The coupled-cluster approach to quantum many-body problem in a three-Hilbert-space reinterpretation

Abstract

The quantum many-body bound-state problem in its computationally successful coupled cluster method (CCM) representation is reconsidered. In conventional practice one factorizes the ground-state wave functions $|Ψ\rangle= e^S |Φ\rangle$ which live in the "physical" Hilbert space ${\cal H}^{(P)}$ using an elementary ansatz for $|Φ\rangle$ plus a formal expansion of $S$ in an operator basis of multi-configurational creation operators. In our paper a reinterpretation of the method is proposed. Using parallels between the CCM and the so called quasi-Hermitian, alias three-Hilbert-space (THS), quantum mechanics, the CCM transition from the known microscopic Hamiltonian (denoted by usual symbol $H$), which is self-adjoint in ${\cal H}^{(P)}$, to its effective lower-case isospectral avatar $\hat{h}=e^{-S} H e^S$, is assigned a THS interpretation. In the opposite direction, a THS-prescribed, non-CCM, innovative reinstallation of Hermiticity is shown to be possible for the CCM effective Hamiltonian $\hat{h}$, which only appears manifestly non-Hermitian in its own ("friendly") Hilbert space ${\cal H}^{(F)}$. This goal is achieved via an ad hoc amendment of the inner product in ${\cal H}^{(F)}$, thereby yielding the third ("standard") Hilbert space ${\cal H}^{(S)}$. Due to the resulting exact unitary equivalence between the first and third spaces, ${\cal H}^{(P)}\sim {\cal H}^{(S)}$, the indistinguishability of predictions calculated in these alternative physical frameworks is guaranteed.