By means of infinite-dimensional nuclear spaces, we generalize important results on the representation of the Weyl commutation relations. For this purpose, we construct a new nuclear Lie group generalizing the groups introduced by Parthasarathy [An Introduction to Quantum Stochastic Calculus (Birkhäuser, 1992)] and Gelfand–Vilenkin [Generalized Functions (Academic Press, 1964)] (see Ref. 15). Then we give an explicit construction of Weyl representations generated from a non-Fock representation. Moreover, we characterize all these Weyl representations in quantum white noise setting.
Abstract We present a pedagogical review of projective representations of finite groups and their physical applications in quantum many-body systems. Some of our physical results are new. We begin with a self-contained introduction to projective representations, highlighting the role of group cohomology, representation theory, and classification of irreducible projective representations. We then focus on a special subset of cohomology classes, known as the Bogomolov multiplier, which consists of cocycles that are symmetric on commuting pairs but remain nontrivial in group cohomology. Such cocycles have important physical implications: they characterize (1+1)D SPT phases that cannot be detected by string order parameters and give rise, upon gauging, to distinct gapped phases with completely broken non-invertible
Rep(G) Rep ( G ) symmetry. We construct explicit lattice models for these phases and demonstrate how they are distinguished by the fusion rules of local order parameters. We show that a pair of completely broken
Rep(G) Rep ( G ) SSB phases host nontrivial interface modes at their domain walls. As an example, we construct a lattice model where the ground state degeneracy on a ring increases from 32 without interfaces to 56 with interfaces.
his graduate-level text develops the aspects of group theory most relevant to physics and chemistry (such as the theory of representations) and illustrates their applications to quantum mechanics. The first five chapters focus chiefly on the introduction of methods, illustrated by physical examples, and the final three chapters offer a systematic treatment of the quantum theory of atoms, molecules, and solids.The formal theory of finite groups and their representation is developed in Chapters 1 through 4 and illustrated by examples from the crystallographic point groups basic to solid-state