Eigenstates of ##\phi^4## theory

[HomogenousCow]

What is known about the eigenstates of the $\phi^4$ theory in QFT? Is there an informal understanding of how these states are organized in the nonperturbative regime? For example, are there known to be any bound states in any dimensions? How does the energy of a multiparticle eigenstate (if anything like this exists) depend on its constituent particle momenta? I realize that there are probably no rigorous answers to these questions, however I would like some kind of intuition as this issue has been bothering me for years, thank you.

 

[hilbert2]

I tried to find if anything has been written even about the inner product of the non-interacting Klein-Gordon vacuum state with the vacuum of the system with $\phi^4$ interaction turned on, but couldn't find anything. I.e., "how many % of the interacting vacuum is the same as the non-interacting one". Not sure if the two ground states are even defined in the same $\mathcal{H}$.

 

[HomogenousCow]

I tried to find if anything has been written even about the inner product of the non-interacting Klein-Gordon vacuum state with the vacuum of the system with $\phi^4$ interaction turned on, but couldn't find anything. I.e., "how many % of the interacting vacuum is the same as the non-interacting one". Not sure if the two ground states are even defined in the same $\mathcal{H}$.

Yes I've had the same experience looking for information on non-perturbative $\phi^4$ theory, the mathematicians claim that they've formulated the theory in $d<4$ dimensions but I've never seen a single result that it looks like it has anything to do with physics.