Quantum Field Theory

Hmmmm. We’ve found a delta-function, evaluated at zero where it has its infinitespike. Moreover, the integral over ω ⃗p diverges at large p. What to do? Let’s start bylooking at the ground state where this infinity first becomes apparent.2.3 The VacuumFollowing our procedure for the harmonic oscillator, let’s define the vacuum |0〉 byinsisting that it is annihilated by all a ⃗p,a ⃗p |0〉 = 0 ∀ ⃗p (2.23)With this definition, the energy E 0 of the ground state comes from the second term in(2.22),[∫H |0〉 ≡ E 0 |0〉 = d 3 p 1 ]2 ω ⃗p δ (3) (0) | 0〉 = ∞ |0〉 (2.24)The subject of quantum field theory is rife with infinities. Each tells us somethingimportant, usually that we’re doing something wrong, or asking the wrong question.Let’s take some time to explore where this infinity comes from and how we should dealwith it.In fact there are two different ∞’s lurking in the expression (2.24). The first arisesbecause space is infinitely large. (Infinities of this type are often referred to as infra-reddivergences although in this case the ∞ is so simple that it barely deserves this name).To extract out this infinity, let’s consider putting the theory in a box with sides oflength L. We impose periodic boundary conditions on the field. Then, taking the limitwhere L → ∞, we get∫ L/2(2π) 3 δ (3) (0) = lim d 3 x e i⃗x·⃗p∣ ∫ L/2∣L→∞ ⃗p=0= lim d 3 x = V (2.25)−L/2L→∞−L/2where V is the volume of the box. So the δ(0) divergence arises because we’re computingthe total energy, rather than the energy density E 0 . To find E 0 we can simply divide bythe volume,E 0 = E ∫0V = d 3 p 1(2π) 3 2 ω ⃗p (2.26)which is still infinite. We recognize it as the sum of ground state energies for eachharmonic oscillator. But E 0 → ∞ due to the |⃗p| → ∞ limit of the integral. This isa high frequency — or short distance — infinity known as an ultra-violet divergence.This divergence arises because of our hubris. We’ve assumed that our theory is validto arbitrarily short distance scales, corresponding to arbitrarily high energies. This isclearly absurd. The integral should be cut-off at high momentum in order to reflectthe fact that our theory is likely to break down in some way.– 25 –