Title: Instantons and Unitarily Inequivalent Quantum Vacua Author: Roman Derco Department: Institute of Particle and Nuclear Physics Supervisor: doc. Alfredo Iorio, Ph.D., Institute of Particle and Nuclear Physics Abstract: In the presented thesis we investigate the relationship between the topologically distinct instantonic vacua and the unitarily inequivalent vacua of the quantum field theory. We focus on quantum mechanical exam- ples, where instantons appear but the complications due to quantum gauge field theory are absent. A model for quantum dissipation and the theory of one particle escaping from a metastable minimum were compared, what led to some observations. A double well system was build from harmonic oscillators and an interaction term to get closer to the quantum dissipation model, where inequivalent representations are involved. We identified the particularly simple model of a quantum particle constrained on a circle to be the ideal toy model for spotting the relation among unitarily inequivalent vacua and topologically distinct vacua we were seeking for. 1
A general theory is proposed for the study of the single-particle description of a nondegenerate finite-particle system in a truncated shell-model space. In this theory, it is established that one can introduce a single-paritcle representation satisfying the requirements that single-particle operators to be defined should obey the Fermion-commutation relation, and that the corresponding free vacuum should also be the lowest eigenstate of the Hamiltonian. It is proved that such a representation based on the true ground state can be derived by the unitary transformation of the usual Fermion operators similarly to the quasi-particle transformation method. The equation is given for the determination of the unitary operator proper to new representation. This approach is proved to be understood as a natural extension of the usual Hartree-Fock theory. The theory is applied to a normal system, and a successive linearization procedure is proposed in order to solve the equation for the unitary operator. The advantages of the theory are discussed in connection with the nuclear shell-model problems.
The many approaches that have been pursued in seeking an understanding of nuclear rotational dynamics are reviewed and reassessed with a view to their development in the light of recent progress and the research tools that are now available. A motivation for this review is the widespread observation of nuclear shape coexistence and sequences of rotational states in all regions of the nuclear periodic table combined with the recognition that the study of the rotational dynamics of quantum fluids has led to significant advances in the quantum theory of many-boson systems. Recent experimental investigations of the rotational dynamics of a low-temperature
6Li gas indicate that its slow rotational flows are likewise the irrotational flows of a superfluid. In this context, the dynamics of rotating nuclei are of fundamental interest because the nucleus is a unique zero-temperature finite many-fermion quantum system. A promising approach is provided by algebraic mean-field theory which, as its name suggests, is a combination of algebraic and mean-field methods. Static mean-field theories play a central role in many-body theory by defining optimal independent-particle and independent quasi-particle basis states for the quantum mechanics of many-fermion systems. Their time-dependent extensions also lead, in the small-amplitude random-phase approximation, to the quantisation of the classical normal-mode vibrations of many-fermion systems about their static equilibrium states. This review shows that mean-field methods become significantly more powerful when combined with algebraic methods and an appropriate coupling scheme for the nuclear shell model.