# LecturesUdine2012-Problems - Problems in Quantum Field...

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Problems in Quantum Field TheoryJ. C. Rom˜aoDepartamento de F´ısica and CFTP, Instituto Superior T´ecnicoAv. Rovisco Pais 1,1049-001 Lisboa, PortugalAbstractWe collect here problems for the four Lectures I gave in Quantum FieldTheory at the 2nd IDPASC School held at Udine, January 23rd to February3rd, 2012.
1. Problems Quantum Mechanics: Lecture 111 Problems Quantum Mechanics: Lecture 11.1Consider the system of units used in high energy physics, that is, where wedefine ¯h= 1,c= 1. In this system all the physical quantities can be expressed inunits of the energy or powers of the energy.a) Express 1 s, 1 Kg and 1 m in MeV.b) Write you weight, height and age in MeV.1.2The lifetimeτof an unstable particle is defined as the time needed for theinitial number of particles to reduced to 1/eof its value, that is,N(t) =N0etτwhereN0is the number of particles att= 0. Knowing that the charged pions have,in their rest frame,τπ= 2.6×108s andmπ= 140 MeV evaluate:a) Theγfactor for a beam of 200 GeV pions.b) The lifetime in the laboratory frame.c) The percentage of pions that have decayed after travelling 300 m in the labo-ratory. If there was no time dilation, what would have been the percentage?1.3Consider the decayπµ+ν, wheremπ= 139.6 MeV,mμ= 105.7 MeVandmν= 0. Determine:a) The linear momenta of theµand of theνin the center of mass frame, thatis, where theπis at rest.b) The linear momenta of theµand of theνin the laboratory frame, assumingthat theνis emitted in the same direction of theπ.c) Repeat b) assuming now that it was theµthat was emitted in the directionof theπ.
1. Problems Quantum Mechanics: Lecture 121.4A photon can be described as a particle of zero mass and 4-momentakα= (ω,vectork)whereω= 2πν= 2π/λand|vectork|=ωh=c= 1). If a photon collides with anelectron with massmeat rest, it will be scattered with an angleθand with energyω(Compton scattering). Show thatλλ= 2λcsin2θ2ondeλc=2πm1.5Consider the electromagnetic field tensorFμν=μAννAμ. From this wecan define thedualtensorFμν=12ǫμνρσFρσ.a) Show that Maxwell equations with sources (Gauss’s and Amp`ere’s Laws) canbe writtenμFμν=Jνb) Show that we haveμFμν= 0Verify that this equation contains the so-called homogeneous Maxwell equa-tions,vector∇ ·vectorB= 0, andvector∇ ×vectorE=vectorB/∂t.Verify that the above relation isequivalent to the more usual form (Bianchi identity)μFνρ+νFρμ+ρFμν= 0c) Express the invariantsFμνFμν,FμνFμνandFμνFμνin terms of the fieldsvectorEandvectorB.d) Show that ifvectorEandvectorBare orthogonal in a reference frame they will remainorthogonal in all reference framese) Consider a reference frame wherevectorEnegationslash= 0 andvectorB= 0. Can we find a referenceframe wherevectorE= 0 evectorBnegationslash= 0? Justify your answer.1.6Use the relationsaμαgμνaνβ=gαβor in matrix formaTga=gto show that for infinitesimal transformationsaνμ=gνμ+ωνμ+· · ·we haveωμν=ωνμ