I was studying this article on consequences of invariance under charge conjugation symmetry(in particular, in positronium decay) where in section IV it says
The state function of any state of positronium can be expanded in terms of free-particle state functions,$$\Psi_\text{positronium}=(\sum_{\vec p,s_1,s_2}c(\vec p,s_1,s_2)a_{s_{1}}^*(\vec p)b_{s_{2}}^*(-\vec p) +\sum_r...a_{r_{1}}^*a_{r_{2}}^*b_{r_{3}}^*b_{r_{4}}^*+...)\Psi_\text{vac}$$ where the second term represents the effect of virtual pair production.
I want to know about the nature of the terms that have been left out and make sure that the claim is actually true. It is confusing to me because, in the introductory lectures, we have been told that the (free) bound states of combinations of some or all possible particles should be taken as a separate sector of the Fock space in the sense, the creation and annihilation operators for these states(positronium atom creation/annihilation operator in this case) are independent of the creation and annihilation operators for the constituent particles. And I guess this is the spirit behind Weinberg's remark(in his book of Quantum Theory of Fields) in section 3.1:
Also, any relevant bound states in the spectrum of $H$ should be introduced into $H_0$ as if they were elementary particles**
**Alternatively, in non-relativistic problems we can include the binding potential in $H_0$. In the application of this method to rearrangement collisions, where some bound states appear in the initial state but not the final state, or vice-versa, one must use a different split of $H$ into $H_0$ and $V$ in the initial and final states.
Another related query is, are positronium decay process and pair annihilation process the same?
Any comment/clarification is welcome!
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