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Causality and Dispersion Relations and the Role of the S-Matrix in the Ongoing Research

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Abstract

The adaptation of the Kramers-Kronig dispersion relations to the causal localization structure of QFT led to an important project in particle physics, the only one with a successful closure. The same cannot be said about the subsequent attempts to formulate particle physics as a pure S-matrix project.

The feasibility of a pure S-matrix approach are critically analyzed and their serious shortcomings are highlighted. Whereas the conceptual/mathematical demands of renormalized perturbation theory are modest and misunderstandings could easily be corrected, the correct understanding about the origin of the crossing property requires the use of the mathematical theory of modular localization and its relation to the thermal KMS condition. These new concepts, which combine localization, vacuum polarization and thermal properties under the roof of modular theory, will be explained and their potential use in a new constructive (nonperturbative) approach to QFT will be indicated. The S-matrix still plays a predominant role but, different from Heisenberg’s and Mandelstam’s proposals, the new project is not a pure S-matrix approach. The S-matrix plays a new role as a “relative modular invariant”.

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Notes

  1. Here and in the following we refer references to the bibliography in [1] wherever it is possible. This monography is a competent and scholarly written account of the subject, though it does not contain the QFT derivation which is based on the Jost-Lehmann-Dyson representation, the latter can be found in [2].

  2. Macro-causality is based on Born-localization and refers primarily to wave functions and their large time propagation. It leads to the concept of velocity of sound in QM and to the velocity v<c for relativistic mechanics (DPI). Micro-causality is an algebraic property of local observables which entails spacelike Einstein causality and the (timelike) causal completion property.

  3. The most prominent opponent against Born’s introduction of the probabilistic aspect to QM was Einstein, even though the resolution of the thermal aspect of the “Einstein-Jordan conundrum” which, similar to the Unruh Gedanken experiment brought thermodynamic probability into zero temperature QFT [7] may have softened his resistance.

  4. The exceptional cases where Haag duality for local observables breaks down are of special interest. For the local algebras generated by the free Maxwell field this happens for \(\mathcal{O}= \mbox{toroidal}\) spacetime region (the full QFT version of the semiclassical Aharonov Bohm effect [8]).

  5. The meaning of primitive causality in [1] is slightly different.

  6. Even in those few cases where they have a Lagrangian name, their existence and the construction of their formfactors cannot be achieved by Lagrangian quantizations.

  7. It was really a crisis resulting from insufficient understandings about QFT.

  8. A perturbative account of dispersion relations and momentum transfer analyticity was presented in [14]. At that time the divergence of the perturbative series was only a suspicion, but meanwhile it is a fact.

  9. The other alternative, namely that the global charge vanishes [16] is related to the Schwinger-Higgs mechanism of charge screening.

  10. As cynics commented, this was the safest way to prevent any further (after its failure in strong interactions) disagreement with observations.

  11. They may be seen as the QFT analogs of the Kepler- or quantum mechanical hydrogen-problem, but whereas integrable systems in the classical or quantum mechanical setting exist in any spacetime dimension, integrability in QFT does not extend beyond d=1+1 [44].

  12. The cardinality of scattering functions obeying the bootstrap principles is bigger than that of renormalizable Lagrangian couplings of free fields.

  13. Also the thermal Hawking effect is localization-caused, in this case the localization boundary is defined in terms of the curved spacetime metric.

  14. Only in factorizing theories their purely elastic S-matrices (only vacuum polarization no on-shell particle creation through scattering) can be computed through the bootstrap project. In higher dimensions there are no theories with only elastic scattering (Áks theorem), real particle creation and vacuum polarization go together and prevent a bootstrap construction.

  15. It is automatically fulfilled (the causal one-particle structure) in microcausal QFT and can be implemented in the appropriately formulated (see below) DPI setting of relativistic QM.

  16. Although Dirac introduced important concepts based on his project of a relativistic particle theory his implementation of a particle-hole theory led to inconsistencies in perturbative orders in which vacuum polarization entered.

  17. In field theoretic terminology this means changing the pointlike field by passing to another (composite) field in the same equivalence class (Borchers class), or in the setting of AQFT by picking another generator from the same local operator algebra.

  18. Contrary to popular opinion it is not the curvature but rather the localization which generates the thermal aspect. The event horizon attributes to the localization in front of the Schwarzschild horizon a physical reality whereas the causal horizon of a Rindler wedge has a more fleeting existence.

  19. For sharp localization it does not correspond to a density matrix; only if one passes to a somewhat fuzzy surface the algebra becomes a standard B(H) algebra and the KMS state passes to a Gibbs density matrix state. We believe that there is a relation to ’t Hooft’s “brick-wall” construction [49]; further work on this point is required.

  20. The localization-induced vacuum polarization may be seen as the metaphor-free aspect of the “broiling vacuum polarization soup” of the books on QFT [7].

  21. An S-matrix does however not distinguish a particular field, rather it associates to a local equivalence class (Borchers class) or more compactly to a unique net of local operator algebras of which those fields are different generators.

  22. Any canonical conformal free field is a free field and the LSZ asymptotic limit of any anomalous dimensional field vanishes [44, 50]. No inclusive cross section construction as used in QED led to an observable which could be interpreted in terms of particles.

  23. The original Lagrangian containing square roots of quadratic form in X μ(σ,τ) leads to the Pohlmeyer invariants which have no relation to string theory [60].

  24. As any field resulting from Lagrangian quantization, the string field is irreducible (the mathematical meaning of “dynamic”) whereas a direct sum is not.

  25. Wigner was hoping that his relativistic representation theory would lead to an intrinsic access (without invoking the quantization of classical structures) to QFT. Whereas the hope was well-founded and found its later realization in the notion of modular localization, the frame-dependent Newton-Wigner localization (the Born localization adapted to the relativistic inner product) was not what he had hoped for.

  26. It is a solution of the old problem of finding dynamical infinite component fields for which prior attempts to generate an representation containing an interesting infinite (m, s) tower spectrum failed [55].

  27. The requirement that a unitary positive energy representation of the Poincaré group acts on the inner symmetry target space of a non-rational chiral sigma model determines a mass/spin spectrum.

  28. Holistic refers to the fact that localization in QFT is, different from QM since it is always accompanied by thermal manifestations and vacuum polarization at the causal horizon of the spacetime localization region. In [7] and [48] the reader finds a more detailed presentation and illustrations of this concepts.

  29. One which is not representable as a line integral over a pointlike field.

  30. The iteratively constructed S(g) is the Stueckelberg-Bogoliubov-Shirkov operator functional which depends on space-time dependent coupling function. The S-matrix (formally for constant g’s) is related to this functional by the “adiabatic limit” whose existence is roughly equivalent to the asymptotic convergence of fields in the LSZ scattering theory, In many theories (e.g. QED) this limit does not exist as a result of the infraparticle phenomenon [9].

  31. Nowadays referred to as the Einstein-Jordan conundrum [22]. It played an important role in Heisenberg’s later discovery of vacuum-polarization, but the understanding of localization thermality had to wait more than 6 decades.

  32. The localization probability of a Schrödinger wave function and its relation to the spectral decomposition of the localization operator was introduced soon after Born defined the scattering probability (cross section) for the Born approximation. by Pauli; it appears as an added footnote in Born’s paper.

  33. For example the tensor factorization formalism known as “thermo-field formalism” breaks down in the thermodynamic limit for the same reasons as the Gibbs density matrix description, i.e. this formalisms is not suited to describe “open systems”.

  34. The possibility of doing this is called “the split property” [63]. Wheres the standard box quantization does not allow to view the boxed system as a subsystem of a system in a larger spacetime, the splitting achieves precisely this at the price of vacuum polarization at the boundary. The physics based on splitting is called “open system” setting.

  35. In [24] it was called the weak Unruh inverse; the terminology strong Unruh inverse being reserved for an isomorphism between a heat bath- and a localization caused-thermal system.

  36. Note that as a result of momenta conservation the dull S-matrices as well as their grazing shot counterparts for integrable models allow a simpler notation. The matrix elements of the n-particle S-matrix is a combinatorial product of two-particle amplitudes [70].

  37. Even if the existence of the Higgs particle (i.e. the particle associated with the real scalar field which survives after the charge of the complex scalar QED has been screened) will be observed, it is always better to have theoretical alternatives since less desperation enhances the credibility of results.

  38. Any Lagrangian in a d>2 with fields whose index space is noncompact does not have a quantum counterpart unless the indices are tensor/spinor indices associated with the Minkowski spacetime on which the field lives. Compact index spaces are allowed and represent inner symmetries [44].

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Acknowledgements

Am indebted to Herch Moyses Nussenzveig for valuable advice concerning historical aspects.

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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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  1. Institut für Theoretische Physik, FU-Berlin, Arnimallee 14, 14195, Berlin, Germany

    Bert Schroer

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To the memory of Jaime Tiomno (1920–2011).

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Schroer, B. Causality and Dispersion Relations and the Role of the S-Matrix in the Ongoing Research. Found Phys 42, 1481–1522 (2012). https://doi.org/10.1007/s10701-012-9676-2

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