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

Abstract

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...

A Hilbert Space Setting for Interacting Higher Spin Fields and the Higgs Issue

“…A gauge theoretic formulation for s > 1 interactions does not seem to be known. On the other hand the higher spin analogs of the linear relation between stringlocal free fields and their pointlike partners in the same localization class are rather straightforward; instead of a single scalar stringlike Stückelberg field one obtains a linear relation involving a family of intrinsic tensor Stückelberg fields for all spins up to s−1 [12]. These are the prerequisites for an extension to higher spin interactions.…”

“…This new concept is not only important for the ongoing research in QFT[11], but it also permits to understand old problems in a new light[12] 13. Based on the standard idea that a click in a counter localizes a particle in the counter (with rapidly decreasing vacuum polarization caused tails).…”

The role of positivity and causality in interactions involving higher spin

“…The important role of the S-matrix for modular localization in wedges has triggered attempts to reconstruct a full causal QFT from its "on-shell footprint" in form of its S-matrix [77,78]. These ideas are presently too weak for constructions in higher dimensions.…”

Quantum Field Theory as a Faithful Image of Nature

“…14 Maybe all these problems can be resolved in the wider context of algebraic quantum field theory by means of the promising idea of modular localization [111]. 1.1.6 Approaches to quantum field theory Figure 1.4 on p. 26 suggests that a large number of particles can be created in a high-energy proton-proton collision.…”

“…Whereas mainstream physics clearly goes with the successful pragmatic approach, the supposed necessity to choose between "no (mathematical) rigor" and "no (practical) relevance" is much harder for philosophers, as they should insist on both (so should physicists). The powerful concept of modular localization, which is based on the Tomita-Takesaki modular theory of operator algebras, has a promising potential for achieving the big goal of bringing the algebraic and pragmatic approaches together, thus even revealing the reasons for the success of the pragmatic approach and at the same time leading to a deeper understanding of the quantization of gauge theories and to a demystification of the Higgs mechanism (for an overview, see [111]). Moreover, the ensemble aspect and statistical mechanics type of probability follow naturally from modular localization, although in a more radical way compared to how we here implement these features by an irreversible contribution to time evolution on a finite space.…”