Question

One of the key elements of any quantum mechanical system is the spectrum of the Hamiltonian. But what about in quantum field theory? It seems as if nobody ever discusses the spectrum of a system at all -- or have I missed something? Just what role does the spectrum of any given potential play in field theory? To put it another way: does knowledge of a spectrum in QM help to understand the analogous problem in QFT?

Answer 1

Because of Lorentz Invariance, the spectrum of the Hamiltonian is always continuous. This does not mean bound state problems are not interesting, but the relevant discrete spectrum to discuss is the spectrum of (Lorentz invariant) masses. For an example think about bound state of electron and positron (positronium), in which they orbit each other. There is a discrete spectrum of masses of this composite object, to do with what is the angular momentum of their orbit around the center of mass. But of course, if you boost the system as a whole the energy of that moving object (which unlike the rest mass is not Lorentz invariance) will change and can take continuous values.

The other reason you hear less about the spectrum in QFT is because the problem of finding the spectrum in bound state problems is a lot harder (to define and to calculate). If you want an entry point to relevant discussions you can look for references for the Bethe-Salpeter equation, and to lattice QCD, two of the many approaches that discuss spectrum in QFT.