Toward a Finite-Dimensional Formulation of Quantum Field Theory

Rules of quantization and equations of motion for a finite-dimensional formulation of quantum field theory are proposed which fulfill the following properties: (a) Both the rules of quantization and the equations of motion are covariant; (b) the equations of evolution are second order in derivatives and first order in derivatives of the spacetime coordinates; and (c) these rules of quantization and equations of motion lead to the usual (canonical) rules of quantization and the (Schrödinger) equation of motion of quantum mechanics in the particular case of mechanical systems. We also comment briefly on further steps to fully develop a satisfactory quantum field theory and the difficuties which may be encountered when doing so.     

Langlands Duality for Finite-Dimensional Representations of Quantum Affine Algebras

We describe a correspondence (or duality) between the q-characters of finite-dimensional representations of a quantum affine algebra and its Langlands dual in the spirit of Frenkel and Hernandez (Math Ann, to appear) and Frenkel and Reshetikhin (Commun Math Phys 197(1):1?C32, 1998). We prove this duality for the Kirillov?CReshetikhin modules and their irreducible tensor products. In the course of the proof we introduce and construct ??interpolating (q, t)-characters?? depending on two parameters which interpolate between the q-characters of a quantum affine algebra and its Langlands dual.     

Combinatorics of q-Characters of Finite-Dimensional Representations of Quantum Affine Algebras

We study finite-dimensional representations of quantum affine algebras using q-characters. We prove the conjectures from [FR] and derive some of their corollaries. In particular, we prove that the tensor product of fundamental representations is reducible if and only if at least one of the pairwise normalized R-matrices has a pole. Received: 16 December 1999 / Accepted: 12 July 2000     

The Formulation of Quantum Mechanics in Terms of Nuclear Algebras

In this work we analyze the convenience ofnuclear barreled b*-algebras as a better mathematicalframework for the formulation of quantum principles thanthe usual algebraic formalism in terms of C*-algebras. Unbounded operators on Hilbert spaces have anabstract counterpart in our approach. The main resultsof the C*-algebra theory remain valid. We demonstrate anextremal decomposition theorem, an adequate functional representation theorem, and anextension of the classical GNS theorem.     

Quantum Implication Algebras

Quantum implication algebras without complementation are formulated with the same axioms for all five quantum implications. Previous formulations of orthoimplication, orthomodular implication, and quasi-implication algebras are analyzed and put in perspective to each other and our results.     

Quantum Quasi-Shuffle Algebras

We establish some properties of quantum quasi-shuffle algebras. They include the necessary and sufficient condition for the construction of the quantum quasi-shuffle product, the universal property, and the commutativity condition. As an application, we use the quantum quasi-shuffle product to construct a linear basis of T(V), for a special kind of Yang–Baxter algebras (V, m, σ).     

Complementarity in Categorical Quantum Mechanics

We relate notions of complementarity in three layers of quantum mechanics: (i) von Neumann algebras, (ii) Hilbert spaces, and (iii) orthomodular lattices. Taking a more general categorical perspective of which the above are instances, we consider dagger monoidal kernel categories for (ii), so that (i) become (sub)endohomsets and (iii) become subobject lattices. By developing a ‘point-free’ definition of copyability we link (i) commutative von Neumann subalgebras, (ii) classical structures, and (iii) Boolean subalgebras.     

A Link between Quantum Logic and Categorical Quantum Mechanics

Abramsky and Coecke (Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, pp. 415–425, IEEE Comput. Soc., New York, 2004) have recently introduced an approach to finite dimensional quantum mechanics based on strongly compact closed categories with biproducts. In this note it is shown that the projections of any object A in such a category form an orthoalgebra Proj  A. Sufficient conditions are given to ensure this orthoalgebra is an orthomodular poset. A notion of a preparation for such an object is given by Abramsky and Coecke, and it is shown that each preparation induces a finitely additive map from Proj  A to the unit interval of the semiring of scalars for this category. The tensor product for the category is shown to induce an orthoalgebra bimorphism Proj  A×Proj  B→Proj (A ⊗ B) that shares some of the properties required of a tensor product of orthoalgebras. These results are established in a setting more general than that of strongly compact closed categories. Many are valid in dagger biproduct categories, others require also a symmetric monoidal tensor compatible with the dagger and biproducts. Examples are considered for several familiar strongly compact closed categories.     

Glafka-2004: Categorical Quantum Gravity

A synopsis-cum-update of work in the past half-decade or so on applying the algebraico-categorical concepts, technology and general philosophy of Abstract Differential Geometry (ADG) to various issues in current classical and quantum gravity research is presented. The exposition is mainly discursive, with conceptual, interpretational and philosophical matters emphasized throughout, while their formal technical-mathematical underpinnings have been left to the original papers. The general position is assumed that Quantum Gravity is in need of a new mathematical, novel physical concepts and principles introducing, framework in which old and current problems can be reformulated, readdressed and potentially retackled afresh. It is suggested that ADG can qualify as such a theoretical framework.Paper version of a talk given at the Glafka–2004: Iconoclastic Approaches to Quantum Gravity international theoretical physics conference, held in Athens, Greece (summer 2004).     

Commutativity of Quantum Family Algebras

Recently, A. A. Kirillov introduced an important notion of classical and quantum family algebras. Here the criterion of commutativity is given. The quantum eigenvalues of are computed.     

Duality Between Quantum Symmetric Algebras

Using certain pairings of couples, we obtain a large class of two-sided non-degenerated graded Hopf pairings for quantum symmetric algebras.      

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.     

Star Products and Quantum Algebras

I show explicitly that the star product on atriangular Poisson Lie group leads to a quantum algebrastructure (triangular Hopf algebra) on the quantizedenveloping algebra of the Lie algebra of the Lie group, and that equivalent star-productsgenerate isomorphic quantum algebras.     

Quantum Groups and Fuss--Catalan Algebras

The categories of representations of compact quantum groups of automorphisms of certain inclusions of finite dimensional ℂ^*-algebras are shown to be isomorphic to the categories of Fuss–Catalan diagrams. Received: 9 March 2001 / Accepted: 12 November 2001     

Non-Commutative Algebras and Quantum Structures

We present a survey on pseudo-effect algebras and pseudo MV-algebras, which generalize effect algebras and MV-algebras by dropping the assumption on commutativity. A non-commutative logic is nowadays used even in programming languages. We show when a pseudo-effect algebra E is an interval in a unital po-group. This is possible, e.g. if E satisfies a Riesz-type decomposition property, i.e. another kind of distributivity with respect to addition. Every pseudo MV-algebra is an interval in a unital ℓ-group. We study a case when compatibility can be expressed by a pseudo MV-structure, i.e. when E can be covered by blocks being pseudo MV-algebras. Finally, we study the state space of such structures.     

Modules-at-Infinity for Quantum Vertex Algebras

This is a sequel to [Li4] and [Li5] in a series to study vertex algebra-like structures arising from various algebras such as quantum affine algebras and Yangians. In this paper, we study two versions of the double Yangian , denoted by DY _q (sl ₂) and with q a nonzero complex number. For each nonzero complex number q, we construct a quantum vertex algebra V _q and prove that every DY _q (sl ₂)-module is naturally a V _q -module. We also show that -modules are what we call V _q -modules-at-infinity. To achieve this goal, we study what we call -local subsets and quasi-local subsets of for any vector space W, and we prove that any -local subset generates a (weak) quantum vertex algebra and that any quasi-local subset generates a vertex algebra with W as a (left) quasi module-at-infinity. Using this result we associate the Lie algebra of pseudo-differential operators on the circle with vertex algebras in terms of quasi modules-at-infinity.     

Quantum Double for Quasi-Hopf Algebras

We introduce a quantum double quasi-triangular quasi-Hopf algebra D(H) associated to any quasi-Hopf algebra H. The algebra structure is a cocycle double cross-product. We use categorical reconstruction methods. As an example, we recover the quasi-Hopf algebra of Dijkgraaf, Pasquier and Roche as the quantum double D(G) associated to a finite group G and group 3-cocycle .     

Quantum Observables and Effect Algebras

We study observables on monotone σ-complete effect algebras. We find conditions when a spectral resolution implies existence of the corresponding observable. We characterize sharp observables of a monotone σ-complete homogeneous effect algebra using its orthoalgebraic skeleton. In addition, we study compatibility in orthoalgebras and we show that every orthoalgebra satisfying RIP is an orthomodular poset.