Bisognano-Wichmann Property Research Articles

Wedge Domains in Compactly Causal Symmetric Spaces

AbstractMotivated by constructions in Algebraic Quantum Field Theory we introduce wedge domains in compactly causal symmetric spaces $M=G/H$, which includes in particular anti-de Sitter space in all dimensions and its coverings. Our wedge domains generalize Rindler wedges in Minkowski space. The key geometric structure we use is the modular flow on $M$ defined by an Euler element in the Lie algebra of $G$. Our main geometric result asserts that three seemingly different characterizations of these domains coincide: the positivity domain of the modular vector field, the domain specified by a KMS-like analytic extension condition for the modular flow, and the domain specified by a polar decomposition in terms of certain cones. In the second half of the article we show that our wedge domains share important properties with wedge domains in Minkowski space. If $G$ is semisimple, there exist unitary representations $(U,{\mathcal {H}})$ of $G$ and isotone covariant nets of real subspaces $\textsf {H}({\mathcal {O}}) \subseteq {\mathcal {H}}$, defined for any open subset ${\mathcal {O}} \subseteq M$, which assign to connected components of the wedge domains a standard subspace whose modular group corresponds to the modular flow on $M$. This corresponds to the Bisognano–Wichmann property in Quantum Field Theory. We also show that the set of $G$-translates of the connected components of the wedge domain provides a geometric realization of the abstract wedge space introduced by the first author and V. Morinelli.

Bisognano–Wichmann Property for Rigid Categorical Extensions and Non-local Extensions of Conformal Nets

Given an (irreducible) Mobius covariant net $\mathcal A$, we prove a Bisognano-Wichmann theorem for its categorical extension $\mathscr E^{\textrm{d}}$ associated to the braided $C^*$-tensor category $\textrm{Rep}^{\textrm{d}}(\mathcal A)$ of dualizable (more precisely "dualized") Mobius covariant $\mathcal A$-modules. As a closely related result, we prove a (modified) Bisognano-Wichmann theorem for any (possibly) non-local extension of $\mathcal A$ obtained by a $C^*$-Frobenius algebra $Q$ in $\textrm{Rep}^{\textrm{d}}(\mathcal A)$. As an application, we discuss the relation between the domains of modular operators and the preclosedness of certain unbounded operators in $\mathscr E^{\textrm{d}}$.

Nets of standard subspaces on Lie groups

Let G be a Lie group with Lie algebra g, h∈g an element for which the derivation ad h defines a 3-grading of g and τG an involutive automorphism of G inducing on g the involution eπiadh. We consider antiunitary representations (U,H) of the Lie group Gτ=G⋊{idG,τG} for which the positive cone CU={x∈g:−i∂U(x)≥0} and h span g. To a real subspace E⊆H−∞ of distribution vectors invariant under exp⁡(Rh) and an open subset O⊆G, we associate the real subspace HE(O)⊆H, generated by the subspaces U−∞(φ)E, where φ∈Cc∞(O,R) is a real-valued test function on O. Then HE(O) generates the complex Hilbert space HE(G):=HE(G)+iHE(G)‾ for every non-empty open subset O⊆G (Reeh–Schlieder property).For the real standard subspace V⊆H, for which JV=U(τG) is the modular conjugation and ΔV−it/2π=U(exp⁡th) is the modular group, we obtain sufficient conditions to be of the form HE(S) for an open subsemigroup S⊆G. If g is semisimple with simple hermitian ideals of tube type, we verify these criteria and obtain nets of cyclic subspaces HE(O), O⊆G, satisfying the Bisognano–Wichmann property in a suitable sense. Our construction also yields such nets on simple Jordan space-times and compactly causal symmetric spaces of Cayley type. By second quantization, these nets lead to free quantum fields in the sense of Haag–Kastler on causal homogeneous spaces whose groups are generated by modular groups and conjugations.

The Bisognano–Wichmann Property on Nets of Standard Subspaces, Some Sufficient Conditions

We discuss the Bisognano-Wichmann property for local Poincar\'e covariant nets of standard subspaces. We give a sufficient algebraic condition on the covariant representation ensuring the Bisognano-Wichmann and Duality properties without further assumptions on the net called modularity condition. It holds for direct integrals of scalar massive and massless representations. We present a class of massive modular covariant nets not satisfying the Bisognano-Wichmann property. Furthermore, we give an outlook in the standard subspace setting on the relation between the Bisognano-Wichmann property and the Split property.

Where Infinite Spin Particles are Localizable

Particles states transforming in one of the infinite spin representations of the Poincar\'e group (as classified by E. Wigner) are consistent with fundamental physical principles, but local fields generating them from the vacuum state cannot exist. While it is known that infinite spin states localized in a spacelike cone are dense in the one-particle space, we show here that the subspace of states localized in any double cone is trivial. This implies that the free field theory associated with infinite spin has no observables localized in bounded regions. In an interacting theory, if the vacuum vector is cyclic for a double cone local algebra, then the theory does not contain infinite spin representations. We also prove that if a Doplicher-Haag-Roberts representation (localized in a double cone) of a local net is covariant under a unitary representation of the Poincar\'e group containing infinite spin, then it has infinite statistics. These results hold under the natural assumption of the Bisognano-Wichmann property, and we give a counter-example (with continuous particle degeneracy) without this property where the conclusions fail. Our results hold true in any spacetime dimension s+1 where infinite spin representations exist, namely s > 1.

Noninteraction of Waves in Two-dimensional Conformal Field Theory

In higher dimensional quantum field theory, irreducible representations of the Poincare group are associated with particles. Their counterpart in two-dimensional massless models are "waves" introduced by Buchholz. In this paper we show that waves do not interact in two-dimensional Moebius covariant theories and in- and out-asymptotic fields coincide. We identify the set of the collision states of waves with the subspace generated by the chiral components of the Moebius covariant net from the vacuum. It is also shown that Bisognano-Wichmann property, dilation covariance and asymptotic completeness (with respect to waves) imply Moebius symmetry. Under natural assumptions, we observe that the maps which give asymptotic fields in Poincare covariant theory are conditional expectations between appropriate algebras. We show that a two-dimensional massless theory is asymptotically complete and noninteracting if and only if it is a chiral Moebius covariant theory.

The Bisognano-Wichmann Theorem for Massive Theories

The geometric action of modular groups for wedge regions (Bisognano-Wichmann property) is derived from the principles of local quantum physics for a large class of Poincare covariant models in d=4. As a consequence, the CPT theorem holds for this class. The models must have a complete interpretation in terms of massive particles. The corresponding charges need not be localizable in compact regions: The most general case is admitted, namely localization in spacelike cones.

GROUP COHOMOLOGY, MODULAR THEORY AND SPACE-TIME SYMMETRIES

The Bisognano-Wichmann property on the geometric behavior of the modular group of the von Neumann algebras of local observables associated to wedge regions in Quantum Field Theory is shown to provide an intrinsic sufficient criterion for the existence of a covariant action of the (universal covering of) the Poincaré group. In particular this gives, together with our previous results, an intrinsic characterization of positive-energy conformal pre-cosheaves of von Neumann algebras. To this end we adapt to our use Moore theory of central extensions of locally compact groups by polish groups, selecting and making an analysis of a wider class of extensions with natural measurable properties and showing henceforth that the universal covering of the Poincaré group has only trivial central extensions (vanishing of the first and second order cohomology) within our class.