Abstract We show that non-abelian quantum statistics can be studied using certain topological invariants which are the homology groups of configuration spaces. In particular, we formulate a general framework for describing quantum statistics of particles constrained to move in a topological spaceX. The framework involves a study of isomorphism classes of flat complex vector bundles over the configuration space ofXwhich can be achieved by determining its homology groups. We apply this methodology for configuration spaces of graphs. As a conclusion, we provide families of graphs which are good candidates for studying simple effective models of anyon dynamics as well as models of non-abelian anyons on networks that are used in quantum computing. These conclusions are based on our solution of the so-called universal presentation problem for homology groups of graph configuration spaces for certain families of graphs.
A geometric framework for describing quantum particles on a possibly curved background is proposed. Natural constructions on certain distributional bundles ("quantum bundles") over the space–time manifold yield a quantum "formalism" along any 1-dimensional timelike submanifold (a "detector"); in the flat, inertial case this reproduces the basic results of the usual quantum field theory, while in general it could be seen as a local, "linearized" description of the actual physics.
The structure of braid groups on topologically nontrivial surfaces is reviewed. The physical meaning of the braid relations and their implications on quantum mechanical properties of anyonic quasiparticles are discussed. These include not only the restrictions on statistics of anyons but also the Aharonov-Bohm effect for the anyons on a surface with holes. Several results on the novel properties of anyons of their states, which were previously derived by microscopic considerations, are reproduced by this seemingly kinematic but topology dependent braid group analysis. It is suggested than in the thermodynamic limit, the global excitations in a system on a surface of nontrivial topology do not interfere with properties of local anyonic quasiparticles.
This thesis investigates parametrized quantum spin systems in the thermodynamic limit from a
C∗-algebraic point of view. Our main physical result is the construction of a phase invariant for one-dimensional quantum spin chains parametrized by a topological space
X. This invariant is constructed using
C∗-algebraic techniques and takes values in degree one Čech cohomology
H1(X;PU(H)), where
PU(H) is the projective unitary group of an infinite-dimensional Hilbert space
H, endowed with the strong operator topology. Using Dixmier-Douady theory [Bry93, DD63, HJJS08] one may equivalently view this as an element of
H3(X;Z). An exactly solvable model of a one-dimensional spin system parametrized by the 3-sphere
X=S3 is presented and it is shown that its invariant is nontrivial [WQB+22]. We also prove several mathematical results on topological aspects of the pure state space
P(A) of a
C∗-algebra
A. We prove that
P(A), endowed with the weak* topology, has trivial fundamental group for every UHF algebra
A [BHM+23]. We review the results of [SMQ+22] that show how the outputs of the GNS representation and Kadison transitivity theorem can be understood to depend continuously on their inputs. These results are expanded upon with the construction of a distinguished Čech class in
H1(P(A);U(1)) that generalizes the principal
U(1)-bundle
SH→PH in a representation independent way, where
SH and
PH are the unit sphere and projective Hilbert space of
H. Finally, we prove a selection theorem in the vein of [SMQ+22] for the weak* topology on
P(A).
The structure of braid groups on topologically nontrivial surfaces is reviewed. The physical meaning of the braid relations and their implications on quantum mechanical properties of anyonic quasiparticles are discussed. These include not only the exotic statistics of anyons but also the Aharonov-Bohm effect for the anyons on a surface with holes. Several results on the novel properties of anyons or their states, which were previously derived by microscopic considerations, are reproduced by this seemingly kinematic and topology-dependent braid group analysis. It is suggested that in the thermodynamic limit, the global excitations in a system on a surface of nontrivial topology do not interfere with properties of local anyonic quasiparticles.
This thesis reviews fibre-bundle theory for the purpose of showing its equivalence to gauge theory and, in particular, utilizing fibre bundles to explain the origin of a quantum mechanical effect named the Aharonov-Bohm (AB) effect. Initially, the mathematical preliminaries necessary to review fibre-bundle theory are presented. This includes topological manifolds, smooth manifolds, vector fields and 1-forms on smooth manifolds as well as Lie groups and their Lie algebras. Thereafter, fibre bundles are introduced. In particular, principal bundles, connections on principal bundles, parallel transport and associated bundles are studied. The electrons in the setup of the AB effect experience the effect of an electromagnetic potential and form an interference pattern on the screen they hit. The AB effect is described by a principal U(1)-bundle over R2\{0} and an associated complex line bundle with bundle space P xρ C, where P is the bundle space in the principal bundle and ρ a representation of U(1) on C defined ρ(x) = xq, for x in U(1) and q in Z. The vector potential A, on a simply connected subset U ⊂ R2\{0}, is given by A = ∇χU, for a smooth real-valued function χU. The complex scalar field, i.e. wave function, ψU: U → C of an electron with charge q has a local solution ψU = eiqχUψ0, where ψ0 satisfies the free Schrödinger equation. Consequentially, the holonomy is eiqΦ, where Φ is the total flux of the magnetic field.