  We investigate how to include bound states in a thermal gas in the context of
quantum field theory (QFT). To this end, we use for definiteness a scalar QFT
with a $\varphi^{4}$ interaction, where the field $\varphi$ represents a
particle with mass $m$. A bound state of the $\varphi$-$\varphi$ type is
created when the coupling constant is negative and its modulus is larger than a
certain critical value. We investigate the contribution of this bound state to
the pressure of the thermal gas of the system by using the $S$-matrix formalism
involving the derivative of the phase-shift scattering. Our analysis, which is
based on an unitarized one-loop resumed approach which renders the theory
finite and well-defined for each value of the coupling constant, leads to
following main results: (i) We generalize the phase-shift formula in order to
take into account within a unique formal approach the two-particle interaction
as well as the bound state (if existent). (ii) \textit{On the one hand}, the
number density of the bound state in the system at a certain temperature $T$ is
obtained by the standard thermal integral; this is the case for any binding
energy, even if it is much smaller than the temperature of the thermal gas.
(iii) \textit{On the other hand}, the contribution of the bound state to the
total pressure is partly -- but not completely -- canceled by the two-particle
interaction contribution to the pressure. (iv) The pressure as function of the
coupling constant is \textit{continuous} also at the critical coupling for the
bound state formation: the jump in pressure due to the sudden appearance of the
bound state is exactly canceled by an analogous jump (but with opposite sign)
of the interaction contribution to the pressure.
