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227
The quantum structure of spacetime at the Planck scale and quantum
, 1995
"... Abstract. We propose uncertainty relations for the different coordinates of spacetime events, motivated by Heisenberg’s principle and by Einstein’s theory of classical gravity. A model of Quantum Spacetime is then discussed where the commutation relations exactly implement our uncertainty relations. ..."
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Cited by 226 (4 self)
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Abstract. We propose uncertainty relations for the different coordinates of spacetime events, motivated by Heisenberg’s principle and by Einstein’s theory of classical gravity. A model of Quantum Spacetime is then discussed where the commutation relations exactly implement our uncertainty relations. We outline the definition of free fields and interactions over QST and take the first steps to adapting the usual perturbation theory. The quantum nature of the underlying spacetime replaces a local interaction by a specific nonlocal effective interaction in the ordinary Minkowski space. A detailed study of interacting QFT and of the smoothing of ultraviolet divergences is deferred to a subsequent paper. In the classical limit where the Planck length goes to zero, our Quantum Spacetime reduces to the ordinary Minkowski space times a two component space whose components are homeomorphic to the tangent bundle TS 2 of the 2–sphere. The relations with Connes’ theory of the standard model will be studied elsewhere. 1.
Liouville Theory Revisited
, 2001
"... This paper focuses on the understanding of Liouville theory on a (spacetime) cylinder with circumference 27r, timecoordinate t and (periodic) spacecoordinate a as a two dimensional quantum field theory in its own right. (Semi)classically the theory is defined by the action 2 1 2 (1) S = dt/d( ..."
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Cited by 94 (12 self)
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This paper focuses on the understanding of Liouville theory on a (spacetime) cylinder with circumference 27r, timecoordinate t and (periodic) spacecoordinate a as a two dimensional quantum field theory in its own right. (Semi)classically the theory is defined by the action 2 1 2 (1) S = dt/d(l((O) (0) 2) /ce)
D.E.: Modular Invariants, Graphs and αInduction for Nets of Subfactors II
 In preparation
"... We analyze the induction and restriction of sectors for nets of subfactors defined by Longo and Rehren. Picking a local subfactor we derive a formula which specifies the structure of the induced sectors in terms of the original DHR sectors of the smaller net and canonical endomorphisms. We also obta ..."
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Cited by 80 (8 self)
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We analyze the induction and restriction of sectors for nets of subfactors defined by Longo and Rehren. Picking a local subfactor we derive a formula which specifies the structure of the induced sectors in terms of the original DHR sectors of the smaller net and canonical endomorphisms. We also obtain a reciprocity formula for induction and restriction of sectors, and we prove a certain homomorphism property of the induction mapping. Developing further some ideas of F. Xu we will apply this theory in a forthcoming paper to nets of subfactors arising from conformal field theory, in particular those coming from conformal embeddings or orbifold inclusions of SU(n) WZW models. This will provide a better understanding of the labeling of modular invariants by certain graphs, in particular of the ADE classification of SU(2) modular invariants.
Operator algebras and conformal field theory  III. Fusion of positive energy representations of LSU(N) using bounded operators
, 1998
"... ..."
The Conformal spin and statistics theorem
 Commun. Math. Phys
, 1996
"... During the recent years Conformal Quantum Field Theory has become a widely studied topic, especially on a low dimensional spacetime, because of physical motivations such as the desire of a better understanding of twodimensional critical phenomena, and also for its rich mathematical structure provi ..."
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Cited by 64 (23 self)
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During the recent years Conformal Quantum Field Theory has become a widely studied topic, especially on a low dimensional spacetime, because of physical motivations such as the desire of a better understanding of twodimensional critical phenomena, and also for its rich mathematical structure providing remarkable connections with different areas such as Hopf algebras, low dimensional topology, knot invariants, subfactors
NonEquilibrium Steady States of Finite Quantum Systems Coupled to Thermal Reservoirs
 COMMUN. MATH. PHYS
, 2001
"... We study the nonequilibrium statistical mechanics of a #level quantum system, # , coupled to two independent free Fermi reservoirs # # , # # , which are in thermal equilibrium at inverse temperatures # # ## # # .Weprove that, at small coupling, the combined quantum system ### # ## # has ..."
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Cited by 46 (8 self)
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We study the nonequilibrium statistical mechanics of a #level quantum system, # , coupled to two independent free Fermi reservoirs # # , # # , which are in thermal equilibrium at inverse temperatures # # ## # # .Weprove that, at small coupling, the combined quantum system ### # ## # has a unique nonequilibrium steady state (NESS) and that the approach to this NESS is exponentially fast. We show that the entropy production of the coupled system is strictly positive and relate this entropy production to the heat uxes through the system. A part of our argument is general and deals with spectral theory of NESS. In the abstract setting of algebraic quantum statistical mechanics we introduce the new concept of # Liouvillean, #, and relate the NESS to zero resonance eigenfunctions of # # . In the specific model ### # ## # we study the resonances of # # using the complex deformation technique developed previously by the authors in [JP1].
Scaling algebras and renormalization group in algebraic quantum field theory
 Rev. Math. Phys
, 1995
"... Abstract: The concept of scaling algebra provides a novel framework for the general structural analysis and classification of the short distance properties of algebras of local observables in relativistic quantum field theory. In the present article this method is applied to the simple example of ma ..."
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Cited by 42 (7 self)
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Abstract: The concept of scaling algebra provides a novel framework for the general structural analysis and classification of the short distance properties of algebras of local observables in relativistic quantum field theory. In the present article this method is applied to the simple example of massive free field theory in s = 1,2 and 3 spatial dimensions. Not quite unexpectedly, one obtains for s = 2,3 in the scaling (short distance) limit the algebra of local observables in massless free field theory. The case s = 1 offers, however, some surprises. There the algebra of observables acquires in the scaling limit a nontrivial center and describes charged physical states satisfying Gauss ’ law. The latter result is of relevance for the interpretation of the Schwinger model at short distances and illustrates the conceptual and computational virtues of the method. 1
An algebraic spin and statistics theorem
 Commun. Math. Phys
, 1995
"... Dedicated to Hans Borchers on the occasion of his seventieth birthday ..."
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Cited by 42 (11 self)
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Dedicated to Hans Borchers on the occasion of his seventieth birthday
Modular invariants from subfactors: Type I coupling matrices and intermediate subfactors
 Commun. Math. Phys
, 2000
"... A braided subfactor determines a coupling matrix Z which commutes with the S and Tmatrices arising from the braiding. Such a coupling matrix is not necessarily of “type I”, i.e. in general it does not have a blockdiagonal structure which can be reinterpreted as the diagonal coupling matrix with r ..."
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Cited by 35 (5 self)
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A braided subfactor determines a coupling matrix Z which commutes with the S and Tmatrices arising from the braiding. Such a coupling matrix is not necessarily of “type I”, i.e. in general it does not have a blockdiagonal structure which can be reinterpreted as the diagonal coupling matrix with respect to a suitable extension. We show that there are always two intermediate subfactors which correspond to left and right maximal extensions and which determine “parent ” coupling matrices Z ± of type I. Moreover it is shown that if the intermediate subfactors coincide, so that Z + = Z − , then Z is related to Z + by an automorphism of the extended fusion rules. The intertwining relations of chiral branching coefficients between original and extended S and Tmatrices are also clarified. None of our results depends on nondegeneracy of the braiding, i.e. the S and Tmatrices need not be modular. Examples from SO(n) current algebra models illustrate that the parents can be different, Z + ̸ = Z − , and that Z need not be related to a type I invariant by such an automorphism. 1