2012Matthew SchwartzI-3: Classical Field Theory1IntroductionWe have now seen how quantum field theory (QFT) is just quantum mechanics with an infinitenumber of oscillators. We saw that QFT can do some remarkable things, such as explain sponta-neous emission. But it also seems to lead to absurdities, such as an infinite shift in the energylevels of the Hydrogen atom (see next lecture). To show that QFT is not absurd, but extremelypredictive, we will have to be very careful about how we do calculations. We’ll begin by goingthrough carefully some of the predictions which QFT gets right without infinities. These arecalled the tree-level processes, which means they are leading order in perturbation theory. Wewill start to study them in this lecture.Before we get started, it’s worth pointing out that the vast majority of calculations peopledo day-to-day using QFT are really just calculations in classical field theory. Mostly, people justsolve Lagrange’s equations of motion for really simple systems in a really complicated way.There is actually a pretty good reason for this: once the seemingly elaborate formalism is set upto do the tree-level (classical) calculations, it’s easier to do the loops (quantum corrections).Although we now have a pretty coherent story about how and why QFT works, and what’sgoing on physically, it took 20 years, from roughly 1930 to 1950 for people to know that QFTcalculations even made sense at all. Hopefully, it will take you less than that to make sense ofthem.2Classical Field TheoryA classical field theory is just a mechanical system with a continuous set of degrees of freedom.Think about something like the density of a fluidρ(x)as a function of position, or the electricfieldE(x). Field theories can be defined in terms of either a Hamiltonian or a Lagrangian whichwe often write as integrals over all space of Hamiltonian or Lagrangian densitiesH=integraldisplayd3xH, L=integraldisplayd3xL(1)We will use a calligraphic script for densities and a roman script for integrated quantities. Theword “density” is almost always omitted, since whether a Lagrangian or Lagrangian density ismeant will always be clear from context.Formally, theHamiltonian(density) is a functional of fields and their conjugate momentaH[φ, π]. TheLagrangian(density) is the Legendre transform of the Hamiltonian (density).Formally it is defined asLbracketleftbigφ,φ˙bracketrightbig=πbracketleftbigφ,φ˙bracketrightbigφ˙− HbracketleftBigφ,πbracketleftbigφ,φ˙bracketrightbigbracketrightBig(2)whereφ˙=∂tφandπbracketleftbigφ,φ˙bracketrightbigis implicitly defined by∂H[φ, π]∂π=φ˙. The inverse transform isH[φ,π] =πφ˙[φ,π]− LbracketleftBigφ,φ˙[φ,π]bracketrightBig(3)whereφ˙[φ,π]is implicitly defined by∂Lbracketleftbigφ, φ˙bracketrightbig∂φ˙=π.