Division Algebras and Quantum Theory

Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of axiomatic approaches. However, there are internal problems with real or quaternionic quantum theory. Here we argue that these problems can be resolved if we treat real, complex and quaternionic quantum theory as part of a unified structure. Dyson called this structure the ‘three-fold way’. It is perhaps easiest to see it in the study of irreducible unitary representations of groups on complex Hilbert spaces. These representations come in three kinds: those that are not isomorphic to their own dual (the truly ‘complex’ representations), those that are self-dual thanks to a symmetric bilinear pairing (which are ‘real’, in that they are the complexifications of representations on real Hilbert spaces), and those that are self-dual thanks to an antisymmetric bilinear pairing (which are ‘quaternionic’, in that they are the underlying complex representations of representations on quaternionic Hilbert spaces). This three-fold classification sheds light on the physics of time reversal symmetry, and it already plays an important role in particle physics. More generally, Hilbert spaces of any one of the three kinds—real, complex and quaternionic—can be seen as Hilbert spaces of the other kinds, equipped with extra structure.     

Differential Representations of SO(4) Dynamical Group

In this paper we present systematic differential representations for the dynamical group SO(4). These representations include the left and the right differential representations and the left and the right adjoint differential representations in both the group parameter space and its coset spaces. They are the generalization of the differential representations of the SO(3) rotation group in the Euler angles. These representations may find their applications in the study of the physical systems with SO(4) dynamical symmetry.     

On the integrability of representations of finite dimensional real Lie algebras

We give an integrability criterion for Lie algebra representations in a reflexive Banach space. Applications are given to skewsymmetric Lie algebra representations in Hilbert spaces and to essential skewadjointness of a sum of two skewadjoint operators.     

On a class of representations of the Virasoro algebra and a conjecture of Kac

We consider a class of representations of the Virasoro algebra that we call bounded admissible representations. For this class, we prove a conjecture of Victor Kac concerning the irreducibility of these representations. Results concerning the center and dimensions of weight spaces are also obtained.     

Coherent states and induced representations

A generalization of the notion of coherent states is given. The following one-to-one correspondences are pointed out: (1) between covariant overcomplete systems of coherent states and a class of covariant semi-spectral measures; (2) between covariant semispectral measures and unitary irreducible subrepresentations of induced representations of Lie groups; (3) between unitary irreducible representations of Lie groups with covariant overcomplete systems of coherent states and unitary irreducible subrepresentations of induced representations, whose representation spaces are reproducing kernel Hilbert spaces.     

Inherent limits on optimization and discovery in physical systems

Topological mapping of a large physical system on a graph, and its decomposition using universal measures are proposed. We find inherent limits to the potential for optimization of a given system and its approximate representations by motifs, and the ability to reconstruct the full system given approximate representations. The approximate representation of the system most suited for optimization may be different from that which most accurately describes the full system.     

The Transition to Unigroups

A unigroup is defined to be a partially orderedabelian group with a distinguished generative universalorder unit. Virtually any structure that has beenproposed for the logic, sharp or unsharp, of a physical system can be represented by the order intervalin a unigroup. Furthermore, probability statescorrespond to positive, normalized, real-valued grouphomomorphisms, and physical symmetries correspond to unigroup automorphisms. We show that thecategory of unigroups admits arbitrary products andcoproducts. A new class of interval effect algebrascalled Heyting effect algebras (HEAs) is introduced andstudied. Among other things, an HEA is both a Heytingalgebra and a BZ-lattice in which the sharp elements areprecisely the central elements. Certain HEAs arisenaturally from partially ordered abelian groupsaffiliated with Stone spaces. Using Stone unigroups, weobtain perspicuous representations for certainmultivalued logics, including the three-valued logic ofconditional events utilized by Goodman, Nguyen, andWalker in their study of logic for expertsystems.     

Induced Representations of the One-Dimensional Quantum Galilei Group

We study the representations of the quantum Galilei group by a suitable generalization of the Kirillov method on spaces of noncommutative functions. On these spaces, we determine a quasi-invariant measure with respect to the action of the quantum group by which we discuss unitary and irreducible representations. The latter are equivalent to representations on ², i.e. on the space of square summable functions on a one-dimensional lattice.     

Decomposition of the Fock Space in Two-Dimensional Square Lattice Systems

We discuss how to decompose the Fock space of a many-fermion system embedded in two-dimensional square lattice. Wefirst notice that the symmetry group inherent in the system is one of the two-dimensional space groups. We shortly review thecorresponding irreducible representations of the group. We then find the characters of the reducible representation of the many-fermion Fock space. Using the characters, we obtain the multiplicity of each irreducible representation contained in the Fock space of a fixed number of fermions. We present specific examples, where we calculate the multiplicities which are the dimensions of the decomposed spaces.     

Classification and Realizations of Type III Factor Representations of Cuntz–Krieger Algebras Associated with Quasi-Free States

We completely classify type III factor representations of Cuntz–Krieger algebras associated with quasi-free states up to unitary equivalence. Furthermore, we realize these representations on concrete Hilbert spaces without using GNS construction. Free groups and their type II₁ factor representations are used in these realizations.      

Geometro-differential conception of extended particles and their quantum theory in de Sitter space

A geometro-differential quantum theory of extended particles is presented. The geometrical selling is that of Hilbert fiber bundles whose base manifolds are pseudo-Riemannian space-times of points which are interpreted as partial aspects of physical reality (the extended particle). The fibers are carrier spaces of induced (internal configuration and momentum) representations of the structural group (the de Sitter group here). Sections of these bundles are seen as physical representations of the particle, and their values in the fibers are interpreted as states of internally localized modes composing the particle. This geometrical structure is analogous to that of a geometro-stochastic quantization developed in recent years. The physical interpretation is a combination of those of an old functional quantum theory and of an induced representation scheme based on an interpretation of intertwining, between configuration and momentum representations, as localization, materialization, and propagation of particles. Our model is applied in two cases: (1) Induced representation is applied in both space-time and internal space: both have de Sitter symmetry whose connection is ignored. Intertwining is considered in both spaces in a composed fashion. (2) For generic spacetimes and when the connection is taken into account, intertwining is applied only in the internal space. Parallel transport is combined with intertwining for a redefinition of localization, materialization, and propagation (the latter in a path integral context inspired from geometro-stochastic quantization).     

Symmetric spaces and star representations: II. Causal symmetric spaces

We construct and identify star representations canonically associated with holonomy-reducible simple symplectic symmetric spaces. This leads a non-commutative geometric realization of the correspondence between causal symmetric spaces of Cayley-type and Hermitian symmetric spaces of tube-type.     

Self-similar random processes and infinite-dimensional configuration spaces

We discuss various infinite-dimensional configuration spaces that carry measures quasi-invariant under compactly supported diffeomorphisms of a manifold M corresponding to a physical space. Such measures allow the construction of unitary representations of the diffeomorphism group, which are important to nonrelativistic quantum statistical physics and to the quantum theory of extended objects in M = ?^d. Special attention is given to measurable structure and topology underlying measures on generalized configuration spaces obtained from self-similar random processes (both for d = 1 and d > 1), which describe infinite point configurations having accumulation points.     

Coherent states and symmetric spaces

Properties of system of the coherent states related to representations of the class I of principal series of the motion groups of symmetric spaces of rank 1 have been studied. It has been proved that such states are given by horospherical kernels and are the generalization of the plane waves for the case of symmetric spaces.     

Algebraic investigations in landau model of structural phase transitions

First we introduce the basic notions of the theory of permutation representations: stabilizers, orbits, stable subsets and strata. Then we consider the relation between permutation and linear representations which lead to some formulae connecting subduction coefficients, Clebsch-Gordan coefficients, and dimensions of stability spaces. This relation also leads to the concept of suborbits. Epikernels, the subgroups which are stabilizers of vectors of irreducible subspaces (either on the complex or on the real field) — are, studied and several theorems about them are proved. Further we consider the relation between epikernels, stability spaces and strata for subspaces irreducible on the real field as compared with subspaces irreducible on the complex field. Finally, the exomorphism is defined with use of permutation representations. The vectors of irreducible subspaces and corresponding epikernels (their stabilizers) for real ireps (representations irreducible on the real) of the classical crystal point groups are given in the appendix.     

Classical Mechanics in Hilbert Space,Part 2

We continue from Part 1. We will illustrate the general theory of Hamiltonian mechanics in the Lie group formalism. We then obtain the Hamiltonian formalism in the Hilbert spaces of square integrable functions on the symplectic spaces. We illustrate this general theory with several concrete examples, two of which are the representations of the Lorentz group and the Poincaré group with interactions.     

Oscillator-like unitary representations of non-compact groups with a jordan structure and the non-compact groups of supergravity

We give a general bosonic construction of oscillator-like unitary irreducible representations (UIR) of non-compact groups whose coset spaces with respect to their maximal compact subgroups are Hermitian symmetric. With the exception of E₇₍₇₎, they include all the non-compact invariance groups of extended supergravity theories in four dimensions. These representations have the remarkable property that each UIR is uniquely determined by an irreducible representation of the maximal compact subgroup. We study the connection between our construction, the Hermitian symmetric spaces and the Tits-Koecher construction of the Lie algebras of corresponding groups. We then give the bosonic construction of the Lie algebra ofE ₇₍₇₎ in SU(8), SO(8) and U(7) bases and study its properties. Application of our method toE ₇₍₇₎ leads to reducible unitary representations.Dedicated to Feza Gürsey on the occasion of his 60th birthdayAlexander von Humboldt Fellow, on leave from Physics Dept., Bogaziçi University, Istanbul/Turkey: work supported in part by TBTAK, The National Science and Technology Council of Turkey     

Classical Mechanics in Hilbert Space,Part 1

We consider the Hamilton formulation as well as the Hamiltonian flows on a symplectic (phase) space. These symplectic spaces are derivable from the Lie group of symmetries of the physical system considered. In Part 2 of this work, we then obtain the Hamiltonian formalism in the Hilbert spaces of square integrable functions on the symplectic spaces so obtained.     

Extensions of representations and cohomology

In this article we study the extensions of Banach space representations of a Lie group G. We introduce different spaces of 1-cohomology on G, or on its Lie algebra $\text{G}$, and make the connection between these spaces and the equivalence (or weak equivalence) classes of extensions.We characterize, from the properties of the 1-cohomology groups, the spaces of differentiable and analytic vectors of an extension and prove a kind of Whitehead's lemma.For Lie groups with a large compact subgroup K, we specialize to K-finite representations, and introduce and study Naimark equivalence of extensions.The results are applied to classify the extensions of the irreducible representations of $\text{G =}\text{SL}\text{(2,}\text{R}\text{)}$.     

Some remarks on quasi-invariant actions of loop groups and the group of diffeomorphisms of the circle

We construct the series of quasi-invariant actions of the group Diff of diffeomorphisms of the circle and loop groups on the functional spaces provided by non-Wiener Gauss measures. We construct some measures which can be considered as analogues of Haar measure for loop groups and the group Diff. These constructions allow us to construct series of representations of these groups including all known types of representations (higest weight representations, energy representations, almost invariant structures, etc.)