In addition, add an extra history \(Y^0=(I-[\Psi_0])\odot I_1\odot I_2\cdots\) so that the sum of all the history projectors together is the identity \(\breve I\) on the Hilbert space \(\HC_0\odot\HC_1\odot\cdots\). (Here, consistent with the notation used elsewhere in this article, \([\Psi_0]\) denotes the projector \(\dya{\Psi_0}\).) Then define a collection of “chain kets” \(\ket{\Phi^\alpha}\),