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Modular Localization and the Foundational Origin of Integrability

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Abstract

The main aim of this work is to relate integrability in QFT with a complete particle interpretation directly to the principle of causal localization, circumventing the standard method of finding sufficiently many conservation laws. Its precise conceptual-mathematical formulation as “modular localization” within the setting of local operator algebras also suggests novel ways of looking at general (non-integrable) QFTs which are not based on quantizing classical field theories.

Conformal QFT, which is known to admit no particle interpretation, suggest the presence of a “partial” integrability, referred to as “conformal integrability”. This manifests itself in a “braid-permutation” group structure which contains in particular informations about the anomalous dimensional spectrum. For chiral conformal models this reduces to the braid group as it is represented in Hecke- or Birman-Wenzl-algebras associated to chiral models.

Another application of modular localization mentioned in this work is an alternative to the BRST formulation of gauge theories in terms of stringlike vectorpotentials within a Hilbert space setting.

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Notes

  1. In the presence of backward scattering and/or inner symmetry indices the scattering function is a matrix function which fulfills the Yang-Baxter equation [13].

  2. The KMS relation is an antic relation which is fulfilled by tracial Gibbs states. It survives in the thermodynamic limit when the tracial characterization is lost [21].

  3. In fact in a private correspondence Heisenberg challenged Jordan to account for a logarithmic divergence resulting from the vacuum polarization cloud at the endpoints of his localization interval [20].

  4. There is an ongoing discussion whether thermal radiation effects in Unruh situations can be seen in subnuclear laboratory experiments.

  5. A general wedge results from W 0={|x 0|<x 1} by applying Poincaré transformations.

  6. The tensor factorization of type I∞ “thermofield theory” breaks down and the algebra changes its type.

  7. The short name which we will use for the (up to isomorphism unique) before-mentioned operator algebra. Besides the standard algebra B(H) of all bounded operators this the only type of operator algebra which one encounters in continuous quantum systems [57].

  8. Born localization entered QM through his famous probabilistic interpretation of (the Born approximation of) the scattering amplitude i.e. the cross section. This was afterwards extended to the position operator and its associated wave functions.

  9. Since positive energy representations are completely reducible this works for all such representations, not only irreducible ones.

  10. According to the Reeh-Schlieder [21] theorem a local algebra \(\mathcal{A}(\mathcal{O})\) in QFT is in standard position with respect to the vacuum i.e. it acts on the vacuum in a cyclic and separating manner. The spatial standardness, which follows directly from Wigner representation theory, is just the one-particle projection of the Reeh-Schlieder property.

  11. In Jordan’s terminology “without classical crutches” [65].

  12. For convenience of notation our spinorial indices are half of the standard ones.

  13. The standard A-B effect refers to QM in an external magnetic field.

  14. Concerning the application of the BRST formalism to massive vectormeson we follow Scharf’s book [45].

  15. The prototype illustration is the derivation of the C-S equation of the massive Thirring model. In this case the vanishing of beta to all orders insures the existence of the massless limit as a conformal QFT [46].

  16. Schwinger’s attempt to find a perturbative realization in spinor QED failed and he instead used two-dimensional QED which became known as the “Schwinger model”. He was apparently not aware that by replacing spinors by complex scalar fields he could have had a perturbative realization of his idea [48].

  17. Besides his formulation of operator gauge invariance Scharf uses only the intrinsic logic of the BRST formalism.

  18. It led Buchholz and Fredenhagen to discover the connection of spectral gaps with the stringlike generating property of superselected charge-carrying fields (as the only possibility which replaces the pointlike localization in case of non-existence of pointlike generators) [21].

  19. The terminology “second quantization” is a misdemeanor since one is dealing with a rigorously defined functor within QT which has little in common with the artful use of that parallelism to classical theory called “quantization”.

  20. Here we consider modular localization as part of Wigner’s theory because the modular localization is constructed within positive energy representations of the Poincaré group [35].

  21. The functor second quantization functor Γ preserves the standardness i.e. maps the spatial one-particle standardness into its algebraic counterpart.

  22. For discrete combinatorial algebraic structures, as one encounters them in lattice theories, also type II1 enters, see remarks in last section.

  23. In QFT any finite energy vector (which of course includes the vacuum) has this property, as well as any nondegenerated KMS state.

  24. For emphasizing the importance of this property for the issue of the cosmological constant we refer to the paper: “Quantum Field Theory Is Not Merely Quantum Mechanics Applied to Low Energy Effective Degrees of Freedom” by Hollands and Wald [59].

  25. If the model has sufficient analyticity properties which allow real/imaginary time Wick-rotations, one can “curl up” a time component by taking the high temperature limit in a KMS state and create a new time direction by Wick rotation. But this is not what the proponents of Kaluza-Klein reductions in QFT have in mind.

  26. The elastic two-particle amplitude in d=1+1 is the only scattering amplitude which cannot be distinguished from the identity contribution by cluster factorization (equality of the product of two particle plane wave inner products with the energy-momentum delta function), by which factorizing models indicate their kinematical proximity to free fields.

  27. These elastic S-matrices were obtained from the classification of solutions of a “bootstrap” project for scattering functions (solutions of the Yang-Baxter equations in case of matrix valued scattering functions) [11].

  28. Unlike quantum fields they are not (operator-valued) Schwartz distributions.

  29. In fact it is easy to see that [18] \(( A_{\mathcal{A}(W)} )^{\ast} \vert 0 \rangle =S_{\mathit{scat}}A\vert 0 \rangle \).

  30. For free fields the operator algebra is generated by exponential Weyl operators.

  31. A similar case in x-space occurs if one extends a d=1+2 Wightman setting to fields with braid.group statistics.

  32. In [66] it was shown that as a consequence of Huygens principle, conformal QFT leads to the uniqueness of the inverse problem for S scat =1.

  33. The consistency of this method with the emulation construction may lead to further restrictions.

  34. It is however incompatible with dual model and string theory constructions and their S-matrix interpretations.

  35. This is different in QM where the interacting Schrödinger equation remains functorially related to its operator Fock space formulation.

  36. To put it bluntly: compact groups arise from quantum causal localization in the presence of mass gaps [21].

  37. The indirect way of interpreting conformal theories as massless limits of perturbatively accessible massive models was only successful in the case of the massive Thirring model for which the perturbative Callan-Symanzik equation comes with a vanishing beta function [46].

  38. Equivalently they can be interpreted as operator-valued distributional sections on M c.

  39. A notable exception is the chiral Ising QFT [68].

  40. In many contemporary articles the fact that the tree-approximation of conformal theory (isomorphic to the classical structure) allow a restriction to a zero mass shell has been used to incorrectly allege that they can describe quantum particles in the sense of scattering theory and the S-matrix.

  41. The most prominent exception is the massive Thirring model. In fact the suspicion that β(g)≡0 which led the derivation of the Callan-Symanzik equation to all orders [46] came from the observation of softness in m→0.

  42. The Hilbert space positivity forces the Källén-Lehmann spectral measure to have a singularity which is milder than a mass-shell delta function.

  43. For semiinteger dimension as they already occur for free spinors it is necessary to take the double covering of M c . These fields fulfill an extended Huygens principle on the double covering.

  44. The analogy works better with squares of charges since the matter-antimatter charge compensation has no counterpart the composition of anomalous dimensions.

  45. Its abundance of degrees of freedom leads to the before-mentioned pathological timelike causality properties and the absence of reasonable thermodynamik behavior.

  46. There are no string-localized infinite spin representation components in the reducible superstring representation.

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Acknowledgement

I am indebted to Jens Mund and Jakob Yngvason for innumerous discussions on various topics which entered this work. Special thanks go to Detlev Buchholz for a critical reading of Sect. 5 which led to its reformulation.

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  1. Bert Schroer

    Present address: CBPF, Rua Dr. Xavier Sigaud 150, 22290-180, Rio de Janeiro, Brazil

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    Bert Schroer

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Dedicated to Raymond Stora on the occasion of his 80th birthday.

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Schroer, B. Modular Localization and the Foundational Origin of Integrability. Found Phys 43, 329–372 (2013). https://doi.org/10.1007/s10701-013-9699-3

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