Deformations of Quantum Field Theories and Integrable Models

Lechner, Gandalf

doi:10.1007/s00220-011-1390-y

Cited by 45 publications

(121 citation statements)

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“…In fact these new local algebraic methods led, for the first time in the history of QFT, to an existence proof in the presence of interactions in a family of d=1+1 models with realistic (noncanonical) short distance behavior 26 . This is the content of pathbreaking work by Gandalf Lechner [58] [72]. In this way the the existence of certain integrable models about which there were already exact computational results for certain formfactors [59] was finally secured and a new direction for future more general attempts was pointed out.…”

Section: Theorem 1 ([24])mentioning

confidence: 95%

“…This was achieved by the use of modular nuclearity in a pathbreaking work by Lechner [58]. These ideas have been extended by deformation theory (deformation of free fields for models without bound states [72]), and meanwhile integrable models which even by experts are considered to be difficult (as the O(N )-model [63]) are in the range of the modular nuclearity arguments [64] which already secured the existence of simpler models.…”

Section: Theorem 1 ([24])mentioning

confidence: 99%

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The Einstein-Jordan conundrum and its relation to ongoing foundational research in local quantum physics

Schroer

2012

EPJ H

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We demonstrate the extraordinary modernity of the 1924/25 "EinsteinJordan fluctuation conundrum", a Gedankenexperiment which led Jordan to his quantization of waves published as a separate section in the famous Born-Heisenberg-Jordan 1926 "Dreimännerarbeit". The thermal nature of energy fluctuations caused by the restriction of the QFT vacuum to a subvolume remained unnoticed mainly because it is not present in QM. In order to understand the analogy with Einstein's fluctuation calculation in a thermal black body system, it is important to expose the mechanism which causes a global vacuum state to become impure on a localized subalgebra of QFT.The present work presents the fascinating history behind this problem which culminated in the more recent perception that "causal localization" leads to thermal manifestations. The most appropriate concept which places this property of QFT into the forefront is "modular localization". These new developments in QFT led to a new access to the existence problem for interacting quantum fields whose solution has remained outside the range of renormalized perturbation theory. It also clarifies open problems about the relation of particles and fields in particular about the incompletely understood crossing property. Last not least it leads to a constructive understanding of integrable versus non-integrable QFTs..

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Section: Theorem 1 ([24])mentioning

confidence: 95%

Section: Theorem 1 ([24])mentioning

confidence: 99%

The Einstein-Jordan conundrum and its relation to ongoing foundational research in local quantum physics

Schroer

2012

EPJ H

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“…Apart from d=1+1 integrable models (section 5), for which rigorous methods of LQP led to existence proofs [38] [55], there is of course renormalized perturbation theory; but since perturbative expansions in the coupling strengths (which are consistent on the level of polynomial relations) inevitably lead to divergent series, they are not the right objects for an intrinsic formulation of QFT. In fact there exists not even a mathematical argument that they define an asymptotic small coupling approximation in the limit of vanishing coupling to an existing model of QFT, although the use of low order perturbative results led in certain cases to quite spectacular agreements with observations.…”

Section: Modular Localization and Its Thermal Manifestationmentioning

confidence: 99%

“…This is a very nontrivial step which has been accomplished with the use of modular nuclearity in the work of Lechner [38]. The same author also showed how (in the absence of boundstates ) one can construct the wedge-algebra generating PFG's in terms of deformations of free fields [55].…”

Section: Theorem 1 ([24])mentioning

confidence: 99%

Modular localization and the holistic structure of causal quantum theory, a historical perspective

Schroer

2014

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Recent insights into the conceptual structure of localization in QFT ("modular localization") led to clarifications of old unsolved problems. The oldest one is the Einstein-Jordan conundrum which led Jordan in 1925 to the discovery of quantum field theory. This comparison of fluctuations in subsystems of heat bath systems (Einstein) with those resulting from the restriction of the QFT vacuum state to an open subvolume (Jordan) leads to a perfect analogy; the globally pure vacuum state becomes upon local restriction a strongly impure KMS state. This phenomenon of localization-caused thermal behavior as well as the vacuum-polarization clouds at the causal boundary of the localization region places localization in QFT into a sharp contrast with quantum mechanics and justifies the attribute "holstic". In fact it positions the E-J Gedankenexperiment into the same conceptual category as the cosmological constant problem and the Unruh Gedankenexperiment. The holistic structure of QFT resulting from "modular localization" also leads to a revision of the conceptual origin of the crucial crossing property which entered particle theory at the time of the bootstrap S-matrix approach but suffered from incorrect use in the S-matrix settings of the dual model and string theory.The new holistic point of view, which strengthens the autonomous aspect of QFT, also comes with new messages for gauge theory by exposing the clash between Hilbert space structure and localization and presenting alternative solutions based on the use of stringlocal fields in Hilbert space. Among other things this leads to a radical reformulation of the Englert-Higgs symmetry breaking mechanism. Contents 1 Introduction 282 Vacuum polarization, area law 383 Modular localization and its thermal manifestation 424 The E-J conundrum, Jordan's model 545 Particle crossing, on-shell constructions from modular setting 58 PrefaceThe subject of this paper grew out of many discussions about Jordan's discovery of quantum field theory (QFT) which I had with the late Jürgen Ehlers. These conversations focussed in particular on events between the publication of Jordan's thesis on quantum aspects of statistical quantum mechanics in 1924 [1], and his discovery of QFT which was published in one section of the famous 1926 "Dreimännerarbeit" [2] together with Born and Heisenberg. This famous paper was in fact conceived as the second paper on quantum mechanics (QM). The resistance of Born and Heisenberg against Jordan's section has its natural explanation in that these two authors felt that Jordan was burdening the conceptual struggle to understand the new quantum mechanics with something which was even further out. I met Jürgen Ehlers the first time around 1957 at the University of Hamburg when he was Jordan's assistant and played the leading role in Jordan's general relativity seminar. Our paths split, after I wrote my diploma thesis on a topic of particle theory at the time when particle physics moved away from the university to the newly constructed high energy la...

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“…It is, however, possible to circumvent this No-Go theorem by considering fields which are only localized in wedge regions in Minkowski space. In this case one can modify the method of multiplicative deformations, established in [16,17], to obtain one-particle generators localized in so-called "paths of wedges" which have anyonic commutation relations and are covariant w.r.t a representation of P + for spin s ∈ R [24]. This is only possible because in the proof of the No-Go theorem one needs to have three mutually spacelike separated localization regions, which is impossible for wedges.…”

Section: Introductionmentioning

confidence: 99%

Local Anyonic Quantum Fields on the Circle Leading to Cone-Local Anyons in Two Dimensions

Plaschke

2015

Lett Math Phys

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Using the method of implementable one-particle Bogoliubov transformations it is possible to explicitly define a local covariant net of quantum fields on the (universal covering of the) circle S1 with braid group statistics. These Anyon fields transform under a representation of U (1) for arbitrary real-valued spin and their commutation relations depend on the relative winding number of localization regions. By taking the tensor product with a local covariant field theory on R 2 one can obtain a (non-boost-covariant) cone-localized field net for Anyons in two dimensions.

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Deformations of Quantum Field Theories and Integrable Models

Cited by 45 publications

References 45 publications

The Einstein-Jordan conundrum and its relation to ongoing foundational research in local quantum physics

The Einstein-Jordan conundrum and its relation to ongoing foundational research in local quantum physics

Modular localization and the holistic structure of causal quantum theory, a historical perspective

Local Anyonic Quantum Fields on the Circle Leading to Cone-Local Anyons in Two Dimensions

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