Quantum Field Theory

means that for experiments at small energies E ≪ Λ, the interaction terms of theform φ n with n > 4 will be suppressed by powers of (E/Λ) n−4 . This is usually asuppression by many orders of magnitude. (e.g for the energies E explored at theLHC, E/M pl ∼ 10 −16 ). It is this simple argument, based on dimensional analysis, thatensures that we need only focus on the first few terms in the interaction: those whichare relevant and marginal. It also means that if we only have access to low-energyexperiments (which we do!), it’s going to be very difficult to figure out the high energytheory (which it is!), because its effects are highly diluted except for the relevant andmarginal interactions. The discussion given above is a poor man’s version of the ideasof effective field theory and Wilson’s renormalization group, about which you can learnmore in the “Statistical Field Theory” course.Examples of Weakly Coupled TheoriesIn this course we’ll study only weakly coupled field theories i.e. ones that can truly beconsidered as small perturbations of the free field theory at all energies. In this section,we’ll look at two types of interactions1) φ 4 theory:L = 1 2 ∂ µφ∂ µ φ − 1 2 m2 φ 2 − λ 4! φ4 (3.5)with λ ≪ 1. We can get a hint for what the effects of this extra term will be. Expandingout φ 4 in terms of a ⃗pand a † ⃗p, we see a sum of interactions that look likea † ⃗p a† ⃗p a† ⃗p a† ⃗pand a † ⃗p a† ⃗p a† ⃗p a ⃗p etc. (3.6)These will create and destroy particles. This suggests that the φ 4 Lagrangian describesa theory in which particle number is not conserved. Indeed, we could check that thenumber operator N now satisfies [H, N] ≠ 0.2) Scalar Yukawa TheoryL = ∂ µ ψ ⋆ ∂ µ ψ + 1 2 ∂ µφ∂ µ φ − M 2 ψ ⋆ ψ − 1 2 m2 φ 2 − gψ ⋆ ψφ (3.7)with g ≪ M, m. This theory couples a complex scalar ψ to a real scalar φ. Whilethe individual particle numbers of ψ and φ are no longer conserved, we do still havea symmetry rotating the phase of ψ, ensuring the existence of the charge Q definedin (2.75) such that [Q, H] = 0. This means that the number of ψ particles minus thenumber of ψ anti-particles is conserved. It is common practice to denote the antiparticleas ¯ψ.– 49 –