Cluster Operator Parameterization

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In search for a state specific ansatz we recognize that the number of reference coefficients matches the number of $\mathbb {P}$ projections naturally ( ord({c_i}) = ord($\mathbb {P}$)). Analyzing the remaining equation ord({t_j}) = ord($\mathbb {Q}$) we state that for every t_j there must be a corresponding |$\beta$$\rangle$ $\in$ $\mathbb {Q}$. Thus as a first guess we try to use exactly this |$\beta$$\rangle$ to index the amplitudes t and write

$$\beta \rangle$$ (18)

with the sign rule

$$\beta \rangle \beta \rangle$$ (19)

applied. In order to insure potential completeness we introduce the phase factor $\varphi$(z) = e^{-i arg z},  z $\in$ $\mathbb {C}$ according to

$$\Psi \rangle \sum_{{\mu}}^{} \mu \left(\vphantom{\varphi (c_\mu) \sum_{\hat \tau_i(\mu) \in \mathbb{T}_\mu} \!\!\! t_{\hat \tau_i(\mu) \vert\mu\> } \hat \tau_i(\mu) }\right. \varphi \mu \sum_{{\hat \tau_i(\mu) \in \mathbb{T}_\mu}}^{} \hat{\tau}_{i} \mu \mu \rangle \hat{\tau}_{i}^{} \mu \left.\vphantom{\varphi (c_\mu) \sum_{\hat \tau_i(\mu) \in \mathbb{T}_\mu} \!\!\! t_{\hat \tau_i(\mu) \vert\mu\> } \hat \tau_i(\mu) }\right) \mu \rangle$$ (20)

It is crucial to insert the phase factor inside the exponential and not outside.