Physics 214 UCSD/225a UCSB Lecture 11 Finish Halzen & Martin Chapter 4
|
|
- Domenic Thomas
- 5 years ago
- Views:
Transcription
1 Physics 24 UCSD/225a UCSB Lecture Finish Halzen Martin Chapter 4 origin of the propagator Halzen Martin Chapter 5 Continue Review of Dirac Equation Halzen Martin Chapter 6 start with it if time permits
2 Origin of propagator When we discussed perturbation theory a few lectures ago, we did what some call old fashioned perturbation theory. It was not covariant We required momentum conservation at vertex but not Energy conservation At second order, we need to consider time ordered products. When you do this more modern Fully covariant 4-momentum is conserved at each vertex However, propagating particles are off-shell This is what you ll learn in QFT!
3 Spinless massive propagator HM has a more detailed discussion for how to go from a time ordered 2nd order perturbation theory to get the propagator. Here, we simply state the result: p + p 2 =! m 2 p 2! m 2 A B For more details see Halzen Martin
4 HM Chapter 5 Review of Dirac Equation Dirac s Quandery Notation Reminder Dirac Equation for free particle Mostly an exercise in notation Define currents Make a complete list of all possible currents Aside on Helicity Operator Solutions to free particle Dirac equation are eigenstates of Helicity Operator Aside on handedness
5 Dirac s Quandery Can there be a formalism that allows wave functions that satisfy the linear and quadratic equations simultaneously: H! = "!" p!" + #m! H 2! = P 2 + m 2! If such a thing existed then the linear equation would provide us with energy eigenvalues that automatically satisfy the relativistic energy momentum relationship
6 Dirac s Quandery 2 Such a thing does indeed exist: Wave function is a 4 component object! i = # " i " i # = I *I α is thus a 3-vector of 4x4 matrices with special commutator relationships like the Pauli matrices. While β is a diagonal 4x4 matrix as shown. "! = " ;! # 2 = i " ;! # i 3 = " ; I = # #
7 The rest is history In the following I provide a very limited reminder of notation and a few facts. If these things don t sound familiar, then I encourage you to work through ch. 5 carefully.
8 Notation Reminder Sigma Matrices: "! = " ;! 2 = i " ;! 3 = " ; I = # # i # # Gamma Matrices: #! = I # ;! j =,2,3 j =,2,3 # = ;! 5 = I "I " j =,2,3 I State Vectors: "..! = ;!! T * =..... #.
9 Notation Reminder 2 Obvious statements about gamma matrices! j =,2,5T =! j =,2,5 ;! j =,3T = "! j =,3! j =,5T * =! j =,5 ;! j =,2,3T * = "! j =,2,3 Probability density!"! =! T *! =##! "! T * # Scalar product of gamma matrix and 4-vector A! " µ A µ = " A # " A # " 2 A 2 # " 3 A 3 Is again a 4-vector
10 Dirac Equation of free particle i! µ " # m = µ i" µ! µ + m = Ansatz:! = e "ipx u p Explore this in restframe of particle: # #! + 2 = 2me "imt ;! " 2 = 2me "imt ; Normalization chosen to describe 2E particles, as usual.
11 Particle vs antiparticle in restframe # #! + 2 = 2me "imt ;! " 2 = 2me "imt ; particle Recall, particle -> antiparticle means E,p -> -E,-p Let s take a look at Energy Eigenvalues: " # mi!mi u = Eu Hu =!!" p!" + "mu = Eu for p= we get this equation to satisfy by the energy eigenvectors It is thus obvious that 2 of the solutions have E <, and are the lower two components of the 4-component object u.
12 Particle Anti-particle! + 2 = 2me "imt # ;! " 2 = 2me "imt # ;! + 2 = 2me "imt # ;! " 2 = 2me "imt # ; Negative energy solution Positive energy solution
13 Not in restframe this becomes: Hu =!!"!" p + "mu = Eu Hu = m #!"!"!" p #!" * p m u = Eu The lower 2 components are thus coupled to the upper 2 via this matrix equation, leading to free particle and antiparticle solutions as follows.
14 Anti-Particle not in restframe! + 2 = Ne "ipx # p E + m ;! " 2 = Ne "ipx # p E + m! + 2 = Ne +ipx "# p E +m ;! " 2 = Ne +ipx "# p E +m I suggest you read up on this in HM chapter 5 if you re not completely comfortable with it.
15 What we learned so far: Dirac Equation has 4 solutions for the same p: Two with E> Two with E< The E< solutions describe anti-particles. The additional 2-fold ambiguity describes spin +-/2. You will show this explicitly in Exercise HM 5.4, which is part of HW next week. We thus have a formalism to describe all the fundamental spin /2 particles in nature.
16 Helicity Operator The helicity operator commutes with both H and P. Helicity is thus conserved for the free spin /2 particle. 2 "!"! # p #!"! # p The unit vector here is the axis with regard to which we define the helicity. For,,, i.e. the Z-axis, we get the desired +-/2 eigenvalues. #! 3 = "
17 Dirac Equation for particle and anti-particle spinors It s sometimes notationally convenient to write the antiparticle spinor solution i.e. the -p,-e as an explicit Antiparticle spinor that satisfies a modified dirac equation:! µ p µ " mu =! µ p µ + mv = particle antiparticle The v-spinor then has positive energy. We won t be using v-spinors in this course. HM Equation 5.33 and 5.34
18 Antiparticles We will stick to the antiparticle description we introduced in chapter 4: e - e - p,e -p,-e Initial state e + e - is an initial state e - e - with the positron being an electron going in the wrong direction, i.e. backwards in time.
19 Some more reflections on γ µ There are exactly 5 distinct γ matrices: #! = I # ;! j =,2,3 j =,2,3 # = ;! 5 = I "I " j =,2,3 I! 5 " i!!! 2! 3 Where Every one of them multiplied with itself gives the unit matrix. As a result, any product of 5 of them can be expressed as a product of 3 of them.
20 Currents Any bi-linear quantity can be a current as long as it has the most general form:! 4x4! By finding all possible forms of this type, using the gamma-matrices as a guide, we can form all possible currents that can be within this formalism.
21 The possible currents!!!" 5!!" µ!!" 5 " µ! scalar pseudo-scalar vector pseudo-vector or axial-vector! i 2 " µ " # " # " µ! tensor
22 Chapter 6 Electrodynamics of Spin-/2 particles.
23 Spinless vs Spin /2! t, x = Ne "ip µ x µ! t, x = u pe "ip µ x µ J µ =!ie" # µ "! µ " # " T fi =!i J µ fi A µ d 4 x + Oe 2 J µ =!e"# µ " T fi =!i J µ fi A µ d 4 x + Oe 2 We basically make a substitution of the vertex factor: p + p! u f i f " u µ µ i And all else in calculating M 2 remains the same.
24 Example: e - e - scattering For Spinless i.e. bosons we showed: " M =!e p 2 A + p C µ p B + p D µ # p A! p C µ p B + p C µ p A! p D + p A + p D 2 2 For Spin /2 we thus get: # M =!e u c" µ u 2 A u D " µ u B p A! p C µ u A u C " µ u B p A! p D! u D" 2 2 Minus sign comes from fermion exchange!!!
25 Spin Averaging The M from previous page includes spinors in initial and final state. In many experimental situations, in particular in hadron collissions, you neither fix initial nor final state spins. We thus need to form a spin averaged amplitude squared before we can compare with experiment: M 2 = 2s A + 2s B +! M 2 = 4 spin! spin M 2 A B C D
26 Spin Averaging in non-relativistic limit Incoming e - : Outgoing e - :! # s= +/ 2 u = 2m# # # " u s=/ 2 = 2m Reminder: #! = I "I #! j =,2,3 j =,2,3 = " j =,2,3 # u! u = 2m if µ = " s i = s f f µ i otherwise
27 Invariant variables s,t,u Example: e- e- -> e- e- s = p A + p B 2 = 4 k 2 + m 2 A C t = p A - p C 2 = -2 k 2 - cosθ B D u = p A - p D 2 = -2 k 2 + cosθ k = k i = k f m = m e θ = scattering angle, all in com frame.
28 Invariant variables s,t,u s = p A + p B 2 = 4 k 2 + m 2 A C B D B,D are antiparticles! p B thus negative, leading to the + in p A +p B. k = k i = k f m = m e θ = scattering angle, all in com frame.
29 M for Different spin combos # M =!e u c" µ u 2 A u D " µ u B t! u D" µ u A u C " µ u B u [u c " µ u A u D " µ u B ] *+* = 4m 2 [u D " µ u A u C " µ u B ] = 4m 2 *+* [u c " µ u A u D " µ u B ] *+* = etc. M 2 = 4 4m2 e 2 " 2 2 * t! # u 2 + t u 2,
30 M for Different spin combos # u M =!e 2 c " µ u A u D " µ u B t! u D" µ u A u C " µ u B u A B C D A B C D t 2 And alike for the other permutations. u 2 A B C D " # t! u 2
31
32
33
H&M Chapter 5 Review of Dirac Equation
HM Chapter 5 Review of Dirac Equation Dirac s Quandary Notation Reminder Dirac Equation for free particle Mostly an exercise in notation Define currents Make a complete list of all possible currents Aside
More informationPhysics 214 UCSD Lecture 7 Halzen & Martin Chapter 4
Physics 214 UCSD Lecture 7 Halzen & Martin Chapter 4 (Spinless) electron-muon scattering Cross section definition Decay rate definition treatment of identical particles symmetrizing crossing Electrodynamics
More informationLecture 4 - Dirac Spinors
Lecture 4 - Dirac Spinors Schrödinger & Klein-Gordon Equations Dirac Equation Gamma & Pauli spin matrices Solutions of Dirac Equation Fermion & Antifermion states Left and Right-handedness Non-Relativistic
More informationDr Victoria Martin, Spring Semester 2013
Particle Physics Dr Victoria Martin, Spring Semester 2013 Lecture 3: Feynman Diagrams, Decays and Scattering Feynman Diagrams continued Decays, Scattering and Fermi s Golden Rule Anti-matter? 1 Notation
More informationQuantum Physics 2006/07
Quantum Physics 6/7 Lecture 7: More on the Dirac Equation In the last lecture we showed that the Dirac equation for a free particle i h t ψr, t = i hc α + β mc ψr, t has plane wave solutions ψr, t = exp
More information129 Lecture Notes More on Dirac Equation
19 Lecture Notes More on Dirac Equation 1 Ultra-relativistic Limit We have solved the Diraction in the Lecture Notes on Relativistic Quantum Mechanics, and saw that the upper lower two components are large
More informationThe path integral for photons
The path integral for photons based on S-57 We will discuss the path integral for photons and the photon propagator more carefully using the Lorentz gauge: as in the case of scalar field we Fourier-transform
More informationParticle Physics Dr M.A. Thomson Part II, Lent Term 2004 HANDOUT V
Particle Physics Dr M.A. Thomson (ifl μ @ μ m)ψ = Part II, Lent Term 24 HANDOUT V Dr M.A. Thomson Lent 24 2 Spin, Helicity and the Dirac Equation Upto this point we have taken a hands-off approach to spin.
More informationMaxwell s equations. electric field charge density. current density
Maxwell s equations based on S-54 Our next task is to find a quantum field theory description of spin-1 particles, e.g. photons. Classical electrodynamics is governed by Maxwell s equations: electric field
More informationLecture 6:Feynman diagrams and QED
Lecture 6:Feynman diagrams and QED 0 Introduction to current particle physics 1 The Yukawa potential and transition amplitudes 2 Scattering processes and phase space 3 Feynman diagrams and QED 4 The weak
More informationParticle Physics WS 2012/13 ( )
Particle Physics WS /3 (3..) Stephanie Hansmann-Menzemer Physikalisches Institut, INF 6, 3. How to describe a free particle? i> initial state x (t,x) V(x) f> final state. Non-relativistic particles Schrödinger
More information3 Quantization of the Dirac equation
3 Quantization of the Dirac equation 3.1 Identical particles As is well known, quantum mechanics implies that no measurement can be performed to distinguish particles in the same quantum state. Elementary
More information3.3 Lagrangian and symmetries for a spin- 1 2 field
3.3 Lagrangian and symmetries for a spin- 1 2 field The Lagrangian for the free spin- 1 2 field is The corresponding Hamiltonian density is L = ψ(i/ µ m)ψ. (3.31) H = ψ( γ p + m)ψ. (3.32) The Lagrangian
More informationLSZ reduction for spin-1/2 particles
LSZ reduction for spin-1/2 particles based on S-41 In order to describe scattering experiments we need to construct appropriate initial and final states and calculate scattering amplitude. Summary of free
More informationREVIEW REVIEW. A guess for a suitable initial state: Similarly, let s consider a final state: Summary of free theory:
LSZ reduction for spin-1/2 particles based on S-41 In order to describe scattering experiments we need to construct appropriate initial and final states and calculate scattering amplitude. Summary of free
More information11 Spinor solutions and CPT
11 Spinor solutions and CPT 184 In the previous chapter, we cataloged the irreducible representations of the Lorentz group O(1, 3. We found that in addition to the obvious tensor representations, φ, A
More informationQFT. Unit 1: Relativistic Quantum Mechanics
QFT Unit 1: Relativistic Quantum Mechanics What s QFT? Relativity deals with things that are fast Quantum mechanics deals with things that are small QFT deals with things that are both small and fast What
More informationlattice QCD and the hadron spectrum Jozef Dudek ODU/JLab
lattice QCD and the hadron spectrum Jozef Dudek ODU/JLab a black box? QCD lattice QCD observables (scattering amplitudes?) in these lectures, hope to give you a look inside the box 2 these lectures how
More informationLecture 7 From Dirac equation to Feynman diagramms. SS2011: Introduction to Nuclear and Particle Physics, Part 2 2
Lecture 7 From Dirac equation to Feynman diagramms SS2011: Introduction to Nuclear and Particle Physics, Part 2 2 1 Dirac equation* The Dirac equation - the wave-equation for free relativistic fermions
More informationLecture 10. The Dirac equation. WS2010/11: Introduction to Nuclear and Particle Physics
Lecture 10 The Dirac equation WS2010/11: Introduction to Nuclear and Particle Physics The Dirac equation The Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist
More informationStandard Model of Particle Physics SS 2012
Lecture: Standard Model of Particle Physics Heidelberg SS 22 Fermi Theory Standard Model of Particle Physics SS 22 2 Standard Model of Particle Physics SS 22 Fermi Theory Unified description of all kind
More informationStandard Model of Particle Physics SS 2013
Lecture: Standard Model of Particle Physics Heidelberg SS 23 Fermi Theory Standard Model of Particle Physics SS 23 2 Standard Model of Particle Physics SS 23 Weak Force Decay of strange particles Nuclear
More informationParticle Physics Dr. Alexander Mitov Handout 2 : The Dirac Equation
Dr. A. Mitov Particle Physics 45 Particle Physics Dr. Alexander Mitov µ + e - e + µ - µ + e - e + µ - µ + e - e + µ - µ + e - e + µ - Handout 2 : The Dirac Equation Dr. A. Mitov Particle Physics 46 Non-Relativistic
More informationPhysics 222 UCSD/225b UCSB. Lecture 3 Weak Interactions (continued) muon decay Pion decay
Physics UCSD/5b UCSB Lecture 3 Weak Interactions (continued) muon decay Pion decay Muon Decay Overview (1) Feynman diagram: µ! " u p ( ) Matrix Element: M = G u ( k )! µ ( 1"! 5 )u p [ ( )] u p'! " u (
More informationQuantum Electrodynamics Test
MSc in Quantum Fields and Fundamental Forces Quantum Electrodynamics Test Monday, 11th January 2010 Please answer all three questions. All questions are worth 20 marks. Use a separate booklet for each
More informationLecture 3. Experimental Methods & Feynman Diagrams
Lecture 3 Experimental Methods & Feynman Diagrams Natural Units & the Planck Scale Review of Relativistic Kinematics Cross-Sections, Matrix Elements & Phase Space Decay Rates, Lifetimes & Branching Fractions
More informationLecture 4 - Relativistic wave equations. Relativistic wave equations must satisfy several general postulates. These are;
Lecture 4 - Relativistic wave equations Postulates Relativistic wave equations must satisfy several general postulates. These are;. The equation is developed for a field amplitude function, ψ 2. The normal
More informationParticle Physics. Michaelmas Term 2011 Prof. Mark Thomson. Handout 2 : The Dirac Equation. Non-Relativistic QM (Revision)
Particle Physics Michaelmas Term 2011 Prof. Mark Thomson + e - e + - + e - e + - + e - e + - + e - e + - Handout 2 : The Dirac Equation Prof. M.A. Thomson Michaelmas 2011 45 Non-Relativistic QM (Revision)
More informationlattice QCD and the hadron spectrum Jozef Dudek ODU/JLab
lattice QCD and the hadron spectrum Jozef Dudek ODU/JLab the light meson spectrum relatively simple models of hadrons: bound states of constituent quarks and antiquarks the quark model empirical meson
More informationLoop corrections in Yukawa theory based on S-51
Loop corrections in Yukawa theory based on S-51 Similarly, the exact Dirac propagator can be written as: Let s consider the theory of a pseudoscalar field and a Dirac field: the only couplings allowed
More informationOutline. Charged Leptonic Weak Interaction. Charged Weak Interactions of Quarks. Neutral Weak Interaction. Electroweak Unification
Weak Interactions Outline Charged Leptonic Weak Interaction Decay of the Muon Decay of the Neutron Decay of the Pion Charged Weak Interactions of Quarks Cabibbo-GIM Mechanism Cabibbo-Kobayashi-Maskawa
More information4. The Standard Model
4. The Standard Model Particle and Nuclear Physics Dr. Tina Potter Dr. Tina Potter 4. The Standard Model 1 In this section... Standard Model particle content Klein-Gordon equation Antimatter Interaction
More informationParticle Physics. experimental insight. Paula Eerola Division of High Energy Physics 2005 Spring Semester Based on lectures by O. Smirnova spring 2002
experimental insight e + e - W + W - µνqq Paula Eerola Division of High Energy Physics 2005 Spring Semester Based on lectures by O. Smirnova spring 2002 Lund University I. Basic concepts Particle physics
More informationLecture 01. Introduction to Elementary Particle Physics
Introduction to Elementary Particle Physics Particle Astrophysics Particle physics Fundamental constituents of nature Most basic building blocks Describe all particles and interactions Shortest length
More informationAttempts at relativistic QM
Attempts at relativistic QM based on S-1 A proper description of particle physics should incorporate both quantum mechanics and special relativity. However historically combining quantum mechanics and
More informationThe Strong Interaction and LHC phenomenology
The Strong Interaction and LHC phenomenology Juan Rojo STFC Rutherford Fellow University of Oxford Theoretical Physics Graduate School course Lecture 2: The QCD Lagrangian, Symmetries and Feynman Rules
More informationParticle Physics WS 2012/13 ( )
Particle Physics WS 2012/13 (6.11.2012) Stephanie Hansmann-Menzemer Physikalisches Institut, INF 226, 3.101 2 2 3 3 4 4 5 Where are we? W fi = 2π 4 LI matrix element M i (2Ei) fi 2 ρ f (E i ) LI phase
More informationOrganisatorial Issues: Exam
Organisatorial Issues: Exam Date: - current date: Tuesday 24.07.2012-14:00 16:00h, gr. HS - alternative option to be discussed: - Tuesday 24.07.2012-13:00 15:00h, gr. HS - Friday 27.07.2012-14:00 16:00h,
More informationThe Dirac Equation. H. A. Tanaka
The Dirac Equation H. A. Tanaka Relativistic Wave Equations: In non-relativistic quantum mechanics, we have the Schrödinger Equation: H = i t H = p2 2m 2 = i 2m 2 p t i Inspired by this, Klein and Gordon
More informationL = e i `J` i `K` D (1/2,0) (, )=e z /2 (10.253)
44 Group Theory The matrix D (/,) that represents the Lorentz transformation (.4) L = e i `J` i `K` (.5) is D (/,) (, )=exp( i / /). (.5) And so the generic D (/,) matrix is D (/,) (, )=e z / (.53) with
More informationLecture 8 Feynman diagramms. SS2011: Introduction to Nuclear and Particle Physics, Part 2 2
Lecture 8 Feynman diagramms SS2011: Introduction to Nuclear and Particle Physics, Part 2 2 1 Photon propagator Electron-proton scattering by an exchange of virtual photons ( Dirac-photons ) (1) e - virtual
More information1 Spinor-Scalar Scattering in Yukawa Theory
Physics 610 Homework 9 Solutions 1 Spinor-Scalar Scattering in Yukawa Theory Consider Yukawa theory, with one Dirac fermion ψ and one real scalar field φ, with Lagrangian L = ψ(i/ m)ψ 1 ( µφ)( µ φ) M φ
More informationElectroweak Physics. Krishna S. Kumar. University of Massachusetts, Amherst
Electroweak Physics Krishna S. Kumar University of Massachusetts, Amherst Acknowledgements: M. Grunewald, C. Horowitz, W. Marciano, C. Quigg, M. Ramsey-Musolf, www.particleadventure.org Electroweak Physics
More informationQM and Angular Momentum
Chapter 5 QM and Angular Momentum 5. Angular Momentum Operators In your Introductory Quantum Mechanics (QM) course you learned about the basic properties of low spin systems. Here we want to review that
More informationQuantum Field Theory Spring 2019 Problem sheet 3 (Part I)
Quantum Field Theory Spring 2019 Problem sheet 3 (Part I) Part I is based on material that has come up in class, you can do it at home. Go straight to Part II. 1. This question will be part of a take-home
More informationWeak interactions, parity, helicity
Lecture 10 Weak interactions, parity, helicity SS2011: Introduction to Nuclear and Particle Physics, Part 2 2 1 Weak decay of particles The weak interaction is also responsible for the β + -decay of atomic
More informationParticle Physics WS 2012/13 ( )
Particle Physics WS 2012/13 (9.11.2012) Stephanie Hansmann-Menzemer Physikalisches Institut, INF 226, 3.101 QED Feyman Rules Starting from elm potential exploiting Fermi s gold rule derived QED Feyman
More informationFermi Fields without Tears
Fermi Fields without Tears Peter Cahill and Kevin Cahill cahill@unm.edu http://dna.phys.unm.edu/ Abstract One can construct Majorana and Dirac fields from fields that are only slightly more complicated
More informationiδ jk q 2 m 2 +i0. φ j φ j) 2 φ φ = j
PHY 396 K. Solutions for problem set #8. Problem : The Feynman propagators of a theory follow from the free part of its Lagrangian. For the problem at hand, we have N scalar fields φ i (x of similar mass
More informationLecture 3 (Part 1) Physics 4213/5213
September 8, 2000 1 FUNDAMENTAL QED FEYNMAN DIAGRAM Lecture 3 (Part 1) Physics 4213/5213 1 Fundamental QED Feynman Diagram The most fundamental process in QED, is give by the definition of how the field
More informationPart I. Many-Body Systems and Classical Field Theory
Part I. Many-Body Systems and Classical Field Theory 1. Classical and Quantum Mechanics of Particle Systems 3 1.1 Introduction. 3 1.2 Classical Mechanics of Mass Points 4 1.3 Quantum Mechanics: The Harmonic
More informationQuantum Field Theory
Quantum Field Theory PHYS-P 621 Radovan Dermisek, Indiana University Notes based on: M. Srednicki, Quantum Field Theory 1 Attempts at relativistic QM based on S-1 A proper description of particle physics
More information129 Lecture Notes Relativistic Quantum Mechanics
19 Lecture Notes Relativistic Quantum Mechanics 1 Need for Relativistic Quantum Mechanics The interaction of matter and radiation field based on the Hamitonian H = p e c A m Ze r + d x 1 8π E + B. 1 Coulomb
More information4. The Dirac Equation
4. The Dirac Equation A great deal more was hidden in the Dirac equation than the author had expected when he wrote it down in 1928. Dirac himself remarked in one of his talks that his equation was more
More informationbe stationary under variations in A, we obtain Maxwell s equations in the form ν J ν = 0. (7.5)
Chapter 7 A Synopsis of QED We will here sketch the outlines of quantum electrodynamics, the theory of electrons and photons, and indicate how a calculation of an important physical quantity can be carried
More informationPHYS 508 (2015-1) Final Exam January 27, Wednesday.
PHYS 508 (2015-1) Final Exam January 27, Wednesday. Q1. Scattering with identical particles The quantum statistics have some interesting consequences for the scattering of identical particles. This is
More informationBeta functions in quantum electrodynamics
Beta functions in quantum electrodynamics based on S-66 Let s calculate the beta function in QED: the dictionary: Note! following the usual procedure: we find: or equivalently: For a theory with N Dirac
More informationThe Klein-Gordon equation
Lecture 8 The Klein-Gordon equation WS2010/11: Introduction to Nuclear and Particle Physics The bosons in field theory Bosons with spin 0 scalar (or pseudo-scalar) meson fields canonical field quantization
More informationQuantum Field Theory
Quantum Field Theory PHYS-P 621 Radovan Dermisek, Indiana University Notes based on: M. Srednicki, Quantum Field Theory 1 Attempts at relativistic QM based on S-1 A proper description of particle physics
More informationQuantum Field Theory
Quantum Field Theory PHYS-P 621 Radovan Dermisek, Indiana University Notes based on: M. Srednicki, Quantum Field Theory 1 Attempts at relativistic QM based on S-1 A proper description of particle physics
More informationIntroduction to Neutrino Physics. TRAN Minh Tâm
Introduction to Neutrino Physics TRAN Minh Tâm LPHE/IPEP/SB/EPFL This first lecture is a phenomenological introduction to the following lessons which will go into details of the most recent experimental
More informationLecture 10: A (Brief) Introduction to Group Theory (See Chapter 3.13 in Boas, 3rd Edition)
Lecture 0: A (Brief) Introduction to Group heory (See Chapter 3.3 in Boas, 3rd Edition) Having gained some new experience with matrices, which provide us with representations of groups, and because symmetries
More informationParticle Notes. Ryan D. Reece
Particle Notes Ryan D. Reece July 9, 2007 Chapter 1 Preliminaries 1.1 Overview of Special Relativity 1.1.1 Lorentz Boosts Searches in the later part 19th century for the coordinate transformation that
More informationBy following steps analogous to those that led to (20.177), one may show (exercise 20.30) that in Feynman s gauge, = 1, the photon propagator is
20.2 Fermionic path integrals 74 factor, which cancels. But if before integrating over all gauge transformations, we shift so that 4 changes to 4 A 0, then the exponential factor is exp[ i 2 R ( A 0 4
More informationParticle Physics I Lecture Exam Question Sheet
Particle Physics I Lecture Exam Question Sheet Five out of these 16 questions will be given to you at the beginning of the exam. (1) (a) Which are the different fundamental interactions that exist in Nature?
More informationPhysics 217 FINAL EXAM SOLUTIONS Fall u(p,λ) by any method of your choosing.
Physics 27 FINAL EXAM SOLUTIONS Fall 206. The helicity spinor u(p, λ satisfies u(p,λu(p,λ = 2m. ( In parts (a and (b, you may assume that m 0. (a Evaluate u(p,λ by any method of your choosing. Using the
More informationWe will also need transformation properties of fermion bilinears:
We will also need transformation properties of fermion bilinears: Parity: some product of gamma matrices, such that so that is hermitian. we easily find: 88 And so the corresponding bilinears transform
More informationSymmetry Groups conservation law quantum numbers Gauge symmetries local bosons mediate the interaction Group Abelian Product of Groups simple
Symmetry Groups Symmetry plays an essential role in particle theory. If a theory is invariant under transformations by a symmetry group one obtains a conservation law and quantum numbers. For example,
More informationExtending the 4 4 Darbyshire Operator Using n-dimensional Dirac Matrices
International Journal of Applied Mathematics and Theoretical Physics 2015; 1(3): 19-23 Published online February 19, 2016 (http://www.sciencepublishinggroup.com/j/ijamtp) doi: 10.11648/j.ijamtp.20150103.11
More informationG : Quantum Mechanics II
G5.666: Quantum Mechanics II Notes for Lecture 5 I. REPRESENTING STATES IN THE FULL HILBERT SPACE Given a representation of the states that span the spin Hilbert space, we now need to consider the problem
More informationVector Spaces for Quantum Mechanics J. P. Leahy January 30, 2012
PHYS 20602 Handout 1 Vector Spaces for Quantum Mechanics J. P. Leahy January 30, 2012 Handout Contents Examples Classes Examples for Lectures 1 to 4 (with hints at end) Definitions of groups and vector
More informationDerivation of Electro Weak Unification and Final Form of Standard Model with QCD and Gluons 1 W W W 3
Derivation of Electro Weak Unification and Final Form of Standard Model with QCD and Gluons 1 W 1 + 2 W 2 + 3 W 3 Substitute B = cos W A + sin W Z 0 Sum over first generation particles. up down Left handed
More information1 Introduction. 1.1 The Standard Model of particle physics The fundamental particles
1 Introduction The purpose of this chapter is to provide a brief introduction to the Standard Model of particle physics. In particular, it gives an overview of the fundamental particles and the relationship
More informationPRINCIPLES OF PHYSICS. \Hp. Ni Jun TSINGHUA. Physics. From Quantum Field Theory. to Classical Mechanics. World Scientific. Vol.2. Report and Review in
LONDON BEIJING HONG TSINGHUA Report and Review in Physics Vol2 PRINCIPLES OF PHYSICS From Quantum Field Theory to Classical Mechanics Ni Jun Tsinghua University, China NEW JERSEY \Hp SINGAPORE World Scientific
More informationDISCRETE SYMMETRIES IN NUCLEAR AND PARTICLE PHYSICS. Parity PHYS NUCLEAR AND PARTICLE PHYSICS
PHYS 30121 NUCLEAR AND PARTICLE PHYSICS DISCRETE SYMMETRIES IN NUCLEAR AND PARTICLE PHYSICS Discrete symmetries are ones that do not depend on any continuous parameter. The classic example is reflection
More informationQuantum Field Theory Example Sheet 4 Michelmas Term 2011
Quantum Field Theory Example Sheet 4 Michelmas Term 0 Solutions by: Johannes Hofmann Laurence Perreault Levasseur Dave M. Morris Marcel Schmittfull jbh38@cam.ac.uk L.Perreault-Levasseur@damtp.cam.ac.uk
More informationStandard Model of Particle Physics SS 2013
Lecture: Standard Model of Particle Physics Heidelberg SS 23 Weak Interactions I Standard Model of Particle Physics SS 23 ors and Helicity States momentum vector in z direction u R = p, = / 2 u L = p,
More informationStatic and covariant meson-exchange interactions in nuclear matter
Workshop on Relativistic Aspects of Two- and Three-body Systems in Nuclear Physics - ECT* - 19-23/10/2009 Static and covariant meson-exchange interactions in nuclear matter Brett V. Carlson Instituto Tecnológico
More informationFirst structure equation
First structure equation Spin connection Let us consider the differential of the vielbvein it is not a Lorentz vector. Introduce the spin connection connection one form The quantity transforms as a vector
More informationIntroduction to particle physics Lecture 6
Introduction to particle physics Lecture 6 Frank Krauss IPPP Durham U Durham, Epiphany term 2009 Outline 1 Fermi s theory, once more 2 From effective to full theory: Weak gauge bosons 3 Massive gauge bosons:
More informationWhat s up with those Feynman diagrams? an Introduction to Quantum Field Theories
What s up with those Feynman diagrams? an Introduction to Quantum Field Theories Martin Nagel University of Colorado February 3, 2010 Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 1
More informationParticle Physics 2018 Final Exam (Answers with Words Only)
Particle Physics 2018 Final Exam (Answers with Words Only) This was a hard course that likely covered a lot of new and complex ideas. If you are feeling as if you could not possibly recount all of the
More informationFundamental Interactions (Forces) of Nature
Chapter 14 Fundamental Interactions (Forces) of Nature Interaction Gauge Boson Gauge Boson Mass Interaction Range (Force carrier) Strong Gluon 0 short-range (a few fm) Weak W ±, Z M W = 80.4 GeV/c 2 short-range
More information7 Quantized Free Dirac Fields
7 Quantized Free Dirac Fields 7.1 The Dirac Equation and Quantum Field Theory The Dirac equation is a relativistic wave equation which describes the quantum dynamics of spinors. We will see in this section
More informationPhysics 557 Lecture 5
Physics 557 Lecture 5 Group heory: Since symmetries and the use of group theory is so much a part of recent progress in particle physics we will take a small detour to introduce the basic structure (as
More informationLecture 3: Propagators
Lecture 3: Propagators 0 Introduction to current particle physics 1 The Yukawa potential and transition amplitudes 2 Scattering processes and phase space 3 Feynman diagrams and QED 4 The weak interaction
More informationChapter 17 The bilinear covariant fields of the Dirac electron. from my book: Understanding Relativistic Quantum Field Theory.
Chapter 17 The bilinear covariant fields of the Dirac electron from my book: Understanding Relativistic Quantum Field Theory Hans de Vries November 10, 008 Chapter Contents 17 The bilinear covariant fields
More information2.4 Parity transformation
2.4 Parity transformation An extremely simple group is one that has only two elements: {e, P }. Obviously, P 1 = P, so P 2 = e, with e represented by the unit n n matrix in an n- dimensional representation.
More informationReview and Notation (Special relativity)
Review and Notation (Special relativity) December 30, 2016 7:35 PM Special Relativity: i) The principle of special relativity: The laws of physics must be the same in any inertial reference frame. In particular,
More informationIntroduction to Elementary Particle Physics I
Physics 56400 Introduction to Elementary Particle Physics I Lecture 16 Fall 018 Semester Prof. Matthew Jones Review of Lecture 15 When we introduced a (classical) electromagnetic field, the Dirac equation
More informationThe Dirac Field. Physics , Quantum Field Theory. October Michael Dine Department of Physics University of California, Santa Cruz
Michael Dine Department of Physics University of California, Santa Cruz October 2013 Lorentz Transformation Properties of the Dirac Field First, rotations. In ordinary quantum mechanics, ψ σ i ψ (1) is
More informationQFT. Unit 11: Cross Sections and Decay Rates
QFT Unit 11: Cross Sections and Decay Rates Decays and Collisions n When it comes to elementary particles, there are only two things that ever really happen: One particle decays into stuff Two particles
More informationSummary of free theory: one particle state: vacuum state is annihilated by all a s: then, one particle state has normalization:
The LSZ reduction formula based on S-5 In order to describe scattering experiments we need to construct appropriate initial and final states and calculate scattering amplitude. Summary of free theory:
More informationTHE FEYNMAN RULES. < f S i >. \ f >
THE FEYNMAN RULES I. Introduction: Very generally, interactions can be viewed as the result of a scattering operator (S), a unitary operator acting on an initial state (\i >) leading to a multitude of
More informationOUTLINE. CHARGED LEPTONIC WEAK INTERACTION - Decay of the Muon - Decay of the Neutron - Decay of the Pion
Weak Interactions OUTLINE CHARGED LEPTONIC WEAK INTERACTION - Decay of the Muon - Decay of the Neutron - Decay of the Pion CHARGED WEAK INTERACTIONS OF QUARKS - Cabibbo-GIM Mechanism - Cabibbo-Kobayashi-Maskawa
More informationSolution Set of Homework # 6 Monday, December 12, Textbook: Claude Cohen Tannoudji, Bernard Diu and Franck Laloë, Second Volume
Department of Physics Quantum II, 570 Temple University Instructor: Z.-E. Meziani Solution Set of Homework # 6 Monday, December, 06 Textbook: Claude Cohen Tannoudji, Bernard Diu and Franck Laloë, Second
More informationWeek 3: Renormalizable lagrangians and the Standard model lagrangian 1 Reading material from the books
Week 3: Renormalizable lagrangians and the Standard model lagrangian 1 Reading material from the books Burgess-Moore, Chapter Weiberg, Chapter 5 Donoghue, Golowich, Holstein Chapter 1, 1 Free field Lagrangians
More informationLecture 8. September 21, General plan for construction of Standard Model theory. Choice of gauge symmetries for the Standard Model
Lecture 8 September 21, 2017 Today General plan for construction of Standard Model theory Properties of SU(n) transformations (review) Choice of gauge symmetries for the Standard Model Use of Lagrangian
More information1.7 Plane-wave Solutions of the Dirac Equation
0 Version of February 7, 005 CHAPTER. DIRAC EQUATION It is evident that W µ is translationally invariant, [P µ, W ν ] 0. W is a Lorentz scalar, [J µν, W ], as you will explicitly show in homework. Here
More informationConstruction of spinors in various dimensions
Construction of spinors in various dimensions Rhys Davies November 23 2011 These notes grew out of a desire to have a nice Majorana representation of the gamma matrices in eight Euclidean dimensions I
More information