Localization and the interface between quantum mechanics, quantum field theory and quantum gravity I: The two antagonistic localizations and their asymptotic compatibility
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 ; 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 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 An inclusion of standard operator algebras is “modular” if and () are standard and acts like a compression on i.e. . A modular inclusion is said to be standard if in addition the relativeWiesbrock, 1993
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2019, Nuclear Physics BCitation Excerpt :This does not exclude the use of the quantum mechanical Newton–Wigner localization to describe the dissipation of wave-packets. One may even construct a macro-causal Poincaré-invariant multi-particle theory which satisfies cluster decomposition properties and leads to a Lorentz-invariant scattering matrix ([43] section 3). Such a construction disproves the conjecture that relativistic particles and cluster properties alone will lead to QFT.
Modular localization and the holistic structure of causal quantum theory, a historical perspective
2015, Studies in History and Philosophy of Science Part B - Studies in History and Philosophy of Modern PhysicsCitation Excerpt :All structural properties of LQP and the resulting general theorems can be expressed in terms of local nets of operator algebras, but the present formulation of renormalized perturbation theory still needs generating fields. A particular radical illustration of the conceptual differences between QFT and QM is the reconstruction of a net of operator algebras from the relative modular position of a finite number of copies of the monad (Schroer, 2010b). For chiral theories on the lightray one needs two monads in a shared Hilbert space in the position of a modular inclusion, for d=1+2 this “modular GPS” construction needs three and in case of d=1+3 seven modular positioned monads are sufficient to create the full reality of a causal quantum matter world, including its Poincaré symmetry (and hence Minkowski spacetime) from the abstract modular groups (Kaehler & Wiesbrock, 2001).
Localization and the interface between quantum mechanics, quantum field theory and quantum gravity II. The search of the interface between QFT and QG
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