lecture-3-ClassicalFieldTheory - 2012 Matthew Schwartz I-3...

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2012 Matthew Schwartz I-3: Classical Field Theory 1 Introduction We have now seen how quantum field theory (QFT) is just quantum mechanics with an infinite number 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 energy levels of the Hydrogen atom (see next lecture). To show that QFT is not absurd, but extremely predictive, we will have to be very careful about how we do calculations. We’ll begin by going through carefully some of the predictions which QFT gets right without infinities. These are called the tree-level processes, which means they are leading order in perturbation theory. We will start to study them in this lecture. Before we get started, it’s worth pointing out that the vast majority of calculations people do day-to-day using QFT are really just calculations in classical field theory. Mostly, people just solve 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 up to 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’s going on physically, it took 20 years, from roughly 1930 to 1950 for people to know that QFT calculations even made sense at all. Hopefully, it will take you less than that to make sense of them. 2 Classical Field Theory A classical field theory is just a mechanical system with a continuous set of degrees of freedom. Think about something like the density of a fluid ρ ( x ) as a function of position, or the electric field E ( x ) . Field theories can be defined in terms of either a Hamiltonian or a Lagrangian which we often write as integrals over all space of Hamiltonian or Lagrangian densities H = integraldisplay d 3 x H , L = integraldisplay d 3 x L (1) We will use a calligraphic script for densities and a roman script for integrated quantities. The word “density” is almost always omitted, since whether a Lagrangian or Lagrangian density is meant will always be clear from context. Formally, the Hamiltonian (density) is a functional of fields and their conjugate momenta H [ φ, π ] . The Lagrangian (density) is the Legendre transform of the Hamiltonian (density). Formally it is defined as L bracketleftbig φ,φ ˙ bracketrightbig = π bracketleftbig φ,φ ˙ bracketrightbig φ ˙ − H bracketleftBig φ,π bracketleftbig φ,φ ˙ bracketrightbig bracketrightBig (2) where φ ˙ = t φ and π bracketleftbig φ,φ ˙ bracketrightbig is implicitly defined by H [ φ, π ] ∂π = φ ˙ . The inverse transform is H [ φ,π ] = πφ ˙ [ φ,π ] − L bracketleftBig φ,φ ˙ [ φ,π ] bracketrightBig (3) where φ ˙ [ φ,π ] is implicitly defined by L bracketleftbig φ, φ ˙ bracketrightbig ∂φ ˙ = π .

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