In d+1 dimensional quantum field theory, the 2-point Green's function will diverge at the same spacetime point when d≥1.
When d=0, ϕ(t)=q(t), that is the case of QM, and 2-point Green's function at the same spacetime point ⟨Ω|T(q(t)q(t))|Ω⟩ is well-defined.
While d≥1, the 2-point Green's function at the same spacetime point ⟨Ω|T(ϕ(x)ϕ(x))|Ω⟩ will diverge.
So what's the physical or mathematical essence of this diverge. I especially want to know the physical picture in the path integral. Why the randomly walking of a particle will be different from that of a string?
Answer
The simple idea is the following. At short distances, you're probing the high-energy regime (UV) of the theory - you're sensitive to very short wavelengths. Take for instance the free boson in the Euclidean: ⟨ϕ(x)ϕ(0)⟩=∫ddp(2π)deip⋅xp2+m2.
You can make a similar argument more rigorously also for an interacting theory, using the Källen-Lehmann spectral representation (see e.g. Peskin-Schroeder).
No comments:
Post a Comment