Quantum Field Theory I - ETH Z

1 Quantum Field Theory IBabis AnastasiouInstitute for Theoretical Physics,ETH Zurich,8093 Zurich, 16, 2020 Contents1 Quantum Field Theory . Why?72 Review of principles of classical and Quantum Time evolution in classical mechanics .. of Poisson brackets .. way to think of classical time evolution .. Time evolution in Quantum mechanics .. Conservation and symmetries in classical mechanics .. classical example .. Symmetries in Quantum mechanics ..153 Theory of Classical Fields from a discretised space (lattice) .. Euler-Lagrange equations for a classical Field from a Lagrangian density . Noether s theorem .. Field symmetry transformations .. symmetry transformations .. tensor .. symmetry transformations and conserved currents .. Field Hamiltonian Density from discretization .. equations for fields .. An example: acoustic waves ..334 Quantisation of the Schr odinger The Schr odinger equation from a Lagrangian density.

2 Symmetries of the Schroedinger Field .. Quantisation of Fields .. Quantised Schr odinger Field .. Particle states from quantised fields .. What is the wave-function in the Field quantisation formalism? ..425 The Klein-Gordon Real Klein-Gordon Field .. solution of the Klein-Gordon equation .. of the real Klein-Gordon Field .. states for the real Klein-Gordon Field .. of particles and normal ordering .. momentum conservation .. of particle states? .. Casimir effect: the energy of the vacuum .. Two real Klein-Gordon fields .. equal-mass real Klein-Gordon fields .. real Klein-Gordon fields = One complex Klein-Gordon Field . Conserved Charges as generators of symmetry transformations .. Can the Klein-Gordon Field be an one-particle wave-function? ..606 Quantisation of the free electromagnetic Maxwell Equations and Lagrangian formulation.

3 Gauge invariance and gauge-fixing .. of the electromagnetic Field .. Quantisation of the Electromagnetic Field .. Massive photons: The Higgs mechanism ..687 The Dirac Mathematical interlude .. matrices and their properties .. product of 2 2 matrices .. Dirac representation of -matrices .. Traces of matrices .. matrices as a basis of 4 4 matrices .. Lagrangian for the Dirac Field ..748 Lorentz symmetry and free Field transformations and representations of the Lorentz group .. representationM( ) = 1 .. representationM( ) = .. Generators of Field representations of Lorentz symmetry of the scalar representation .. of the vector representation .. algebra of continuous groups .. Spinor representation .. Lorentz Invariance of the Dirac Lagrangian .. General representations of the Lorentz group .. Weyl spinors .. Majorana equation .. Lagrangian and Majorana equation in a four-dimensionalspinor notation*.

4 919 Classical solutions of the Dirac Solution in the rest frame .. Lorentz boost of rest frame Dirac spinor along the z-axis .. Solution for an arbitrary vector .. A general solution ..9610 Quantization of the Dirac One-particle states .. Particles and anti-particles .. Particles and anti-particles of spin-12.. Fermions .. Quantum symmetries .. Lorentz transformation of the quantized spinor Field .. Transformation of the quantized Dirac Field .. Parity .. Other discrete symmetries .. 10911 Propagation of free Transition amplitude for the Schr odinger Field .. Transition amplitude for the real Klein-Gordon Field .. Time Ordering and the Feynman-St uckelberg propagator for the real Klein-Gordon Field .. Feynman propagator for the complex Klein-Grodon Field .. Feynman propagator for the Dirac Field .. Feynman propagator for the photon Field .

5 Wick s theorem: time-ordering, normal-ordering and propagation .. Wick s theorem for Dirac fermion fields .. Wick s theorem for Majorana fermions* .. 12312 Scattering Theory (S-matrix) Propagation in a general Field Theory .. A special case: free scalar Field Theory .. Typical interacting scalar Field Theory .. Spectral assumptions in scattering Theory .. In and Out states .. Scattering Matrix-Elements .. S-matrix and Green s functions .. The LSZ reduction formula .. Truncated Green s functions .. Cross-sections .. 13713 Perturbation Theory and Feynman Time evolution operator in the interaction picture .. Field operators in the interacting and free Theory .. The ground state of the interacting and the free Theory .. Feynman Diagrams for 4theory .. Feynman rules in momentum space .. Truncated Green s functions in perturbation Theory .

6 14914 Loop The simplest loop integral. Wick rotation .. Dimensional Regularization .. Angular Integrations .. Properties of the Gamma function .. Radial Integrations .. Feynman Parameters .. 157315 Quantum Gauge invariance .. Perturbative QED .. Dimensional regularization for QED .. Gamma-matrices in dimensional regularization .. Tensor loop-integrals .. The electron propagator at one-loop .. Electron propagator at all orders .. The electron mass .. The photon propagator at one-loop .. Ward identity .. Photon propagator at all orders .. 17716 Renormalisation of Running of the QED coupling constant and the electron mass .. 181A Special Proper time .. Subgroups of Lorentz transformations .. Time dilation .. Doppler effect .. Particle dynamics .. Energy and momentum.

7 The inverse of a Lorentz transformation .. Vectors and Tensors .. Currents and densities .. Energy-Momentum tensor .. Relativistic formulation of Electrodynamics .. Energy-Momentum Tensor in the presence of an electromagnetic field1984 Bibliography[1] The Quantum Theory of Fields, Volume I Foundations, Steven Weinberg, CambridgeUniversity Press.[2] An introduction to Quantum Field Theory , M. Peskin and D. Schroeder, Addison-Wesley[3] Quantum Field Theory in a nutshell, A. Zee, Princeton University Press.[4] Quantum Field Theory , Mark Srednicki, Cambridge University Press.[5] An introduction to Quantum Field Theory , George Sterman, Cambridge UniversityPress.[6] Classical mechanics , Goldstein, Poole and Safko, Addison-Wesley[7] Lectures On Qed And Qcd: Practical Calculation And Renormalisation Of One-And Multi-loop Feynman Diagrams, Andrea Grozin, World Scientific5 Conventions for Special RelativityOur metric convention isg = diag (1, 1, 1, 1).

8 (1)A contravariant position four-vector isx = (x0,x1,x2,x3) (ct,x,y,z) = (ct,~x).(2)A covariant position four-vector isx =g x ,(3)which givesx = (x0,x1,x2,x3) (ct, x, y, z) = (ct, ~x).(4)Space-time derivatives form four-vectors, x =( x0, x1, x2, x3)=(1c t, x, y, z)=(1c t,~ )(5) x =( x0, x1, x2, x3)=(1c t, x, y, z)=(1c t, ~ )(6)The D Alambert scalar second order differential operator is 2 =1c2 2 t2 ~ 2.(7)6 Chapter 1 Quantum Field Theory . Why?The goal of this lecture series is to introduce a synthesis of Quantum mechanics and specialrelativity into a unified Theory , the Theory of quantised fields. Rendering the two theoriesconsistent with each other was a challenge for the physicists of the last century. Naivegeneralisations of the Schr odinger equation to incorporate relativity were giving non-physical results, such as particles with negative kinetic energies. Quantum Field theoryprovided the solution to this and other Field Theory allows us to tackle deep questions.

9 What is a particle? Whyparticles have a spin? Why particles carry electric charge? What types of charge mayexist beyond the electric charge? Why do particles have mass?Novel phenomena emerge at high energies in collider experiments. As an example,which has been exhaustively studied at the Large-Electron-Positron (LEP) collider inGeneva, think of the production of a muon and its anti-particle out of the annihilationof an electron and a positron:e +e+ + +.( )Such a common reaction cannot be explained with Quantum mechanics as we have knownit so far. While we could assign a wave-function for the electron/positron system beforethe reaction takes place and similarly a different wave-function for the muon/anti-muonsystem, the Schrodinger equation does not predict that the latter is the evolution of fields, on the other hand, allow for the annihilation of particles and thecreation of others, as long as this is consistent with symmetries and the correspondingconservation laws.

10 Quantum Field Theory is a predictive framework. Together with sym-metries, it tells us precisely how particles may interact at the shortest distances andhigher energies that we have explored so far in 2 Review of principles of classical andquantum mechanicsBefore we introduce Quantum Field Theory , it will be useful to recall how we describedthe dynamics of simple mechanical systems in classical and Quantum physics. In QFT,we will postulate principles that we have already seen there, such as the principle ofleast action and canonical quantization. Conservation theorems derived in classical andquantum mechanics will also apply to Time evolution in classical mechanicsConsider, for simplicity, a one-dimensional mechanical system whose dynamical behavioris encoded in a LagrangianL(x(t), x(t)).The Lagrangian depends on the position and the velocity, which are functions of will focus on energy conserving systems, in which the Lagrangian acquires all of itstime dependence through these functions and has no other explicit time dependence, L t= 0.