In QFT, specifically, QCD, as I understand it stable hadrons are energy eigenstates of the hamiltonian (and 3 momentum) operator(s). For example with the pion (in the context of pure QCD the pion is stable) would satisfy the relations $$\hat{H}| \pi( \boldsymbol{p}) \rangle = E(\boldsymbol{p})| \pi( \boldsymbol{p}) \rangle $$ for $$E(\boldsymbol{p}) = \sqrt{p^2 + m^2_{\pi}} .$$
Now we see unstable particles through propagator functions. For example, a spin 0 propagator in Fourier space might look like $$\Delta(p)^{-1} = p^2 -m^2 + \Sigma(p^2)$$If $\Sigma(p)$ has an imaginary component we identify this with the propagation of a particle of mass $$m_{phy}^2 = m^2 - \Re \Sigma(p^2)|_{p^2 = m_{phy}^2}$$ and decay time proportional to $$\Im \Sigma(p^2)|_{p^2 = m_{phy}^2}.$$
They have a 'mass' in the complex plane and through the connection of propagator functions to the $S$-matrix, we can infer the existence of unstable particles by their effect scattering cross-sections.
My question is is there a way to represent unstable particles as states in the hilbert space? If so what do they look like?
I have seen some literature claiming unstable particles and energy eigenstates with complex energy (ie complex valued eigenvalues). This can't be as its breaks unitarity and the hamiltonian is hermitian by construction.
