa(x) δϕa(x) · Γ, Γ ,Γ and Γ • eff ren int In our case, the four symbols Γ, Γeff ,Γren and Γint are often used. The Γ denotes the generating functional of vertex functions; the Γeff denotes the effective action, which is needed to discuss the structure of vacuum in models with spontaneously broken sym- metries; the Γeff can be obtained by calculating Γ because they are different faces of the same thing. The Γren denotes the local part of the Γeff (or Γ) is obtained by considering contributions only from tree Feynman diagrams, and is applied to generate insertions of normal products in the quantum action principle in the BPHZ renormalization. The Γint denotes a non-free part of the Γren, namely interaction terms in the classical action Γcl plus all counter terms, and is used to generate all possible vertices in the Gell-Mann– Low formulae. However, in [38], [45] and [46], for a perturbative quantum field theory defined by the algebraic renormalization procedure, the symbols Γ, Γeff , and Γint are always used: the Γeff plays the same role as the Γren in our case and it is always finite in the BPHZ renor- malization. Moreover, for a perturbative quantum field theory defined by the dimensional regularization and the dimensional renormalization in [32], the symbols Γ, Γeff ,Γren and Γint are used and they can be introduced similar to those in our case. But the Γren is not finite since it includes counter terms with poles, and is also not a local part of Γ(Γeff ) since the latter one is always finite. Notes The above notations have been almost presented in [45] and [46]. But there are some differences. In [45], the Euclidean space-time is taken from the begin to the end, and in [46], the Planck constant ~ is not shown explicitly in all formulas. In addition, here, it is specially emphasized that insertions of quantum composite operators can be treated by introducing external fields. 2 Introduction The BPHZ renormalization denotes a procedure developed by Bogoliubov, Parasiuk, Hepp and Zimmermann in a series papers, see [1], [2], [3], [4] and [5], which equips a perturbative quantum field theory with exact rules to compute the renormalized Green functions and derive relations among them. It can be divided into the two parts: BPH renormalization and Zimmermann’s recipe. Although the BPH renormalization is by itself a rigorous and complete renormalization scheme, in this article the BPHZ renormalization will be introduced based on the Zimmermann’s papers. In the following, a historical development of the BPHZ renormalization will be firstly reviewed in a simple way, and then naturally induces an outline of the article. Secondly, as an example, a massive ϕ4 model will be defined by applying the BPHZ renormalization. 6 2.1 The history of the BPHZ renormalization In the BPHZ renormalization, the Zimmermann’s forest formulae in [3], the momentum subtraction scheme, and the normal product algorithm in [4] and [5], are the most fun- damental elements, from which the Zimmermann’s identities in [5], the Callan–Symanzik equation in [6] and the quantum action principle in [7] and [8], can be derived. Before Zimmermann’s original work on the forest formulae, at least the three things had been well prepared: in the practical calculation it was realized that the momentum subtraction procedure makes the divergent Feynman integral convergent, see [9], [10] and [1]; the power counting theorem had been proved in the Euclidean space-time, see [11], in order to obtain the asymptotic behaviour of the Feynman integral in the high energy- momenta domain; the R operation completely solved the nonlocal problem induced by overlapping divergences and could be used to define the finite S-matrix, see [1] and [2]. Therefore, first of all, the proof for the power counting theorem would be simplified in [12] and had to be generalized to be also right in the Minkowski’s space-time in [13]. Second, a solution of the R operation without a recursive recipe was found out and had to be proved to make the divergent Feynman integral convergent by applying the momentum subtraction procedure to the unrenormalized Feynman integrand, see [3]. Third, the later progress showed that this type of solution together with other regularization schemes was also successful, for example see the case of the dimensional regularization and the dimensional renormalization in [14] and [15]. Hence in a common sense they are called Zimmermann’s forest formulae. The BPHZ renormalization is different from others such as the dimensional renor- malization mainly because of the momentum subtraction scheme. It is a normal Taylor expansion up to the order specified by the subtraction degree. On the one hand, therefore, no any regularization schemes are involved, and then only finite coefficients appear in the renormalized action Γren. That is to say that only physical parameters will be treated and the bare ones are not needed. Naturally, those physical ones are fixed by the renormaliza- tion conditions. On the other hand, furthermore, it is controlled by the subtraction degree being a positive number not less than the superficial divergence degree. This suggests that the oversubtraction problem can be well treated. This results in the Zimmermann’s iden- tities, see [4], the best formulae relating the case of the subtraction and that of the over subtraction. The normal product algorithm is combining the momentum subtraction scheme with the forest formulae to define the insertion of the renormalized composite operators into the renormalized Green function in a rigorous way from the mathematical point, see [4] and [5]. With its help, the equations of motion could be expressed in a renormalized version, the Zimmermann’s identities was derived, and the operator product expansions could be proved, see [16]. It was improved in [17] and [18] and used to derive the Callan–Symanzik equation in [6]. Furthermore, in [6], [17] and [18] it was realized that derivations of pa- rameters or linear transformations of fields on the vertex functional Γ are corresponding to local insertions of normal products into Γ, which had been concluded in the quantum action principle represented by several differential equations. Moreover, the Ward identi- ties for linear transformations, such as the U(1) gauge transformation in QED, could be constructed to all orders of the Planck constant ~ in the BPHZ renormalization, see [19], [20] and [21], as shed a light on the subject of “renormalization and symmetries”. The quantum action principle for non-linear transformations was proved in [7, 8], which suggested that symmetries represented by non-linear transformations could be well treated in the BPHZ renormalization. For example, in the non-abelian gauge field theory with 7 the spontaneously broken symmetries, the Slavnov–Taylor identity can be constructed to all orders of ~, see [22] and [23]: the point is applying the quantum action principle represented by the normal product algorithm to change the problem of “renormalization and symmetries” into the algebraic one of how to solve the cohomology of local inte- grals with constraints from the power counting theorem, discrete symmetries and so on. Later progress shows that the quantum action principle could be also derived in other renormalization schemes, see [14], [24], [25], [26]. Hence the approach of constructing the renormalized Slavnov–Taylor identity or the renormalized Ward identity could be argued to be independent of renormalization procedures and regularization schemes, and then called in a common sense by the algebraic renormalization procedure. In addition, via the LSZ reduction procedure in [27], charges responsible for the BRST transformations was constructed and the unitarity of the S-matrix could be proved in [28] and [29]. The BPHZ renormalization invented in [5] was mainly devised for massive models and based on the momenta subtraction at the zero-momenta subtraction point. Its generalized version in [30] was also suitable for massless models, called by the BPHZL renormaliza- tion. It had been also applied in supersymmetrical field theories, in [31]. Afterwards, it didn’t make a big progress for a long time. There are at least two types of reasons. The algebraic renormalization procedure is more interesting since physical results have to be independent of choices of renormalization schemes. Furthermore, for the high energy phenomenology physics, the momenta subtraction scheme can’t compete with the dimen- sional regularization in [32]. However, until now, the BPHZ renormalization is still the best approach of showing the quantum action principle by the normal product algorithm. Recently, moreover, Connes and Kreimer discovered the Hopf algebra behind the Zimmer- mann’s forest formulae, see [33] and [34], which means that a lot of things in the BPH or BPHZ renormalization need to be revived. The outline of the article is given in the following. Firstly, as an example, the massive ϕ4 model will be defined by the BPHZ renormalization, Secondly, the forest formulae will be explained by examples in two-loop. Thirdly, the quantum action principle will be introduced and its proof in [7] will be sketched. Then the Zimmermann’s identities will be introduced by examples and used to derive the Callan–Symanzik equation. Fourthly, the algebraic renormalization procedure will be introduced in a common way, current constructions and charge constructions via the LSZ reduction will be given in detail. Lastly, the Connes–Kreimer Hopf algebra will be reviewed in the Zimmermann’s sense. 2.2 The massive ϕ4 model via the BPHZ renormalization As an example, the well-defined massive ϕ4 model in the BPHZ renormalization is intro- duced. Namely, the renormalized action, the renormalization conditions, the Zimmerman- n’s forest formula, the renormalized Green function, the renormalized Green function with the insertions of the normal products, the quantum action principle and the Zimmermann identities are presented. The renormalized action Γren can be regarded as the sum of the free part Γ0 and the interaction part Γint, Γ = Γ + Γ = z ∆ a∆ ρ ∆ . (23) ren 0 int − 1 − 2 − 4 In the tree approximation, the coefficients z, a, ρ are specified by z(0) = 1, a(0) = m2, ρ(0) = λ, (24) 8 where the upper indices denote the power counting of ~ in this section. The normal products ∆1, ∆2, ∆4 are given by 4 1 ∆1 = d x 2 ϕ(x)2ϕ(x) , (25) [∫ ]4 4 1 2 ∆2 = d x 2 ϕ (x) , (26) [∫ ]4 1 ∆ = d4x ϕ4(x) . (27) 4 4! [∫ ]4 The free part Γ is given by ∆ m2∆ determining the propagator. The renormalized 0 − 1 − 2 Lagrangian density is chosen to be 1 = 1 z ϕ2ϕ 1 a ϕ2 ρ ϕ4, (28) Lren − 2 − 2 − 4! but it can be changed by adding total derivatives. On the other hand, the renormalized action Γren is also the sum of the classical action Γcl and all the possible local counterterms Γcounter, namely, 1 2 1 4 Γren = Γcl + Γcounter = ( ϕ(2 + m )ϕ + λϕ ) + (~). (29) − 2 4! O ∫ In higher orders, the coefficients z, a, ρ are decided by the renormalization conditions Γ (p, p) 2 2 = 0, (30) 2 − |p =m ∂ 2 Γ2(p, p) 2 2 = 1, (31) p e − |p =µ Γ4(p1, p2, p3, p4) Q = λ, (32) e | − where m is the physical mass, eµ is the normalization mass denoting the renormalization scale, λ is the physical coupling constant and Q is the set given by 4 Q = p p2 = µ2, (p + p )2 = µ2; i = j; i, j = 1, 2, 3, 4 . (33) { i | i i j 3 ̸ } Γ (p, p) and Γ (p , p , p , p ) are the two-point 1PI Green function and the four-point 2 − 4 1 2 3 4 1PI Green function respectively. Here the rule of the Fourier transformation of an ordinary functione F , suche as a Green function or an 1PI Green function, between the momentum space and the coordinate space has been chosen as n 4 4 F (p1, p2, , pn) = (2 π) δ pi F (p1, p2, , pn) ··· ( ) ··· ∑i=1 4 4 4 e i n p x = d x d x d x e j=1 j · j F (x , x , , x ). (34) 1 2 ··· n 1 2 ··· n ∫ ∑ The Zimmermann’s forest formula was proposed in [3]. It denotes a procedure for obtaining a renormalized Feynman integrand RΓ(p, k) from a unrenormalized Feynman integrand IΓ(p, k), namely R (p, k) = S ( tδγ S ) I (p, k), (35) Γ Γ − γ Γ U γ U ∑∈F ∏∈ 9 where U is a forest, is a set of all possible renormalization forests, S or S are sub- F Γ γ stitution operators, tδγ is a Taylor subtraction operator cut off by a subtraction degree δγ, the argument p denotes a set of external momenta and the argument k denotes a set of independent internal momenta, and in the product ( tδγ S ) an element on the γ U − γ right hand side is either disjoint with or a subset of one on∈ the left hand side. ∏ A renormalized Green function in the BPHZ renormalization was defined in [4] and [5]. A unrenormalized Green function is given in the Gell-Mann– Low formulation, and its renormalized version is directly defined as its finite part in the BPHZ renormalization, namely, T ϕ(x )ϕ(x ) ϕ(x ) ⟨ 1 2 ··· n ⟩ := BPHZ finite part of i Γ0 i Γ0 T ϕ (x )ϕ (x ) ϕ (x )e } int / T e } int . (36) ⟨ 0 1 0 2 ··· 0 n ⟩ ⟨ ⟩ A renormalized Green function with insertions of normal products is given by N [Q (y )] G (x , x , , x ) δi i i · n 1 2 ··· n ∏i := T N [Q (y )] ϕ(x )ϕ(x ) ϕ(x ) ⟨ δi i i 1 2 ··· n ⟩ ∏i = BPHZ finite part of 0 i Γ0 i Γ0 T N Q (y ) ϕ (x )ϕ (x ) ϕ (x )e } int / T e } int , (37) ⟨ δi i i 0 1 0 2 ··· 0 n ⟩ ⟨ ⟩ i ∏ [ ] 0 where a normal product Nδi Qi (yi) denotes vertices in Feynman graphs to be treated with a subtraction degree δ . The symbols with the upper index 0 are defined in free quantum i [ ] field theory: the Gell-Mann– Low formula is calculated by an approach of diagram-by- diagram, and each diagram is corresponding to a unrenormalized Feynman integral by ordinary Feynman rules. For convenience, the subtraction degree of a normal product will not be explicitly shown in some cases. The quantum action principle was derived in the BPHZ renormalization, see [6], [7] and [8]. It means that variations of parameters or fields of a Green function can be represented by appropriate local insertions of normal products. Its differential formalism is given in the following way, ∂ Γ = [∂ Γ ] Γ,A = m, µ, λ, (38) A A ren 4 · δΓ δΓ ϕ(x) = ϕ(x) ren Γ, (39) δϕ(x) δϕ(x) · [ ]4 where the lower indices of normal products are the subtraction degrees used in the BPHZ renormalization. The Zimmermann’s identities were proved in [4] and [5]. They relate the subtraction and the over-subtraction in the BPHZ renormalization. They are given by N [Q] Γ = u N [Q ] Γ, (40) δ · i χ i · ∑i where the sum is over all possible normal products of the over-subtraction degree χ with the same quantum numbers, and δ is the subtraction degree, with χ>δ. The coefficients ui are determined by normalization conditions on insertions of composite operators. The involved notations in the article are given in the appendix. 10 Notes The BPH renormalization has been reviewed in [35]. The BPHZ renormalization was reviewed in [16], [36] and [37], and was recently introduced in [38]. Besides the BPH renormalization and the BPHZ renormalization, there are also oth- er renormalization procedures: the Epstein- Glaser renormalization in [39]; the analytic renormalization in [24]; the dimensional regularization and the dimensional renormaliza- tion in [32], reviewed in [40]; the differential renormalization in [41]; and the renormal- ization group differential equation approach in [42]. They are all invented for defining a perturbative quantum field theory, but the first four schemes are based on the expansion of diagram-by-diagram and the last one is on the expansion of scale-by-scale. 3 The Zimmermann’s forest formula The Zimmermann’s forest formula plays the key role in the BPHZ renormalization. In this section, it will be firstly introduced by an example and then some necessary remarks are given. Furthermore, a general procedure applying it to calculate a renormalized Feynman integral is presented, as will be explained in details by examples in two-loop. In addition, only connected graphs are considered. 3.1 A simple introduction The Zimmermann’s forest formula can be observed in a heuristic way, because it has to solve problems in non-overlapping divergent cases and is a solution of the R-operation in the BPHZ renormalization. Firstly, a given divergent Feynman graph without overlapping divergences can be made convergent by a natural strategy in [9], namely by making its all divergent subgraphs convergent and then making itself finite. For example, a divergent Feynman graph γ, in the massive ϕ4 model in four-dimension, will be treated and it has two disjoint divergent subgraphs γ1 and γ2, see Figure 1. Figure 1: A graph γ and its two disjoint divergent subgraphs γ1 and γ2 Its unrenormalized Feynman integral is given by 4 4 d k1d k2 Dp k1 Dk1 Dp k2 Dk2 , (41) (2π)8 − − ∫∫ and the corresponding convergent Feynman integral is obtained by 4 4 d k1d k2 dγ (1 tp ) (2π)8 − ∫∫ dγ dγ 1 γ γ 2 γ γ [(1 tpγ1 )Dp 1 k1 Dk1 ]p 1 =p[(1 tpγ2 )Dp 2 k2 Dk2 ]p 2 =p, (42) × − − − − 11 where the symbols dγ, dγ1 and dγ2 respectively denote the superficial divergent degrees of the corresponding graphs γ, γ1 and γ2, and the action of the Taylor subtraction operator dγ tp acting on an arbitrary rational function f(p) is given by, dγ dγ µ p dγ tp f(p) = f(0) + p ∂pµ f(0) + + ∂ f(0), (43) ··· (dγ)! dγ1 dγ2 similarly for tpγ1 and tpγ2 . Secondly, expanding the above convergent integral (42) partly, the result is 4 4 d k1d k2 dγ1 dγ2 dγ1 dγ2 [1 + ( t γ ) + ( t γ ) + ( t γ )( t γ )] (2π)8 − p 1 − p 2 − p 1 − p 2 ∫∫ γ γ γ γ Dp 1 k1 Dk1 Dp 2 k2 Dk2 p 1 =p 2 =p × − − | dγ dγ1 dγ2 dγ1 dγ2 +( tp ) [1 + ( t γ ) + ( t γ ) + ( t γ )( t γ )] − − p 1 − p 2 − p 1 − p 2 γ γ γ γ Dp 1 k1 Dk1 Dp 2 k2 Dk2 p 1 =p 2 =p, (44) × − − | which suggests a recursive procedure in the BPH renormalization, namely, the R-operation: obtaining a divergent graph without subdivergences by subtracting overall divergences of its all divergent subgraphs; obtaining an overall divergences of a graph by firstly making it free from subdivergences and then taking a remaining divergent part; and obtaining a finite graph by subtracting its overall divergence, see [10] and [1]. Thirdly, expanding the above convergent Feynman integral (44) completely, the corresponding renormalized Feynman integral is obtained by dγ dγ1 dγ2 1 + ( tp )S + S ( t γ ) + S ( t γ ) { − γ γ − p 1 γ − p 2 dγ dγ1 dγ dγ2 dγ1 dγ2 +( tp )S ( t γ ) + ( tp )S ( t γ ) + S ( t γ )( t γ ) − γ − p 1 − γ − p 2 γ − p 1 − p 2 dγ dγ dγ 1 2 γ γ +( tp )Sγ( tpγ1 )( tpγ2 ) Dp 1 k1 Dk1 Dp 2 k2 Dk2 , (45) − − − } − − where the substitution operator Sγ is defined by S : pγ1 pγ1 (p), pγ2 pγ2 (p), (46) γ −→ −→ namely, γ1 γ2 Sγ f(p ) = f(p),Sγ f(p ) = f(p). (47) Introducing a forest set as ∅, γ , γ1 , γ2 , γ1, γ , γ2, γ , γ1, γ2 , γ1, γ2, γ , (48) { } { } { } { } { } { } { } the Zimmermann’s forest formula comes out and its general formalism is given by R (p, k) = S ( tδγ S ) I (p, k), (49) Γ Γ − γ Γ U γ U ∑∈F ∏∈ which was firstly presented in [3]. It denotes a procedure of obtaining a renormalized Feynman integrand RΓ(p, k) from a unrenormalized Feynman integrand IΓ(p, k). To apply the Zimmermann’s forest formula in the calculation, several things have to be made clear. First, in the BPHZ renormalization, a graph is defined by a set of its all lines, and so its subgraph is obtained by taking one subset. However, in the BPH renormalization in [1] and the Epstein- Glaser renormalization in [39], a graph is defined by a set of its all vertices. Naturally for the same graph, the number of subgraphs in the 12 latter case is less than in the BPHZ renormalization. Second, a subtraction degree of a composite operator in a graph is different from that in its subgraph. In a 1PI subgraph, some of all lines connecting a composite operator in a graph become external and are not involved in a renormalization procedure, see [4] and [7]. Third, a substitution operator has to be chosen to obey the admissible substitution rules in the next section, which is required by the proof applying both the forest concept and the momenta subtraction procedure, see [3]. 3.2 A general calculation procedure In order to calculate a renormalized Feynman integral corresponding to a given Feynman diagram in the BPHZ renormalization, the following steps have to be carried out. Step I. Specify a formula calculating subtraction degrees; • Step II. Find out all possible forests; • Step III. Calculate admissible momenta assignments; • Step VI. Specify substitution operators satisfying the admissible rule; • Step V. Apply the Zimmermann’s forest formula to obtain a renormalized Feynman • integrand; Step VI. Calculate a renormalized Feynman integral. • For the massive ϕ4 model in the BPHZ renormalization, a subtraction degree of a connected Feynman graph γ with N external legs, is given by δ = 4 N + (δ 4), (50) γ − i − ∑i where the symbol δi denotes a subtraction degree of a normal product at the i-th insertion point as a vertex. In the case without insertions of normal products, it is simplified as the superficial divergence degree dγ, namely, δ = d = 4 N. (51) γ γ − Furthermore, when subtraction degrees of normal products are their mass dimensions, the subtraction degree δγ is also the same as the superficial divergence degree dγ. Moreover, a divergent graph means its non-negative subtraction degree. A forest denoted by U is a set in which each element is one divergent 1PI subgraph and every two elements are not overlapping. This shows that two 1PI graphs µ and τ in the forest U are either disjointed or nested, namely, µ τ = ∅, or µ τ, or µ γ. (52) ∩ ⊂ ⊃ Two special forests of the divergent graph γ are the empty set ∅ and the graph itself, γ. The set of all possible forests denoted by is involved in the Zimmermann’s forest F formula. Obtaining is a big task: first, all possible divergent 1PI subgraphs have to F be found out; second, all possible non-overlapping combinations among them have to be worked out. For example, in cases of two-loop, the involved forest sets have been obtained, see the following subsection. 13 In the following, an algorithm of calculating admissible momenta assignments of all divergent graphs in a forest is given. Firstly, notations will be specified. Secondly, how to obtain admissible momenta assignments is explained. Thirdly, the admissible substitution rule is defined, and then an algorithm calculating substitution operators is presented. Once an oriented Feynmangraph γ is given, its admissible momenta assignment is e also fixed. It has Vγ vertexes and Vγ external vertexes connecting external lines. It has γ m independent oriented loops, denoted respectively by C ,C , C . The symbol pa γ 1 2 ··· mγ denotes the external momentum flow at the external vertex Va, which is a variable of γ a Taylor subtraction operator. The symbol labσ denotes the σ-th oriented line from the vertex Va to the vertex Vb. It also represents the momentum flow in the line and can be γ γ divided into a sum of the external line momentum pabσ and the no-source momentum kabσ, γ γ γ labσ = pabσ + kabσ. (53) In the oriented graph γ, the symbol lγ takes lγ . The “no-source” means that any abσ − baσ vertex is not the source of this type of momenta, namely kabσ = 0, (54) ∑b,σ γ for a fixed vertex point a. Furthermore, every internal line labσ is assigned the resistance 4 variable rabσ, taking values of zero or one. In addition, in the massive ϕ model, the maximum of the symbol σ is three since vacuum graphs are not involved. γ γ Determining the admissible momenta assignment is to calculate pabσ and kabσ. The γ external line momenta flow pabσ are obtained by solving the following equations, γ γ e pa = b,σ pabσ, a = 1, 2, ,Vγ 1, γ ··· − . (55) 0 = lγ C rabσ pabσ, i = 1, 2, mγ, rabσ = 0 or 1. { ∑ abσ∈ i ··· ∑ in which it is best to set rabσ zero on the grounds that in a loop the resistance of at least one line is 1, and once rabσ fixed its value holds for all graphs including the line labσ, so γ that it doesn’t need the upper index γ. The no-source momenta flow kabσ are chosen in a different way. Each loop has its own momentum flow, namely, kγ, kγ, , kγ , (56) 1 2 ··· mγ γ which are integral variables in the measure of a Feynman integral. The kabσ can be constructed by kγ = kγ kγ. (57) abσ i − j lγ C lγ C abσ∑∈ i baσ∑∈ j Similarly, an admissible momenta assignment of a subgraph τ is calculated by the same procedure, that is to say that the formulas (55), (56) and (57) are available except that the upper index γ is replaced with τ. For a divergent graph γ and its divergent subgraph τ, the substitution operator Sγ is defined by S : pτ pγ , kτ kγ , τ γ. (58) γ abσ −→ abσ abσ −→ abσ ⊂ The admissible substitution rule in the Zimmermann’s recipe requires that the substitution operator has to satisfy kτ = kτ (kγ), τ γ, (59) abσ abσ ⊂ 14 which means that a momenta assignment of a graph can’t be chosen in an arbitrary way. Based on the fact, γ τ labσ = labσ, (60) τ γ γ the external momenta flow pa is a function of variables p , k , namely, τ τ γ pa = labσ = labσ b,σ l τ ∑ abσ∑∈ = pγ lγ = pτ (pγ, kγ), (61) a − abσ a l γ τ abσ∑∈ − where the vertex a is fixed and the γ τ denotes a difference of two sets. Hence the − admissible substitution operator Sγ can be constructed in the following way, pτ = pτ (pτ ) = pτ (pγ, kγ) S : abσ abσ abσ . (62) γ kτ = lγ pτ = kτ (kγ) { abσ abσ − abσ abσ Afterwards, a renormalized Feynman integral can be computed from a renormalized Feynman integrand, see the following subsection. 3.3 Examples in two-loop To explain the procedure of applying the Zimmermann’s forest formula to obtain a renor- malized Feynman integral proposed in the above subsection, seven examples will be given as follows. They treat 1PI self-energy two-loop Feynman graphs with an insertion of a normal product or without in the massive ϕ4 model in the BPHZ renormalization. The 1 2 1 2 normal product is either 2 [ϕ ]2 or 2 [ϕ ]4, and the latter one is with an over-subtraction degree. As the first example, the sunset diagram γ is a best choice because it is a typical two-loop Feynman diagram with overlapping divergences. It consists of three internal lines, l121, l122, l123, connecting two external vertexes V1, V2, and has three 1PI divergent subgraphs: γ1, γ2, γ3, see Figure 2. l121 γ1 γ2 γ3 l122 l121 l121 l122 £ ¤ ¢γ1 γ2 γ3 l123 l123 l122 l123 ¡ Figure 2: A sunset diagram γ and its divergent subgraphs γ1, γ2, γ3 The forest set is found out by trying all possible non-overlapping combinations of F four divergent 1PI graphs γ, γ1, γ2, γ3, and the result is obtained by ∅, γ1 , γ2 , γ3 , γ , γ1, γ , γ2, γ , γ3, γ . (63) { } { } { } { } { } { } { } The resistance of each line is chosen as r121 = 0, r122 = 1, r123 = 1. (64) 15 The external momentum flow at the vertex V1 is noted by the symbol p. The admissible momenta assignments of the graph γ and its subgraphs γ1, γ2, γ3 are obtained by solving the corresponding equations (55) and using the construction (57), and the results are given as follows, l = p + k + k lγ1 = pγ1 + 1 (k k ) 121 1 2 122 122 2 1 − 2 l122 = k2 ; ; − γ γ l = k l 1 = p 1 1 (k k ) 123 − 1 123 123 − 2 1 − 2 lγ2 = pγ2 + k lγ3 = pγ3 + k 121 121 1 121 121 2 ; , (65) lγ2 = k lγ3 = k 123 − 1 122 − 2 which have considered the following substitution operator Sγ given by S : kγ1 = 1 (k k ), kγ2 = k , kγ3 = k , (66) γ 122 2 1 − 2 121 1 121 2 showing that the given momenta assignments are indeed admissible. The unrenormalized Feynman integrands in the forest formula are given by Iγ = Dp+k1+k2 Dk1 Dk2 , γ γ Iγ1 Iγ/γ = Dp+k1+k2 D 1 D 1 , 1 l122 l123 γ Iγ2 Iγ/γ2 = D k2 Dp 2 +k D k1 , − 122 1 − γ Iγ3 Iγ/γ3 = D k1 Dp 3 +k Dk1 . (67) − 121 2 The renormalized Feynman integrand is obtained by, 3 2 0 R = (1 t ) S (I + ( t γ )I I ) γ − p γ γ − p i γi γ/γi i=1 2 ∑ 2 = (1 tp)Dp+k1+k2 [Dk2 Dk1 D 1 ], (68) (k1 k2) − − 2 − 0 2 where the Taylor subtraction operators tpγi and tp are defined in (43). The renormalized Feynman integral is calculated by 4 4 1 2 d k1d k2 (~λ) Rγ 6 (2π)8 ∫∫ 4 4 1 2 d k1d k2 = (~λ) 6 (2π)8 ∫∫ 1 µ ν (Dp+k3 Dk3 p p ∂µ∂νDk3 )Dk1 Dk3 k1 , (69) − − 2 − where in the second line the integral variable k3 replaces k1 + k2 and some calculation has been already carried out. The second example is the sunset diagram γ with an insertion of the normal product 1 2 2 N[ϕ ]2, in order to show that a divergent Feynman diagram with insertions are treated the same as one without insertions. It is specified by the four internal lines, l131, l321, l122, l123, which connects three internal vertex points, V1, V2, V3, and the vertex V3 is the insertion point, see Figure 3. The subtraction degrees of the graph γ and its divergent 1PI subgraph γ1 are calculated by applying (50), δγ = dγ = 0, δγ1 = dγ1 = 0. (70) 16 l131 l321 γ1 l122 l122 ¢γ1 l123 l123 ¡ 1 2 Figure 3: A sunset graph with an insertion 2 N[ϕ ]2 and its divergent subgraph γ1 Then the forest set is given as follows, F ∅, γ1 , γ , γ1, γ . (71) { } { } { } The external momentum flows at the vertex V1 and the vertex V3 are respectively denoted by the symbols p and q. The resistance of each line is assumed to be r131 = r321 = 0, r122 = r123 = 1. (72) Then the admissible momenta assignment is obtained by applying the given algorithm and is given by l = p + k + k lγ1 = pγ1 + 1 (k k ) 131 1 2 122 122 2 1 − 2 l = p + q + k + k (73) 321 1 2 l = k , l = k ; lγ1 = pγ1 1 (k k ), 122 − 2 123 − 1 123 123 − 2 1 − 2 which determines the renormalized Feynman integral given by 4 4 1 2 d k1d k3 2 (i~λ) (Dp+k3 Dp+q+k3 Dk )Dk1 Dk3 k1 6 (2π)8 − 3 − ∫∫ 1 d4k d4k (i λ)2 1 2 (D D D2 )D2 (74) ~ 8 p+k1+k2 p+q+k1+k2 k1+k2 1 − 6 (2π) − (k1 k2) ∫∫ 2 − 1 2 1 2 Furthermore, replacing the local insertion 2 [ϕ ]2 with the local integral insertion 2 [ϕ ]2, the corresponding renormalized Feynman integral will be the first term in the above ex- ∫ pression, the variable q being zero. 1 2 In the third example, the sunset graph with an insertion of the normal product 2 [ϕ ]4. Its subtraction degree is the same as in the first example, and hence they have the same forest set. Its admissible momenta assignment is the same as in the second example. The renormalized Feynman integral is given by 4 4 1 2 d k1d k3 2 (i~λ) (1 tp,q)Dp+k3 Dp+q+k3 Dk1 Dk3 k1 . (75) 6 (2π)8 − − ∫∫ In the fourth example, the scoop diagram γ will be treated. It has three internal lines, connecting one internal vertex and one external vertex. It has two bubbles, and the one is constructed by an internal line connecting an internal vertex to itself, called by a tadpole. It has two 1PI subgraphs γ1, γ2. See Figure 4. The subtraction degrees of the graph γ and its subgraphs γ1, γ2 are given by dγ = 2, dγ1 = 0, dγ2 = 2. (76) 17 The forest set is the same as in (48). The external momenta flow at the external vertex F of γ is zero due to the conservation of the energy-momentum, and the same holds for γ1 and γ2. Then the admissible momenta assignment can be obtained by only taking loop momenta k1, k2, without considering external momenta. The renormalized Feynman integral is obtained by applying the forest formula, 4 4 1 2 d k1d k2 2 0 2 2 (~λ) (1 t )Sγ(1 t γ1 )(1 t γ2 )D Dk2 = 0, (77) 4 (2π)8 − p − p − p k1 ∫∫ which suggests that in the BPHZ renormalization procedure all Feynman diagrams with tadpoles do not make contributions. The reason is that the momenta subtraction scheme hires external momenta which are not needed in tadpoles. This is consistent with results from an ordinary normal ordering recipe in quantum field theory. γ γ2 £ ¢γ1 ¡ Figure 4: A scoop diagram γ and its two divergent subgraphs γ1, γ2 The fifth example will be chosen as the scoop diagram with the insertion of a normal 1 2 1 2 product, either 2 [ϕ ]2 or 2 [ϕ ]4, and the insertion point is not standing in a tadpole. Hence the final result is still zero. See Figure 5. γ1 γ2 Figure 5: A scoop diagram with an insertion and its one subgraph being a tadpole The sixth example is the scoop diagram with the insertion of the normal product 1 2 2 [ϕ ]2 but without a tadpole. The graph γ is constructed by four internal lines l121, l122, l231, l232, connecting two external vertexes V1, V3 and one internal vertex V2. It has two 1PI subgraphs γ1 and γ2. See Figure 6. The subtraction degrees of the graph γ and its subgraph γ1, γ2 are given by dγ = dγ1 = dγ2 = 0. (78) The forest set is the same as in the fourth example. The external momentum flow at F the vertex V1 is denoted by the symbol p. The resistance of each line is chosen as r121 = r231 = 0, r122 = r232 = 1. (79) 18 γ2 γ1 l231 l121 l121 l231 γ γ1 γ2 l122 l232 γ1 l122 lγ2 ¢ £ ¡ 232 Figure 6: A scoop diagram with an insertion and no subgraphs being tadpoles The admissible momenta assignment containing the substitution rules are given by l = p + k = lγ1 l = p + k = lγ2 121 1 121 ; 231 2 231 , (80) l = k = lγ1 l = k = lγ2 { 122 − 1 121 { 232 − 2 232 and the renormalized Feynman integral is calculated by 4 4 1 2 d k1d k2 0 (i~λ) (1 t )Sγ 4 (2π)8 − p 0∫ 0 γ γ (1 tpγ1 )(1 tpγ2 )Dp 1 +k1 D k1 Dp 2 +k2 D k2 × − − − − 4 4 1 2 d k1d k2 = (i~λ) 4 (2π)8 ∫∫ (Dp+k1 D k1 Dk1 D k1 )(Dp+k2 D k2 Dk2 D k2 ). (81) − − − − − − In the end, all the above examples in the subsection are taken from [44]. Notes The Zimmermann’s forest formula had been reviewed in [16], [36], [37] and [38]. Here explaining how to apply it in the calculation is a point. 4 The quantum action principle The quantum action principle is always represented by differential equations in which variations of parameters or fields on the Green function are related to local insertions of normal products. In the BPHZ renormalization, the first type of differential equations are given by ∂ Γ = [∂ Γ ] Γ,A = m, µ, λ, (82) A A ren 4 · while the second type of differential equations are δΓ δΓ δϕ = [δϕ ren ] Γ, (83) δϕ δϕ 4 · where δϕ describes all general transformations. First, δϕ being constants, such as space- time translations, (83) denotes the equation of motion in quantum field theory. Second, δϕ being linear functionals of ϕ, such as the conformal transformations, (83) denotes linear 19 transformations of the Green function. Third, δϕ being non-linear functionals of ϕ, such as the BRST transformation or supersymmetric transformations in the Wess–Zumino gauge, (83) denotes non-linear transformations of the Green function. In the first subsection, the one differential equation representing the quantum action principle will be related to the equation of motion in quantum field theory. In the second subsection, the strategy of proving the quantum action principle will be presented and delicate points will be explained by simple examples. 4.1 The equation of motion As an example, the differential equation will be treated in detail, δΓ δΓ = [ ren ] Γ, (84) δϕ(x) δϕ(x) · which relates the variation of a classical field ϕ(x) on the vertex functional Γ to a local insertion into the vertex functional Γ. It can be firstly checked in the classical approximation. The vertex functional Γ is also the effective action and its classical approximation is the classical action Γcl, Γ = Γcl + (~), (85) O and thus its left hand side is given by δΓ δΓcl = + (~). (86) δϕ(x) δϕ(x) O Its right hand side is given by δΓren δΓren [ ] Γ = + (~), (87) δϕ(x) · δϕ(x) O due to the fact that in the classical approximation the local contribution of the insertion is the composite operator itself in the classical approximation. With the generating functional Z[J] of the general Green functions, (84) is represented by δΓ [ ren ] Z[J] + J(x)Z[J] = 0. (88) δϕ(x) · which has a unrenormalized version by δΓ [ cl ] Z + J(x)Z[J] = 0. (89) δϕ(x) · The above equation can be derived by applying the action principle to the path integral representation of the generating functional Z[J] i Z[J] = ϕ exp Γ [ϕ, J], (90) D cl ∫ ~ in which the symbol Γcl[ϕ, J] is given by 4 Γcl[ϕ, J] = Γcl[ϕ] + d x J(x)ϕ(x). (91) ∫ 20 The generating functional Z[J] being invariant by infinitesimally shifting the integral variable ϕ(x), ϕ′ = ϕ + δϕ, ϕ′ = ϕ, Z′[J] = Z[J]. (92) D D it induces the equation of motion in a quantum sense, δΓcl[ϕ, J] δ Z[J] = 0, (93) ϕ= i~ δϕ(x) | − δJ(x) which becomes (89) when the insertion is given by δΓcl[ϕ] δΓcl[ϕ] δ Z[J] = Z[J]. (94) ϕ= i~ δϕ(x) | − δJ(x) δϕ(x) · With the exponential factor Γcl replaced with Γren, (93) is a renormalized version of the equation of motion. Furthermore, in the canonical quantization, the Dyson- Schwinger equation determines dynamics. Its popular unrenormalized version is given by 2 2 2 λ 3 λ δ ϕcl(x) λ δϕcl(x) (2x + m )ϕcl(x) = J(x) ~ ϕ + + i ~ ϕcl(x) , (95) − 3! cl 3! δ2J(x) 2 J(x) in which the classical field ϕcl(x) is defined by δZ [J] ϕ (x) = C . (96) cl δJ(x) With the general Green functions instead of the connected Green functions, the Dyson- Schwinger equation has a simpler form, 3 2 δ λ 3 δ (2x + m )( i~ )Z[J] = J(x)Z[J] ( i~) Z[J], (97) − δJ(x) − 3! − δ3J(x) which is the same as (93). Therefore, the quantum action principle indeed takes root in the action principle. 4.2 The renormalized version in higher orders In this subsection, the proof of the quantum action principle will be sketched. The first type of differential equations (82) can be represented by the Green functions, i~ ∂AGn = [∂AΓren]4 Gn,A = m, µ, λ. (98) − · When A is parameters µ and λ, the above equations can be proved in the following way, i~ ∂AGn = i~ ∂A BPHZ finite part of − − { i Γ0 i Γ0 T ϕ (x )ϕ (x ) ϕ (x )e } int / T e } int ⟨ 0 1 0 2 ··· 0 n ⟩ ⟨ ⟩} = BPHZ finite part of { 0 i Γ0 i Γ0 T ∂ Γ ϕ (x )ϕ (x ) ϕ (x )e } int / T e } int ⟨ A int 0 1 0 2 ··· 0 n ⟩ ⟨ ⟩} = BPHZ finite part of { 0 i Γ0 i Γ0 T ∂ Γ ϕ (x )ϕ (x ) ϕ (x )e } int / T e } int ⟨ A ren 0 1 0 2 ··· 0 n ⟩ ⟨ ⟩} = [∂ Γ ] G , (99) A ren 4 · n 21 in which the point is that the involved differential operations commute with the momenta subtraction procedure in the BPHZ renormalization. When A is the physical mass, in addition, the differentiation operation acting on propagators have to be involved. For the second type of differential equations sedieq, the proof in [7, 8] will be sketched. The given differential equation can be represented by the general Green functions, 2 0 [δϕ(x)(2x + m )ϕ(x)]ϕ(x1) ϕ(xn) 0 −⟨ |Tn ··· | ⟩ 4 ˇ i~ δ (x xi) 0 [δϕ(x)]ϕ(x1)ϕ(x2) ϕ(xi) ϕ(xn) 0 − − ⟨ |T ··· ··· | ⟩ ∑i=1 = (a m2) 0 [ϕ(x)δϕ(x)]ϕ(x ) ϕ(x ) 0 − ⟨ |T 1 ··· n | ⟩ + (z 1) 0 [δϕ(x)2xϕ(x)]ϕ(x1) ϕ(xn) 0 ρ − ⟨ |T ··· | ⟩ + 0 [δϕ(x)ϕ3(x)]ϕ x ) ϕ(x ) 0 . (100) 3!⟨ |T ( 1 ··· n | ⟩ To prove the above the equality in the BPHZ renormalization, the strategy of perturbative expansion on diagram-by-diagram is used. The general Feynman diagram is a research object. Here, On its both sides, two general Feynman diagrams are considered and have to be proved to have the same unrenormalized Feynman integrals, the same forest sets and the same external momenta assignments. When δϕ being constant, the corresponding differential equation in the general Green functions is given by 0 [(2 + m2)ϕ(x)]ϕ x ) ϕ(x ) 0 −⟨ |T x ( 1 ··· n | ⟩ n 4 ˇ i~ δ (x xi) 0 ϕ(x1)ϕ(x2) ϕ(xi) ϕ(xn) 0 − − ⟨ |T ··· ··· | ⟩ ∑i=1 = (a m2) 0 ϕ(x)ϕ(x ) ϕ(x ) 0 − ⟨ |T 1 ··· n | ⟩ + (z 1) 0 [2xϕ(x)]ϕ(x1) ϕ(xn) 0 ρ − ⟨ |T ··· | ⟩ + 0 [ϕ3(x)]ϕ x ) ϕ(x ) 0 (101) 3!⟨ |T ( 1 ··· n | ⟩ where the symbol ϕˇ means that ϕ(xi) is not included. On its left hand side, one connected Feynman graph is picked up and it includes an external vertex point x. When an external line connecting the point x and the other external point xi is amputated, the factor δ4(x x ) comes from the calculation, − i 2 (2x + m )∆F (x xi) (ˇxi) = i~ δ(x xi) (ˇxi) (102) − − G − G 1 4 When an external line connecting the point x and the interaction vertex 4! ϕ (z) in four 1 3 equivalent ways is amputated, see Figure 7, a new interaction vertex 3! ϕ (z) is created by i d4z 1 (2 + m2)( ) ∆ (x z) (z) = (x), (103) − x − 3! F − G 3!G ~ ∫ where is a Feynman integral excluding the propagator from x to z. G Considering renormalized interaction vertexes for counter terms, the differential equa- tion (101) can be proved in the level of the unrenormalized Feynman integral. Furthermore, applying the forest formulae, it is also proved because a line connecting the point x and the other is always external which does not affect the power counting of divergence degree. When δϕ being linear transformations of ϕ, the corresponding differential equation δZ[J] δΓ J(x) = [ϕ(x) ren ] Z[J], (104) δJ(x) δϕ(x) 4 · 22 1 3 Figure 7: A new vertex 3! ϕ (z) generated by amputating one external line is represented in the general Green function, 0 [ϕ(x)(2 + m2)ϕ(x)] ϕ x ) ϕ(x ) 0 −⟨ |T x 4 ( 1 ··· n | ⟩ n 4 ˇ i~ δ (x xi) 0 ϕ(x)ϕ(x1)ϕ(x2) ϕ(xi) ϕ(xn) 0 − − ⟨ |T ··· ··· | ⟩ ∑i=1 = (a m2) 0 [ϕ2(x)] ϕ(x ) ϕ(x ) 0 − ⟨ |T 4 1 ··· n | ⟩ +(z 1) 0 [ϕ(x)2xϕ(x)]4ϕ(x1) ϕ(xn) 0 ρ − ⟨ |T ··· | ⟩ + 0 [ϕ4(x)] ϕ x ) ϕ(x ) 0 . (105) 3!⟨ |T 4 ( 1 ··· n | ⟩ Similar to what was done in the above example, both sides of (105) have the same un- renormalized Feynman integral. Involving the forest formula, several points have to make clear. First, the oversubtraction is necessary for defining the amputation. The Feynman 2 2 graph with the insertion [ϕ (x)]4 instead of [ϕ (x)]2 has the same forest set as the one with the insertion [ϕ(x)2xϕ(x)]4. Second, two types of the Feynman graphs on the left hand side must make no contribution because no corresponding parts on the right hand side. The first case is that two or more than two lines connect the vertex x and the other one. Its counter part contains the tadpole and vanishes in the BPHZ renormalization. The second case is that there is a renormalized 1PI subgraph including the vertex point x and the other one but excluding the line between them. This renormalized subgraph plus that line is still a renormalized one so both will be mapped to the same part after the amputation, which makes ambiguous. Third, choosing the resistance of an amputated line zero, the external momentum at the vertex point x will only be in the amputated line, then both sides have the equivalent admissible momenta assignments. In the following, two examples are given to declare (105). The first example is used to emphasize the importance of the oversubtraction. It is on the four-point graph in one- loop with one insertion. Obviously, without the oversubtraction, the amputation can’t be carried out in the renormalized level. See Figure 8, the corresponding Feynman integral is given by 4 4 i 2 L.H.S = ( λ) ( i~) ∆F (xi) exp ipix − − ~ i=1 i=1 4 ∏ ∑ 2 d k (i~) (Dk Dp1+p2 k Dk Dk). (106) (2π)4 − − ∫ The second example treats the case of two lines connecting the vertex point x and the other one. It is natural to obtain a tadpole. See Figure 9, its unrenormalized Feynman 23 x x + + + = x x ¡ ¢ £ ¤ ¥ x Figure 8: An example to emphasize the importance of the oversubtraction integral is given by 1 i ( λ) d4z ∆ (x z)∆ (x z)∆ (x z)(2 + m2)∆ (x z) S − F 1 − F 2 − F − x F − L ~ ∫ 1 = ( λ)∆F (x1 x)∆F (x2 x)∆F (x x), (107) SL − − − − where the symmetry factors SL and SR can be calculated by 1 1 = 1, = 2. (108) SL SR x x + + = x ¢ £ ¤ ¡ x Figure 9: A tadpole obtained by amputating one external line Notes The quantum action principle had been proved in [7, 8] in the BPHZ renormalization; in [14], [43] in the dimensional regularization and the dimensional renormalization scheme; in [24] in the analytical renormalization; in [25] in the BPH renormalization; in [26] in the Wilson-Polchinski renormalization. 5 The Callan–Symanzik equation The Callan–Symanzik equation was originally obtained in [53, 54]. It denotes broken scale invariance, the generator of scale transformation being represented by m∂m +µ∂µ. Then it can be also derived by applying both the quantum action principle and the Zimmermann identities, see [17]. In the first subsection, the Zimmermann identities will be introduced by examples. In the second subsection, the Lownstein’s recipe of deriving the Callan– Symanzik equation in the BPHZ renormalization will be sketched. In the third subsection, the Callan–Symanzik equations in different formalisms will be shown up. 24 5.1 The Zimmermann identities In the BPHZ renormalization, the same composite operator can be assigned different subtraction degrees. Applying the forest formulae, the normal subtraction and the over- subtraction can be related in the Zimmermann identity, expanding one insertion with the lower subtraction into linear combination of all admissible insertions with the same higher subtraction degree. Here many examples are given in detail to show the Zimmermann identities. One Zimmermann identity used in the article is given by ∆ Γ = ∆ Γ + u ∆ Γ + t ∆ Γ, (109) d · 2 · 1 · 4 · where ∆d denotes the same compositee operatore ase ∆2 but hase the subtraction degree 2, 1 2 1 2 ∆2 = [ 2 ϕ ]2, ∆d = [ 2 ϕ ]4. (110) ∫ ∫ Its coefficients u, t are decided either by using the normalization conditions of insertions of composite operators or by testing special Feynman diagrams. Here, the latter method will be applied to determine coefficients up to orders of ~. Determining u(0), t(0). • First, the zero order values of the coefficients, u(0), t(0), comes out by testing tree Feyn- man diagrams or convergent ones. In order to determine u(0), the following Zimmermann identity can be chosen, (∆ Γ)(0) = (∆ Γ)(0) + u(0) (∆ Γ)(0) + t(0) (∆ Γ)(0), (111) d · 2 2 · 2 1 · 2 4 · 2 where the upper indicese in the bracketse denote thee power countingse of ~. The simple analysis of the relevant tree Feynman diagrams results in (∆ Γ)(0) = (∆ Γ)(0) = 1, (∆ Γ)(0) = p2, (∆ Γ)(0) = 0, (112) d · 2 2 · 2 1 · 2 − 4 · 2 (0) (0) which suggestse that u eis zero. In order toe determine t , the followinge Zimmermann identity can be chosen, (∆ Γ)(0) = (∆ Γ)(2) + u(0) (∆ Γ)(0) + t(0) (∆ Γ)(0), (113) d · 4 2 · 4 1 · 4 4 · 4 (0) which implies that t e is zero by thee results from treee diagram calculation,e (∆ Γ)(0) = (∆ Γ)(2) = (∆ Γ)(0) = 0, (∆ Γ)(0) = 1. (114) d · 4 2 · 4 1 · 4 4 · 4 (0) (0) In order to determinee u , t e at the samee time, the followinge Zimmermann identity up to ~ can be considered, (∆ Γ)(1) = (∆ Γ)(1) + u(0) (∆ Γ)(1) + u(1) (∆ Γ)(0) d · 6 2 · 6 1 · 6 1 · 6 (0) (1) (1) (0) + t (∆4 Γ) + t (∆4 Γ) . (115) e e · 6 e · 6 e The relevant Feynman diagram given in Figuree 10, it showse that the subtraction degrees of (∆ Γ)(1) and (∆ Γ)(1) are respectively given by 2 and 4, then the above identity d · 6 d · 6 − − can be simplified as e e (∆ Γ)(1) = (∆ Γ)(1). (116) d · 6 2 · 6 e 25 e Figure 10: Diagram 10. By simple calculation, the values of the involved insertions are given by (∆ Γ)(0) = (∆ Γ)(0) = 0, (∆ Γ)(1) = 0, (∆ Γ)(1) = 0, (117) 4 · 6 1 · 6 4 · 6 ̸ 1 · 6 ̸ (0) (0) which means thate u and t e have to be zero. e e Determining u(1), t(1). • In order to determine u(1), the following Zimmermann identity up to one-loop can be chosen, (∆ Γ)(1) = (∆ Γ)(1) + u(1) (∆ Γ)(0) + t(1) (∆ Γ)(0), (118) d · 2R 2 · 2R 1 · 2 4 · 2 where the symbol R denotes the renormalized contribution. The involved typical Feynman e e e e diagram is shown in Figure 11. Figure 11: Since it is similar to the tadpole, it vanishes under the BPHZ renormalization, (∆ Γ)(1) = (∆ Γ)(1) = 0, (119) d · 2R 2 · 2R which suggests e e u(1) = 0. (120) In order to determine t(1), the following Zimmermann identity up to one-loop can be chosen as (∆ Γ)(1) = (∆ Γ)(1) + t(1) (∆ Γ)(0), (121) d · 4 2 · 4R 4 · 4 in which (∆ Γ)(1) is given by d · 4 e e e 2 3 4 e (1) λ d k (∆ Γ) = i (D2 D + D D2 ), (122) d 4 ~ 4 k Pi k k Pi k · 2 (2π) − − ∑i=1 ∫ e 26 where P1, P2, P3 are specified by P = p + p ,P = p p ,P = p p . (123) 1 1 2 2 1 − 3 3 1 − 4 Its diagrammatic representation is shown in Figure 12. ∆d ∆2 = + t(1) ¡ ¢ £ Figure 12: Therefore, t(1) is obtained by 4 2 (1) 2 d k 3 3λ 1 t = 3 i~λ D = ~ . (124) (2π)4 k 2 × 16 m2 π2 ∫ Determining u(2). • In order to determine u(2), the following Zimmernmann identity up two orders can be chosen, (∆ Γ)(2) = (∆ Γ)(2) + u(2) (∆ Γ)(0) + t(1) (∆ Γ)(1), (125) d · 2R 2 · 2R 1 · 2 4 · 2 which is simplified as e e e e (∆ Γ)(2) = (∆ Γ)(2) + u(2) ( p2), (126) d · 2R 2 · 2R − since (∆ Γ)(1) only has the tadpole contribution. The typical Feynman diagram is shown 4 · 2 e e in Figure 13. e Figure 13: The corresponding Feynman integral for (∆ Γ)(2) given by d · 2R 4 4 (2) 1 2 d k1 d k3e 2 2 (∆d Γ) = (i~λ) (Dp+k Dk )Dk3 k1 Dk1 , (127) · 2R 2 (2π)4 (2π)4 3 − 3 − ∫ ∫ the coefficient u(2)ecan be calculated by 4 4 (2) 1 2 d k1 d k3 2 u = (~λ) (2k3 Dk )Dk3 k1 Dk1 . (128) 16 (2π)4 (2π)4 3 − ∫ ∫ 27 In addition, the local Zimmermann identity is most used to derive the conformal trans- formations of the S-matrix in [44]. Here, two relevant examples are calculated. The first one is given by 1 ([ 1 ϕ2(x)] Γ)(1) = ([ 1 ϕ2(x)] Γ)(1) + t(1)([ ϕ4(x)] Γ)(0), (129) 2 2 · 4R 2 4 · 4R 4! 4 · 4 whose integral version has been shown in (121). The second one is given by ([ 1 ϕ2(x)] G)(1) = ([ 1 ϕ2(x)] G)(1) + v(1)([ 1 2 ϕ2(x)] G)(0), (130) 2 2 · 2R 2 4 · 2R k 2 x 4 · 2 (1) where vk is obtained by simple calculation, (1) λ λ 1 v = i~ (a 4 b) = ~ k 2 − 6 × 32 m2 π2 d4k 1 d4k k k a = ; bη = µ ν . (131) (2π)4 (k2 m2 + iϵ)3 µν (2π)4 (k2 m2 + iϵ)4 ∫ − ∫ − Its diagrammatic representation is shown in Figure 14. 1 2 1 2 1 2 [ φ ]2 [ φ ]4 [ 2φ ]4 2 2 (1) 2 = + vk ¡ ¢ £ Figure 14: The Zimmermann identity is a special property owned by the BPHZ renormalization procedure. This means that in other regularization or renormalization schems, the similar thing is impossible or difficult to obtain. It solves the problem of how to relate the normal subtraction and the oversubtraction. The main reason is that in the momenta subtrac- tion procedure the subtraction is exactly controlled by the subtraction degree. It can be proved by applying the forest formulae to insertions of involved composite operators in the momentum space and carefully treating additional forests due to the oversubtraction. Notes The Zimmermann identity can be used to prove the OPE(operator product expansion) in a perturbative but rigorous sense, see [5]. It is also applied to study anomalies or breakings in abelian gauge theories, see in [19, 20, 21], but in complicated field theories such as non-abelian gauge field theories, the algebraic renormalization procedure is more suitable to apply, see [22, 23]. 5.2 Deriving the Callan–Symanzik equation In the BPHZ renormalization, the Callan–Symanzik equation can be derived in a rigorous sense. Combing the quantum action principle and the Zimmermann identity, the following 28 five equations comes out, m∂ Γ = [ m∂ z ∆ m∂ a ∆ m∂ ρ ∆ ] Γ; m − m 1 − m 2 − m 4 · µ∂ Γ = [ µ∂ z ∆ µ∂ a ∆ µ∂ ρ ∆ ] Γ; µ − µ 1 − µ 2 − µ 4 · ∂ Γ = [ ∂ z ∆ ∂ a ∆ ∂ ρ ∆ ] Γ; (132) λ − λ 1 − λ 2 − λ 4 · 1 Γ = [ z ∆ a ∆ 2ρ ∆ ] Γ; 2 N − 1 − 2 − 4 · ∆d Γ = [∆2 + u ∆1 + t ∆4] Γ; · · where the parameters u, t are decided by normalization conditions on the insertions of the relevant composite operators. Since there only have three independent insertions ∆1, ∆2, ∆4, one constraint equation, the Callan–Symanzik equation, is obtained by m∂ + β ∂ 1 γ Γ = α ∆ Γ, (133) m λ λ − 2 N m d · where the symbol “m∂m” denotes( m∂m+µ∂µ, the) classical approximation of the parameter (0) 2 α is given by αm = 2m and , ∆ are respectively denoted by m − N d δ = d4x ϕ(x) , ∆ = d4x 1 ϕ2(x) . (134) N δϕ(x) d 2 ∫ [∫ ]2 Furthermore, the following three constraint conditions have been imposed, α = 2 a β ∂ a + γ a, m − − λ λ α u = β ∂ z + γ z, (135) m − λ λ α t = β ∂ ρ + 2γ ρ. m − λ λ There are two ways to determine βλ, γ and αm, either solving the constraint equations order by order or test special Feynman diagrams. Both will be used in order to give the Callan–Symanzik equation a solid grounding. (0) (0) (0) To calculate βλ , γ and αm , the following information is needed, a(0) = m2, u(0) = t(0) = 0, z(0) = 1, ρ(0) = λ, (136) which are applied to the constraint equations to obtain (0) (0) (0) (0) αm = 2 a + γ a , −(0) 0 = γ (137) (0) 0 = β + 2γ(0) λ − λ Solving the above equations, the results are given by α(0) = 2 a(0) = 2 m2, β(0) = γ(0) = 0. (138) m − − λ On the other way, they are also decided by choosing the suitable 1PI Green functions. For (0) example, applying the Callan–Symanzik equation to Γ2 given by (0) Γ = p2 m2, e (139) 2 − (0) (0) (0) the values of αm , γ are obtained.e Testing with Γ4 given by (0) Γ = λ, e (140) 4 − (0) the value of βλ will be known. e 29 (1) (1) (1) To calculate βλ , γ and αm , the input is given by a(1) = z(1) = u(1) = 0, ρ(1) = 0, t(1) = 0. (141) ̸ ̸ It induces the constraint equations, (1) (1) (0) αm = γ a , (1) 0 = γ , (142) (0) (1) (1) (1) αm t = β + 2γ λ. − λ which are solved to give (1) 2 (1) (1) (1) βλ = 2 m t , γ = αm = 0. (143) (1) (1) (1) Similarly, testing with Γ2 which is zero, αm has to vanish. Testing with Γ4R given by 2 3 4 e(1) λ d k 2 e Γ = i~ (Dk DPi k Dk), (144) 4R − 2 (2π)4 − − ∑i=1 ∫ e the corresponding Callan–Symanzik equation up one-loop is m∂ Γ(1) + β(1) (∂ Γ)(0) 2γ(1) Γ(0) = α(0) (∆ Γ)(1). (145) m 4R λ λ 4 − 4 m d · 4R Due to the equation e e e e m∂ Γ(1) = α(0) (∆ Γ)(1), (146) m 4R m 2 · 4R the known information in the above subsection about the Zimmermann identity can be (1) e (1) e applied to get the values of βλ and γ . (2) (2) (2) In order to get βλ , γ and αm , the constraint equations up to two-loop have to be used, (2) (2) (2) (0) αm = 2 a + γ a , (0) (2) −(2) αm u = γ , (147) (0) (2) (1) (1) (2) αm t = β ∂ ρ + 2γ λ. − λ λ (2) Testing Γ2R, the Callan–Symanzik equation up two-loop comes out, e m∂ Γ(2) γ(2) Γ(0) = α(0) (∆ Γ)(2) + α(2). (148) m 2R − 2 m d · 2R m Applying again the relevante Zimmermanne identity ande the equation m∂ Γ(2) = α(0) (∆ Γ)(2), (149) m 2R m 2 · 2R (2) (2) the values of γ and αm are givene by e γ(2) = 2 m2 u(2), α(2) = 2 m4 u(2), (150) − m − which induce one result a(2) = 0. (151) (2) (2) (2) About the values of βλ and t , Γ4R has to be tested. In the quantum field theory, there are two types of differential equations: the one is the renormalization group equatione and the other is the Callan–Symanzik equation. The 30 first one is a homogeneous equation, and the latter one is inhomogeneous. They both treat scaling transformations, but the first one is related to the renormalization scale and the second is with the physical mass. They both use the beta function and the gamma function but give them different meanings in massive models. The Callan–Symanzik equation was derived in several ways. In [53] it was realized by applying the skeleton expansions of the Green functions to study broken scale invari- ance. In [54] it was obtained by adding one mass counter term with a finite coefficient in the action equivalent to changing the renormalization scale. The additional mass term relates two renormalized Green functions by the wavefunction renormalization, but one is independent of its coefficient and the other one is not. Taking the differentiation on this additional parameter then setting zero, the original Callan–Symanzik equation comes out after dividing one normalization factor. Both are very intuitive and indeed based on previous research on the renormalization group differential equation. In models defined by regularization schemes, see [40], physical parameters and bare parameters are needed, and they are related by the wavefunction renormalization. The theory represented by the bare parameters is independent of the renormalization scale µ. Taking the differentiation of µ, the renormalization group differential equation is obtained. In [6], the Callan–Symanzik equation can be derived in some simpler models defined by the BPHZ renormalization. But in models such as non-abelian gauge field theories, the typical algebraic renormalization strategy is suitable: expanding the insertion given by the differential operator m∂m into equivalent basis which are chosen by constraint conditions from the Slavnov–Taylor identity, the gauge fixing condition and the ghost equation, see [45]. Notes The Callan–Symanzik equation is helpful to understand the quantum field theory. It was used to study scaling breakings in [53], and the topic of the small distance behaviour, OPE, the asymptotic expansion in [54]. It was also used to simplify the renormalizability proof by combining it with the Weinberg power-counting theorem in [47]. Furthermore, it was applied to prove the non-renormalization theorem in [48]. 5.3 The different representations of the Callan–Symanzik equation The Callan–Symanzik equation can be shown up in different ways. It can be represented by the general Green function or the connected Green function or the 1PI Green function. It is necessary to derive the transformation laws among the different formalisms. In addition, the skeleton expansion of the general Green function is used in the proofs. Among Γ, Z and Z • C The Callan–Symanzik equation is denoted by the generating functional of 1PI Green functions, m∂ Γ + β ∂ Γ 1 γ Γ = α ∆ Γ. (152) m λ λ − 2 N m d · By using the Legendre transformation (9) and definitions of insertions (18), (19), the Callan–Symanzik equation denoted by the generating functional of the connected Green functions, is obtained by δ m∂ Z + β ∂ Z + 1 γ d4x J(x) Z = α ∆ Z . (153) m C λ λ C 2 δJ(x) C m d · C ∫ 31 By using the definition (14) and definitions of insertions (17), (18), the Callan– Symanzik equation denoted by the generating functional of the general Green functions, is given by δ i m∂ Z + β ∂ Z + 1 γ d4x J(x) Z = α ∆ Z . (154) m λ λ 2 δJ(x) m d · ∫ ~ Between G and Γ • 2 2 Assuming the Callan–Symanzik on the two-point general Green function G2 given by i m∂mG2(x, y) + βλ∂λG2(x, y) + γG2(x, y) = αm∆d G2(x, y), (155) ~ · we have to prove that the Callan–Symanzik on the two-point general Green function Γ2 is by m∂ Γ (x, y) + β ∂ Γ (x, y) γΓ (x, y) = α ∆ Γ (x, y). (156) m 2 λ λ 2 − 2 m d · 2 4 4 Proofs are as follows. From (10) we derive d z G2(x, z)Γ2(z, y) = i~δ (x y). Using − the differential operator m∂ + β ∂ acting on both sides of the above equation, we get m λ λ ∫ d4z G (x, z)(m∂ Γ (x, z) + β ∂ Γ (x, z) γΓ (z, y)) 2 m 2 λ λ 2 − 2 ∫ i = d4z ∆ G (x, z)Γ (z, y). (157) − d · 2 2 ~ ∫ Then applying the relation between ∆ G and ∆ Γ , d · 2 d · 2 ∆ G (x, z) = d4z d4z G (x, z )G (z, z )∆ Γ (z , z ), (158) d · 2 1 2 2 1 2 2 d · 2 1 2 ∫ ∫ we get the result same as (156). Between G and Γ • n n Similar to what we did, we have to prove that the Callan–Symanzik on the n-point 1PI Green function Γn is given by m∂ Γ + β ∂ Γ nγΓ = α ∆ Γ , (159) m n λ λ n − n m d · n Assuming that the Callan–Symanzik on the n-point general Green function Gn is given by i m∂mGn + βλ∂λGn + nγGn = αm∆d Gn (160) ~ · Proofs are sketched as follows. With our notations, we expand G and ∆ G respec- n d · n tively as i Gn = G2 G2 G2 Γn + , (161) ~ ··· ··· ∫ n ∆ G = G |G {zG ∆} Γ d · n 2 2 ··· 2 d · n ∫ n i + n| {z(∆d G} 2) G2 G2 Γn + (162) ~ · ··· ··· ∫ n 1 − | {z } 32 where the symbol denotes the multi-variables integration and the symbol denotes the ··· other unwritten terms which do not affect our proof. Applying the differential operator ∫ m∂m + βλ∂λ acting on both sides of the equation (161), using the Callan–Symanzik equation (160) and the equation (162), then comparing the terms with the same order of i , we get the result like (159). ~ In the above, the transformation rules are obtained by applying the Legendre trans- formation to the generating functionals Γ, ZC [J] and Z[J]. The two things are used. The insertion is defined by introducing the external field; the skeleton expansion is needed. In fact, changing rules are carried out on the variation of the external field, instead of the insertion itself. The different formalism of the Callan–Symanzik equation are all useful when the alge- braic renormalization procedure is applied. First, the standard model requires spontaneous symmetry breakings, which prefers loop expansions instead of perturbative expansions of coupling constants. In the loop expansion of a 1PI Green function, the power counting of the Planck constant is the same as the counting of the loop number, as can simplify the algebraic renormalization procedure which constructs the Slavnov–Taylor to all orders in a perturbative sense. Second, the Feynman rules are invented particular for the general Green functions. In calculation, the formalisms given by the 1PI Green functions have to be transformed to those by the general Green functions. Third, the quantum action principle represented by the 1PI Green functions is clearer, but is proved in the general Green functions. Notes For example, the different formalisms of the Callan–Symanzik equation are used often in [44]. 6 The algebraic renormalization procedure In this section, the algebraic renormalization procedure will be introduced based on [49]. It has several things different from other reviews on the same subject, for example [45, 46]. First, it aims at presenting a general review independent of specific models. Second, it tries to introduce the linearized Slavnov–Taylor operator in a natural way from the SΓ beginning, which is argued from the non-linear property of the Slavnov–Taylor identity, and plays the key role in the construction. Third, it gives a complete procedure of the recursive construction, which consists of checking in first order, assumption in n-th order and construction in n + 1-th order. The algebraic renormalization procedure was invented to treat the problem related to renormalization and symmetry in the perturbative quantum field theory. A given regu- larization procedure or subtraction scheme ensures one theory finite but may break its symmetries. However, in the physical sense, some symmetries such as gauge ones rep- resented by the Slavnov–Taylor identities, must survive under renormalization. Further- more, in some models such as chiral gauge field theories or supersymmetry gauge field theories, no known approaches preserve both renormalizability and symmetries at the same time. Therefore, one type of recipe is adding non-invariant counter terms to cancel breaking terms induced by the renormalization procedure, in other words, the algebraic renormalization procedure is the program of how to constructing suitable non-invariant terms. It makes a judgement whether the anomaly exists, and obtains its name by chang- 33 ing the problem to the algebraic one, namely, solving the cohomology of a given nilpotent operator. The first subsection will make a sketch of how to construct of the Slavnov–Taylor identity to all orders of ~. The second subsection will give an example of how to apply the algebraic constraint. 6.1 The construction of the Slavnov–Taylor identity Assuming the Slavnov–Taylor identity is in the classical level, one question is proposed as Can the Slavnov–Taylor identity survive the quantization? Before construction, assume that the answer has been reached, namely, (Γ) = 0. (163) S Involving additional quantum correction, the new vertex functional Γε expands in the form Γε = Γ + ε ∆, (164) where ϵ is an infinitesimal constant. Replacing Γε with Γ in the Slavnov–Taylor identity, it results in (Γ ) = (Γ) + ε ∆ + (ε2), (165) S ε S SΓ O where the linearized Slavnov–Taylor operator is introduced. The following constraint SΓ condition by ∆ = 0. (166) SΓ means the Slavnov–Taylor identity invariant under the new quantum correction at least in first order of ϵ Furthermore, the linearized Slavnov–Taylor operator satisfies two equations given SΓ by (Γ) = 0, Γ, (167) SΓ S ∀ = 0, Γ, if (Γ) = 0. (168) SΓ SΓ ∀ S The first identity is proved by using the statistics conventions in non-abelian gauge field theories. The second one denotes the nilpotency of the functional operator , and it is SΓ proved by both applying the spin-statistics theorem and the Slavnov–Taylor identity. For example, in the classical case, there are b (Γ ) = 0, b2 = 0, (Γ ) = 0, (169) S cl S cl where the symbol b is defined by b := . (170) SΓcl In the following, the construction starts. As a first step, the Slavnov–Taylor identity has to be realized in one-loop order. Applying the quantum action principle, the possible breaking of the Slavnov–Taylor identity is (Γ) = ∆ Γ, (171) S · 34 which is independent of choices of regularization schemes and renormalization procedures in a nonperturbative sense. The insertion ∆ Γ is at least in one-loop order since the · classical Slavnov–Taylor identity is given. The linearized Slavnov–Taylor operator being a local functional, a integral of x-dependent terms, the insertion ∆ Γ can be represented · by ∆ Γ = d4 x ∆ (x) Γ, (172) · I · ∫ where ∆I (x) is a local polynomial. In a perturbative quantum field theory, the insertion ∆ Γ has an expansion in ~, · 2 ∆ Γ = ~ ∆class + (~ ), (173) · O where the ∆class is an integral of x-dependent local field polynomial and the second term denotes non-local contributions at least including one loop. Hence, the possible breaking of the Slavnov–Taylor identity is 2 (Γ) = ~ ∆class + (~ ). (174) S O Applying on both sides of the above equation, the constraint equation or the con- SΓ sistency condition on the ∆class is b ∆class = 0, (175) where the approximation is used Γ = b + (~). (176) S O Due to the nilpotency of the operator b, one type of solutions of the consistency condition (175) have the form 4 ∆class = b ∆class′ = b d x∆I′ (x), (177) ∫ where ∆I′ (x) is the local field polynomial with the dimension 4, the positive parity and the zero ghost number in the ordinary gauge field theory. Shifting the given vertex functional Γ like the following, 2 Γ Γ ~ ∆′ + (~ ) (178) −→ − class O by adding ~ ∆class′ into the renormalized action Γren, the Slavnov–Taylor identity is con- structed up to first order of ~, namely, 2 2 (Γ ~ ∆′ ) = ~ ∆class ~ b ∆′ + (~ ) = (~ ), (179) S − class − class O O where the equation (165) has been used. However, it is possible that solutions like (177) are not general, that is to say that there may exist one solution which is not the action of the operator b on a local polynomial. In the latter case, the construction fails. In the second step of the construction, the recursive procedure is used. Assuming the n 1 problem has been solved to order of ~ − , n (Γ) = (~ ), (180) S O the construction of the Slavnov–Taylor identity up to order of ~n is equivalent to looking for one suitable local counter terms ∆˜ satisfying, n n+1 (Γ ~ ∆)˜ = (~ ). (181) S − O Hence, a general procedure is summarized in the following. 35 1. With the quantum action principle, the possible breaking of the Slavnov–Taylor identity is given by the insertion of the local functional, n n n+1 (Γ) = ~ ∆ Γ = ~ ∆ + (~ ). (182) S · O 2. The consistency condition is obtained by b ∆ = 0, (183) by applying on both sides of (182) and taking the approximation formula (176). SΓ 3. Solve the consistency condition (183) to get the general solution like ∆ = b ∆˜ + r , (184) A where the coefficient r is a number and is not the solution of the b-variation type. A 4. In the case that either no solutions are like the term or symmetries force r to A vanish, the vertex functional Γ is shifted to Γ ~n ∆,˜ which satisfies − n n n+1 n+1 (Γ ~ ∆)˜ = (Γ) ~ ∆ + (~ ) = (~ ). (185) S − S − O O Therefore, the Slavnov–Taylor identity is constructed up to order of ~n. Otherwise, the Slavnov–Taylor identity is broken by anomaly. In the algebraic renormalization procedure, an insertion has to be an integral of a local field polynomial, and has suitable quantum numbers required by symmetries or physical ansatz and has to satisfy the dimensional constraint due to the power-counting renormalizability. They are enough to solve the consistency condition. Without knowing the insertion exactly, therefore, the Slavnov–Taylor identity can be constructed to all orders. In addition, in simpler models such as abelian gauge theories, an insertion can be calculated by using the renormalized action Γren. 6.2 The consistency condition to all orders In simpler models defined by the BPHZ renormalization, an insertion can be exactly calculated. To realize symmetries or consistency conditions in this case, the procedure is rather different from the construction of the Slavnov–Taylor identity to all orders. One example is given in the following. In the massive scalar field theory treated in [49], the Callan–Symanzik operator and C the improved Ward identity operator ˆ K , are commutative in the classical approximation, W (0) = m∂ , ( ˆ K )(0) = K , [m∂ , K ] = 0. (186) C m W W m W One problem is how to keep the commutativity to all orders of ~, namely, under quantum corrections there is the commutator by [ , ˆ K ]Γ = 0. (187) C W Assuming it has been realized in order of ~(n), one constraint equation on coefficients in the renormalized action Γren has been known. It collects all relevant coefficients up to 36 order of ~n. Considering the constraint equation in order of ~(n+1) and expanding it in orders of ~, the following equation is obtained by n+1 n+1 K (n+1 k) (k) (n+1 k) K (k) ( ˆ ) − ( Γ) = − ( ˆ Γ) . (188) W C C W ∑k=0 ∑k=0 Applying the quantum action principle in the BPHZ renormalization, n (n+1) (n+1) (k) (n+1 k) ( Γ) = ( Γ ) + ([ Γ ] Γ) − C C ren C ren · k=0 ∑n (n+1) (n+1) (k) (n+1 k) ( ˆ Γ) = ( ˆ Γ ) + ([ ˆ Γ ] Γ) − , W W ren W ren · ∑k=0 (189) the above equation (188) is collecting all the coefficients up to order of ~(n+1). So there are two equations: one is known; the other needs verified. The strategy is applying the known one to the other to get the one new constraint equation on the coefficients in order of ~(n+1). In the case that there are some parameters to be decided, at least one parameter can be fixed to all orders by using the new constraint one. Notes In [22, 23], the algebraic renormalization procedure was used to construct non-abelian gauge field theories with spontaneous symmetry breakings. In [50], it was used to construct supersymmetrical field theories defined in the superspace and in [55, 56] to construct supersymmetrical field theories in the Wess–Zumino gauge. In [51], it was applied to construct the standard model to all orders of ~. 7 Current constructions and charge constructions In the above several sections, the basic ingredients of the BPHZ renormalization procedure have been introduced. Besides them, the LSZ reduction procedure and the algebraic renormalization procedure are also used to treat the conformal transformations of the S-matrix, see [44]. In this section, the charge constructions via the LSZ reduction are treated in detail. The first subsection presents the conformal algebra in the classical case. The second subsection shows how to construct the currents responsible for the conformal transformations and the charges for the Poincar´etransformations. 7.1 The conformal algebra The generators of the conformal algebra in the four dimension space-time are denoted by Pµ for the translation transformation, Lµν for the Lorentz transformation, D for the dilatation transformation and Kµ for the special conformal transformation, which satisfy the following Lie algebra, [Pµ,Pν] = 0, (190) [P ,L ] = η P η P , (191) µ ρσ µρ σ − µσ ρ [Pµ,D] = Pµ, (192) 37 [Pµ,Kλ] = 2 ηµλ D + 2 Lλµ, (193) [L ,L ] = η L η L + η L η L , (194) µν ρσ νρ µσ − µσ ρν µρ νσ − νσ µρ [D,Lµν] = 0, (195) [L ,K ] = η K η K , (196) µν λ νλ µ − µλ ν [D,D] = 0, (197) [D,Kλ] = Kλ, (198) [Kσ,Kλ] = 0. (199) The infinitesimal transformations generated by the above algebra are denoted by a β a δϕα (x) = Σα (x, ∂x)ϕβ (x), (200) which are shown in detail respectively by T a µ a δ ϕα (x) = a ∂µϕα (x); (201) δLϕ a(x) = ωµν(x ∂ x ∂ )ϕ a(x) i ωµν[Σ ] βϕ a(x); (202) α µ ν − ν µ α − µν α β D a µ a δ ϕα (x) = ϵ(d + x ∂µ)ϕα (x); (203) δK ϕ a(x) = αµ (2x x η x2)∂ν + 2 d x ϕ a(x) α µ ν − µν µ α 2i αµxν[Σ ] βϕ a(x), (204) − ( µν α β ) β a µ µν where [Σµν]α is the spin matrix; d is the dimension of the field ϕα (x); and a , ω , ϵ, αµ are the infinitesimal constant parameters. 7.2 Current constructions and charge constructions the S-matrix can be defined in the LSZ reduction procedure. With this approach, the information on the operator formalism can be recovered, such as current constructions, charge constructions and quantum transformations of quantum fields. For simplification, the moment constructions of the local Ward identity operators can be chosen as follows, δ 1 δ wT (x) = ∂ ϕ(x) ∂ ϕ(x) , (205) µ µ δϕ(x) − 4 µ δϕ(x) ( ) wL (x) = x wT (x) x wT (x), (206) eµν µ ν − ν µ D µ T w (x) = x wµ (x), (207) e K eµ µ 2e T wν (x) = (2x xν ην x )wµ (x). (208) e e − Applying the quantume action principle in thee BPHZ renormalization procedure, the energy-momentum tensor Tµν can be calculated from wT (x) Γ = [∂ν T (x)] Γ. (209) µ · − µν · Then the breaking of the conformale invariance are controlled by the insertion of the trace of the energy-momentum tensor, namely, D ν ν w Γ = ∂ [Dν] Γ + [Tν ] Γ, (210) K − µ · · µ wν Γ = ∂ [Kµν] Γ + 2xν Tµ Γ, (211) e − · · [ ] e 38 µ where the current Dν is x Tµν for the dilatation transformation and the current Kµν is ζ ζ 2 (2x x ηνx )T for the special conformal transformation. ν − µζ Define the local Ward identity for the space-time translation in the generating func- tional Z[J] by δ 1 δ wT (x)Z[J] := J(x)∂ ∂ (J(x) ) Z[J] µ µ δJ(x) − 4 µ δJ(x) ( ) i ν e = ∂ [Tµν] Z[J]. (212) ~ · It can be realized in the Green function via n wT (x)G = δ(x x )∂xG (x, x , , xˇ , , x ) µ n − l µ n 1 ··· l ··· n l=1 ∑ n e 1 ∂x (δ(x x )G (x, x , , xˇ , , x )) −4 µ − l n 1 ··· l ··· n ∑l=1 i ν = ∂ [Tµν] Gn(x1, , xn), (213) ~ · ··· wherex ˇl indicates thatx ˇl is missing in the string of variables. Furthermore, the local Ward identity for space-time translations in the momentum space is given by n 1 wT (p)G = i (p + p ) p G (p + p , p , , pˇ , , p ) µ n − µ l,µ − 4 µ n l 1 ··· l ··· n ∑l=1 ( ) e i ν = ( ip )[Tµν(p)] Gn(p1, , , pn), (214) ~ − · ··· ··· which can be transfered into the form i ν ( ip )[Tµν(p)] Sn(p1, , , pn) ~ − · ··· ··· n n 1 1 n 2 2 = i (p + p ) p ( ir− 2 ) (p m ) − µ l,µ − 4 µ − j − ∑l=1 ( ) j∏=1 G (p + p , p , , pˇ , , p ) . (215) × n l 1 ··· l ··· n |P In the on-shell limit and in case of p being nonzero and the right hand side being zero, the conservation of the energy-momentum tensor is obtained by ν p Tˆµν(p) = 0, (216) which is given in coordinate space by ν ∂ Tˆµν = 0. (217) Hence the four-momentum charge Pˆµ is defined as 3 Pˆµ := d x Tˆµ0. (218) ∫ Here all involved operators are defined in the asymptotic Hilbert space satisfying H = = . (219) H Hin Hout 39 Similarly, for the other conformal transformations, the corresponding expression can be also set up a ∂ Mˆ µνa = 0, µ ˆ ˆν ∂ Dµ = Tν , µ ˆ ˆµ ∂ Kµν = 2xνTµ , (220) where Mˆ µνa, Dˆµ, and Kˆνµ are respectively given by Mˆ = x Tˆ x Tˆ , Dˆ = xνTˆ , Kˆ = (2x xζ ηζ x2)Tˆ . (221) µνa µ νa − ν µa µ µν νµ ν − ν µζ 3 Then the charge Mµν for the Lorentz rotations is denoted by d xMˆ µν0 . But the charges for both the dilatation transformation and for the special conformal transformation cannot ∫ be easily found out. To derive the quantum transformations of the quantum field operator Φ,ˆ the LSZ reduction procedure can be applied on the both sides of the local Ward identity (214), namely, n 1 i n 2 2 ν ( ir− 2 ) (pj m )( ip )[Tµν(p)] Gn(p1, , , pn) P ~ − − − · ··· ··· | j∏=2 n n 1 1 n 2 2 = i (p + p ) p ( ir− 2 ) (p m ) − µ l,µ − 4 µ − j − ∑l=1 ( ) j∏=2 G (p + p , p , , pˇ , , p ) . (222) × n l 1 ··· l ··· n |P In the on-shell limit, the above formalism is related to 1 i ν i (pµ + pl,µ) pµ Φ(ˆ p + p1) = ( ip ) Tˆµν(p)Φ(ˆ p1) , (223) − − 4 ~ − T ( ) ( ) where the symbol denotes the time-ordering defined in the coordinate space. Then the T action of the local Ward identity operator for space-time translations on the quantum field operator Φ(ˆ x) in coordinate space is given by 1 wT (x)Φ(ˆ x ) := ∂xΦ(ˆ x)δ4(x x ) ∂x(Φ(ˆ x)δ4(x x )) µ 1 µ − 1 − 4 µ − 1 i ν ˆ ˆ e = ∂ Tµν(x)Φ(x1) . (224) ~ T ( ) The similar equations for the other conformal transformations can be also obtained by T T ˆ i a ˆ ˆ (xµwν (x) xνwµ (x))Φ(x1) = ∂ Mµνa(x)Φ(x1) , (225) − ~ T ( ) µ T ˆ i µ ˆ ˆ e x ewµ (x)Φ(x1) = ∂ Dµ(x)Φ(x1) ~ T ( ) i ˆν ˆ e Tν (x)Φ(x1) , (226) −~T ( ) µ µ 2 T ˆ i µ ˆ ˆ (2xνx ην x )wµ (x)Φ(x1) = ∂ Kµν(x)Φ(x1) − ~ T ( ) i ˆµ ˆ e 2xν Tµ (x)Φ(x1) . (227) −~ T ( ) 40 The quantum transformations of the quantum field Φˆ for space-time translations and Lorentz rotations are obtained by integrating x0+ε 0 3 dx1 d x1 (228) x0 ε ∫ − ∫ on both sides of the above equations (224) and (225), namely, T ˆ ˆ i ˆ ˆ δµ Φ := ∂µΦ = [Pµ, Φ], ~ L ˆ ˆ i ˆ ˆ δµν Φ := (xµ∂ν xν∂µ)Φ = [Mµν, Φ]. (229) − ~ In the cases of the dilatation transformation and the special conformal transformation, the quantum transformations in the free(or asymptotic free) field theory are constructed by D i δ ϕˆin(x) := (1 + x∂x)ϕˆin(x) = [D,ˆ ϕˆin(x)], ~ K ˆ µ µ 2 ˆ i ˆ ˆ δν ϕin(x) := (2xνx ην x )∂µ + 2xν ϕin(x) = [Kν, ϕin(x)], (230) − ~ ( ) where Dˆ is the charge for the dilatation transformation and Kˆν is the charge for the special conformal transformation. The charge constructions are important in the perturbative quantum field theory de- fined by the algebraic renormalization procedure. There are two types of reasons. The one is fundamental. As in quantum mechanics there are both the operator formulation and the wavefunction one, the quantum field theory also has its representation of operators in the canonical quantization procedure and the formulation of ordinary functionals in the path integral theory. Operator is convenient to present the interpretation of the quantum physics, and functional is suitable to practical calculation. The algebraic renormalization procedure only deals with Green functions, so no operators are involved. The other one is pragmatic. Since the algebraic renormalization procedure aims at keeping symmetries under renormalization, the charges responsible for the symmetries can be constructed in principle. Notes In [28, 29], charges responsible for BRST transformations were constructed and used to prove the unitarity of the S-matrix. In [52], with the above method, supersymmetry transformation of quantum fields are derived. 8 The Connes–Kreimer Hopf algebra This section is taken from [57]. The axioms of the Hopf algebra will be firstly introduced, and then the Connes–Kreimer will be sketched in Zimmermann’s sense. 8.1 The axioms of the Hopf algebra The Hopf algebra is a bialgebra with a linear antipode map. The bialgebra is both an algebra and a coalgebra in a compatible way. The coalgebra is a vector space with a 41 coassociative linear coproduct map. The algebra is a vector space with an associative linear product map and a linear unit map. The vector space is a set over a field together with a linear addition map. We denote the set by H, the field by , the addition by +, the product by m, the F unit map by η, the coproduct by ∆, the counit by ε, the antipode by S, the identity map by Id, and the tensor product by . Then the vector space is given in terms of the triple ⊗ (H, +; ) where the field with unit 1. F F Let (H, +, m, η, ∆, ε, S; ) be a Hopf algebra over . The Hopf algebra has to satisfy F F the following seven axioms, (1) m(m Id) = m(Id m), ⊗ ⊗ (2) m(Id η) = Id = m(η Id), ⊗ ⊗ m : H H H, (231) ⊗ → m(a b) = ab, a, b H, ⊗ ∈ η : H, F → which denote the associative product m and the linear unit map η in the algebra (H, +, m, η; ) F over respectively; F (3) (∆ Id)∆ = (Id ∆)∆, ⊗ ⊗ (4) (Id ϵ)∆ = Id = (ϵ Id)∆, ⊗ ⊗ (232) ∆ : H H H, → ⊗ ε : H , → F which denote the coassociative coproduct ∆ and the linear counit map ε in the coalgebra (H, +, ∆, ε; ) over and where we used H=H or H =H; F F F ⊗ ⊗ F (5) ∆(ab) = ∆(a)∆(b), (6) ε(ab) = ε(a)ε(b), (233) ∆(e) = e e, ε(e) = 1, a, b, e H, ⊗ ∈ which are the compatibility conditions between the algebra and the coalgebra in the bial- gebra (H, +, m, η, ∆, ε; ) over , claiming that the coproduct ∆ and the counit ϵ are F F homomorphisms of the algebra (H, +, m, η; ) over with the unit e; F F (7) m(S Id)∆ = η ϵ = m(Id S)∆, (234) ⊗ ◦ ⊗ which is the antipode axiom that can be used to define the antipode. With the algebra (H, +, m, η; ) and the coalgebra (H, +, ∆, ε; ), a convolution ⋆ F F can be defined in the vector space End(H, H), consisting of all linear maps from H to H, namely, f ⋆ g := m(f g)∆, f, g End(H, H). (235) ⊗ ∈ (End(H, H), +, ⋆, η ε; ) is an algebra over . The convolution can also be defined by ◦ F F the character ϕ of algebra. The character ϕ is a nonzero linear functional over the algebra and is a homomorphism of the algebra satisfying ϕ(ab)=ϕ(a)ϕ(b). The corresponding convolution between two characters ϕ and φ, is defined by ϕ ⋆ φ := m(ϕ φ)∆, ϕ, φ are characters. (236) ⊗ The Hopf algebra is said to be commutative if we have the equality m = m P , ◦ where the flip operator P is defined by P (a b)=b a. Similarly, the Hopf algebra ⊗ ⊗ (H, m, ∆, η, ϵ, S) is not cocommutative if P ∆ =∆. ◦ ̸ An algebra is graded if it can be written as the vector sum of subsets: H = H H 0 ⊕ 1 ⊕ H . If h H , we denote the grading of h by grad(h)=m. The grading must be 2 ⊕ · · · ∈ m compatible with the product: if h H and h H , then (h h ) H . 1 ∈ m 2 ∈ n 1 · 2 ∈ m+n 42 8.2 An example: the Connes–Kreimer Hopf algebra As an example, we sketch the Connes–Kreimer Hopf algebra in Zimmermann’s sense given in [3], in order to show the general procedure of realizing the Hopf algebra in perturbative quantum field theory. First of all, some notations have to be declared. The connected Feynman graph is constructed from vertices, external lines and internal lines between the two vertices. Denote the set of all the lines of the Feynman graph Γ by (Γ), the set of all external lines of Γ by e(Γ), and the set of all internal lines of Γ by L L i(Γ), then L (Γ) = e(Γ) i(Γ). (237) L L ∪ L Denote the set of all vertices of Γ by (Γ), where only the vertices related to at least two V lines are included. In the case of BPHZ, the Feynman graph Γ is completely fixed by (Γ) L and (Γ) is the set of all vertices attaching to the lines in (Γ). V L Subgraph. The subgraph γ is determined by (γ), where (γ) (Γ). The connected L L ⊆L subgraph γ is expanded by (γ) and (γ), where (γ) (Γ). L V V ⊆V i i Disjoint subgraphs. The graphs γ and γ′ are disjoint if (γ) (γ′)=∅. Denote L ∩ L the disjoint subgraphs γ , γ γ by 1 2 ··· c s = (γ , γ γ ), (238) 1 2 ··· c where every two graphs are disjoint. The set of all possible s in Γ is denoted by , S i i = s (γj) (γk) = ∅, (γj) (Γ), j = k; j, k = 1, 2 c , (239) S { | L ∩ L L ⊆ L ̸ ··· } where =∅ is allowed. The symbol s denotes the summation over all possible dis- S ∈ S joint connected subgraphs and the symbol s′′ denotes the same except for s=(Γ) and ∑ ∈ S s=∅. The symbol ′ denotes the same except for s=(Γ). Under renormalization, only s ∑ divergent graphs and divergent∈ S 1PI subgraphs are involved. ∑ Reduced subgraph. Denote the disjoint union among connected subgraphs by δ = γ γ γ . (240) 1 2 ··· c The reduced subgraph Γ/δ is obtained by contracting each connected parts of δ to a point, namely, (Γ/δ) = (Γ) i(δ), (241) L L \L (Γ/δ) = (Γ) (δ) V , V V , (242) V V \V ∪ { 1 2 ··· c} where V , i=1, 2 c, are the new vertices from the contraction. Here A B denotes the i ··· \ difference between two sets A and B. And Γ/δ denotes the reduced diagram Γ/γ γ γ . 1 2 ··· c Feynman rules. The Feynman rules are applied to get unrenormalized Feynman integrand, or unrenormalized Feynman integral from the Feynman graph. Construct the Feynman rules as a map, from the set of Feynman graphs to the set of the unrenormalized Feynman integrand(integral), ϕ :Γ ϕ(Γ), (243) −→ where the external lines do not contribute to the map. For the disjoint union γ1γ2, we have a homomorphism, ϕ(γ1γ2) = ϕ(γ1)ϕ(γ2), (244) 43 thus the Feynman rules can be regarded as a character of the algebra of Feynman graphs. Feynman rules are unchanged for the subgraph, but for the reduced graph, the factor of 1 has to be arranged to the new vertices from the contraction. In order to see the translation between the BPHZ and the Connes–Kreimer Hopf algebra, we sketch the recursive formalism of BPHZ. Γ Γ Γ For the unrenormalized Feynman integrand(integral) IΓ(K , q ), where K denote internal momenta and qΓ denote external momenta in the Feynman graph Γ. The global divergence is defined by OΓ = (R ), (245) OΓ −R Γ where is the renormalization map to extract the relevant divergent part from the R , R Γ given by Γ Γ RΓ = IΓ(K , q ) + c ′′ γτ γτ IΓ/γ1γ2 γc (K, q) γτ (K , q ), (246) ··· O s S τ=1 ∑∈ ∏ in which RΓ does not contain any subdivergences and it is defined by the recursive procedure. The renormalized Feynman integrand(integral) related to IΓ is obtained by R = R + = R (R ), (247) Γ Γ OΓ Γ − R Γ namely, Γ Γ RΓ = IΓ(K , q ) + c ′ γτ γτ IΓ/γ1γ2 γc (K, q) γτ (K , q ) ··· O s S τ=1 ∈ ∑ ∏c γτ γτ = IΓ/γ1γ2 γc (K, q) γτ (K , q ). (248) ··· O s S τ=1 ∑∈ ∏ Now let us come to the Connes–Kreimer Hopf algebra denoted by (H, +, m, η, ∆, ε, S; ). (249) F Its all elements are given by H : the set generated by 1PI Feynman diagrams; + : the linear combination; : the complex number C with the unit 1; F m : the disjoint union, m(Γ1 Γ2) = Γ1 Γ2; ⊗ ∪ e : the empty set: and η(1) = e; ∅ ∆ : ∆(Γ) = Γ e + e Γ+ ⊗ ⊗ ′′ s S γ1γ2 γc Γ/γ1γ2 γc; ∈ ··· ⊗ ··· ε : ε(e) = e; ε(Γ) = 0, if Γ = e; ∑ ̸ ′′ S : S(Γ) = Γ s S S(γ1γ2 γc)Γ/γ1γ2 γc. − − ∈ ··· ··· ∑ 44 The following axioms need to be checked, (1) m(m Id) = m(Id m), ⊗ ⊗ (2) m(Id η) = Id = m(η Id), ⊗ ⊗ (3) (∆ Id)∆ = (Id ∆)∆, ⊗ ⊗ (4) (Id ϵ)∆ = Id = (ϵ Id)∆, ⊗ ⊗ (250) (5) ∆(Γ Γ ) = ∆(Γ )∆(Γ ), 1 2 1 2 ∆(e) = e e, Γ , Γ H, ⊗ 1 2 ∈ (6) ε(Γ1Γ2) = ε(Γ1)ε(Γ2), ε(e) = 1, Γ1, Γ2 H, ∈ (7) m(S Id)∆ = η ϵ = m(Id S)∆. ⊗ ◦ ⊗ Axioms (1) and (2) are satisfied by the choice of the disjoint union as the product, for it is commutative. Axiom (7) is automatically satisfied, since it was used to define the antipode. The proof for the coassociativity of the coproduct is sketched as follows. Proof. The coassociativity means that we have to prove (∆ Id)∆(Γ) ∆(Γ) e = ⊗ − ⊗ (Id ∆)∆(Γ) ∆(Γ) e, (251) ⊗ − ⊗ namely we have to prove γ′ γ/γ′ Γ/γ = 1 ⊗ 1 ⊗ γ Γ γ γ ∑⊂ ∑1′ ⊆ γ′ γ′′ (Γ/γ′ )/γ′′. (252) 2 ⊗ ⊗ 2 γ Γ γ Γ/γ ∑2′ ⊂ ′′∑⊆ 2′ Then we come to the proof: 1. The divergent subgraph γ′ of the divergent subgraph γ of Feynman graph Γ is still the divergent subgraph in Γ, γ Γ. Then we can choose γ =γ . Denote γ , γ by γ . ⊂ 1′ 2′ 1′ 2′ ′ 2. The reduced graph γ/γ′ obtained by contracting γ′ to a point in the divergent subgraph γ of the Feynman graph Γ is still the divergent subgraph of Γ/γ′. Then we can choose γ′′=γ′ to obtain Γ/γ=(Γ/γ2′ )/γ′′. From the above axioms, we can get S(γ1γ2) = S(γ2)S(γ1), (253) then the antipode can be written as S(Γ) = Γ ′′S(γ )S(γ ) S(γ )Γ/γ γ γ . (254) − − 1 2 ··· c 1 2 ··· c s S ∑∈ Combining the Feynman rules, the character ϕ of the Hopf algebra and the antipode, we get a new character ϕ S, namely, ◦ ϕ S = ϕ(Γ) ′ϕ S(γ γ γ )ϕ(Γ/γ γ γ ). (255) ◦ − − ◦ 1 2 ··· c 1 2 ··· c s S ∑∈ Define the convolution between two characters ϕ and ϕ S by ◦ (ϕ S ⋆ ϕ)(Γ) := m(ϕ ϕ)(S Id)∆(Γ), (256) ◦ ⊗ ⊗ which satisfies (ϕ S ⋆ ϕ)(Γ)=0 provided that Γ=e. ◦ ̸ 45 Introduce the renormalization map to deform the character ϕ S, we get, R ◦ S (Γ) = [ϕ(Γ)] R −R − ′′S (γ γ γ )ϕ(Γ/γ γ γ ) . (257) R R 1 2 ··· c 1 2 ··· c [s S ] ∑∈ SR is also a character, SR(γ1γ2) = SR(γ1)SR(γ2), (258) if the renormalization map satisfies the requirements R (γ γ ) + (γ ) (γ ) = (γ (γ )) + ( (γ )γ ), (259) R 1 2 R 1 R 2 R 1R 2 R R 1 2 where (γ)= ϕ(γ). The corresponding proof can be realized by an iterative procedure. R R ◦ It is remarkable that S (Γ) = (ϕ S)(Γ), (260) R ̸ R ◦ which implies that SR is not a trivial character and we can not use the Hopf algebra directly in problems involving renormalization. Now we are able to construct the translation between the BPHZ and the Connes– Kreimer Hopf algebra with the help of S (γ), (261) OΓ ⇐⇒ R R S ⋆ ϕ(Γ). (262) Γ ⇐⇒ R In order to relate the Connes–Kreimer Hopf algebra to BPHZ in Zimmermann’s sense, we choose the renormalization map (γ) as γ(γ) and specify it as R R γ = ( tγ)S , (263) R − γ where tγ is the Taylor subtraction operator cut off by the divergence degree of the Feynman graph γ, and Sγ is the substitution operator defined by S : Kγ Kγ(Kµ), µ −→ qγ qγ(Kµ, qµ), −→ for γ µ, (264) ⊂ S : Kγ Kγ(KΓ), Γ −→ qγ qγ(KΓ, qΓ), −→ for γ Γ. (265) ⊂ Since the case γ1γ2 (γ γ )= γ1 (γ ) γ2 (γ ) is explicit, we can also define the deformed R 1 2 R 1 R 2 character S . But we still have S (Γ) = (ϕ S)(Γ) due to the fact that R R ̸ R ◦ Γ γ(γ)I = γ(γ) Γ/γ(Γ/γ). (266) R R Γ/γ ̸ R R ( ) Notes Recent development on the subject of the Connes–Kreimer Hopf algebra can be found in [58], [59] and [60]. 46 9 Acknowledgements I thank Dirk Kreimer, Manfred Salmhofer, Klaus Sibold and Raimer Wulkenhaar for helpful discussions on renormalization theories. The DFG is acknowledged for the financial support. 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Kreimer, “New mathematical structures in renormalizable quantum field the- ories”, Annals Phys. 303 (2003) 179, [Erratum-ibid. 305 (2003) 79], [arXiv:hep- th/0211136]. 50 Part III Path Integrals and Non-Abelian Gauge Field Theories 245 Part IV The Standard Model and Particle Physics 246 Part V Integrable Field Theories and Conformal Field Theories 247 Part VI Quantum Field Theories in Condense Matter Physics 248 Part VII General Relativity, Cosmology and Quantum Gravity 249 Part VIII Supersymmetries, Superstring and Supergravity 250 Part IX Quantum Field Theories on Quantum Computer 251 Part X Quantum Gravity on Quantum Computer 252