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## “Elementary Particles” Lecture 4

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**“Elementary Particles”Lecture 4**Niels Tuning Harry van der Graaf Niels Tuning (1)**Thanks**• Ik ben schatplichtig aan: • Dr. Ivo van Vulpen (UvA) • Prof. dr. ir. Bob van Eijk (UT) • Prof. dr. Marcel Merk (VU) Niels Tuning (2)**Exercises Lecture 3:**Niels Tuning (4)**Exercises Lecture 3: mass of Ω and decay of π0**π0→γ1γ2→e+e-e+e- 1232 MeV 1385 MeV 1533 MeV 1680 MeV Gell-Mann and Zweig predicted the Ω- Niels Tuning (5)**Exercises Lecture 3:**Niels Tuning (6)**Exercises Lecture 3:**(1) (3) (2) Niels Tuning (7)**Exercises Lecture 3:**Niels Tuning (8)**Exercises Lecture 3:**(a) (c) Niels Tuning (9)**Plan**11 Feb • Intro: Relativity and accelerators • Basis • Atom model, strong and weak force • Scattering theory • Hadrons • Isospin, strangeness • Quark model, GIM • Standard Model • QED • Parity, neutrinos, weak inteaction • QCD • e+e- and DIS • Higgs and CKM 1900-1940 18 Feb 1945-1965 4 Mar 1965-1975 18 Mar 1975-2000 22 Apr 2000-2013 13 May Niels Tuning (10)**Outlinefortoday: Interactions**• Gauge invariance, and the Lagrangian • Electro-magnetic interaction • QED • Weak interaction • Parity violation • Strong interaction • QCD Niels Tuning (11)**Lecture 1: Accelerators & Relativity**• Theory of relativity • Lorentz transformations (“boost”) • Calculate energy in collissions • 4-vector calculus • High energies needed to make (new) particles Niels Tuning (13)**Lecture 2: Quantum Mechanics & Scattering**• Schrödinger equation • Time-dependence of wave function • Klein-Gordon equation • Relativistic equation of motion of scalar particles • Dirac equation • Relativistically correct, and linear • Equation of motion for spin-1/2 particles • Prediction of anti-matter Niels Tuning (14)**Lecture 2: Quantum Mechanics & Scattering**• Scattering Theory • (Relative) probability for certain process to happen • Cross section • Fermi’s Golden Rule • Decay: “decay width” Γ • Scattering: “cross section” σ Scattering amplitude in Quantum Field Theory Classic a → b + c a + b → c + d Niels Tuning (15)**Lecture 3: Quarkmodel & Isospin**• “Partice Zoo” not elegant • Hadrons consist of quarks • Observed symmetries • Same mass of hadrons: isospin • Slow decay of K, Λ: strangeness • Fermi-Dirac statistics Δ++,. Ω: color • Combining/decaying particles with (iso)spin • Clebsch-Gordan coefficients Niels Tuning (16)**Model elementary particles**Tools Atom Strong force Mesons Leptons Quark model Baryons Quantum numbers Conservation Laws Standard Model Particles Forces Quantum- field theory & Local gauge invariance quarks/leptons ElectromagneticWeak Strong**Electro-magnetism**• Towards a particle interacting with photon • Quantum Electro Dynamics, QED • Start with electric and magnetic fields J.C. Maxwell**Start: Classical electro-magnetism**Maxwell equations • We wish to work relativistically • Can we formulate this in Lorentz covariant form? • Introduce a mathematical tool: Scalar potential also called φ:**Start: Classical electro-magnetism**Maxwell equations • We wish to work relativistically • Can we formulate this in Lorentz covariant form? • Introduce a mathematical tool: • Note: • Choose: Scalar potential also called φ:**Start: Classical electro-magnetism**Maxwell equations • We wish to work relativistically • Can we formulate this in Lorentz covariant form? • Introduce a mathematical tool: • Note: • Choose: Then automatically: Scalar potential also called φ:**Rewrite Maxwell**Maxwell equations • Maxwell eqs. can then be written quite economically…:**Rewrite Maxwell**Maxwell equations • Maxwell eqs. can then be written quite economically…: • Electromagnetic tensor • Unification of electromagnetism**Gauge invariance: Classical**Maxwell equations Physical Fields: Potentials: A = Vector potential ϕ= Scalar potential Invariant under: Gauge transformations Maxwell invariant gauge symmetry**Gauge invariance: Classical**Maxwell equations Physical Fields: Potentials: A = Vector potential ϕ= Scalar potential Invariant under: Gauge transformations Maxwell invariant gauge symmetry**Not unique! Can choose extra constraints:**Coulomb-gauge: Lorenz-gauge: Advantage: Lorentz-invariant**Not unique! Can choose extra constraints:**Coulomb-gauge: Lorenz-gauge: (Ludvig) (Hendrik) Advantage: Lorentz-invariant Lorentz Lorenz Total confusion: Lorentz-Lorenz formula**Gauge invariance : QM**Charged particle moving in electro-magnetic field: Classical A,φ Schrödinger eq. Pauli eq. (spin-1/2 in EM-field) Theory gauge invariant: (same physics with A, φ as with A’, φ’) ?**Rewrite and**Wave equation (A,φ) Theory gauge invariant: (same physics with A, φ as with A’, φ’) ?**Rewrite and**Wave equation (A,φ) Theory gauge invariant: (same physics with A, φ as with A’, φ’) ? Wave equation (A’,φ’) Yes! If**Local gauge symmetry**Schrödinger equation (time-independent): Ψ’ stays solution of Schrödinger eq! Global phase:**Local gauge symmetry**Schrödinger equation (time-independent): Ψ’ stays solution of Schrödinger eq! Global phase: Local phase: Ψ’ solution of Schrödinger eq?**Local gauge symmetry**Schrödinger equation (time-independent): Ψ’ stays solution of Schrodinger eq! Global phase: Local phase: Ψ’ solution of Schrodinger eq? No: • (How) can you keep the Schrödinger equation invariant ?**Before going to Quantum Field Theory,**lets remind ourselves of the Lagrange formalism**Euler-Lagrange**• Euler-Lagrange equations: • Least-action, or Hamilton’s principle: Niels Tuning (38)**Lagrangian and Equation of motion: Example**1) Lagrangiaan 2) Euler-Lagrange vergelijking Equation of motion q: generalized coordinates pendulum**Lagrangian in Field Theory**• Replace Lagrangian by a Lagrangian density in terms of field φ(x): • Least-action principle: • Euler-Lagrange equations: (classic) Niels Tuning (40)**Lagrangian Equation of motion**• spin-0 particles (Klein-Gordon) • spin-1/2 fermions (Dirac) • Photons Klein-Gordon equation Dirac equation Maxwell equations Niels Tuning (41)**Local gauge symmetry**Schrödinger equation (time-independent): Ψ’ stays solution of Schrödinger eq! Global phase: Local phase: Ψ’ solution of Schrödinger eq? No: • (How) can you keep the Schrödinger equation invariant ?**Phase Invariance**• Quantum Mechanics: • Expectation value: • Observation is invariant under transformation: • But what happens to the Lagrangian density, ? • Depends also on derivative..., so: Niels Tuning (45)**Phase Invariance**• Quantum Mechanics: • Expectation value: • Observation is invariant under transformation: • But what happens to the Lagrangian density, ? • Depends also on derivative..., so: • Replace by “gauge-covariant derivative”: Niels Tuning (46)**Phase Invariance**• Quantum Mechanics: • Expectation value: • Observation is invariant under transformation: • But what happens to the Lagrangian density, ? • Depends also on derivative..., so: • Replace by “gauge-covariant derivative”: • And with : Niels Tuning (47)**Gauge Invariance**We started globally: • Assumesymmetryψ→ψ’=ψeiα(x) • Keep Eqsvalid • Covariantderivative • Arbitrarygauge • Keep Eqsvalid • Needψ→ψ’=ψeiα Then we went local: Niels Tuning (48)**Gauge Invariance**We started globally: • Assumesymmetryψ→ψ’=ψeiα(x) • Keep Eqsvalid • Covariantderivative • Arbitrarygauge • Keep Eqsvalid • Needψ→ψ’=ψeiα Then we went local: Why would you want to do that? Niels Tuning (49)**Quantum Electro Dynamics - QED**• Let’s replace the derivative in the Dirac equation by the gauge-covariant derivative: • Gauge invariance leads to: gauge fields, and their interactions! Niels Tuning (50)