Quantum Field Theory

which allows us to express all dimensionful quantities in terms of a single scale whichwe choose to be mass or, equivalently, energy (since E = mc 2 has become E = m).The usual choice of energy unit is eV , the electron volt or, more often GeV = 10 9 eV orTeV = 10 12 eV . To convert the unit of energy back to a unit of length or time, we needto insert the relevant powers of c and . For example, the length scale λ associated toa mass m is the Compton wavelengthλ = mcWith this conversion factor, the electron mass m e = 10 6 eV translates to a length scaleλ e = 2 × 10 −12 m.Throughout this course we will refer to the dimension of a quantity, meaning themass dimension. If X has dimensions of (mass) d we will write [X] = d. In particular,the surviving natural quantity G has dimensions [G] = −2 and defines a mass scale,G = cM 2 p= 1M 2 p(0.2)where M p ≈ 10 19 GeV is the Planck scale. It corresponds to a length l p ≈ 10 −33 cm. ThePlanck scale is thought to be the smallest length scale that makes sense: beyond thisquantum gravity effects become important and it’s no longer clear that the conceptof spacetime makes sense. The largest length scale we can talk of is the size of thecosmological horizon, roughly 10 60 l p .ObservableUniverse~ 20 billion light yearsEarth 10 10 cmCosmologicalConstantAtoms−810 cmNuclei−1310 cmLHCPlanck Scale−3310cmlengthEnergy−3310eV−3 10eV10 1112−10eV= 1 TeV10 2819eV= 10 GeVFigure 2: Energy and Distance Scales in the UniverseSome useful scales in the universe are shown in the figure. This is a logarithmic plot,with energy increasing to the right and, correspondingly, length increasing to the left.The smallest and largest scales known are shown on the figure, together with otherrelevant energy scales. The standard model of particle physics is expected to hold up– 5 –