Modular Localization and Wigner Particles

Brunetti, Romeo; Guido, Daniele; Longo, Roberto

doi:10.1142/s0129055x02001387

Cited by 142 publications

(371 citation statements)

References 40 publications

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“…[20]). They also used the suggestive name of free net as in [14, Example 8.3.1] (see also [12,48]), since the construction avoids the use of quantum fields as 'coordinates' of the corresponding net.…”

Section: Discussionmentioning

confidence: 99%

“…Indeed, in Theorem 4.2 we give a detailed proof of the mentioned continuity for the embedding associated to the Weyl case and that uses explicitly the section defined in Eq. (20). This ensures that the 2-point distributions defined by the corresponding massless quantum fields are tempered.…”

Section: Remarkmentioning

confidence: 99%

“…[53]). We conclude this paper commenting on the relations of the free net construction in Section 3 with the construction given in [20] which is based on the notion of modular localisation (see also [27]). …”

Section: Introductionmentioning

confidence: 95%

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Massless Relativistic Wave Equations and Quantum Field Theory

Lledó

2004

Ann. Henri Poincaré

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We give a simple and direct construction of a massless quantum field with arbitrary discrete helicity that satisfies Wightman axioms and the corresponding relativistic wave equation in the distributional sense. We underline the mathematical differences to massive models. The construction is based on the notion of massless free net (cf. Section 3) and the detailed analysis of covariant and massless canonical (Wigner) representations of the Poincaré group. A characteristic feature of massless models with nontrivial helicity is the fact that the fibre degrees of freedom of the covariant and canonical representations do not coincide. We use massless relativistic wave equations as constraint equations reducing the fibre degrees of freedom of the covariant representation. They are characterised by invariant (and in contrast with the massive case non reducing) one-dimensional projections. The definition of one-particle Hilbert space structure that specifies the quantum field uses distinguished elements of the intertwiner space between E(2) (the two-fold cover of the 2-dimensional Euclidean group) and E(2).We conclude with a brief comparison between the free nets constructed in Section 3 and a recent alternative construction that uses the notion of modular localisation.

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Section: Discussionmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

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Massless Relativistic Wave Equations and Quantum Field Theory

Lledó

2004

Ann. Henri Poincaré

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“…The study of the relation between modular operator theory and causal localization in LQP started a decade later [19], and its first application consisted in a more profound understanding [20] of the Unruh Gedankenexperiment [21]. The terminology "modular localization" is more recent and marks the beginning of a new constructive strategy in QFT based on the modular aspects of localization of states and algebras [44] [6]. In mathematics the theory was decisive instrument which led to Connes closure of the Murray-von Neumann project of classifying von Neumann factor algebras.…”

Section: A(r) = B(h(r)) a ≡ B(h) = A(r) ⊗ A(r ⊥ )mentioning

confidence: 99%

“…These localized subspaces can be defined in a intrinsic way [44] i.e. without quantization, only using operators from the positive energy representation U of the proper Poincaré group P + (det = +1) on the direct sum of two copies of the Wigner representation u of the connected component (proper orthochronous P ↑ + ) on the one-particle space H 1 .…”

Section: Modular Localization and Its Thermal Manifestationmentioning

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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Algebraic Quantum Field Theory and Causal Symmetric Spaces

Neeb

Ólafsson

2023

Trends in Mathematics

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Modular Localization and Wigner Particles

Cited by 142 publications

References 40 publications

Massless Relativistic Wave Equations and Quantum Field Theory

Massless Relativistic Wave Equations and Quantum Field Theory

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

Algebraic Quantum Field Theory and Causal Symmetric Spaces

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