Motivated by topological quantum field theory, we investigate the geometric aspects of unitary 2-representations of finite groups on 2-Hilbert spaces, and their 2-characters. We show how the basic ideas of geometric quantization are ‘categorified’ in this context: just as representations of groups correspond to equivariant line bundles, 2-representations of groups correspond to equivariant gerb...

A theory of non-unitary-invertible as well as unitary canonical transformations is formulated in the context of Weyl's phase space representations. Exact solutions of the transformation kernels and the phase space propagators are given for the three fundamental canonical maps as linear, gauge and contact (point) transformations. Under the nonlinear maps a phase space representation is mapped to...

This paper is an exploration of relationships between the Jones polynomial and quantum computing. We discuss the structure of the Jones polynomial in relation to representations of the Temperley Lieb algebra, and give an example of a unitary representation of the braid group. We discuss the evaluation of the polynomial as a generalized quantum amplitude and show how the braiding part of the eva...

A theory of non-unitary-invertible as well as unitary canonical transformations is formulated in the context of Weyl's phase space representations. Exact solutions of the transformation kernels and the phase space propagators are given for the three fundamental canonical maps as fractional-linear, gauge and contact (point) transformations. Under the nonlinear maps a phase space representation i...

We characterize all simple unitarizable representations of the braid group B3 on complex vector spaces of dimension d ≤ 5. In particular, we prove that if σ1 and σ2 denote the two generating twists of B3, then a simple representation ρ : B3 → GL(V) (for dim V ≤ 5) is unitarizable if and only if the eigenvalues λ1, λ2,. .. , λ d of ρ(σ1) are distinct, satisfy |λi| = 1 and µ (d) 1i > 0 for 2 ≤ i ...

We characterize all simple unitarizable representations of the braid group B 3 on complex vector spaces of dimension d ≤ 5. In particular, we prove that if σ 1 and σ 2 denote the two generating twists of B 3 , then a simple representation ρ : B 3 → GL(V) (for dim V ≤ 5) is unitarizable if and only if the eigenvalues λ 1 , λ 2 ,. .. , λ d of ρ(σ 1) are distinct, satisfy |λ i | = 1 and µ (d) 1i >...