Reviews in Mathematical Physics

For any densely defined, lower semi-continuous trace τ on a C*-algebra A with mutually commuting C*-subalgebras A₁, A₂, … A_n, and a convex function f of n variables, we give a short proof of the fact that the function (x₁, x₂, …, x_n)→ τ (f (x₁, x₂, …, x_n)) is convex on the space . If furthermore the function f is log-convex or root-convex, so is the corresponding trace function. We also introduce a generalization of log-convexity and root-convexity called ℓ-convexity, show how it applies to traces, and give some examples. In particular we show that the Kadison–Fuglede determinant is concave and that the trace of an operator mean is always dominated by the corresponding mean of the trace values.

We present two types of result for approximately inner one-parameter automorphism groups (referred to as AI flows hereafter) of separable C*-algebras. First, if there is an irreducible representation π of a separable C*-algebra A such that π(A) does not contain non-zero compact operators, then there is an AI flow α such that π is α-covariant and α is far from uniformly continuous in the sense that α induces a flow on π(A) which has full Connes spectrum. Second, if α is an AI flow on a separable C*-algebra A and π is an α-covariant irreducible representation, then we can choose a sequence (h_n) of self-adjoint elements in A such that α_t is the limit of inner flows Ad e^ith_n and the sequence π(e^ith_n) of one-parameter unitary groups (referred to as unitary flows hereafter) converges to a unitary flow which implements α in π. This latter result will be extended to cover the case of weakly inner type I representations. In passing we shall also show that if two representations of a separable simple C*-algebra on a separable Hilbert space generate the same von Neumann algebra of type I, then there is an approximately inner automorphism which sends one into the other up to equivalence.

We prove the central limit theorem for Gibbs states and ground states of quasifree Fermions (bilinear Hamiltonians) and those of the off critical XY model on a one-dimensional integer lattice.

This paper discusses entropy production in nonequilibrium steady states for infinite quantum spin systems. Rigorous results have been obtained recently in this area, but a physical discussion shows that some questions of principle remain to be clarified.

Modular invariants satisfy remarkable fusion rules. Let Z be a modular invariant associated to a braided subfactor N ⊂ M. The decomposition of the non-normalized modular invariants ZZ* and Z*Z into sums of normalized modular invariants is related to the decomposition of the full induced M-M system of sectors.

The set of equivalence classes of Longo's Q-systems is shown to serve as a right subfactor analogue of the second cohomology. This "cohomology" is computed for several classes of subfactors, which tells us if a subfactor is uniquely determined up to inner conjugacy by the associated canonical endomorphism. As a byproduct of our analysis, two different groups of order 64 are shown to possess the same representation category as an abstract tensor category. Thus, this category has two permutation symmetries with non-isomorphic groups via the Doplicher–Roberts duality.

We propose a framework for the free field construction of algebras of local observables which uses as an input the Bisognano–Wichmann relations and a representation of the Poincaré group on the one-particle Hilbert space. The abstract real Hilbert subspace version of the Tomita–Takesaki theory enables us to bypass some limitations of the Wigner formalism by introducing an intrinsic spacetime localization. Our approach works also for continuous spin representations to which we associate a net of von Neumann algebras on spacelike cones with the Reeh–Schlieder property. The positivity of the energy in the representation turns out to be equivalent to the isotony of the net, in the spirit of Borchers theorem. Our procedure extends to other spacetimes homogeneous under a group of geometric transformations as in the case of conformal symmetries and of de Sitter spacetime.

Let H be a compact quantum group with faithful Haar measure and bounded counit. If α is an action of H on a C*- algebra A, we show that A is nuclear if and only if the fixed-point subalgebra A^α is nuclear. As a consequence H is a nuclear C*-algebra.

'ax + b' is the group of affine transformations of the real line R. In quantum version ab = q²ba, where q² = e^{-i ℏ} is a number of modulus 1. The main problem of constructing quantum deformation of this group on the C*-level consists in non-selfadjointness of Δ(b) = a ⊗ b + b ⊗ I. This problem is overcome by introducing (in addition to a and b) a new generator β commuting with a and anticommuting with b. β (or more precisely β ⊗ β) is used to select a suitable selfadjoint extension of a ⊗ b + b ⊗ I. Furthermore we have to assume that , where k = 0,1,2, ·. In this case, q is a root of 1.

To construct the group, we write an explicit formula for the Kac–Takesaki operator W. It is shown that W is a manageable multiplicative unitary in the sense of [3,19]. Then using the general theory we construct a C*-algebra A and a comultiplication Δ ∈ Mor (A,A ⊗ A). A should be interpreted as the algebra of all continuous functions vanishing at infinity on quantum 'ax + b'-group. The group structure is encoded by Δ. The existence of coinverse also follows from the general theory [19].

A general form of the "Wick rotation", starting from imaginary-time Green functions of quantum-mechanical systems in thermal equilibrium at positive temperature, is established. Extending work of H. Araki, the rôle of the KMS condition and of an associated anti-unitary symmetry operation, the "modular conjugation", in constructing analytic continuations of Green functions from real- to imaginary times, and back, is clarified.

The relationship between the KMS condition for the vacuum with respect to Lorentz boosts, on one hand, and the spin-statistics connection and the PCT theorem, on the other hand, in local, relativistic quantum field theory is recalled.

General results on the reconstruction of local quantum theories in various non-trivial gravitational backgrounds from "Euclidian amplitudes" are presented. In particular, a general form of the KMS condition is proposed and applied, e.g., to the Unruh- and the Hawking effects.

The assignment of the local observables in the vacuum sector, fulfilling the standard axioms of local quantum theory, is known to determine uniquely a compact group G of gauge transformations of the first kind together with a central involutive element k of G, and a complete normal algebra of fields carrying the localizable charges, on which k defines the Bose/Fermi grading.

We show here that any such pair {G, k}, where G is compact metrizable, does actually appear. The corresponding model can be chosen to fulfill also the split property.

This is not a dynamical phenomenon: a given {G, k} arises as the gauge group of a model where the local algebras of observables are a suitable subnet of local algebras of a possibly infinite product of free field theories.

We study derivatives on an interval of length ℓ (or the associated circle of the same length), and certain pseudo-differential operators that arise as their fractional powers. We compare different translations across the interval (around the circle) that are characterized by a twisting angle. These results have application in

Using a recent construction of observables characterizing the time of occurence of an effect in quantum theory, we present a rigorous derivation of the standard time-energy uncertainty relation. In addition, we prove an uncertainty relation for time measurements alone.