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Instituto Superior T´ecnico Lectures in Quantum Field Theory Lecture 1 Jorge C. Rom˜ao Instituto Superior T´ecnico, Departamento de F´ısica & CFTP A. Rovisco Pais 1, 1049-001 Lisboa, Portugal January 2014 Jorge C. Rom˜ao IDPASC School Braga – 1 IST Summary of Lectures Summary ❐ Part 1 – Relativistic Quantum Mechanics (Lecture 1) Introduction Klein-Gordon Eq. ◆ The basic principles of Quantum Mechanics and Special Relativity Dirac Equation Covariance Dirac Eq. ◆ Klein-Gordon and Dirac equations Free Particle Solutions ◆ Gamma matrices and spinors Minimal Coupling NR Limit Dirac Eq. ◆ Covariance E < 0 Solutions ◆ Solutions for the free particle Charge Conjugation Massless Spin 1/2 ◆ Minimal coupling ◆ Non relativistic limit and the Pauli equation ◆ Charge conjugation and antiparticles ◆ Massless spin 1/2 particles Jorge C. Rom˜ao IDPASC School Braga – 2 IST Summary ... Summary ❐ Part 2 – Quantum Field Theory (Two Lectures) Introduction Klein-Gordon Eq. ◆ QED as an example (Lecture 2) Dirac Equation Covariance Dirac Eq. ■ QED as a gauge theory Free Particle Solutions ■ Propagators and Green functions Minimal Coupling ■ NR Limit Dirac Eq. Feynman rules for QED E < 0 Solutions ■ Example 1: Compton scattering Charge Conjugation − + − + ■ Example 2: e + e µ + µ in QED Massless Spin 1/2 → ◆ Non Abelian Gauge Theories (NAGT) (Lecture 3) ■ Radiative corrections and renormalization ■ Non Abelian gauge theories: Classical theory ■ Non Abelian gauge theories: Quantization ■ Feynman rules for a NAGT ■ Example: Vacuum polarization in QCD Jorge C. Rom˜ao IDPASC School Braga – 3 IST Summary ... Summary ❐ Part 3 – The Standard Model (Lecture 4) Introduction Klein-Gordon Eq. ◆ Gauge group and particle content of the massless SM Dirac Equation Covariance Dirac Eq. ◆ Spontaneous Symmetry Breaking and the Higgs mechanism Free Particle Solutions ◆ The SM with masses after Spontaneous Symmetry Breaking Minimal Coupling NR Limit Dirac Eq. ◆ Interactions dictated by the gauge symmetry group E < 0 Solutions ◆ Example 1: The decay Z ff in the SM Charge Conjugation → + − + − Massless Spin 1/2 ◆ Example 2: e e µ µ in the SM → ◆ Example 3: Muon decay ◆ How the gauge symmetry corrects for the bad high energy behaviour in + − νeνe W W → L L ◆ The need for the Higgs to correct the bad high energy behaviour in W +W − W +W − L L → L L Jorge C. Rom˜ao IDPASC School Braga – 4 IST Basic Ideas of Quantum Mechanics and Special Relativity Summary ❐ In this lecture we will put together the ideas of Quantum Mechanics and Introduction • QM Special Relativity • Special Relativity Klein-Gordon Eq. ❐ This will lead to the substitution of Schr¨odinger equation by the Dirac Equation Klein-Gordon (spin 0) and Dirac equations (spin 1/2) Covariance Dirac Eq. Free Particle Solutions ❐ The idea of describing Quantum Physics by an equation will be abandoned Minimal Coupling in favour of a description in terms of a variable number of particles allowing NR Limit Dirac Eq. for the creation and annihilation of particles E < 0 Solutions Charge Conjugation ❐ However, the formalism that we will develop in this first lecture will be very Massless Spin 1/2 useful for the Quantum Field Theory that we will address in the following lectures Jorge C. Rom˜ao IDPASC School Braga – 5 IST Principles of Quantum Mechanics We list the basic principles of Quantum Mechanics: Summary ❐ For a given physical state there is a state function Φ that contains all the Introduction | i • QM possible information about the system • Special Relativity Klein-Gordon Eq. ◆ In many cases we will deal with the representation of state Φ in terms | i Dirac Equation of the coordinates, the so-called wave function Ψ(qi,si,t). Covariance Dirac Eq. 2 ◆ Ψ(qi,si,t) 0 has the interpretation of a density of probability of Free Particle Solutions | | ≥ Minimal Coupling finding the system in a state with coordinates qi, internal quantum NR Limit Dirac Eq. numbers si at time t E < 0 Solutions Charge Conjugation ❐ Physical observables are represented by hermitian linear operators, as Massless Spin 1/2 ∂ ∂ pi i¯h , E i¯h → − ∂qi → ∂t ❐ A state Φn is an eigenstate of the operator Ω if | i Ω Φn = ωn Φn | i | i where Φn is the eigenstate that corresponds to the eigenvalue ωn. If Ω is | i hermitian then ωn are real numbers. Jorge C. Rom˜ao IDPASC School Braga – 6 IST Principles of Quantum Mechanics ... ❐ For a complete set of commuting operators Ω1, Ω2,... , there exists a Summary { } complete set of orthonormal eigenfunctions, Ψn. An arbitrary state (wave Introduction • QM function) can expanded in this set • Special Relativity Klein-Gordon Eq. Ψ = anΨn Dirac Equation n Covariance Dirac Eq. X ❐ The result of the measurement of the observable Ω is any of the eigenvalues Free Particle Solutions 2 ωn with probability an . The average value of the observable is Minimal Coupling | | NR Limit Dirac Eq. E < 0 Solutions ∗ 2 < Ω >Ψ= dq1...Ψ (qi,si,t)ΩΨ(qi,si,t) = an ωn Charge Conjugation | | s Z n Massless Spin 1/2 X X ❐ The time evolution of a system is given by ∂Ψ i¯h = HΨ (The Hamiltonian H is a linear and hermitian operator) ∂t ❐ Linearity implies the superposition principle and hermiticity leads to the conservation of probability d i dq Ψ∗Ψ = dq [(HΨ)∗Ψ Ψ∗(HΨ)] = 0 dt 1 ¯h 1 s ··· s ··· − X Z X Z Jorge C. Rom˜ao IDPASC School Braga – 7 IST Principles of Special Relativity Special relativity is based on the principles of relativity and of the constancy of the Summary speed of light in all reference frames. For our purposes it is enough to recall: Introduction • QM ❐ The coordinates of two reference frames are related by • Special Relativity Klein-Gordon Eq. ′µ µ ν Dirac Equation x = a ν x , µ,ν = 0, 1, 2, 3 Covariance Dirac Eq. Free Particle Solutions Minimal Coupling ❐ The invariance of the interval NR Limit Dirac Eq. 0 2 µ ν µ E < Solutions ds = gµνdx dx = dx dxµ Charge Conjugation Massless Spin 1/2 where the metric is diagonal and given by gµν = diag(+ ), restricts the µ − −− coefficients a ν to obey µ ν α β α β gµνa αa βdx dx = gαβdx dx which implies µ ν T a αgµν a β = gαβ or in matrix form a ga = g Jorge C. Rom˜ao IDPASC School Braga – 8 IST Principles of Special Relativity ... ❐ The matrices that obey aT ga = g constitute the Lorentz group, designated Summary Introduction by O(3, 1) • QM • Special Relativity ❐ We can easily verify that Klein-Gordon Eq. Dirac Equation det a = 1 Covariance Dirac Eq. ± Free Particle Solutions ❐ Transformations with det a = +1 constitute the proper Lorentz group a Minimal Coupling subgroup of the Lorentz group. They can be built from infinitesimal NR Limit Dirac Eq. E < 0 Solutions transformations. Examples are rotations and Lorentz transformations Charge Conjugation Massless Spin 1/2 1 0 00 γ γβ 0 0 0 cos θ sin θ 0 γβ− γ 0 0 a = , a = 0 sin θ cos θ 0 −0 0 10 − 0 0 01 0 0 01 where 1 V γ = ; β = 1 β2 c − and V is thep relative velocity of reference frame S′ with respect to S Jorge C. Rom˜ao IDPASC School Braga – 9 IST Principles of Special Relativity ... Summary ❐ Examples of transformations with det a = 1 are the space (Parity) and Introduction − • QM time (Time Reversal) inversions. For instance • Special Relativity Klein-Gordon Eq. Dirac Equation 10 0 0 1 0 0 0 0 1 0 0 −0 100 Covariance Dirac Eq. a = − a = Free Particle Solutions 0 0 1 0 0 010 − Minimal Coupling 0 0 0 1 0 001 − NR Limit Dirac Eq. 0 E < Solutions Parity Time Reversal Charge Conjugation Massless Spin 1/2 ❐ The transformations with det a = 1 do not form a subgroup of the full Lorentz group (they do not contain− the identity) Jorge C. Rom˜ao IDPASC School Braga – 10 IST The Klein-Gordon Equation We start with the free particle: Summary Introduction ❐ In non-relativistic quantum mechanics we get Schr¨odinger equation from the Klein-Gordon Eq. fundamental equation Dirac Equation Covariance Dirac Eq. ∂ i¯h ψ = Hψ Free Particle Solutions ∂t Minimal Coupling NR Limit Dirac Eq. using the non-relativistic free particle Hamiltonian E < 0 Solutions Charge Conjugation p2 H = Massless Spin 1/2 2m and making the substitution ~p i¯h~ . We get → − ∇ ∂ψ ¯h2 i¯h = 2ψ ∂t −2m∇ ❐ The idea is to use the relativistic form of H = E, the energy of the free particle. Jorge C. Rom˜ao IDPASC School Braga – 11 IST The Klein-Gordon Equation ... ❐ In special relativity the energy and momentum are related through Summary Introduction µ 2 2 µ E Klein-Gordon Eq. pµp = m c , p , ~p ≡ c Dirac Equation Covariance Dirac Eq. giving Free Particle Solutions Minimal Coupling E2 = p2c2 + m2c4 E = p2c2 + m2c4 NR Limit Dirac Eq. → ± 0 E < Solutions p Charge Conjugation ❐ Classically we require energies to be positive, so we could choose Massless Spin 1/2 H = p2c2 + m2c4 p ❐ We are lead to interpret the square root of an operator. To avoid this problem we can find an equation for H2. Iterating the original equation and ∂ noticing that i¯h ∂t ,H = 0, we get 2 2 2 ∂ 2 2 ~2 2 4 mc µ ¯h ψ =( ¯h c + m c )ψ + ψ = 0, = ∂µ∂ − ∂t2 − ∇ → ⊔⊓ ¯h ⊔⊓ Jorge C. Rom˜ao IDPASC School Braga – 12 IST The Klein-Gordon Equation ... ❐ Now we have no problem with the operators but we reintroduce the negative Summary energy solutions. We will see that the negative energy solutions can not be Introduction Klein-Gordon Eq. avoided and its interpretation will be related to the antiparticles.
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