Quantum Field Theory

2.7 PropagatorsWe could ask a different question to probe the causal structure of the theory. Preparea particle at spacetime point y. What is the amplitude to find it at point x? We cancalculate this:∫ d 3 p d 3 p ′ 1〈0|φ(x)φ(y) |0〉 = √ 〈0|a(2π) 6 ⃗p a † ⃗p| 0〉 e −ip·x+ip′·y4E⃗p E ′ ⃗p ′∫=d 3 p(2π) 3 12E ⃗pe −ip·(x−y) ≡ D(x − y) (2.90)The function D(x −y) is called the propagator. For spacelike separations, (x −y) 2 < 0,one can show that D(x − y) decays likeD(x − y) ∼ e −m|⃗x−⃗y| (2.91)So it decays exponentially quickly outside the lightcone but, nonetheless, is non-vanishing!The quantum field appears to leak out of the lightcone. Yet we’ve just seen that spacelikemeasurements commute and the theory is causal. How do we reconcile these twofacts? We can rewrite the calculation (2.89) as[φ(x), φ(y)] = D(x − y) − D(y − x) = 0 if (x − y) 2 < 0 (2.92)There are words you can drape around this calculation. When (x − y) 2 < 0, thereis no Lorentz invariant way to order events. If a particle can travel in a spacelikedirection from x → y, it can just as easily travel from y → x. In any measurement, theamplitudes for these two events cancel.With a complex scalar field, it is more interesting. We can look at the equation[ψ(x), ψ † (y)] = 0 outside the lightcone. The interpretation now is that the amplitudefor the particle to propagate from x → y cancels the amplitude for the antiparticleto travel from y → x. In fact, this interpretation is also there for a real scalar fieldbecause the particle is its own antiparticle.2.7.1 The Feynman PropagatorAs we will see shortly, one of the most important quantities in interacting field theoryis the Feynman propagator,∆ F (x − y) = 〈0| Tφ(x)φ(y) |0〉 ={D(x − y) x 0 > y 0D(y − x) y 0 > x 0 (2.93)– 38 –