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 $|\Psi ⟩=e^S|\Phi ⟩$ which live in the "physical" Hilbert space ${\mathcal{H}}^{(P)}$ using an elementary ansatz for $|\Phi ⟩$ 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 ${\mathcal{H}}^{(P)}$, 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 ${\mathcal{H}}^{(F)}$. This goal is achieved via an ad hoc amendment of the inner product in ${\mathcal{H}}^{(F)}$, thereby yielding the third ("standard") Hilbert space ${\mathcal{H}}^{(S)}$. Due to the resulting exact unitary equivalence between the first and third spaces, ${\mathcal{H}}^{(P)}\sim {\mathcal{H}}^{(S)}$, the indistinguishability of predictions calculated in these alternative physical frameworks is guaranteed.

15 pages

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• The ODE/IM Correspondence
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Cited by in corpus (5)

• Dimensionality reduction of many-body problem using coupled-cluster sub-system flow equations: classical and quantum computing perspective
• Non-Hermitian coupled cluster method for non-stationary systems and its interaction-picture reinterpretation
• Wheeler-DeWitt equation and the applicability of crypto-Hermitian interaction representation in quantum cosmology
• Theory of response to perturbations in non-Hermitian systems using five-Hilbert-space reformulation of unitary quantum mechanics
• Factorized Hilbert-space metrics and non-commutative quasi-Hermitian observables