The Multi-form Generalised Gleason Theorem

Arising from investigations of decoherence functionals in the “histories” approach to quantum mechanics, the following result is given by the methods of [14]. Let A ₁ and A ₂ be von Neumann algebras without Type I ₂ direct summands and let P(E _j ), (j=1,2) be their lattices of projections. Let m\colon; P(A ₁)×(P(A ₂)→ℂ be a bounded quantum bi-measure. Then there is a unique bounded bilinear functional M on A ₁×A ₂ which extends m. In this note we use a different approach to establish a generalisation of this result to k-fold, vector valued, quantum multi-measures. This tool is needed for further investigations of decoherence functionals in quantum theory. Received: 10 April 1998 / Accepted: 17 April 1998     

On Gleason’s Theorem without Gleason

The original proof of Gleason’s Theorem is very complicated and therefore, any result that can be derived also without the use of Gleason’s Theorem is welcome both in mathematics and mathematical physics. In this paper we reprove some known results that had originally been proved by the use of Gleason’s Theorem, e.g. that on the quantum logic ℒ(H) of all closed subspaces of a Hilbert space H, dim H≥3, there is no finitely additive state whose range is countably infinite. In particular, if dim H=n, then on ℒ(H) there is a unique discrete state, namely m(A)=dim A/dim H, A∈ℒ(H). Dedicated to Pekka J. Lahti on the occasion of his 60th birthday. The paper has been supported by the Center of Excellence SAS–Physics of Information–I/2/2005, the grant VEGA No. 2/6088/26 SAV, by Science and Technology Assistance Agency under the contract APVV-0071-06, Bratislava, Slovakia.     

Homogeneous Decoherence Functionals¶in Standard and History Quantum Mechanics

General history quantum theories are quantum theories without a globally defined notion of time. Decoherence functionals represent the states in the history approach and are defined as certain bivariate complex-valued functionals on the space of all histories. However, in practical situations – for instance in the history formulation of standard quantum mechanics – there often is a global time direction and the homogeneous decoherence functionals are specified by their values on the subspace of homogeneous histories. In this work we study the analytic properties of (i) the standard decoherence functional in the history version of standard quantum mechanics and (ii) homogeneous decoherence functionals in general history theories. We restrict ourselves to the situation where the space of histories is given by the lattice of projections on some Hilbert space ℋ. Among other things we prove the non-existence of a finitely valued extension for the standard decoherence functional to the space of all histories, derive a representation for the standard decoherence functional as an unbounded quadratic form with a natural representation on a Hilbert space and prove the existence of an Isham–Linden–Schreckenberg (ILS) type representation for the standard decoherence functional. Received: 26 November 1998 / Accepted: 2 December 1998     

Quantum measures and states on Jordan algebras

A problem of Mackey for von Neumann algebras has been settled by the conjunction of the early work of Gleason and the recent advances of Christensen and Yeadon. We show that Mackey's conjecture holds in much greater generality. LetA be a JBW-algebra and letL be the lattice of all projections inA. A quantum measure onL is a countably additive map,m, fromL into the real numbers. Our results imply thatm always has a unique extension to a bounded linear functional onA, provided thatA has no TypeI ₂ direct summand.     

The Variational Principle for a Class of Asymptotically Abelian C*-Algebras

Let (A,α) be a C^*-dynamical system. We introduce the notion of pressure P _α(H) of the automorphism α at a self-adjoint operator H∈A. Then we consider the class of AF-systems satisfying the following condition: there exists a dense α-invariant *-subalgebra ? of A such that for all pairs a,b∈? the C^*-algebra they generate is finite dimensional, and there is p=p(a,b)∈ℕ such that [α ^j (a),b]= 0 for |j|≥p. For systems in this class we prove the variational principle, i.e. show that P _α(H) is the supremum of the quantities h _φ(α) −φ(H), where h _φ(α) is the Connes–Narnhofer–Thirring dynamical entropy of α with respect to the α-invariant state φ. If H∈A, and P _α(H) is finite, we show that any state on which the supremum is attained is a KMS-state with respect to a one-parameter automorphism group naturally associated with H. In particular, Voiculescu's topological entropy is equal to the supremum of h _φ(α), and any state of finite maximal entropy is a trace. Received: 19 April 2000 / Accepted: 14 June 2000     

Dynamicalr-matrices on the affinizations of arbitrary self-dual Lie algebras

We associate a dynamicalr-matrix with any such subalgebraL of a finite dimensional self-dual Lie algebraA for which the scalar product ofA remains nondegenerate onL and there exists a nonempty open subsetĽ ⊂L so that the restriction of (ad λ)εEnd(A) toL is invertible ∨λεĽ. Thisr-matrix is also well-defined ifL is the grade zero subalgebra of an affine Lie algebraA obtained from a twisted loop algebra based on a finite dimensional self-dual Lie algebraG. Application of evaluation homomorphisms to the twisted loop algebras yields spectral parameter dependentG ⊗G-valued dynamicalr-matrices that are generalizations of Felder’s ellipticr-matrices. Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001. This work was supported in part by the Hungarian National Science Fund (OTKA) under T034170.     

A Remark on Lp-Boundedness of Wave Operators¶for Two Dimensional Schrödinger Operators

Let H=−Δ+V be a two dimensional Schr?dinger operator with a real potential V(x) satisfying the decay condition , δ > 6. Let H ₀=−Δ. We show that the wave operators are bounded in L ^p (R ²) under the condition that H has no zero resonances or bound states. In this paper the condition , imposed in a previous paper (K. Yajima, Commun. Math. Phys. 208, 125–152 (1999)), is removed. Received: 13 September 2001 / Accepted: 15 October 2001     

Quantum Chains of Hopf Algebras with Quantum Double Cosymmetry

Given a finite dimensional C ^*-Hopf algebra H and its dual Ĥ we construct the infinite crossed product and study its superselection sectors in the framework of algebraic quantum field theory. is the observable algebra of a generalized quantum spin chain with H-order and Ĥ-disorder symmetries, where by a duality transformation the role of order and disorder may also appear interchanged. If is a group algebra then becomes an ordinary G-spin model. We classify all DHR-sectors of – relative to some Haag dual vacuum representation – and prove that their symmetry is described by the Drinfeld double . To achieve this we construct localized coactions and use a certain compressibility property to prove that they are universal amplimorphisms on . In this way the double can be recovered from the observable algebra as a universal cosymmetry. Received: 4 September 1995\,/\,Accepted: 3 December 1996     

Reduction of Quantum Systems with Arbitrary First Class Constraints and Hecke Algebras

We propose a method for reduction of quantum systems with arbitrary first-class constraints. An appropriate mathematical setting for the problem is the homology of associative algebras. For every such algebra A and subalgebra B with augmentation ɛ there exists a cohomological complex which is a generalization of the BRST one. Its cohomology is an associative graded algebra Hk ^*(A,B) which we call the Hecke algebra of the triple (A,B,ɛ). It acts in the cohomology space H ^*(B,V) for every left A module V. In particular the zeroth graded component $Hk^{0}(A,B)$ acts in the space of B invariants of $V$ and provides the reduction of the quantum system. Received: 15 June 1998 / Accepted: 25 January 1999     

L p -Boundedness of Wave Operators for¶Two Dimensional Schrödinger Operators

Let be the two dimensional Schr?dinger operator with the real valued potential V which satisfies the decay condition at infinity for . We show that the wave operators , , are bounded in for any 1<p<∞ under the condition that H has no zero bound states or zero resonance, extending the corresponding results for higher dimensions. As W _± intertwine H ₀ and the absolutely continuous part H P _ac of H : f(H)P _ac=W _± f(H ₀ )W _± ^* for any Borel function f on ℝ¹, this reduces the various L ^p -mapping properties of f(H)P _ac to those of f(H)₀), the convolution operator by the Fourier transform of the function f(ξ²). Received: 5 April 1999 / Accepted: 26 May 1999     

Pauli Operator and Aharonov–Casher Theorem¶ for Measure Valued Magnetic Fields

We define the two dimensional Pauli operator and identify its core for magnetic fields that are regular Borel measures. The magnetic field is generated by a scalar potential hence we bypass the usual A∈L ² _loc condition on the vector potential, which does not allow to consider such singular fields. We extend the Aharonov–Casher theorem for magnetic fields that are measures with finite total variation and we present a counterexample in case of infinite total variation. One of the key technical tools is a weighted L ² estimate on a singular integral operator. Received: 14 May 2001 / Accepted: 5 September 2001     

Möbius Transformations in Noncommutative Conformal Geometry

We study the projective linear group PGL ₂(A) associated with an arbitrary algebra A and its subgroups from the point of view of their action on the space of involutions in A. This action formally resembles M?bius transformations known from complex geometry. By specifying A to be an algebra of bounded operators in a Hilbert space H, we rediscover the M?bius group μ _ev (M) defined by Connes and study its action on the space of Fredholm modules over the algebra A. There is an induced action on the K-homology of A, which turns out to be trivial. Moreover, this action leads naturally to a simpler object, the polarized module underlying a given Fredholm module, and we discuss this relation in detail. Any polarized module can be lifted to a Fredholm module, and the set of different lifts forms a category, whose morphisms are given by generalized M?bius tranformations. We present an example of a polarized module canonically associated with the differentiable structure of a smooth manifold V. Using our lifting procedure we obtain a class of Fredholm modules characterizing the conformal structures on V. Fredholm modules obtained in this way are a special case of those constructed by Connes, Sullivan and Teleman. Received: 2 October 1997 / Accepted: 11 August 1998     

Partial Dynamical Systems and the KMS Condition

 Given a countably infinite 0–1 matrix A without identically zero rows, let 𝒪 _A be the Cuntz–Krieger algebra recently introduced by the authors and 𝒯 _A be the Toeplitz extension of 𝒪 _A , once the latter is seen as a Cuntz–Pimsner algebra, as recently shown by Szymański. We study the KMS equilibrium states of C ^* -dynamical systems based on 𝒪 _A and 𝒯 _A , with dynamics satisfying for the canonical generating partial isometries s _x and arbitrary real numbers N _x > 1. The KMS_β states on both 𝒪 _A and 𝒯 _A are completely characterized for certain values of the inverse temperature β, according to the position of β relative to three critical values, defined to be the abscissa of convergence of certain Dirichlet series associated to A and the N(x). Our results for 𝒪 _A are derived from those for 𝒯 _A by virtue of the former being a covariant quotient of the latter. When the matrix A is finite, these results give theorems of Olesen and Pedersen for 𝒪 _n and of Enomoto, Fujii and Watatani for 𝒪 _A as particular cases. Received: 21 November 2000 / Accepted: 31 May 2002 Published online: 22 November 2002 RID="*" ID="*" Partially supported by CNP_q RID="**, ***"     

Renormalization in Quantum Field Theory and the Riemann--Hilbert Problem II: The β-Function, Diffeomorphisms and the Renormalization Group

We showed in Part I that the Hopf algebra ℋ of Feynman graphs in a given QFT is the algebra of coordinates on a complex infinite dimensional Lie group G and that the renormalized theory is obtained from the unrenormalized one by evaluating at ɛ= 0 the holomorphic part γ₊(ɛ) of the Riemann–Hilbert decomposition γ₋(ɛ)^{− 1}γ₊(ɛ) of the loop γ(ɛ)∈G provided by dimensional regularization. We show in this paper that the group G acts naturally on the complex space X of dimensionless coupling constants of the theory. More precisely, the formula g ₀=gZ ₁ Z ₃ ^−3/2 for the effective coupling constant, when viewed as a formal power series, does define a Hopf algebra homomorphism between the Hopf algebra of coordinates on the group of formal diffeomorphisms to the Hopf algebra ℋ. This allows first of all to read off directly, without using the group G, the bare coupling constant and the renormalized one from the Riemann–Hilbert decomposition of the unrenormalized effective coupling constant viewed as a loop of formal diffeomorphisms. This shows that renormalization is intimately related with the theory of non-linear complex bundles on the Riemann sphere of the dimensional regularization parameter ɛ. It also allows to lift both the renormalization group and the β-function as the asymptotic scaling in the group G. This exploits the full power of the Riemann–Hilbert decomposition together with the invariance of γ₋(ɛ) under a change of unit of mass. This not only gives a conceptual proof of the existence of the renormalization group but also delivers a scattering formula in the group G for the full higher pole structure of minimal subtracted counterterms in terms of the residue. Received: 21 March 2000 / Accepted: 3 October 2000     

A diffusion model of chain-branching reaction of the explosive decomposition of heavy metal azides

A diffusion model of a solid-phase chain reaction of explosive decomposition of heavy metal azides was developed. The dimensional effects of initiation of the reaction were examined: the dependence of the critical fluence of initiation on the microcrystal size H(R) and on the irradiated zone diameter H(d). It was demonstrated that the diffusion model of the chain reaction closely describes the measured H(R) dependence at diffusion coefficients of D ∼ 0.2–0.3 cm²/s, values that correspond to experimentally measured mobility of electronic charge carriers of μ ∼ 10 cm²/(V s). To account for the measured H(d) dependence and the reaction front propagation velocity (V = 1.2 km/s), it is necessary that the diffusion coefficient be three orders of magnitude higher than the experimentally determined value. That the H(R) and H(d) dependences cannot be quantitatively described simultaneously is indicative of the underlying mechanisms of energy transfer being different.     

Sequential Product and Jordan Product of Quantum Effects

The quantum effects for a physical system can be described by the set E(H)\mathcal{E(H)} of positive operators on a complex Hilbert space H\mathcal{H} that are bounded above by the identity operator I. We denote the set of sharp effects by P(H){\mathcal{P(H) }}. For A,B ? E(H)A,B\in\mathcal{E(H)}, the operation of sequential product A°B=A^\frac12BA^\frac12A\circ B=A^{\frac{1}{2}}BA^{\frac{1}{2}} was proposed as a model for sequential quantum measurements. Denote by A*B=\fracAB+BA2A\ast B=\frac{AB+BA}{2} the Jordan product of A,B ? E(H)A,B\in\mathcal{E(H)}. The main purpose of this note is to study some of the algebraic properties of the Jordan product of effects. Many of our results show that algebraic conditions on A∗B imply that A and B commute for the usual operator product. And there are many common properties between Jordan product and sequential product of effects. For example, if A∗B satisfies certain associative laws, then AB=BA. Moreover, A*B ? P(H)A\ast B\in{\mathcal{P(H) }} if and only if A°B ? P(H)A\circ B\in{\mathcal{P(H)}}.     

On Majorana Representations of A6 and A7

The Majorana representations of groups were introduced in Ivanov (The Monster Group and Majorana Involutions, 2009) by axiomatising some properties of the 2A-axial vectors of the 196 884-dimensional Monster algebra, inspired by the sensational classification of such representations for the dihedral groups achieved by Sakuma (Int Math Res Notes, 2007). This classification took place in the heart of the theory of Vertex Operator Algebras and expanded earlier results by Miyamoto (J Alg 268:653–671, 2003). Every subgroup G of the Monster which is generated by its intersection with the conjugacy class of 2A-involutions possesses the (possibly unfaithful) Majorana representation obtained by restricting to G the action of the Monster on its algebra. This representation of G is said to be based on an embedding of G in the Monster. So far the Majorana representations have been classified for the groups G isomorphic to the symmetric group S ₄ of degree 4 (Ivanov et al. in J Alg 324:2432–2463, 2010), the alternating group A ₅ of degree 5 (Ivanov AA, Seress á in Majorana Representations of A ₅, 2010), and the general linear group GL ₃(2) in dimension 3 over the field of two elements (Ivanov AA, Shpectorov S in Majorana Representations of L ₃(2), 2010). All these representations are based on embeddings in the Monster of either the group G itself or of its direct product with a cyclic group of order 2. The dimensions and shapes of these representations are given in the following table:     

The number of independent traces and supertraces on symplectic reflection algebras

It is shown that A:= H₁, _η (G), the sympectic reflection algebra over ?, has T_G independent traces, where T_G is the number of conjugacy classes of elements without eigenvalue 1 belonging to the finite group G ? Sp(2N) ? End(?^2N) generated by the system of symplectic reflections.

Simultaneously, we show that the algebra A, considered as a superalgebra with a natural parity, has S_G independent supertraces, where S_G is the number of conjugacy classes of elements without eigenvalue -1 belonging to G.

We consider also A as a Lie algebra A^L and as a Lie superalgebra A^S.

It is shown that if A is a simple associative algebra, then the supercommutant [A^S, A^S] is a simple Lie superalgebra having at least S_G independent supersymmetric invariant non-degenerate bilinear forms, and the quotient [A^L, A^L]/([A^L, A^L] ∩ ?) is a simple Lie algebra having at least T_G independent symmetric invariant non-degenerate bilinear forms.     

Controllability Issues for Continuous-Spectrum Systems and Ensemble Controllability of Bloch Equations

We study the controllability of the Bloch equation, for an ensemble of non interacting half-spins, in a static magnetic field, with dispersion in the Larmor frequency. This system may be seen as a prototype for infinite dimensional bilinear systems with continuous spectrum, whose controllability is not well understood. We provide several mathematical answers, with discrimination between approximate and exact controllability, and between finite time or infinite time controllability: this system is not exactly controllable in finite time T with bounded controls in L ²(0, T), but it is approximately controllable in L ^∞ in finite time with unbounded controls in L^￥_loc([0,+￥)){L^{\infty}_{loc}([0,+\infty))}. Moreover, we propose explicit controls realizing the asymptotic exact controllability to a uniform state of spin + 1/2 or −1/2.