Exact Results in Quantum Theory

Given a system of many interacting fermions described by a time-independent Hamiltonian Ĥ, the Coupled-Cluster (CC) Ansatz introduces an exponential representation of the Hamiltonian eigenstates: Ψ = exp(T̂) Φ, where Φ is some indpendent-fermion wave function, and T̂ is the so-called cluster operator which produces excitations out of Ψ. The CC Ansatz has turned out to be a very effective computational tool in quantum chemistry and nuclear theory.In my talk I will present a general mathematical model of many-fermion systems based on the notion of quasiparticles and the CC Ansatz. This model, called the quasiparticle Fock-space coupled-cluster (QFSCC) theory, was introduced in 1985 by me and Henk Monkhorst.The departing point is the second-quantization formalism based on the algebraic approximation: one chooses a finite basis of single-fermion states (the spin orbitals), and builds the fermionic Fock space which represent all possible antisymmetric many-fermion states of a given system. A necessary algebraic machinery is provided by the algebra of linear operators acting in the Fock space, generated by the fermion creation and annihilation operators. The physics is governed by the Fock-space Hamiltonian operator Ĥ, commuting with the fermion-number operator N̂, that determines the system stationary states and their energies.The first step into the QFSCC theory is to apply the Bogoliubov-Valatin transformation, which converts the primary fermionic particles into some fermionic quasiparticles. Then one arrives at a new representation of the Fock basis: the new hierarchy of states is built upon the quasiparticle vacuum instead of the physical vacuum. The new creation and annihilation operators (which now correspond to the quasiparticles) fulfill the same anticommutation relations as the old (original) ones. By rewriting the Hamiltonian Ĥ with the help of the new quasiparticle operators, one immediately realizes that Ĥ does not commute with the quasiparticle-number operator N̂q.The crucial step in the QFSCC theory is to enforce the quasiparticle-number conservation principle. This is done by applying a special similarity transformation which converts Ĥ into the quasiparticle Hamiltonian Ĝ that explicitly commutes with N̂q. The corresponding transformation operator is built by employing the CC Ansatz, which has to be substantially generalized for this purpose. Now the spectrum of Ĝ corresponding to 0, 1, 2,... quasiparticles can be easily found -- this is equivalent to calculating a part of the spectrum of Ĥ corresponding to some N, N±1, N±2,... fermion states (where the ground state of the system, represented by the quasiparticle vacuum, is a N-fermion state).The QFSCC theory requires a substantial reformulation of the traditional second-quantization language, by making a full use of the algebraic properties of the Fock space and its operator algebra. In particular, the role of operators not conserving the number of particles (or quasiparticles) is emphasized. These "algebraic preparations" will be presented in detail in Part I of my talk.In Part II of my talk a step-by-step construction of the QFSCC theory will be performed. It will be seen that the emerging quasiparticle model of many-fermion systems offers useful physical insights and computational effectiveness, provided a natural truncation/decoupling scheme. I will also introduce a terse diagrammatic language to write down the working CC equations.