Quantum Field Theory

The scalar Yukawa theory has a slightly worrying aspect: the potential has a stablelocal minimum at φ = ψ = 0, but is unbounded below for large enough −gφ. Thismeans we shouldn’t try to push this theory too far.A Comment on Strongly Coupled Field TheoriesIn this course we restrict attention to weakly coupled field theories where we can useperturbative techniques. The study of strongly coupled field theories is much moredifficult, and one of the major research areas in theoretical physics. For example, someof the amazing things that can happen include• Charge Fractionalization: Although electrons have electric charge 1, underthe right conditions the elementary excitations in a solid have fractional charge1/N (where N ∈ 2Z + 1). For example, this occurs in the fractional quantumHall effect.• Confinement: The elementary excitations of quantum chromodynamics (QCD)are quarks. But they never appear on their own, only in groups of three (in abaryon) or with an anti-quark (in a meson). They are confined.• Emergent Space: There are field theories in four dimensions which at strongcoupling become quantum gravity theories in ten dimensions! The strong couplingeffects cause the excitations to act as if they’re gravitons moving in higherdimensions. This is quite extraordinary and still poorly understood. It’s calledthe AdS/CFT correspondence.3.1 The Interaction PictureThere’s a useful viewpoint in quantum mechanics to describe situations where we havesmall perturbations to a well-understood Hamiltonian. Let’s return to the familiarground of quantum mechanics with a finite number of degrees of freedom for a moment.In the Schrödinger picture, the states evolve asi d|ψ〉 Sdtwhile the operators O S are independent of time.= H |ψ〉 S(3.8)In contrast, in the Heisenberg picture the states are fixed and the operators changein timeO H (t) = e iHt O S e −iHt|ψ〉 H= e iHt |ψ〉 S(3.9)– 50 –