Abstract

The different roles and natures of spacetime appearing in a quantum field theory and in classical physics are analyzed implying that a quantum theory of gravitation is not necessarily a quantum theory of curved spacetime. Developing an alternative approach to quantum gravity starts with the postulate that inertial energy-momentum and gravitational energy-momentum need not be the same for virtual quantum states. Separating their roles naturally leads to the quantum gauge field theory of volume-preserving diffeomorphisms of an inner four-dimensional space. The classical limit of this theory coupled to a quantized scalar field is derived for an on-shell particle where inertial energy-momentum and gravitational energy-momentum coincide. In that process the symmetry under volume-preserving diffeomorphisms disappears and a new symmetry group emerges: the group of coordinate transformations of four-dimensional spacetime and with it General Relativity coupled to a classical relativistic point particle.

Highlights

  • Spacetime is a basic ingredient in the construction of any Quantum Field Theory (QFT) of microscopic interactions such as the electro-magnetic, weak and strong forces in the Standard Model (SM)

  • Looking at the experimental information we have about microscopic interactions of elementary particles which originates from scattering experiments, what we observe is a number of incoming particles—typically two characterized by their masses, four-momenta, electric charges etc. transitioning with some probability into a number of outgoing particles again characterized by their masses, four-momenta, electric charges etc

  • On the other hand a new symmetry group emerges: the group of coordinate transformations of four-dimensional spacetime and with it General Relativity coupled to a classical relativistic point particle

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Summary

Introduction

Spacetime is a basic ingredient in the construction of any Quantum Field Theory (QFT) of microscopic interactions such as the electro-magnetic, weak and strong forces in the Standard Model (SM). To establish a model for what happens in the unobservable black box, one links the experimental information to the machinery of an appropriate QFT and its S-matrix by abstracting the incoming and outgoing particles as non-interacting asymptotic quantum states and employing the LSZ-reduction formalism to express the scattering amplitudes as Fourier-transformed, amputed, on-shell vacuum expectation values of time-ordered products of quantum field operators [1]-[4]. Instead any approach should be worthwhile to develop which respects the various conditions for a viable QFT such as causality, renormalizability and the validity of conservation laws such as for energy-momentum and which yields a classical limit respecting the Equivalence Principle, geometrizing gravity at the classical level One such approach based on Minkowski space as the idealized spacetime embedded in the QFT model for gravity exists. It is understood that the physical limit has to be taken whenever aiming for the calculation of observable quantities

Classical Limit of a Scalar Quantum Field Coupled to Quantum Electrodynamics
Gauge Theory of Volume-Preserving Diffeomorphisms Revisited
Conclusions
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