Having difficulty comprehending a problem in Peskin's text

[kof9595995]

In his "an introduction to quantum field theory", problem 5.4 (c), he describes a bound state of positronium as |B(k)\rangle=\sqrt{2M}\int{\frac{d^{3}p}{(2\pi)^{3}}\psi_{i}(p)a^{\dagger}_{p+\frac{k}{2}}\Sigma^{i}b^{\dagger}_{-p+\frac{k}{2}}|0\rangle}
where \psi_{i}(p) are the p-orbtal wavefunctions in momentum space(i=1,2,3), a^{\dagger}and b^{\dagger} are electron and positron creation operator, \Sigma^{i} is some 2 by 2 matrix. I don't understand where this \Sigma^{i} comes from. LHS of the equation is just a ket, in this case shouldn't RHS be a superposition of kets? What should I make of \Sigma^{i}?

 

[azntoon]

The wavefunction of the positronium bound state is a two-particle wavefunction, which is a function of both electron and positron coordinates. The matrix $\Sigma^i$ is used to represent the two-particle wavefunction in the momentum space representation. It is essentially a two by two matrix whose elements contain the momentum space wavefunction $\psi_i(p)$. The matrix $\Sigma^i$ is used to construct the two-particle wavefunction in the momentum space representation of the positronium bound state.