We consider an invariant quantum Hamiltonian H = −ΔLB + V in the L2 space based on a Riemannian manifold ˜M with a discrete symmetry group Γ. To any unitary representation Λ of Γ one can relate another operator on M = ˜M /Γ, called HΛ, which formally corresponds to the same differential operator as H but which is determined by quasi-periodic boundary conditions. As originally observed by Schulman in theoretical physics and Sunada in mathematics, one can construct the propagator associated with HΛ provided one knows the propagator associated with H. This approach is reviewed and demonstrated on a quantum model describing a charged particle on the plane with two Aharonov-Bohm vortices. The construction of the propagator is explained in full detail including all substantial intermediate steps.
We consider an invariant quantum Hamiltonian H = −ΔLB + V in the L2 space based on a Riemannian manifold ˜M with a discrete symmetry group Γ. To any unitary representation Λ of Γ one can relate another operator on M = ˜M /Γ, called HΛ, which formally corresponds to the same differential operator as H but which is determined by quasi-periodic boundary conditions. As originally observed by Schulman in theoretical physics and Sunada in mathematics, one can construct the propagator associated with HΛ provided one knows the propagator associated with H. This approach is reviewed and demonstrated on a quantum model describing a charged particle on the plane with two Aharonov-Bohm vortices. The construction of the propagator is explained in full detail including all substantial intermediate steps.
We consider a spin-1/2 charged particle in the plane under the influence of two idealized Aharonov–Bohm fluxes. We show that the Pauli operator as a differential operator is defined by appropriate boundary conditions at the two vortices. Further we explicitly construct a basis in the deficiency subspaces of the symmetric operator obtained by restricting the domain to functions with supports separated from the vortices. This construction makes it possible to apply the Krein’s formula to the Pauli operator.
We develop a general technique for finding self-adjoint extensions of a symmetric operator that respects a given set of its symmetries. Problems of this type naturally arise when considering two- and three-dimensional Schrödinger operators with singular potentials. The approach is based on constructing a unitary transformation diagonalizing the symmetries and reducing the initial operator to the direct integral of a suitable family of partial operators. We prove that symmetry preserving self-adjoint extensions of the initial operator are in a one-to-one correspondence with measurable families of self-adjoint extensions of partial operators obtained by reduction. The general scheme is applied to the three-dimensional Aharonov-Bohm Hamiltonian describing the electron in the magnetic field of an infinitely thin solenoid. We construct all self-adjoint extensions of this Hamiltonian, invariant under translations along the solenoid and rotations around it, and explicitly find their eigenfunction expansions.
This paper is the second in a series of papers considering symmetry properties of bosonic quantum systems over 2D graphs, with continuous spins, in the spirit of the Mermin-Wagner theorem. In the model considered here the phase space of a single spin is ℋ1=L2(M), where M is a d -dimensional unit torus M=ℝd/ℤd with a flat metric. The phase space of k spins is ℋk=L2sym(Mk) , the subspace of L2(Mk) formed by functions symmetric under the permutations of the arguments. The Fock space H =⊕k=0,1,… ℋk yields the phase space of a system of a varying (but finite) number of particles. We associate a space H≃H(i) with each vertex i∈Γ of a graph (Γ,ℰ) satisfying a special bidimensionality property. (Physically, vertex i represents a heavy “atom” or “ion” that does not move but attracts a number of “light” particles.) The kinetic energy part of the Hamiltonian includes (i) -Δ/2 , the minus a half of the Laplace operator on M , responsible for the motion of a particle while “trapped” by a given atom, and (ii) an integral term describing possible “jumps” where a particle may join another atom. The potential part is an operator of multiplication by a function (the potential energy of a classical configuration) which is a sum of (a) one-body potentials U(1)(x) , x∈M , describing a field generated by a heavy atom, (b) two-body potentials U(2)(x,y) , x,y∈M , showing the interaction between pairs of particles belonging to the same atom, and (c) two-body potentials V(x,y) , x,y∈M , scaled along the graph distance d(i,j) between vertices i,j∈Γ , which gives the interaction between particles belonging to different atoms. The system under consideration can be considered as a generalized (bosonic) Hubbard model. We assume that a connected Lie group G acts on M , represented by a Euclidean space or torus of dimension d'≤d , preserving the metric and the volume in M . Furthermore, we suppose that the potentials U(1) , U(2) , and V are G -invariant. The result of the paper is that any (appropriately defined) Gibbs states generated by the above Hamiltonian is G -invariant, provided that the thermodynamic variables (the fugacity z and the inverse temperature β ) satisfy a certain restriction. The definition of a Gibbs state (and its analysis) is based on the Feynman-Kac representation for the density matrices.
The Aharonov–Bohm effect in general gauge theories, for particles in gauge-curvature-free regions, is studied using the quantum mechanical propagator in the form of a Feynman sum over paths. Following Schulman [L. S. Schulman, Techniques and Applications of Path Integration (Wiley, New York, 1981)], such paths are divided into their homotopy equivalence classes, and the contributions from each class of paths of the Feynman sum are identified with propagators of a wave equation in the universal covering manifold of M, resulting in a simple form for the propagator on M. A group homomorphism from ℋ, the fundamental homotopy group of M, to the gauge group G is shown to characterize possible Aharonov–Bohm effects, which can be divided into two types, Abelian and non-Abelian, according to whether ℋ*, the image of this homomorphism, is Abelian or non-Abelian. For a non-Abelian Aharonov–Bohm effect, it is necessary that both ℋ and G be non-Abelian. Simple examples illustrate the theory.
In this work, we make a review of the literature that the so-called Berry’s geometric phase. A recent issue, which has been employed in various areas of physics. For this, we use, in a number of references, in which many were critical to the theoretical foundation of this work. At first, we focus on the study of the time evolution of quantum systems governed by Hamiltonians operators whose time dependence is manifested by a set of parameters called classical fields. The adiabatic approximation, is justified mathematically when demonstrated the adiabatic theorem. Under these conditions, we obtain geometric phases - Berry’s phase - associated with these systems in different representations, each displaying its particularity. Finally, as a particular case of geometric phase, we studi from the viewpoint of duality transformations of Maxwell’s equations, the dual Aharonov-Bohm effect, extracting the geometric Berry’s phase associated with the dynamics of the magnetic monopole in the presence of the electric vector potential
The Aharonov-Bohm effect is investigated in the Feynman path-integral formulation of quantum mechanics. We consider an idealized situation with an electron moving in a magnetic-field-free region outside a solenoid whose radius to length ratio is very small. The nonvanishing vector potential term in the Lagrangian is written as an angular-velocity-dependent potential. In order to account for a singularity due to the presence of the solenoid itself, a periodic constraint is imposed on the path integral. The propagator can then be evaluated using the polar-coordinate methods of Peak and Inomata. It is found that the propagator has the general form $K({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}\ensuremath{'}},{\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}};\ensuremath{\tau})=\ensuremath{\Sigma}{n}^{}{\ensuremath{\chi}}_{n}{K}_{n}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}\ensuremath{'}},{\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}};\ensuremath{\tau})$ where the sum is taken over all classes of homotopic paths and the ${\ensuremath{\chi}}_{n}$ are a one-dimensional representation of the homotopy group. This is the form of the propagator as conjectured by Schulman.