The physical interpretation of partial traces: Two nonstandard views

Mixed states are introduced in physics to express our ignorance about the actual state of a physical system and are represented in standard quantum mechanics by density operators. Such operators also appear if we consider a (pure) entangled state of a compound system Ω and take partial traces on the projection operator representing it. But because the coefficients in the convex sums expressing them never bear the ignorance interpretation in this case, they represent not mixed states (proper mixtures) but improper mixtures of the subsystems. Hence, states cannot be attributed to the subsystems of a compound physical system in an entangled state (the subentity problem). We discuss two alternative proposals that can be developed in the Brussels and the Lecce approaches. We firstly summarize the general framework provided by the Brussels approach, which suggests that improper mixtures can be regarded as new pure states. We then show that improper mixtures can also be regarded as true (but nonpure) states according to the Lecce approach. Despite their different terminologies, the two proposals seem compatible. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 2, pp. 248–264, August, 2007.     

Free group representations from vector-valued multiplicative functions,I

Let Γ denote a noncommutative free group and let Ω stand for its boundary. We construct a large class of unitary representations of Γ. This class contains many previously studied representations, and is closed under several natural operations. Each of the constructed representations is in fact a representation of Γ ⋉_λ C(Ω). We prove here that each of them is irreducible as a representation of Γ ⋉_λ C(Ω). Actually, as will be shown in further work, each of them is irreducible as a representation of Γ, or is the direct sum of exactly two irreducible, inequivalent Γ-representations. This research was supported by the Italian CNR.     

A quantum spin system with random interactions I

We study a quantum spin glass as a quantum spin system with random interactions and establish the existence of a family of evolution groups {τ_t(ω)}_ω∈/Ω of the spin system. The notion of ergodicity of a measure preserving group of automorphisms of the probability space Ω, is used to prove the almost sure independence of the Arveson spectrum Sp(τ(ω)) of τ_t(ε). As a consequence, for any family of (τ(ω),β) — KMS states {ρ(ω)}, the spectrum of the generator of the group of unitaries which implement τ(ω) in the GNS representation is also almost surely independent of ω.     

Space of <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript>-smooth skew products of maps of an interval

Using the notions of an Ω-function and of functions suitable for an Ω-function, we show that the space of C ¹ -smooth skew products of maps of an interval such that the quotient map of each is Ω-stable in the space of C ¹ -smooth maps of a closed interval into itself and has a type ≻ 2 ^∞ (i.e., contains a periodic orbit with the period not equal to a power of 2) can be represented as a union of four nonempty pairwise nonintersecting subspaces. We give examples of maps belonging to each of the identified subspaces.     

Orbit equivalence and permutation groups defined by unordered relations

For a set Ω an unordered relation on Ω is a family R of subsets of Ω. If R is such a relation we let G(R)\mathcal{G}(R) be the group of all permutations on Ω that preserve R, that is g belongs to G(R)\mathcal{G}(R) if and only if x∈R implies x ^g ∈R. We are interested in permutation groups which can be represented as G=G(R)G=\mathcal{G}(R) for a suitable unordered relation R on Ω. When this is the case, we say that G is defined by the relation R, or that G is a relation group. We prove that a primitive permutation group ≠Alt(Ω) and of degree ≥11 is a relation group. The same is true for many classes of finite imprimitive groups, and we give general conditions on the size of blocks of imprimitivity, and the groups induced on such blocks, which guarantee that the group is defined by a relation.     

Nonexistence of Proper Holomorphic Maps Between Certain Classical Bounded Symmetric Domains

The author,motivated by his results on Hermitian metric rigidity,conjectured in [4] that a proper holomorphic mapping f：Ω→Ω′from an irreducible bounded symmetric domainΩof rank≥2 into a bounded symmetric domainΩ′is necessarily totally geodesic provided that r′：=rank（Ω′）≤rank（Ω）：=r.The Conjecture was resolved in the affirmative by I.-H.Tsai [8].When the hypothesis r′≤r is removed,the structure of proper holomorphic maps f：Ω→Ω′is far from being understood,and the complexity in studying such maps depends very much on the difference r′-r,which is called the rank defect.The only known nontrivial non-equidimensional structure theorems on proper holomorphic maps are due to Z.-H.Tu [10],in which a rigidity theorem was proven for certain pairs of classical domains of type I,which implies nonexistence theorems for other pairs of such domains.For both results the rank defect is equal to 1,and a generaliza- tion of the rigidity result to cases of higher rank defects along the line of arguments of [10] has so far been inaccessible. In this article, the author produces nonexistence results for infinite series of pairs of （Ω→Ω′） of irreducible bounded symmetric domains of type I in which the rank defect is an arbitrarily prescribed positive integer. Such nonexistence results are obtained by exploiting the geometry of characteristic symmetric subspaces as introduced by N. Mok and L-H Tsai [6] and more generally invariantly geodesic subspaces as formalized in [8]. Our nonexistence results motivate the formulation of questions on proper holomorphic maps in the non-equirank case.     

Quantum hyperbolic invariants for diffeomorphisms of small surfaces

An earlier article [Bonahon, F., Liu, X. B.: Representations of the quantum Teichmüller space and invariants of surface diffeomorphisms. Geom. Topol., 11, 889-937 (2007)] introduced new invariants for pseudo-Anosov diffeomorphisms of surface, based on the representation theory of the quantum Teichmu¨ller space. We explicitly compute these quantum hyperbolic invariants in the case of the 1-puncture torus and the 4-puncture sphere.     

Categorified Quantum sl(2) and Equivariant Cohomology of Iterated Flag Varieties

A 2-category was introduced in that categorifies Lusztig’s integral version of quantum sl(2). Here we construct for each positive integer N a representation of this 2-category using the equivariant cohomology of iterated flag varieties. This representation categorifies the irreducible (N + 1)-dimensional representation of quantum sl(2).     

Conjugation of injections by permutations

Let Ω be a countably infinite set, Inj(Ω) the monoid of all injective endomaps of Ω, and Sym(Ω) the group of all permutations of Ω. Also, let f,g,h∈Inj(Ω) be any three maps, each having at least one infinite cycle. (For instance, this holds if f,g,h∈Inj(Ω)∖Sym(Ω).) We show that there are permutations a,b∈Sym(Ω) such that h=afa ⁻¹ bgb ⁻¹ if and only if |Ω∖(Ω)f|+|Ω∖(Ω)g|=|Ω∖(Ω)h|. We also prove a generalization of this statement that holds for infinite sets Ω that are not necessarily countable.     

D p spaces on bounded symmetric domains ofC n

In this paper, the spaceD ^p(Ω) of functions holomorphic on bounded symmetric domain ofC ⁿ is defined. We prove thatH ^p(Ω)⊂D ^p(Ω) if 0<p≤2, andD ^p(Ω)⊂H ^p(Ω) ifp≥2, and both the inclusions are proper. Further, we find that some theorems onH ^p(Ω) can be extended to a wider classD ^p(Ω) for 0<p≤2.     

Characterization of standard embeddings between complex Grassmannians by means of varieties of minimal rational tangents

In 1993,Tsal proved that a proper holomorphic mapping f:Ω→Ω' from an irreducible bounded symmetric domainΩof rank≥2 into a bounded symmetric domainΩ' is necessarily totally geodesic provided that r':=rank(Ω')≤rank(Ω):= r,proving a conjecture of the author's motivated by Hermitian metric rigidity.As a first step in the proof,Tsai showed that df preserves almost everywhere the set of tangent vectors of rank 1.Identifying bounded symmetric domains as open subsets of their compact duals by means of the Borel embedding,this means that the germ of f at a general point preserves the varieties of minimal rational tangents(VMRTs). In another completely different direction Hwang-Mok established with very few exceptions the Cartan- Fubini extension priniciple for germs of local biholomorphisms between Fano manifolds of Picard num- ber 1,showing that the germ of map extends to a global biholomorphism provided that it preserves VMRTs.We propose to isolate the problem of characterization of special holomorphic embeddings between Fano manifolds of Picard number 1,especially in the case of classical manifolds such as ratio- nal homogeneous spaces of Picard number 1,by a non-equidimensional analogue of the Cartan-Fubini extension principle.As an illustration we show along this line that standard embeddings between com- plex Grassmann manifolds of rank≤2 can be characterized by the VMRT-preserving property and a non-degeneracy condition,giving a new proof of a result of Neretin's which on the one hand paves the way for far-reaching generalizations to the context of rational homogeneous spaces and more generally Fano manifolds of Picard number 1,on the other hand should be applicable to the study of proper holomorphic mappings between bounded domains carrying some form of geometric structures.     

Limit algebras of differential forms in non-commutative geometry

Given a C*-normed algebra A which is either a Banach *-algebra or a Frechet *-algebra, we study the algebras Ω_∞ A and Ω_ε A obtained by taking respectively the projective limit and the inductive limit of Banach *-algebras obtained by completing the universal graded differential algebra Ω*A of abstract non-commutative differential forms over A. Various quantized integrals on Ω_∞ A induced by a K-cycle on A are considered. The GNS-representation of Ω_∞ A defined by a d-dimensional non-commutative volume integral on a d ⁺-summable K-cycle on A is realized as the representation induced by the left action of A on Ω*A. This supplements the representation A on the space of forms discussed by Connes (Ch. VI.1, Prop. 5, p. 550 of [C]).     

Quantum volterra model and the universalR-matrix

We prove that an integrable system solved by the quantum inverse scattering method can be described by a purely algebraic object (universal R-matrix) and a proper algebraic representation. For the quantum Volterra model, we construct the L-operator and the fundamental R-matrix from the universal R-matrix for the quantum affine algebra U_q(ŝl₂) and the q-oscillator representation for it. Thus, there is an equivalence between an integrable system with the symmetry algebra A and the representation of this algebra. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 3, pp. 384–396, December, 1997.     

Measure free martingales and martingale measures

Let T ⊂ ℝ be a countable set, not necessarily discrete. Let f _t , t ∈ T, be a family of real-valued functions defined on a set Ω. We discuss conditions which imply that there is a probability measure on Ω under which the family f _t , t ∈ T, is a martingale.     

Existence of Universal Taylor Series for Nonsimply Connected Domains

It is known that, for any simply connected proper subdomain Ω of the complex plane and any point ζ in Ω, there are holomorphic functions on Ω that possess “universal” Taylor series expansions about ζ; that is, partial sums of the Taylor series approximate arbitrary polynomials on arbitrary compacta in ℂ\Ω that have connected complement. This paper shows, for nonsimply connected domains Ω, how issues of capacity, thinness and topology affect the existence of holomorphic functions on Ω that have universal Taylor series expansions about a given point.     

Interactive epistemology II: Probability

Formal Interactive Epistemology deals with the logic of knowledge and belief when there is more than one agent or “player.” One is interested not only in each person's knowledge and beliefs about substantive matters, but also in his knowledge and beliefs about the others' knowledge and beliefs. This paper examines two parallel approaches to the subject. The first is the semantic, in which knowledge and beliefs are represented by a space Ω of states of the world, and for each player i, partitions ℐ_i of Ω and probability distributions π_i(·; ω) on Ω for each state ω of the world. The atom of ℐ_i containing a given state ω represents i's knowledge at that state – the set of those other states that i cannot distinguish from ω; the probability distributions π_i(·; ω) represents i's beliefs at the state ω. The second is the syntactic approach, in which beliefs are embodied in sentences constructed according to certain syntactic rules. This paper examines the relation between the two approaches, and shows that they are in a sense equivalent.  In game theory and economics, the semantic approach has heretofore been most prevalent. A question that often arises in this connection is whether, in what sense, and why the space Ω, the partitions ℐ_i, and the probability distributions π_i(·; ω) can be taken as given and commonly known by the players. An answer to this question is provided by the syntactic approach.     

On representation of the solution of the Neumann problem in a domain with peak by the harmonic simple layer potential

The problem of finding a solution of the Neumann problem for the Laplacian in the form of a simple layer potential Vρ with unknown density ρ is known to be reducible to a boundary integral equation of the second kind to be solved for density. The Neumann problem is examined in a bounded n-dimensional domain Ω⁺ (n > 2) with a cusp of an outward isolated peak either on its boundary or in its complement Ω⁻ = R ⁿ \Ω⁺. Let Γ be the common boundary of the domains Ω^±, Tr(Γ) be the space of traces on Γ of functions with finite Dirichlet integral over R ⁿ , and Tr(Γ)* be the dual space to Tr(Γ). We show that the solution of the Neumann problem for a domain Ω⁻ with a cusp of an inward peak may be represented as Vρ⁻, where ρ⁻ ∈ Tr(Γ)* is uniquely determined for all Ψ⁻ ∈ Tr(Γ)*. If Ω⁺ is a domain with an inward peak and if Ψ⁺ ∈ Tr(Γ)*, Ψ⁺ ⊥ 1, then the solution of the Neumann problem for Ω⁺ has the representation u ⁺ = Vρ⁺ for some ρ⁺ ∈ Tr(Γ)* which is unique up to an additive constant ρ₀, ρ₀ = V ⁻¹(1). These results do not hold for domains with outward peak.     

Massless Interacting Scalar Quantum Fields in deSitter Spacetime

We generalize key aspects of gr-qc/1010.5367 (and also gr-qc/1010.5327) to the case of massless lf²ⁿ{\lambda \phi^{2n}} quantum field theory on deSitter spacetime. As in that paper, our key objective is to derive a suitable “Mellin-Barnes-type” representation of deSitter correlation functions in a deSitter-invariant state, which holds to arbitrary orders in perturbation theory, and which incorporates renormalization. The representation is suitable for the study of large distance/time properties of correlation functions. It is arrived at via an analytic continuation from the corresponding objects on the sphere, and, as in the massive case, relies on the use of graph-polynomials and their properties, as well as other tools. However, the perturbation expansion is organized somewhat differently in the massless case, due to the well-known subtleties associated with the “zero-mode” of the quantum field. In particular, the correlation functions do not possess a well-defined limit as the self-coupling constant of the field goes to zero, reflecting the well-known non-existence of a deSitter invariant state in the free massless scalar theory. We establish that generic correlation functions cannot grow more than polynomially in proper time for large time-like separations of the points. Our results thus leave open the possibility of quantum-induced IR-instabilities of deSitter spacetime on very large time-scales.