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On the \(K\)-theoretic classification of topological phases of matter. (English) Zbl 1344.81144
The dynamics of a quantum mechanical system are specified by a Hamiltonian. The Hamiltonians for many physical systems have a spectral gap. Even free particles may have interesting phases when various symmetries are present. In is now generally agreed that there are 10 different classes of free Fermion phases in each dimension. This model uses operator K-theory to provide a mathematical explanation of these phases. The quantum system is modeled by a \(C^\ast\) algebra with extra structure to take possible Hamiltonian symmetry and time-preserving or reversing symmetry into account. The author does a nice job explaining both the mathematics and physical motivation.

MSC:
81T70 Quantization in field theory; cohomological methods
81V70 Many-body theory; quantum Hall effect
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
46L60 Applications of selfadjoint operator algebras to physics
19K56 Index theory
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