<PERSON> in his text on field quantisation tries to justify this exact same point in example 10.2:
It is very difficult to find an exact description of the bound states of the $e^+e^-$ system since this would amount to solving the relativistic two-body problem...
According to the general principle of Lorentz invariance the state vector of positronium $|Ps\rangle$ can be classified by the eigenvalues of the operators $P^\mu$, $\mathbf J^2$ and $J^z$. In addition we have the operators of space inversion P and charge conjugation C, which commute with the set of kinematic operators. The corresponding eigenvalues are $$P|Ps\rangle = \pi_P |Ps\rangle, C|Ps\rangle = \pi_C |Ps\rangle\tag{1}$$ where $\pi_P = ±1$ denotes space parity and $\pi_C = ±1$ denotes charge parity. Now we make the following ansatz for the state vector of positronium in the center-of-mass system $(P = 0)$: $$|Ps\rangle=\int d^3\vec p\sum_{s,s'}R(\vec p,s,s')b^\dagger_{\vec p,s}d^\dagger_{-\vec p,s'}|0\rangle\tag{2}$$ where $R(\vec p,s,s')$ is the wave function in momentum space. 66
Here $s$ and $s'$ are the projection of the electron and positron spins onto the z axis. The state (2) contains one electron-positron pair with a combined momentum of zero and is an approximation to the true bound state since the particle number in QED is not a conserved quantiy. In general higher multi-pair configurations with total charge zero, such as $b^\dagger b^\dagger d^\dagger d^\dagger|0\rangle$ etc., can contribute to the state vector. In positronium, however, such admixtures are very small, owing to the essentially nonrelativistic nature of this system. In any case, for the purpose of classifying the bound states it is sufficient to use the ansatz (2); any complicated higher-order admixtures would have the same symmetry properties.
For now, I'll take this as an explanation which invalidates my previous understanding
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.
Since now the creation operator of positronium atom, for the positron and for the electron don't commute pairwise.
Any additional remarks/clarification/answers are welcome! 976