Is QFT linear with respect to superposition of multi-particle states?

Question

I saw other posts such as this one but I don't think it's quite the same question, or even if it is, the answer employs the operator formalism and I'm not sure I follow it. I'm wondering, if you have two multiparticle states - a multiparticle state being, in my mind, a complex probability amplitude for each possible configuration of particles, as a function of time - then, is a normalized linear combination of these two states still a valid multiparticle state? Keep in mind they are both functions of time that obey the equations of QFT, so the linear combination is also a function of time, and I'm asking if it still obeys the equations of QFT.

I'm trying to think about it in terms of Feynman diagrams. In particular, I'm pretty sure you can linearly combine two multiparticle states at one time with no problem - you just get a superposition rather than a pure state. And since the amplitude for a final configuration is essentially the sum of all propagators from the initial state, well, this sounds linear enough to me. I think you would just sum up the independent contributions from all the pure states of which the initial state was composed. What else could it be?

What throws me for a loop is that I've seen several posts here and elsewhere talking about the inherent non-linearity of QFT. But I think they might be talking about linearity with respect to combination of single particle states. I'm not worried about that, however, since I can see that multiple identical particles really form a single mathematical entity (a product rather than a sum, e.g. the Slater determinant for fermions), so linearity wouldn't have much meaning in this context anyway. Still, the whole thing appears rather murky so I'd really like to clear up this point.

To put it another way, I know that there is interaction within the evolution of a pure multiparticle state, and this leads to entanglement, which, mathematically, is just the inability of the final state to be factored into a single tensor product of one-particle states. But, is it fair to say there is no interaction between the pure multiparticle components of a mixed multiparticle state? (at least, as long as we don't classify summation of complex amplitudes as "interaction")

Answer 1

There is no distinction between quantum field theory and quantum mechanics in terms of how to answer this question. You can think of QFT as a specific application of quantum mechanics here.

The Schrodinger equation of QM is always linear. If you assign time evolution to the states, then the time evolution of the states is given by a linear operator. A linear superposition of physical states is also a physical state (modulo superselection rules).

The Heisenberg equations of motions of QM are only linear for a harmonic oscillator, collection of harmonic oscillators, or free field(s). For any interesting system, the Heisenberg equations of motion are nonlinear. If you assign time evolution to the operators, then it's only generally possible to exactly solve for operators as a function of time in a free (linear) theory.

It is very important to distinguish these two uses of the concept of linearity in QM.