Algebraic Constructive Quantum Field Theory: Integrable Models and Deformation Techniques

Lechner, Gandalf

doi:10.1007/978-3-319-21353-8_10

Cited by 32 publications

(28 citation statements)

References 135 publications

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“…An illustration of the power of this relatively new concept is the proof of existence of a class of two-dimensional models starting from the observations that certain algebraic structures in integrable d = 1 + 1 models can be used to construct modular localized wedge algebras [14]. In the work of Lechner and others this led to existence proofs for integrable models with nontrivial short distance behavior together with a wealth of new concepts (see the recent reviews [15,16] and the references therein).…”

Section: Wigner's Infinite Spin Representation and String-localizationmentioning

confidence: 99%

Wigner’s infinite spin representations and inert matter

Schroer

2017

Eur. Phys. J. C

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Positive energy ray representations of the Poincaré group are naturally subdivided into three classes according to their mass and spin content: m > 0, m = 0 finite helicity and m = 0 infinite spin. For a long time the localization properties of the massless infinite spin class remained unknown, until it became clear that such matter does not permit compact spacetime localization and its generating covariant fields are localized on semi-infinite spacelike strings. Using a new perturbation theory for higher spin fields we present arguments which support the idea that infinite spin matter cannot interact with normal matter and we formulate conditions under which this also could happen for finite spin s > 1 fields. This raises the question of a possible connection between inert matter and dark matter. Wigner's infinite spin representation and string-localizationWigner's famous 1939 theory of unitary representations of the Poincaré group P was the first systematic and successful attempt to classify relativistic particles according to the intrinsic principles of relativistic quantum theory [1]. As we know nowadays, his massive and massless spin/helicity class of positive energy ray representations of P does not only cover all known particles, but their "covariantization" [2] leads also to a complete description of all covariant pointlocal (pl) free fields. For each covariant Lorentz transformation property compatible with physical spin or helicity there exists a pl field.It is not necessary that these free fields permit a characterization as Euler-Lagrange fields; in fact the attempt to describe higher spin fields in this way and pass to quantum fields by canonical quantization or use other ways to Dedicated to the memory of Robert Schrader. a e-mail: schroer@zedat.fu-berlin.de start from a classical field creates problems of their own. On the other hand the construction of free fields starting from Wigner's representation theory does not require one to rely on any "quantization parallelism" to classical structures.The idea that a more fundamental theory should not rely on structural properties of a less basic one can also be upheld in the presence of interactions. Defining a first order Lorentz invariant interaction densities in terms of products of free fields and using the Epstein-Glaser [3] formulation of renormalization theory permits a perturbative cutoff-and regularization-free higher order inductive implementation of the causal localization principle in a positivity-obeying Wigner-Fock Hilbert space without a reference to the classical field formalism. Renormalizable theories in this setting are those for which this higher order induction increases the model-defining first order parameters by an at most finite number of "counter term" parameters. When higher spin s ≥ 1 fields participate in the interaction there are no positivity-obeying renormalizable couplings. The traditional way out has been to skip positivity in the calculations and save it by restricting the result. A total abandonment would have...

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Section: Wigner's Infinite Spin Representation and String-localizationmentioning

confidence: 99%

Wigner’s infinite spin representations and inert matter

Schroer

2017

Eur. Phys. J. C

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“…The idea that the avoidance of singular pl or sl fields may be important for existence proofs of models of QFT is corroborated by the "top-down" construction of certain two-dimensional integrable models in which one starts with their known S-matrix which contains information as regards the algebraic structure of the wedge-localized algebras. By showing the existence of nontrivial intersections corresponding to compact localized algebras one arrives at the construction of a system of local algebras which fulfills all the properties of an algebraic QFT [12]. The question of whether it contains generating pl or sl fields remains open.…”

Section: Iim( F G) = [A( F ) A(g)] F G Real Test Functions (24)mentioning

confidence: 99%

“…In contrast to perturbation theory based on fields, these "top-down" constructions do not (yet?) arrive at generating fields for these algebras [12].…”

Section: Introductory Remarks On Origin and Scope Of String Localizationmentioning

confidence: 99%

Beyond gauge theory: positivity and causal localization in the presence of vector mesons

Schroer¹

2016

Eur. Phys. J. C

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The Hilbert space formulation of interacting s = 1 vector-potentials stands is an interesting contrast with the point-local Krein space setting of gauge theory. Already in the absence of interactions the Wilson loop in a Hilbert space setting has a topological property which is missing in the gauge-theoretic description (Haag duality, Aharonov-Bohm effect); the conceptual differences increase in the presence of interactions. The Hilbert space positivity weakens the causal localization properties of interacting fields, which results in the replacement of the gauge-variant point-local matter fields in Krein space by string-local physical fields in Hilbert space. The gauge invariance of the perturbative S-matrix corresponds to its independence of the space-like string direction of its interpolating fields. In contrast to gauge theory, whose direct physical range is limited to a gauge-invariant perturbative S-matrix and local observables, its Hilbert space string-local counterpart is a full-fledged quantum field theory (QFT). The new setting reveals that the Lie structure of self-coupled vector mesons results from perturbative implementation of the causal localization principles of QFT. Introductory remarks on origin and scope of string localizationIt is well known that the use of point-local massless vector potentials is incompatible with the positivity of Hilbert space. One usually resolves this problem by abandoning the positivity requirement of quantum field theory (QFT) while maintaining the point-local field formalism which leads to gauge theory (GT). The price to pay is well known from quantum electrodynamics (QED) in the standard indefinite metric (Gupta-Bleuler) gauge setting: positivity can be recoveredDedicated to the memory of Raymond Stora and John Roberts.a e-mail: schroer@zedat.fu-berlin.de for local observables, whereas charge-carrying fields remain outside its physical range. The separation between physical and unphysical quantum fields is done in terms of gauge symmetry, which is not a physical symmetry but a formal device to extract a physical subtheory. Although the standard gauge formalism is restricted to the presence of vector potentials, the clash between zero mass point-local fields and positivity is a general phenomenon for all s ≥ 1 zero mass tensor potentials. It does not affect the corresponding field strengths, but the short-distance dimension of the latter (d sd = 2 for s = 1) prevents their direct use in renormalizable interactions.The alternative option is to accept the weaker localization required by positivity. The tightest covariant localization beyond point-local consistent with positivity turns out to be causal localization on semi-infinite space-like "strings" 1 x + R + e, e 2 = −1. It is easy to construct m > 0 covariant string-localized fields (x, e) in terms of semi-infinite line integrals on point-local fields [1]. The immediate advantage of such a construction is that one obtains two improvements in one stroke, the lowering of the short-distance dimension and the e...

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“…The Zamolodchikov-Faddeev algebras [ZZ79,SF78] are a class of quadratic exchange algebras of "creation" and "annihilation" operators which generalize the familiar CCR and CAR algebras [BR79] and are closely related to Wick algebras that allow normal ordering [JSW95]. These algebras and their Hilbert space representations are of central importance in integrable quantum field theory (see, for example, [Smi92,AAR01,Lec15]) as well as in other fields such as q-deformations [MN08,SBGD04] or anyonic statistics.…”

Section: Introductionmentioning

confidence: 99%

Fock representations of ZF algebras and R-matrices

Lechner

Scotford

2020

Lett Math Phys

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A variation of the Zamolodchikov-Faddeev algebra over a finite dimensional Hilbert space H and an involutive unitary R-Matrix S is studied. This algebra carries a natural vacuum state, and the corresponding Fock representation spaces F S (H) are shown to satisfyand R (on K ⊗ K). This analysis generalises the well-known structure of Bose/Fermi Fock spaces and a recent result of Pennig.It is also discussed to which extent the Fock representation depends on the underlying R-matrix, and applications to quantum field theory (scaling limits of integrable models) are sketched.

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Algebraic Constructive Quantum Field Theory: Integrable Models and Deformation Techniques

Cited by 32 publications

References 135 publications

Wigner’s infinite spin representations and inert matter

Wigner’s infinite spin representations and inert matter

Beyond gauge theory: positivity and causal localization in the presence of vector mesons

Fock representations of ZF algebras and R-matrices

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