Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page

LQG for the Bewildered

We present a pedagogical introduction to the notions underlying the connection formulation of General Relativity - Loop Quantum Gravity (LQG) - with an emphasis on the physical aspects of the framework. We begin by reviewing General Relativity and Quantum Field Theory, to emphasise the similarities between them which establish a foundation upon which to build a theory of quantum gravity. We then explain, in a concise and clear manner, the steps leading from the Einstein-Hilbert action for gravity to the construction of the quantum states of geometry, known as \emph{spin-networks}, which provide the basis for the kinematical Hilbert space of quantum general relativity. Along the way we introduce the various associated concepts of \emph{tetrads}, \emph{spin-connection} and \emph{holonomies} which are a pre-requisite for understanding the LQG formalism. Having provided a minimal introduction to the LQG framework, we discuss its applications to the problems of black hole entropy and of quantum cosmology. A list of the most common criticisms of LQG is presented, which are then tackled one by one in order to convince the reader of the physical viability of the theory. An extensive set of appendices provide accessible introductions to several key notions such as the \emph{Peter-Weyl theorem}, \emph{duality} of differential forms and \emph{Regge calculus}, among others. The presentation is aimed at graduate students and researchers who have some familiarity with the tools of quantum mechanics and field theory and/or General Relativity, but are intimidated by the seeming technical prowess required to browse through the existing LQG literature. Our hope is to make the formalism appear a little less bewildering to the un-initiated and to help lower the barrier for entry into the field.Comment: 87 pages, 15 figures, manuscript submitted for publicatio

Orbit structure on the Silov boundary of a tube domain and the Plancherel decomposition of a causally compact symmetric space, with emphasis on the rank one case

We construct a G-equivariant causal embedding of a compactly causal symmetric space G/H as an open dense subset of the Silov boundary S of the unbounded realization of a certain Hermitian symmetric space G1/K1 of tube type. Then S is an Euclidean space that is open and dense in the flag manifold G1/P\u27, where P\u27 denotes a certain parabolic subgroup of G1. The regular representation of G on L2(G/H) is thus realized on L2(S), and we use abelian harmonic analysis in the study thereof. In particular, the holomorphic discrete series of G/H is being realized in function spaces on the boundary via the Euclidean Fourier transform on the boundary. Let P\u27=L1N1 denote the Langlands decomposition of P\u27. The Levi factor L1 of P\u27 then acts on the boundary S, and the orbits O can be characterized completely. For G/H of rank one we associate to each orbit O the irreducible representation L2Oi:={fεL2(S,dx)|supp fcOi} of G1 and show that the representation of G1 on L2(S) decompose as an orthogonal direct sum of these representations. We show that by restriction to G of the representations L2Oi, we thus obtain the Plancherel decomposition of L2(G/H) into series of unitary irreducible representations, in the sense of Delorme, van den Ban, and Schlichtkrull

Dynamical quantum ergodicity from energy level statistics

Ergodic theory provides a rigorous mathematical description of classical dynamical systems including a formal definition of the ergodic hierarchy. Closely related to this hierarchy is a less-known notion of cyclic approximate periodic transformations [see, e.g., I. Cornfield, S. Fomin, and Y. Sinai, Ergodic theory (Springer-Verlag New York, 1982)], which maps any "ergodic" dynamical system to a cyclic permutation on a circle and arguably represents the most elementary notion of ergodicity. This paper shows that cyclic ergodicity generalizes to quantum dynamical systems, and this generalization is proposed here as the basic rigorous definition of quantum ergodicity. It implies the ability to construct an orthonormal basis, where quantum dynamics transports an initial basis vector to all other basis vectors one by one, while maintaining a sufficiently large overlap between the time-evolved initial state and a given basis state. It is proven that the basis, maximizing the overlap over all cyclic permutations, is obtained via the discrete Fourier transform of the energy eigenstates. This relates quantum cyclic ergodicity to level statistics. We then show that the near-universal Wigner-Dyson level statistics implies quantum cyclic ergodicity, but the reverse is not necessarily true. For the latter, we study irrational flows on a 2D torus and prove that both the classical and quantum flows are cyclic ergodic. However, the corresponding level statistics is neither Wigner-Dyson nor Poisson. Finally, we use the cyclic construction to motivate a quantum ergodic hierarchy of operators and argue that under the additional assumption of Poincare recurrences, cyclic ergodicity is a necessary condition for such operators to satisfy the eigenstate thermalization hypothesis. This work provides a general framework for transplanting some rigorous concepts of ergodic theory to quantum dynamical systems.Comment: 42+11 pages, 9+1 figures; v2: updated definition of aperiodicity, analytical results for tori, improved presentation and some new figure

An anthology of non-local QFT and QFT on noncommutative spacetime

Ever since the appearance of renormalization theory there have been several differently motivated attempts at non-localized (in the sense of not generated by point-like fields) relativistic particle theories, the most recent one being at QFT on non-commutative Minkowski spacetime. The often conceptually uncritical and historically forgetful contemporary approach to these problems calls for a critical review the light of previous results on this subject.Comment: 33 pages tci-latex, improvements of formulations, shortening of sentences, addition of some reference