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.

  • 1
    $\begingroup$ What was the source that you saw that claims that? $\endgroup$
    – octonion
    Jul 5, 2022 at 17:39
  • $\begingroup$ Well, pseudounitarity is, indeed, violated in particle decay: the probability of existence of the particle is not conserved, as it gradually "leaks" out of its state and into those of the decay products, via time-dependent perturbation theory. The conceit is your Hilbert space is that of the particle, not including its decay products. Including everything, of course, involves unusable hermitian hamiltonians and operators... $\endgroup$ Jul 5, 2022 at 19:11
  • $\begingroup$ @octonion Maybe I ask about which claims in particular? $\endgroup$ Jul 5, 2022 at 21:57
  • $\begingroup$ @SheldonCooper, You said "I have seen some literature claiming unstable particles and energy eigenstates with complex energy" $\endgroup$
    – octonion
    Jul 5, 2022 at 22:49
  • $\begingroup$ If you pine for rigor, instead of intuition, Sudarshan et al, 1978 should get you there... $\endgroup$ Jul 6, 2022 at 20:50


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