Localization and the interface between quantum mechanics, quantum field theory and quantum gravity I: The two antagonistic localizations and their asymptotic compatibility

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Abstract

It is shown that there are significant conceptual differences between QM and QFT which make it difficult to view the latter as just a relativistic extension of the principles of QM. At the root of this is a fundamental distiction between Born-localization in QM (which in the relativistic context changes its name to Newton–Wigner localization) and modular localization which is the localization underlying QFT, after one separates it from its standard presentation in terms of field coordinates. The first comes with a probability notion and projection operators, whereas the latter describes causal propagation in QFT and leads to thermal aspects of locally reduced finite energy states. The Born–Newton–Wigner localization in QFT is only applicable asymptotically and the covariant correlation between asymptotic in and out localization projectors is the basis of the existence of an invariant scattering matrix.

In this first part of a two part essay the modular localization (the intrinsic content of field localization) and its philosophical consequences take the center stage. Important physical consequences of vacuum polarization will be the main topic of part II. The present division into two semi-autonomous essays is the result of a partition and extension of an originally one-part manuscript.

Section snippets

Introductory remarks

Ever since the discovery of quantum mechanics (QM), the conceptual differences between classical theory and QM have been the subject of fundamental investigations with profound physical and philosophical consequences. But the conceptual relation between quantum field theory (QFT) and QM, which is at least as challenging and rich of surprises, has not received the same amount of attention and scrutiny, and often the subsuming of QFT under “relativistic QM” nourished prejudices and prevented a

Historical remarks on the interface between QM and QFT

Shortly after the discovery of field quantization in the second half of the 1920s, there were two opposed viewpoints about its content and purpose represented by Dirac and Jordan (Darrigol, 1986). Dirac's maintained that quantum theory should stand for quantizing a real classical reality which meant field quantization for electromagnetism and quantization of classical mechanics for particles. Jordan, on the other hand, proposed an uncompromising field quantization point of view; his guiding

Direct particle interactions, relativistic QM

The Coester–Polyzou theory of direct particle interactions (DPI) (where “direct” means “not field-mediated”) is a relativistic theory in the sense of representation theory of the Poincaré group which, among other things, leads to a Poincaré invariant S-matrix. Every property which can be formulated in terms of particles, as the cluster factorization into systems with a lesser number of particles and other timelike aspects of macrocausality, can be implemented in this setting. The S-matrix does

First brush with the intricacies of the particles-field problems in QFT

In contrast to QM (Schrödinger-QM or relativistic DPI), interacting QFT does not admit a particle interpretation at finite times.11 If it would not be for the asymptotic scattering interpretation in terms of incoming/outgoing particles associated with the free in/out

More on Born versus covariant localization

In this section it will be shown that the difference between QM and LQP in terms of their localization result in a surprising distinction in their notions of entanglement. We will continue to use the word Born localization for the probability density of the x-space Schrödinger wave function p(x)=|ψ(x)|2; whereas its adaptation to the invariant inner product of relativistic wave functions which was done by Newton and Wigner (1949) and will referred to as BNW localization. Being a bona fide

Modular localization

Previously it was mentioned on several occasions that the localization underlying QFT can be freed from the contingencies of field coordinatizations. This is achieved by a physically as well as mathematically impressive, but for historic and sociological reasons little known theory. Its name “modular theory” is of mathematical origin and refers to a vast generalization of the (uni)modularity encountered in the relation between left/right Haar measure in group representation theory. In the

Algebraic aspects of modular theory

A net of real subspaces K(O)H1 for a finite spin (helicity) Wigner representation can be “second quantized”31 via the CCR (Weyl) respectively CAR quantization functor; in

String-localization and gauge theory

Zero mass fields of finite helicity play a crucial role in gauge theory. Whereas in classical gauge theory a pointlike massless vectorpotential is a perfectly acceptable concept, the situation changes in QT as a consequence of the Hilbert space positivity which for massless unitary representations leads to the loss of many spinorial realizations as expressed in the second line of (19), in particular to that of the vector-potential without which it is hardly possible to formulate perturbative

Building LQP via positioning of monads in a Hilbert space

We have seen that modular localization of states and algebras is an intrinsic i.e. field-coordinatization-independent way to formulate the kind of localization which is characteristic for QFT. It is deeply satisfying that it also leads to a new constructive view of QFT.

Definition

Wiesbrock, 1993

An inclusion of standard operator algebras (AB,Ω) is “modular” if (A,Ω) and (B,Ω) are standard and ΔBit acts like a compression on A i.e. (AdΔBitAA). A modular inclusion is said to be standard if in addition the relative

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