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Abstract:

Abstract We introduce the idea of the path integral and use it to study the properties of identical particles under exhange. We explain why the topology of paths through space-time, the fundamental group of the configuration space (the braid group in 2+1 dimensions and the permutation group in 3+1 dimensions) determines the possible particle types. We argue that in 3+1 dimensions one only has bosons and fermions but in 2+1 dimensions other particles, known as anyons, can exist. We describe both the abelian and nonabelian type of anyon and argue why nonabelian anyons could be useful for quantum computation. We introduce the notion of the “quantum dimension” of an anyon which describes the size of the many anyon Hilbert space.

Abstract:

Abstract We return to study planar diagram algebras as in chapters 8 and 9, but now more formally. We define diagrams as being operators that can be "stacked" on top of each other. We pay particular attention to properties of diagrams such as completeness of states and orthonormality of basis vectors, and we again review the use of F-matrices for change of basis. We discuss the implications of causality for these diagrams, and to what extent diagrams can be deformed in the plane. We give explicit detailed rules for evaluating planar diagrams.

Abstract:

Abstract The mathematical structure known as a Quantum Group can be thought of as a deformation of a classical Lie group. The (deformed) representation theory of quantum groups gives a natural way to generate data (F-matrices, and R-matrices) for anyon theories. In many cases, this approach gives data equivalent to Chern-Simons theory based on the corresponding Lie group.

Abstract:

Abstract Topologically ordered matter is famously robust to small perturbations. If a Hamiltonian has a topologically ordered (TQFT) ground state with a gap, then adding a small pertrubation (on the scale of the gap) to the Hamiltonian cannot change its TQFT properties. We explore this robustness with the example of the toric code. We show that the robustness is guaranteed by the fact that the toric code has a protected code space. We further discuss how the properties of the quasiparticles are also unchanged under perturbations of the Hamiltonian. These properties might be unsurprising, given the rigidity of the properties of TQFTs. We define the notion of topological order and give a definition of a topological phase of matter.