Lecture Notes on Quantum Field Theory

Transcription

1 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 Zhang 1 School of Physics and Technology, Wuhan University (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. They are mainly based on lecture notes of Sidney Coleman from Harvard and lecture notes of Michael Luke from Toronto and Peskin & Schroeder s standard textbook, 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 Zhang 2, Graduate student (Id: ), Participant in the Spring semester, 2012 and 2014 Yu-Jiang Bi 3, Graduate student (Id: ), Participant in the Spring semester, 2012 and 2014 The fourth version of these notes are typeset by Kun Zhang, Graduate student (Id: ), Participant in the Spring semester, 2012 and yong zhang@whu.edu.cn 2 kun zhang@whu.edu.cn 3 byujiang@hotmail.com 1

2 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 persistence 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

3 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

4 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 a quantum computer. 4

5 Contents I Quantum Field Theory (I): Basic Concepts and Feynman Diagrams 10 1 Overview of Quantum Field Theory Definition of quantum field theory Introduction to particle physics and Feynman diagrams Feynman diagrams in the pseudo-nucleon meson interaction Feynman diagrams in the Yukawa interaction Feynman diagrams in quantum electrodynamics The canonical quantization procedure Research projects in this course From Classical Mechanics to Quantum Mechanics Classical particle mechanics Lagrangian formulation of classical particle mechanics Hamiltonian formulation of classical particle mechanics Noether s theorem: symmetries and conservation laws Exactly solved problems Perturbation theory Scattering theory Why talk about classical particle mechanics in detail Advanced quantum mechanics The canonical quantization procedure Fundamental principles of quantum mechanics The Schröedinger, Heisenberg and Dirac picture Non-relativistic quantum many-body mechanics Compatible observable The Heisenberg uncertainty principle Symmetries and conservation laws Exactly solved problems in quantum mechanics Simple harmonic oscillator Time-independent perturbation theory Time-dependent perturbation theory Angular momentum and spin Identical particles Scattering theory in quantum mechanics

6 3 Classical Field Theory Special relativity Lorentz transformation Four-vector calculation Causality Mass-energy relation Lorentz group Classification of particles Classical field theory Concepts of fields Electromagnetic fields Lagrangian, Hamiltonian and the action principle A simple string theory Real scalar field theory Complex scalar field theory Multi-component scalar field theory Electrodynamics General relativity Symmetries and conservation laws Symmetries play a crucial role in field theories Noether s theorem in field theory Space-time symmetries and conservation laws Space-time translation invariance and energy-momentum tensor Lorentz transformation invariance and angular-momentum tensor Internal symmetries and conservation laws Definitions Noether s theorem SO(2) invariant real scalar field theory U(1) invariant complex scalar field theory Non-Abelian internal symmetries Discrete symmetries Parity Time reversal Charge conjugation Constructing Quantum Field Theory Quantum mechanics and special relativity Natural units Particle number unfixed at high energy (Special relativity) No position operator at short distance (Quantum mechanics) Micro-Causality and algebra of local observables A naive relativistic single particle quantum mechanics Comparisons of quantum mechanics with quantum field theory Definition of Fock space Rotation invariant Fock space Fock space Occupation number representation (ONR) Hints from SHO (Simple Harmonic Oscillator)

7 5.4.4 The operator formalism of Fock space in a cubic box Drop the box normalization and take the continuum limit Lorentz invariant Fock spce Lorentz group Definition of Lorentz invariant normalized states Lorentz invariant normalized state Notation Canonical quantization of classical field theory Classical field theory Canonical quantization Basic calculation on Hamiltonian Vacuum energy and normal ordered product Micro-causality (Locality) Remarks on canonical quantization Symmetries and conservation laws U(1) invariant quantum complex scalar field theory Discrete symmetries Return to non-relativistic quantum mechanics Feynman Propagator, Wick s Theorem and Dyson s Formula Retarded Green function Real scalar field theory with a classical source The retarded Green function Analytic formulation of D R (x y) Canonical quantization of a real scalar field theory with a classical source Advanced Green function The Feynman Propagator D F (x y) Definition with contour integrals Definition of Feynman propagator with iε 0 + prescription Definition of Feynman propagator with time-ordered product Definition of Feynman propagator with contractions Wick s theorem Theorem Example for Wick s theorem n = Interaction Picture (Dirac Picture) Motivation Dirac picture Dyson s formula Unitary evolution operator Time dependent permutation theory Dyson s formula Examples Scattering Matrix, Cross Section & Decay Width Scattering matrix (operator) Ideal model for scattering process Scattering operator S Calculation of two- nucleon scattering amplitude

8 7.2.1 Model Two- nucleon scattering matrix Computing methods Feynman diagrams Main theorem Correspondence between algebra and diagrams Conventions of drawing external lines Conventions on diagrams vertices & internal lines Example Feynman rules in coordinate space-time (F R A ) Feynman rules in momentum space (F R B ) Calculation hint Feynman rules in momentum space Feynman rules C Calculation hint Feynman rules C (simplified F R B ) Application of F R C Remarks on Feynman propagator Cross sections & decay widths measurable quantities in high energy physics QFT in a box with volume V = L Calculation of differential probability Cross section & decay width Calculation of Cross-sections and Decay-widths Dirac Fields The Dirac equation and Dirac algebra The Dirac algebra Two widely used representations of the Dirac algebra The Lagrangian formulation Plane wave solutions of Dirac equation in Dirac representation Dirac equation and Clifford algebra The Clifford algebra Dirac equation with γ µ matrices Plane wave solutions of Dirac equation with γ µ matrices Lorentz transformation and parity of Dirac bispinor fields Classification of quantities out of Dirac bispinor fields under Lorentz transformation and parity Explicit formulation of D(Λ) Canonical quantization of free Dirac field theory Canonical quantization Fock space with the Fermi-Dirac statistics Symmetries and conservation laws: energy, momentum and charge Fermion propagator Feynman diagrams and Feynmans rules for fermion fields FDs & FRs for vertices Internal lines External lines Combinational factor (Symmetry factor) Special Feynman rules for fermions lines

9 8.7.1 Mapping from matrix to number Feynman rules for a single fermion line from initial state to the final state Relative sign between FDs Minus sign from a loop of fermion lines Examples First order Second order Fourth order Spin-sums and cross section Spin-sums Examples Quantum Electrodynamics Model Representative scattering processes in QED The calculation of the differential cross section of the Bhabha scattering in second order of coupling constant II Renormalization and Symmetries Notes on BPHZ Renormalization 167 III Path Integrals and Non-Abelian Gauge Field Theories 218 IV The Standard Model and Particle Physics 219 V Integrable Field Theories and Conformal Field Theories 220 VI Quantum Field Theories in Condense Matter Physics 221 VII General Relativity, Cosmology and Quantum Gravity 222 VIII Supersymmetries, Superstring and Supergravity 223 IX Quantum Field Theories on Quantum Computer 224 X Quantum Gravity on Quantum Computer 225 9

10 Part I Quantum Field Theory (I): Basic Concepts and Feynman Diagrams 10

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. Preliminaries to QFT Quantum Mechanics Special Relativity Electrodynamics Classical Mechanics Special relativity Gravity General relativity Quantum mechanics Quantum gravity Quantum field theory Quantum gravity Superstring theory 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 Non-relativistic QFT: Condensed matter physics Particle # un-fixed Relativistic QFT: 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. 11

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 E = hν = ω, p = h λ = k, (1.1.1) 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 assumption. 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 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 Integrals. 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. 12

13 The model is characterized by the Lagrangian L = L 0 + L int, (1.2.1) where L 0 is the free part included the free nucleon L(ψ) and meson L(φ), L 0 (ψ) = 1 2 µψ µ ψ m 2 ψ ψ; (1.2.2) L 0 (φ) = 1 2 µφ µ φ µ 2 φ 2. (1.2.3) And the interaction part has the formalism L int = gψ ψφ. The parameter g is named as coupling constant. Note that we also have the Hamiltonian formalism H 0 = L 0 and H int = gψ ψφ. Table 1.1: Nucleon -meson interaction Nucleon Anti- Nucleon Meson Pseudo-nucleon Anti-pseudo-nucleon N N φ something something π 0 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 The decay amplitude is calculated as k q N (1.2.5) S(φ N + N ) = (2π) 4 δ 4 (k p q) ia, (1.2.6) where the delta function is required for energy-momentum conservation. In the way of computing the amplitude ia, we can apply the time-dependent perturbation theory with Dyson s series to derive the Feynman diagrams and the associated Feynman rules. The amplitude ia can be expanded as ia = ia (1) + ia (2) + + ia (n) +, (1.2.7) 13

14 where the label n stands for the number of constant and unusually set as = 1. In the first order ia (1), it is related to the Feynman diagram (1.2.5). The Feynman 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 ia (1) = ( ig) = ig. (1.2.8) E.g.2. Nucleon - nucleon scattering. N (p) + N (q) N (p ) + N (q ). (1.2.9) p 1 p 1 p 2 p 2 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 S( p 1, p 2 p 1, p 2 ) = (2π) 4 δ(p 1 + p 2 p 1 p 2 )ia. (1.2.10) In interaction picture, the initial and final state take the form ψ(t = ) = p 1, p 2, ψ(t = + ) = p 1, p 2. (1.2.11) Then applying the time-dependent perturbation to derive the Dyson s series U I = + n=0 the scattering operation (matrix) can be expressed as S = U I (+, ) = U (n) (t f, t i ), (1.2.12) + n=0 S( p 1, p 2 p 1, p 2 ) = p 1, p 2 p 1, p 2 + S (n) = n=1 + n=1 S (n), (1.2.13) p 1, p 2 S (n) p 1, p 2. (1.2.14) The expansion of the amplitude can be calculated ia (0) = ia (1) = 0 and (2π) 4 δ(p 1 + p 2 p 1 p 2 )ia (2) = p 1p 2 S (2) p 1 p 2. (1.2.15) With Wick s theorem, Feynman diagrams and Feynman rules can be determined. 14

15 q N q N ia (2) = q q + (p q ) N p p N The internal line stands for the Feynman propagator for meson, and contributes the factor i (p 1 p 1 )2 µ 2. ia (2) i = 1 1 ( ig) (p 1 p 1 )2 µ 2 ( ig) (p 1 p 1) ( ) = ( ig) 2 i (p 1 p 1 )2 µ 2 + i (p 1 p 2 )2 µ 2. (1.2.16) E.g.3. Anti- nucleon -anti- nucleon scattering. N (p) + N (q) N (p ) + N (q ). (1.2.17) q N q N q q + (p q ) N p p N E.g.4. Nucleon -anti- nucleon scattering. N (p) + N (q) N (p ) + N (q ). (1.2.18) p N p N p p + N p p + q p N N N q q N q q N E.g.5. Nucleon -meson scattering. N (p) + φ(q) N (p) + φ(q ). (1.2.19) φ q p + q q φ + (q q ) N p p N 15

16 E.g.6. Anti- nucleon -meson scattering. N (p) + φ(q) N (p) + φ(q ). (1.2.20) φ q p + q q φ + (q q ) N p p N E.g.7. Nucleon-anti-nucleon annihilation. N (p) + N (q) φ(p ) + φ(q ). (1.2.21) p N p φ p p + (p q ) N q q φ E.g.8. Meson-meson scattering. φ(p) + φ(q) φ(p ) + φ(q ). (1.2.22) p p k + p k k + p p k q q q Remark: about two-particle scattering at order O(g 2 ). 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) 16

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 π 0 -meson spin- 1 2 spin- 1 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 p + q φ particle with four momentum p + q incoming. p, r outgoing nucleon with four momentum p and spin r, r = 1 2, 1 2. q, s outgoing anti-nucleon with four momentum q and spin s, s = 1 2, 1 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(p, r ) + N(q, s ). (1.2.30) N q, s q, s N q q + (q, s p, r ) N p, r p, r N 17

18 E.g.3. Anti-nucleon-anti-nucleon scattering. N(p, r) + N(q, s) N(p, r ) + N(q, s ). (1.2.31) N q, s q, s N q q + (q, s p, r ) N p, r p, r N E.g.4. Nucleon-anti-nucleon scattering. N(p, r) + N(q, s) N(p, r ) + N(q, s ). (1.2.32) N N p, r q, s p + q p, r N N q, s + (p, r q, s ) N N p, r p, r N p p N q, s q, s E.g.5. Nucleon-meson scattering. N(p, r) + φ(q) N(p, r ) + φ(q ). (1.2.33) φ q p + q q φ + (q q ) N p, r N p, r E.g.6. Anti-nucleon-meson scattering. N(p, r) + φ(q) N(p, r ) + φ(q ). (1.2.34) φ q p + q q φ + (q q ) N p, r N p, r E.g.7. Nucleon -anti- nucleon annihilation. N(p, r) + N(q, s) φ(p ) + φ(q ). (1.2.35) 18

19 N p, r p φ p p + (p q ) N q, s q φ E.g.8. Meson-meson scattering. φ(p) + φ(q) φ(p ) + φ(q ). (1.2.36) p p k + p k k + p p k q q q 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) Yukawa interaction Nucleon (spin- 1, spinor ) meson (spinless, real scalar) 2 QED electron (spin- 1, spinor) photon (spin-1, vector) 2 Example: typical Feynman diagrams in quantum electrodynamics. a). The Moller scattering between two electrons. e + e e + e. (1.2.37) e p, s µ ν p, s e p p (p, s q, r ) e q, r q, r e 19

20 The on-line arrow represents an electron and the out-line for momentum labeled by p, q, p and q, which have the form (E p, p). Parameters r, s, r and s represents for the spin degree of the electron. For example, r = 1 2 stands for spin up and r = 1 2 for 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 (p, s ) and (q, r ) 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) e + p, s µ ν p, s e + p p (p, s q, r ) e + q, r q, r e + b). The Bhabha scattering between an electron and a positron. e + e + e + e +. (1.2.39) e p, s p, s e p p (q, r p, s ) e p, s p + q p, s e e + e + q, r q, r e + q, r q, r e + c). The pair annihilation between an electron and a positron into two photons. e + e + 2γ. (1.2.40) e e + p, s q, r ν µ p, ε ν 1 γ p p γ q, ε µ 2 + (p, ε 1 q, ε 2 ) d). The Compton scattering between an electron and a photon. e + γ e + γ. (1.2.41) 20

21 q, ε ν 1 γ q, ε µ 2 γ ν p + q µ + (q, ε 1 q, ε 2 ) e p, s p, s e 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)). Quantization QM Ĥ QFT Ĥ H H CPM CFT Continuum limit Note 1: In classical mechanics, we both have Lagrangian formalism L(q, q) and Hamiltonian formalism H(q, p) to characterize the equation of motion. Note 2: Non-relativistic quantum mechanics prefers the Hamiltonian formulation, for example the Schrödinger equation 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 circumstances. 1.4 Research projects in this course 1. Conventional projects in quantum field theories Traditional research in (high energy physics) HEP: 21

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 theoretical formulation of the Standard Model. 3. New projects in quantum field theory Nowadays quantum mechanics has been updated with quantum computation and information. Note 1: Quantum information and computation (QIC) represents a further development 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 mechanics). Question:?=Special Relativity+ Quantum Information and Computation. 22

23 Lecture 2 From Classical Mechanics to Quantum Mechanics 2.1 Classical particle mechanics Lagrangian formulation of classical particle mechanics In a system of n-particles, the state is characterized by the generalized coordinates q 1, q 2,, q n and their time derivations q 1, q 2,, q n. The Lagrangian is defined as And the action is defined as L(q 1 q n, q 1 q n ) = T V. (2.1.1) S = t2 Note: L does not have explicit time dependence, L t which is associated with the conservation of energy. t 1 dtl. (2.1.2) = 0, (2.1.3) Action principle which has another names including Hamilton s principle, the principle of least action and variational principle: for arbitrary variation q a q a (t) + δq a (t) with fixed boundaries δq a (t 1 ) = δq a (t 2 ) = 0, we have stationary action, namely δs = 0. with we have δs = δs = t2 t 1 ( L dt δq a + L ) δ q a q a q a L δ q a = d ( ) L δq a d q a dt q a dt t2 t 1 ( L dt d q a dt ( L q a Due fixed boundary condition and action principle, we can derive (2.1.4) ) δq a, (2.1.5) ( )) L δq a + L t 2 δq a q a q a. (2.1.6) t 1 L q a = d dt ( L ), (2.1.7) q a which is the equation of motion, also named as Euler-Lagrangian equation. 23

24 2.1.2 Hamiltonian formulation of classical particle mechanics Canonical momentum: Hamiltonian: p a = L q a, a = 1, 2,, n. (2.1.8) H(q 1 q n, p 1 p n ) = a p a q a L(q 1 q n, q 1 q n ). (2.1.9) Note: Hamiltonian is not a function of q a, and it is a function of q a and p a. Proof. dh = a = a = a ( p a d q a + q a dp a L ( q a dp a L dq a q a ( q a dp a L dq a q a dq a L ) d q a q a q a ) + a ( p a d q a L q a d q a ). (2.1.10) ) Hamilton s equations: q a = H p a, ṗ a = H q a. (2.1.11) Optional problem: Derive Hamilton s equations. Optional problem: Derive Newtonian mechanics in both Lagrangian formulation and Hamiltonian formulation Noether s theorem: symmetries and conservation laws We have two different variations: δq a : variation for deriving EoM; Dq a : variation for symmetries without specifying EoM. Def A symmetry is a transformation to keep physics (EoM) unchanged. A transformation Dq a is called symmetry iff DL = df dt for some F (q a, q a, t) with arbitrary q a (t) which may not satisfy the EoM. (2.1.12) Remark: DS = t2 dtdl = F (t 2 ) F (t 1 ), (2.1.13) S = S + DS, (2.1.14) δs = δs + δ(ds) = δs + δf (t 2 ) δf (t 1 ), (2.1.15) because the boundary terms at t 1 and t 2 are fixed, therefore the EoM is unchanged. 24

25 Thm (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 q a, q a. 2). On the another hand, with EoM and canonical momentum, DL = a = a ( L q a Dqa + L q a D qa ) (ṗ a Dq a + p a D q a ) (2.1.16) = d dt ( a p a Dq a ), we can derive the conserved quantity denoted as Q Q = a p a Dq a F, (2.1.17) d dt ( a p a Dq a F ) = dq dt = 0. (2.1.18) E.g.1. Space translation invariance // momentum conservation. L = 1 2 m 1 q m 2 q 2 2 V (q 1 q 2 ). (2.1.19) The space translation with infinitesimal constant α is defined as q 1 q 1 + α, Dq 1 = α; q 2 q 2 + α, Dq 2 = α. (2.1.20) And the Lagrangian is unchanged under the space translation, namely L = 1 2 m q m q2 2 V (q 1 q 2 ) = L, (2.1.21) thus DL = 0 F = 0. (2.1.22) Therefore, the conserved quantity can be defined as Q = p 1 Dq 1 + p 2 Dq 2 = α(p 1 + p 2 ), (2.1.23) d dt (p 1Dq 1 + p 2 Dq 2 ) = dp dt = 0, (2.1.24) where P is the total momentum which is conserved, i.e., P = p 1 + p 2. 25

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) q a (t) q a (t + α), (2.1.26) L(t) L(t + α), (2.1.27) thus Dq a = q a (t + α) q a = α dqa dt + O(α2 ), (2.1.28) DL = L(t + α) L(t) = α dl dt + O(α2 ) F = αl. (2.1.29) Therefore, the conserved quantity Q is given by Q = p 1 Dq 1 + p 2 Dq 2 αl = α(p 1 q 1 + p 2 q 2 L) = αh, (2.1.30) dq dt = αdh dt namely the Hamiltonian (energy) is conserved Exactly solved problems. The Kepler problem with the inverse-square law of force. where V (r) is the potential energy denoted as = 0 = H = const, (2.1.31) L = 1 2 m(ṙ2 + r 2 θ2 ) V (r), (2.1.32) V (r) = α r (2.1.33) and α is a constant.. The simple Harmonic oscillator. from which the EoM can be derived as L = 1 2 m q2 1 2 kq2, (2.1.34) m q + kq = 0 (2.1.35) or where w = k m. q + w 2 q = 0, (2.1.36) 26

27 2.1.5 Perturbation theory In perturbation theory, the Hamiltonian can be written as H(q a, p a, t) = H 0 (q a, p a, t) + H int(q a, p a, t), (2.1.37) where H 0 is the Hamiltonian that the EoM can be exactly solved and H int is the small perturbation term. Therefore, the total Hamiltonian H can be solved in perturbation theory Scattering theory The procedures of describing scattering phenomena are the same whether the mechanics is classical or quantum. Goldstein The quantity j in, 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 j out, it is the number of scattering particles per unit time. The total cross section σ describes the scattering process in the way and note that the total cross section has the dimension of area. The differential cross section dσ dω is defined as hence djout dω j in σ = j out, (2.1.38) j in dσ dω = dj out dω, (2.1.39) 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 out = j in 2πbdb = j in 2π sin θdθ, (2.1.41) 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: In attractive interaction: dσ dω = bdb sin θdθ ; (2.1.42) dσ dω = bdb sin θdθ. (2.1.43) 27

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 students and quantum field theory for graduate students 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} a with which the EoM can be rewritten as Note that where A is the observable other than H.. Canonical quantization procedure. ( f g f ) g, (2.2.1) p a q a q a p a q a (t) = {H, q a } = H p a ; (2.2.2) ṗ a (t) = {H, p a } = H q a. (2.2.3) da dt = A + {H, A}, (2.2.4) t Step 1: (q a (t), p a (t)) (ˆq a (t), ˆp a (t)). (2.2.5) Replace classical variables (q a (t), p a (t)) with operator-valued function of time satisfying the commutation relations [ˆq a (t), ˆq b (t)] = 0 = [ˆp a (t), ˆp a (t)], (2.2.6) [ˆq a (t), ˆp b (t)] = iδ ab = i δ ab. (2.2.7) 28

29 Step 2: H(q, p) Ĥ(ˆq, ˆp). (2.2.8) Quantum Hamiltonian Ĥ has the same form of classical Hamiltonian H except that it is a function of ˆq a and ˆp b. Note: The canonical quantization suffers from the ordering ambiguity, for instance ˆp 2ˆq ˆpˆqˆp ˆqˆp 2, but they are the same in the classical physics limit, namely 0, such as ˆp 2ˆq = ˆpˆqˆp = ˆp 2 ˆq = p 2 q. (2.2.9) Step 3: Heisenberg s equation of motion: dˆq a (t) dt dˆp a (t) dt = i[ĥ, ˆq a(t)] = H ˆp a, (2.2.10) = i[ĥ, ˆp a(t)] = H ˆq a. (2.2.11) For the time dependent observable Â(t): dâ(t) dt = Â t + i[ĥ, Â]. (2.2.12) Step 4: Prove the Hamiltonian Ĥ 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 Fundamental principles of quantum mechanics.. Static part ψ : State. Â: Observable. H : Hilbert space. Dynamic part Unitary evolution: ψ(t) = U(t) ψ(0), i t ψ(t) = H ψ(t). Non-unitary evolution: Quantum measurement, wave function collapse, ψ ψ. The standard quantum mechanics: System: Hilbert space H. Note: A quantum binary digit is a two-dimensional Hilbert space. 29

30 State: A closed system is described by a pure vector ψ H obeying the linear superposition principle, namely a ψ 1 + b ψ 2 H, a, b C. (2.2.13) Note: Quantum computer is powerful mainly due to the superposition principle which allows a kind of parallel computation on a single quantum computer. Observable Ô. Hermitian operator: Ô = Ô. Unitary evolution of a state vector in a closed system: ψ(t) = U(t) ψ(0), U (t)u(t) = Id. (2.2.14) E.g. A unitary evolution of a closed system is governed by the Schrödinger equation with H = H. i ψ(t) = H ψ(t) (2.2.15) t Note: A quantum gate is a unitary transformation acting on quantum binary digits. Non-unitary evolution of state vector, also named as quantum measurement, quantum 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., H A and H B, is described by the tensor product H A H B, namely ψ A ψ B H A H B. Note: Quantum entanglement arises in the quantum composite system, which distinguishes classical physics from quantum physics and it plays the crucial role in quantum information & computation The Schröedinger, 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.. QM: Shrödinger picture 3 ψ S (0) ψ S (t). Ô S (t) = ÔS(0).. QFT: Heisenberg picture 3 ψ H (0) = ψ H (t). Ô H (0) ÔH(t). 30

31 Transition amplitude is unchanged in any picture, ψ S (t) ÔS ψ S (t) = ψ H (t) ÔH ψ H (t). (2.2.16) The Schrödinger picture Ô S (t) = ÔS(t 0 ); (2.2.17) ψ S (t) = U(t, t 0 ) ψ S (t 0 ). (2.2.18) The Heisenberg picture ψ H (t) = ψ H (t 0 ) ; (2.2.19) Ô H (t) = U (t, t 0 )ÔH(t 0 )U(t, t 0 ). (2.2.20) Table 2.1: The Shrödinger, Heisenberg and Dirac picture Schrödinger Heisenberg Dirac (Interaction) time-dependent state vector time-dependent time-independent H = H 0 + H int i t ψ(t) S = H ψ(t) S ψ(t) H = ψ(0) H i ψ(t) I = HI ψ(t) I t H I(t) = e ih 0t H inte ih 0t observable initial value time-independent time-dependent time-dependent O S(t) = O S(0) i d dt OH = [OH, H] i d OI = [OI, H0] dt O S(0) = O H(0) = O I(0) ψ S(0) S = ψ S(0) H = ψ S(0) I probability amplitude S ψ(t) O S ψ(t) S = H ψ(t) O H ψ(t) H = I ψ(t) O I ψ(t) I Note 1: Non-relativistic quantum mechanics prefers the Schrödinger picture. 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. ψ(t) S = e ih(t t 0) ψ(t 0 ) S ; (2.2.21) O H (t) = e iht O H (0)e iht ; (2.2.22) O I (t) = e ih 0t O I (0)e ih 0t ; (2.2.23) ψ(t) S = e ih 0t ψ(t) I, (2.2.24) where the free Hamiltonian H 0 can be exactly solved. Prove i d dt ψ(t) I = H I (t) ψ(t) I. (2.2.25) 31

32 Proof. i d dt ψ(t) I = i d dt ( e ih 0 t ψ(t) S ) = H 0 e ih 0t ψ(t) S + e ih 0t i d dt ψ(t) S = H 0 ψ(t) I + e ih 0t (H 0 + H int) ψ(t) S = H 0 ψ(t) I + e ih 0t H 0 ψ(t) S + e ih 0t H int ψ(t) S = e ih0t H inte ih0t ψ(t) I = H I (t) ψ(t) I. (2.2.26) In the interaction picture, the equation of time evolution i d dt ψ(t) I = H I (t) ψ(t) I (2.2.27) can be solved in the Dyson s series. Introduce the unitary operator U I (t, t 0 ) satisfying U I U I = Id and U I (t 0, t 0 ) = 1. And the equation of time evolution can be rewritten as Integrate out above equation, we can obtain i d dt U I(t, t 0 ) = H I (t)u I (t, t 0 ). (2.2.28) U I (t, t 0 ) = U I (t 0, t 0 ) + ( i ) t and then we can perform the iteration method to obtain the relation U I (t, t 0 ) = n=1 ( i )n t t 0 dt 1 t1 t 0 dt 2 tn 1 t 0 dt 1 H I (t 1 )U I (t 1, t 0 ), (2.2.29) t 0 dt n H I (t 1 )H I (t 2 ) H I (t n ). (2.2.30) Note: In QFT, the Dyson s series have another compact formulation which is used to derive Feynman diagrams 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 i x j ), 1 < j < k < N. The equation of motion has the form with the Hamiltonian H given by H = i t ψ(t, x 1,, x n ) = Hψ(t, x 1,, x n ), (2.2.31) N j=1 ( 2 2m 2 j + U(x j )) + N j 1 V ( x j x k ). (2.2.32) j=1 k=1 32

33 Note 1: Particle number is fixed, namely N is a conserved quantity, [H, N] = 0. Note 2: Position operators ˆX 1, ˆX 2,, ˆX n are well defined; Note 3: Momentum operators ˆP 1, ˆP 2,, ˆP n are well defined and other observables are independent coordinates x 1, x 2,, x n. However, quantum field theory tells another story, 1. Particle number is not-fixed; 2. Position operators ˆX 1, ˆX 2,, ˆX n do not exist; 3. Observable Ô(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 ˆXi well-defined No ˆX i because particles can be destroyed Position parameter x i well-defined x i well-defined as labels or parameters Momentum operator ˆPi well-defined ˆPi well-defined Observable independent of coordinate ˆ,O(t) dependent on (t, x), Ô(t, x) Special reality No Yes Lorentz transformation No Yes ψ(t, x 1, x 2,, x n) wave function (first quantization) quantum field operator (second quantization) Compatible observable Observables A and B are called compatible when [A, B] = AB BA = 0, (2.2.33) and incompatible when [A, B] 0. Note: The concept of compatible observable is associated with micro-causality in QFT The Heisenberg uncertainty principle The Heisenberg s uncertainty relation ( A) 2 ( B) [A, B] 2, (2.2.34) where A = A A, (2.2.35) ( A) 2 = A 2 A 2. (2.2.36) E.g. x p x 4. When x 0, p x +, which implies p x +. Small distance 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 x 4 is meaningful in QM, but questionable in QFT. 33

34 2.2.7 Symmetries and conservation laws Symmetry in quantum mechanics is a transformation D, iff [D, H] = 0. The conserved charge is denoted as Q, which is a constant of the motion, namely dq dt = 0, then [Q, H] = 0. E.g. D is a unitary transformation, and we can construct the transformation as D = 1 i 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 Schrödinger picture, the equation of time evolution has the form i ψ(t) = H ψ(t). (2.2.38) t Because the symmetry transformation D preserves the EoM, we have i (D ψ(t) ) = H(D ψ(t) ). (2.2.39) t If D 1 exists and D is irreverent of time, then which implies [D, H] = 0. i t ψ(t) = D 1 HD ψ(t) = H ψ(t), (2.2.40) 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 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. 34

35 2.2.9 Simple harmonic oscillator The Hamiltonian of the simple harmonic oscillator takes the form ˆ P 2 Ĥ = 2m mω2 ˆX2, ω 0. (2.2.41) where the position operator ˆX i and momentum operator ˆP j obey the commutative relation Introduce the Canonical transformation: p = [ ˆX i, ˆP j ] = i δ ij. (2.2.42) P mω, q = mωx, (2.2.43) and rewrite the Hamiltonian as H = ω 2 (p2 + q 2 ), (2.2.44) 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 a = q + ip 2, a = q ip 2, (2.2.45) with the Hamiltonian can be rewritten as [a, a ] = 1, (2.2.46) H = ω(a a + 1 ). (2.2.47) 2 The number operator is defined as thus N = a a, (2.2.48) H = ω(n + 1 ). (2.2.49) 2 Due to ψ a a ψ 0, define the ground state 0 as a 0 = 0, and the first excited state as 1 = a 0. H 0 = ω(n ) 0 = 1 2 ω 0 = E 0 1, (2.2.50) H 1 = ω(n ) 1 = 3 2 ω 1 = E 1 1. (2.2.51) The gap of eigenvalue is The eigenstate n is denoted as E 1 E 0 = ω 0. (2.2.52) n = 1 n! (a ) n 0 (2.2.53) with the eigenvalue E n = (n ) ω, namely H n = E n n. Note: Free quantum field theory (QFT without interaction) can be regarded as a collection of an infinite number of simple harmonic oscillator. 35

36 Time-independent perturbation theory The full Hamiltonian can be expanded as two parts H = H 0 + λh int, (2.2.54) where λ is a small real parameter in the range 0 < λ < 1 and H int is timeless. For the free Hamiltonian H 0, it has the eigenstate n (0) with the eigenvalue E n (0), namely And for the full Hamiltonian H, we define H 0 n (0) = E (0) n n (0). (2.2.55) H n = E n n. (2.2.56) We can expand the eigenstate and eigenvalue of the full Hamiltonian in terms of the power of parameter λ where the energy shift n is defined as n = n (0) + λ n (1) + λ 2 n (2) + ; (2.2.57) E n = E n (0) + λ (1) n + λ 2 (2) n +. (2.2.58) n E n E (0) n. (2.2.59) In the perturbative theory, the result formula of the eigenstate and eigenvalue of full Hamiltonian show as n = n (0) + λ k 0 k0 H int n(0) k n E n (0) E (0) + ; (2.2.60) k n (0) H int k(0) 2 E n = E (0) n + λ n (0) H int n (0) + λ 2 k n E (0) n E (0) k +. (2.2.61) Note: When E n (0) E (0) k is very small, the high energy physics arises naturally where QFT has to be considered Time-dependent perturbation theory The full Hamiltonian can be split as where H int (t) is time-dependent. H = H 0 + H int(t), (2.2.62) In the interaction (Dirac) picture, the observable obeys the equation do I (t) dt and the state follows the Shrödinger-like equation = 1 i [O I, H 0 ], (2.2.63) i d dt ψ I(t) = H I(t) ψ I (t) (2.2.64) 36

37 with H I (t) = eih 0t H int e ih 0t. The state ψ I (t) can be expanded in terms of the basis of free Hamiltonian, namely n with H 0 n = E n (0) n, ψ I (t) = C n (t) n, (2.2.65) n where C n (t) = n ψ I (t). Note: ψ S (t) = e ih 0t ψ I (t) = n (0) C n (t)e ie n t n ; (2.2.66) i d dt C n(t) = m n H int m e iωnmt C m (t) (2.2.67) with ω nm = E(0) n E m (0) = ω mn. (2.2.68) Note: The equation of time evolution of state vector in interaction picture, i.e., i d dt ψ I(t) = H I (t) ψ I(t), can be solved in Dyson s series in time-dependent perturbative theory which is used to derive Feynman diagrams in time-dependent perturbative QFT. Denote the initial state ψ I (t 0 ) as i and introduce the time evolution operator U I (t, t 0 ), ψ I (t) = U I (t, t 0 ) i = n n n U I (t, t 0 ) i, (2.2.69) where U I (t, t 0 ) = 1 i t t 0 H I (t 1 )dt 1 + ( i )2 t t 0 dt 1 t1 t 0 dt 2 H I (t 1 )H I (t 2 ) +. (2.2.70) C n (t) as the matrix element of time evolution operator, namely C n (t) = n U I (t, t 0 ) i, can take the perturbation expansion: where C n (t) = C (0) n + C (1) n + C (2) n +, (2.2.71) C (0) n = δ ni ; (2.2.72) C (1) n = i = i t t 0 n H I (t 1 ) i dt 1 t t 0 e iω nit 1 n H int i dt 1 ; (2.2.73) C (2) n =. (2.2.74) The transition probability can be calculated as P ( i n ) = C n (1) + C n (2) + 2 (2.2.75) for n i. 37

38 Angular momentum and spin The angular momentum operator denoted as J = (J 1, J 2, J 3 ) satisfies the commutative relation [J i, J j ] = i ε ijk J k, (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. * Spin- 1 2 system, j = 1 2 (electron, position, proton, neutron): 1 2, 1 2 stands for spin-up state; 1 2, 1 2 stands for spin-down state. * Spin-0 system: j = 0, 00. * Spin-1 system: j = 1, 1, 1, 1, 0, 1, 1. 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 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 ij N bosons = + N bosons, (2.2.77) where P ij interchanges between arbitrary pair. Fermions: The system of fermions are totally anti-symmetrical under the exchange of any pair P ij N fermions = N fermions. (2.2.78) E.g.1: two-boson system: k k, k k, 1 2 ( k k + k k ). E.g.2: two-fermion system: 1 2 ( k k k k ). 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. 38