The quantum mechanical propagators of the linear automorphisms of the two-torus (cat maps) determine a projective unitary representation of the theta group Γ θ ⊂ SL(2, Z). We prove that there exists an appropriate choice of phases in the propagators that defines a proper representation of Γ θ. We also give explicit formulae for the propagators in this representation.

The unitary Cayley graph Xn has vertex set Zn = {0, 1,…, n-1} and vertices u and v are adjacent, if gcd(uv, n) = 1. In [A. Ilić, The energy of unitary Cayley graphs, Linear Algebra Appl. 431 (2009) 1881–1889], the energy of unitary Cayley graphs is computed. In this paper the Wiener and hyperWiener index of Xn is computed.

In a recent preprint by Deutsch et al. 5] the authors suggest the possibility of polynomial approximability of arbitrary unitary operations on n qubits by 2-qubit unitary operations. We address that comment by proving strong lower bounds on the approximation capabilities of g-qubit unitary operations for xed g. We consider approximation of unitary operations on subspaces as well as approximatio...

We study space-time symmetries in scalar quantum field theory on an arbitrary static space-time. We first consider Euclidean quantum field theory, and show that the isometry group is generated by one-parameter subgroups which have either selfadjoint or unitary quantizations. We then show that the self-adjoint semigroups thus constructed can be analytically continued to one-parameter unitary gro...

We devise a ternary operation for combining three quantum states: it consists of permuting the input systems in a continuous fashion and then discarding all but one of them. This generalizes a binary operation recently studied by Audenaert et al. [ADO16] in the context of entropy power inequalities. Our ternary operation continuously interpolates between all such nested binary operations. Our c...

The proposed project is set in pure mathematics within the areas of infinite-dimensional Lie theory and geometric group theory. Its goal is to contribute to the structure theory of unitary forms (i.e., centralisers of Chevalley involutions) of Kac–Moody algebras and of Kac–Moody groups of indefinite type. The main emphasis of this project will be on finite-dimensional representations and on ide...

It is well-known that one of the most important ways of constructing cohomology for an arithmetic quotient Γ\X, where X is the symmetric space associated with a semi-simple Lie group G with finite center and Γ is a discrete subgroup of G, is through the use of Matsushima’s formula, namely by exhibiting irreducible unitary representations with non-zero continuous cohomology which occur in L2(Γ\G...

A representation of su q (n) , which diverges in the limit of q → 1, is investigated. This is an infinite dimensional and a non-unitary representation , defined for the real value of q, 0 < q < 1. Each irreducible representation is specified by n continuous variables and one discrete variable. This representation gives a new solution of the Yang-Baxter equation, when the R-matrix is evaluated. ...

Consider a closed subgroup H of a locally compact group G together with a strongly continuous unitary representation u of H on a Hilbert space K. The construction of the induced representation uses these three ingredients to supply a new strongly continuous unitary representation ρ of the larger group G on a new Hilbert space K. Induced group representations for finite groups where first introd...

In this note we address various algorithmic problems that arise in the computation of the operator norm in unitary representations of a group on Hilbert space. We show that the operator norm in the universal unitary representation is computable if the group is residually finite-dimensional or amenable with decidable word problem. Moreover, we relate the computability of the operator norm on the...