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 groundstate wave functions |Ψ) = e^S |Φ) which live in the “physical” Hilbert space H^(P) using an elementary ansatz for |Φi plus a formal expansion of S in an operator basis of multi-configurational creation operators C⁺. In our paper a reinterpretation of the method is proposed. Using parallels between the CCM and the so called quasi-Hermitian, <em>alias</em> three-Hilbert-space (THS), quantum mechanics, the CCM transition from the known microscopic Hamiltonian (denoted by usual symbol <em>H</em>), which is self-adjoint in <em>H^(P)</em>, to its effective lower-case isospectral avatar <em>h = e^−SHe^S</em>, 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 <em>h</em>, which only appears manifestly non-Hermitian in its own (“friendly”) Hilbert space <em>H^(F)</em>. This goal is achieved via an ad hoc amendment of the inner product in <em>H^(F)</em>, thereby yielding the third (“standard”) Hilbert space <em>H^(S)</em>. Due to the resulting exact unitary equivalence between the first and third spaces, <em>H^(F)</em> ∼ <em>H^(S)</em>, the indistinguishability of predictions calculated in these alternative physical frameworks is guaranteed.