Classical and Quantum Nonlocality

The following sections are included:

• PREFACE

• CONTENTS

Local general relativity theory and non-local Mach-Einstein mechanics are extremely opposite attempts to realise the equivalence of inertia and gravitation. In the first case one is missing certain telescopic elements in the sense of Mach's principle, in the second one arrives at a mechanics that can hardly recover the results of the successful local physics. Therefore, in order to introduce telescopic elements into gravitational theory, one should choose a way between these two extremes. – It will be shown that, in the framework of tetrad theories of gravitation stemming from a Lagrangian built from Weitzenboeck's (anholonomic) invariants, one can harmonise Mach's principle with the locality principle by restricting general relativity. These theories satisfy a (local) field theoretical version of the Mach-Einstein doctrine saying that, via gravitation, the motion of matter described in concordance with the principle of equivalence determines the reference systems. It holds an interesting “complementarity” between strong general relativity of GRT and Mach-Einstein doctrine. In GRT, via Einstein's equations, the covariant and Lorentz-invariant Riemann-Einstein structure of the space-time defines the dynamics of matter. Otherwise, in all other cases, i.e., for the theories here under consideration the matter determines, via gravitation, the reference systems up to global Lorentz transformations.

In its core, the non-locality–locality debate is a discussion about the concept of the physical reality. The difference between the measurement bases of classical and quantum mechanics is often interpreted as a loss of reality arising in quantum mechanics. In this paper it is shown that this apparent loss occurs only if one believes that everyday experience determines the Euclidean space, instead of considering this space, both in classical and quantum mechanics, as a theoretical construction needed for measurement and representing one part of a dualistic space conception. From this point of view, Einstein's program of a unified field theory can be interpreted as the attempt to find a physical theory that is less dualistic. However, if one regards this dualism as resulting from the requirements of measurements, one can hope for a weakening of the dualism but not expect to remove it completely. Further, it follows from the requirements of measurements that the Heisenberg cut cannot be arbitrarily far moved towards the observer.

Epistemological discussions on quantum mechanics yield two insights: (i) To acquire physical knowledge, means of cognition are always necessary, and measurement provides this means. (ii) The space of measurement and the space of the representation of physical dynamics must be distinguished conceptually; both coincide factually in classical physics, but not so in quantum mechanics.

Noethers theorem will be used to derive the differential conservation laws which result from the invariance of an action with respect to infinitesimal space-time symmetries. This will be illustrated for the electromagnetic field and then applied to the Einstein theory. We will see that the structure of the differential conservation laws will lead to a description of the global quantities in asymptoticaly Minkowskian space-times. In particular, we shall discuss gravitational radiation at null infinity. Next we will discuss the concept of a quasilocal energy which was originally introduced by Penrose. While this will be discussed, particular attention will be given to an energy definition proposed by Hawking since it is closely related to a definition for quasi-local energy which I will propose.

Nonlocality, although qualitatively intuitively clear, is not a very well defined quantitative concept for physical theories. We discuss several valid, but inequivalent, quantitative expressions of this concept, both for classical and quantum theories, and apply them in important situations. In particular we exhibit a polarization procedure for construction a large class of nontrivial, nonlocal observables for linear field theories and derive their invariant mappings. An analysis of the physics of the various inequivalent choices of boundary conditions for field quantization throws significant light on a critical question associated with the quantization program for gravitation. It also clarifies the nature of some of the physical assumptions underlying the choice of boundary conditions, as well as indicating the existence, in principle, of experimentally verifiable differences implied by such choices.

We view the final remark of the EPR paper : “one would not be able to arrive at the above conclusion if two or more physical quantities are regarded as simultaneous elements of reality only when they can be simultaneously measured or predicted”. On this point of view, because of the collapse of the wave function of the combined system, the EPR entangled relations (or correlations) of momenta and positions belonging to two spatially separated particles considered in the first part of the paper do not hold good simultaneously for two separate measurements. This case does not reconcile with the locality principle and hence exhibits quantum nonlocality. Moreover, it is quite compatible with uncertainty principle. It is concluded that the EPR argument does not exhibit any paradox.

We will see if it is possible to treat some geometrical objects like the curvature and torsion as conjugate variables for which one can write uncertainty and commutation relations that indirectly have to do with the problem of nonlocality.

Starting from the possibility that curvature and torsion be conjugate variables satisfying uncertainty relations, we consider a quadratic Hamiltonian in torsion and curvature, in order to give a better basis for that conjecture. Due to closure failure (i.e. to space-time defects) we are led (through torsion) to position fluctuations while curvature gives rise to momentum fluctuations. Expressing both torsion and curvature in real space-time and noticing that a quadratic Lagrangian in curvature and torsion describes directly the quantum fluctuations at the Planck scale of space-time points (i.e., for instance, in the early universe) as harmonic oscillators, we show that spin fluctuations are of primary importance and they include also mass fluctuations.

A short historical review is made of some recent literature in the field of noncommutative geometry, especially the efforts to add a gravitational field to noncommutative models of space-time and to use it as an ultraviolet regulator. An extensive bibliography has been added containing reference to recent review articles as well as to part of the original literature.

The introduction of non-locality into elementary particles is to solve two fundamental problems, that is, the divergence difficulty and the unified description of elementary particles. Yukawa's theory is referred to as a typical theory of non-local fields.

As a geometrization of Yukawa's theory of non-local fields, the theory of fields in Finsler spaces is developed. Finsler spaces are spaces whose metric tensor g_μν depends on directional variables y^λ as well as coordinates X^λ. In Finsler spaces, any geometrical quantities depend on the directional variables as well as the coordinates, and, therefore, the internal freedom is introduced in the fields of elementary particles and the effects equivalent to those of the non-locality can be expected. Besides the microscopic world, in the macroscopic world, the geometrical structure of space-time is modified by the directional dependence of metric and the cosmological observable effects originating from the anisotropy of spaces and the torsion is expected to exist.

The removal of divergence is discussed in connection with the causality. Spaces of the special theory of relativity do not have a capacity to accept the extended particles and the metric fundamental form should be modified so as to change micro-causality, leaving macro-causality intact. The indefinite metric in Hilbert space presents itself as the result of the probability distribution of the metric fundamental form, the fluctuation of space-time. The variety of elementary particles is connected with the variety of distribution.

The possibility that quantum fluctuations in the structure of space-time at the Planck scale might be subject to experimental probes is discussed. The effects of space-time foam in an approach inspired by string theory, in which solitonic D-brane excitations are taken into account when considering the ground state are studied. The properties of this medium are described by analyzing the recoil of a D-particle which is induced by the scattering of a closed-string state. This recoil causes an energy-dependent perturbation of the background metric, which in turn induces an energy-dependent refractive index in vacuo, and stochastic fluctuations of the light cone. Distant astrophysical sources such as Gamma-Ray Bursters (GRBs) may be used to test this possibility, and an illustrative analysis of GRBs whose redshifts have been measured is presented. The propagation of massive particles through such a quantum space-time foam is also discussed.

The quantum string emission by Black Holes is computed in the framework of the “string analogue model” (or thermodynamical approach), which is well suited to combine QFT and string theory in curved backgrounds (particulary here, as black holes and strings posses intrinsic thermal features and temperatures). The QFT-Hawking temperature T_H is upper bounded by the string temperature T_S in the black hole background. The black hole emission spectrum is an incomplete gamma function of (T_H – T_S). For T_H ≪ T_S, it yields the QFT-Hawking emission. For T_H → T_S, it shows highly massive string states dominate the emission and undergo a typical string phase transition to a microscopic “minimal” black hole of mass M_min or radius r_min (inversely proportional to T_S) and string temperature T_S.

The semiclassical QFT black hole (of mass M and temperature T_H) and the string black hole (of mass M_min and temperature T_S) are mapped one into another by a “Dual” transform which links classical/QFT and quantum string regimes.

The string back reaction effect (selfconsistent black hole solution of the semiclassical Einstein equations with mass M₊ (radius r₊) and temperature T₊) is computed. Both, the QFT and string black hole regimes are well defined and bounded: r_min ≤ r₊ ≤ r_S, M_min ≤ M₊ ≤ M, T_H ≤ T₊ ≤ T_S.

The string “minimal” black hole has a life time .

In this paper we introduce the developments and develop our point of view in the field of the entanglement, nonlocality and quantum computation. It seems that the recently theoretical and experimental developments on entanglement and nonlocality are solid and important for our understanding the implication of quantum mechanics and that the developments on quantum computers are very important, but further developments will not be easy, therefore will be wonderful and attractive.

The following sections are included:

• SUBJECT INDEX