Beyond the Quantum

PREFACE

In Memoriam: Walter Philipp (1936–2006)

CONTENTS

If there exists a classical, i.e. deterministic theory underlying quantum mechanics, an explanation must be found of the fact that the Hamiltonian, which is defined to be the operator that generates evolution in time, is bounded from below. The mechanism that can produce exactly such a constraint is identified in this paper. It is the fact that not all classical data are registered in the quantum description. Large sets of values of these data are assumed to be indistinguishable, forming equivalence classes. It is argued that this should be attributed to information loss, such as what one might suspect to happen during the formation and annihilation of virtual black holes.

The nature of the equivalence classes is further elucidated, as it follows from the positivity of the Hamiltonian. Our world is assumed to consist of a very large number of subsystems that may be regarded as approximately independent, or weakly interacting with one another. As long as two (or more) sectors of our world are treated as being independent, they all must be demanded to be restricted to positive energy states only. What follows from these considerations is a unique definition of energy in the quantum system in terms of the periodicity of the limit cycles of the deterministic model.

The goal of the article is to provide some glimpses into the challenges and successes of quantum gravity. After a general introduction, for concreteness I focus on a specific approach which goes under the name loop quantum gravity. The underlying ideas are first summarized and recent advances are then illustrated by applying these ideas to cosmology. Quantum effects of geometry resolve the big-bang singularity of classical general relativity. Quantum physics does not break down at the big-bang. In simple models where details have been fully worked out, there is a pre-big-bang branch joined to the current post-big-bang branch by well-defined quantum evolution.

The EPR argument points to the existence of additional variables that are necessary to complete standard quantum theory. It was dismissed by Bohr because it attributes physical reality to isolated microscopic systems, independently of the macroscopic measurement apparatus. Here, we transpose the EPR argument to macroscopic systems, assuming that they are in spatially extended Fock spin states and subject to spin measurements in remote regions of space. Bohr's refutation of the EPR argument does not seem to apply in this case, since the difference of scale between the microscopic measured system and the macroscopic measuring apparatus can no longer be invoked.

In dilute atomic gases at very low temperatures, Bose–Einstein condensates are well described by a large population occupying a single-particle state; this corresponds, in the many particle Hilbert space, to a Fock state (or number state) with large number N. The situations we consider involve two such Fock states associated to two different internal states of the particles. The two internal states can conveniently be seen as the two eigenstates of the Oz component of a fictitious spin. We assume that the two condensates overlap in space and that successive measurements are made of the spins of single particles along arbitrary transverse directions (perpendicular to Oz).

In standard quantum mechanics, Fock states have no well defined relative phase: initially, no transverse spin polarization exists in the system. The theory predicts that it is only under the effect of quantum measurement that the states acquire a well defined relative phase, giving rise to a transverse polarization. This is similar to an EPR situation with pairs of individual spins (EPRB), where spins acquire a well defined spin direction under the effect of measurement - except that here the transverse polarization involves an arbitrary number of spins and may be macroscopic. We discuss some surprising features of the standard theory of measurement in quantum mechanics: strong effect of a small system onto an arbitrarily large system (amplification), spontaneous appearance of a macroscopic angular momentum in a region of space without interaction (non-locality at a macroscopic scale), reaction onto the measurement apparatus and angular momentum conservation (angular momentum version of the EPR argument). Bohr's denial of physical reality for microscopic systems does not apply here, since the measured system can be arbitrarily large. Since here we limit our study to very large number of particles, no Bell type violation of locality is obtained.

The measurement of a spin-½ is modeled by coupling it to an apparatus, that consists of an Ising magnetic dot coupled to a phonon bath. Features of quantum measurements are derived from the dynamical solution of the measurement, regarded as a process of quantum statistical mechanics. Schrödinger cat terms involving both the system and the apparatus, die out very quickly, while the registration is a process taking the apparatus from its initially metastable state to one of its stable final states. The occurrence of Born probabilities can be inferred at the macroscopic level, by looking at the pointer alone. Apparent non-unitary behavior of the measurement process is explained by the arisal of small many particle correlations, that characterize relaxation.

It is the purpose of the present contribution to demonstrate that the generalization of the concept of a quantum mechanical observable from the Hermitian operator of standard quantum mechanics to a positive operator-valued measure is not a peripheral issue, allegedly to be understood in terms of a trivial nonideality of practical measurement procedures, but that this generalization touches the very core of quantum mechanics, viz. complementarity and violation of the Bell inequalities.

We review two topics of quantum optics that shed new light on the effect of state reduction by a quantum measurement. One topic is the observation of quantum jumps switching on and off the fluorescence of a trapped atomic ion. The other one is the spontaneous decay of a single atom, described by the method of quantum trajectories. This method is based on the decomposition of the density matrix of an open system into an ensemble of time-dependent pure state vectors. Here we consider single histories of the spontaneously emitting atom. It is shown that in both cases the evolution is affected by a detection with a null result.

Fluctuations in the Bose-Einstein condensate (BEC) remain a rich field of study even in the ideal gas limit. We here present the laser master equation approach to the problem in the spirit of Eugene P. Wigner who said: “With classical thermodynamics, one can calculate almost everything crudely; with kinetic theory, one can calculate fewer things, but more accurately; and with statistical mechanics, one can calculate almost nothing exactly.” The combination of kinetic theory plus statistical mechanics proves to be a powerful combination for the calculation of essentially exact BEC equilibrium results.

We present some of our results on multi-mode scattering of entangled photon pairs. We describe these scattering processes in terms of trace-preserving and non-trace-preserving local quantum maps. We show that non-trace preserving local maps can lead to apparent violations of causality, when the two-photon states are post-selected by coincidence measurements.

It is shown that the full unknown state of a spin-½ system, S, described by its density matrix, can be determined with a simultaneous measurement with the help of another system A, called assistant, whose initial state is known. The idea is to let S and A interact with each other in a known way during a proper interaction time τ, and then to measure simultaneously two observables, one of S and one of A and then to determine their averages and their correlation. One thus determines the three unknown components of the polarization vector of S by means of repeated experiments using a unique stetting. In this way one can measure all the non-commutative observables of S at the same time, which may seem prohibited in quantum mechanics.

The quite different behaviors exhibited by microscopic and macroscopic systems with respect to quantum interferences suggest the existence of a borderline beyond which quantum systems loose their coherences and can be described classically. Gravitational waves, generated within our galaxy or during the cosmic expansion, constitute a universal environment susceptible to lead to such a quantum decoherence mechanism. We assess this idea by studying the quantum decoherence due to gravitational waves on typical microscopic and macoscopic systems, namely an atom interferometer (HYPER) and the Earth-Moon system. We show that quantum interferences remain unaffected in the former case and that they disappear extremely rapidly in the latter case. We obtain the relevant parameters which, besides the ratio of the system's mass to Planck mass, characterize the loss of quantum coherences.

We discuss the current status of the black hole information loss paradox and propose a plan for its solution based on analogies with solid state physics and the irreversibility problem. In a recent paper Hawking has argued that there is no information loss in black holes in asymptotically AdS spacetimes. We remind that there are several types of information (entropy) in statistical physics – fine grained (microscopic) and coarse grained (macroscopic) ones which behave differently under unitary evolution. We suggest that the coarse grained information of the rest of the Universe is lost while fine grained information is preserved. A possibility to develop in quantum gravity an analogue of the Bogoliubov derivation of the irreversible Boltzmann and Navier - Stokes equations from the reversible mechanical equations is discussed.

Recent investigations of correlations within compound physical systems are considered in the broad context, which includes “superquantum” correlations, namely, those that are stronger than predicted by standard quantum mechanics. Although the significance of these results in the search for deeper principles underlying quantum physics remains uncertain, the results do improve our understanding of quantum correlations.

The interpretation of quantum mechanics (or, for that matter, of any physical theory) consists in answering the question: How can the world be for the theory to be true? That question is especially pressing in the case of the long-distance correlations predicted by Einstein, Podolsky and Rosen, and rather convincingly established during the past decades in various laboratories. I will review four different approaches to the understanding of long-distance quantum correlations: (i) the Copenhagen interpretation and some of its modern variants; (ii) Bohmian mechanics of spin-carrying particles; (iii) Cramer's transactional interpretation; and (iv) the Hess–Philipp analysis of extended parameter spaces.

We compare quantum mechanics as a theory involving probabilities to the framework of Kolmogorov's probability theory with emphasis on the connections of these theories to actual experiments. We find crucial differences in the way incompatible experiments are defined and treated in these two approaches and show that these differences are the origin for difficulties and apparent contradictions that are encountered when considering so called no-go proofs particularly that of John Bell. For example, Bell was convinced that in a theory in which parameters are added to quantum mechanics to determine the results of individual measurements, violations of Einstein-locality must occur. Based on our comparative study, we show that rather the opposite is true and that a precise space-time treatment based on relativity uncovers contradictions in the assumptions for Bell's no-go proof and resolves the difficulties.

We present some recent results of a new statistical analysis of the optical EPR experiment performed by Weihs et al in Innsbruck 1997-1998. Under the commonly used assumption of fair sampling, we show that the coincidences counts exhibit a small and anomalous non-signalling component, which seems impossible to explain by using conventional quantum mechanics, and we discuss some possible interpretations of this phenomenon.

The mathematical formulation of Quantum Mechanics is derived from purely operational axioms based on a general definition of experiment as a set of transformations. The main ingredient of the mathematical construction is the postulated existence of faithful states that allows one to calibrate the experimental apparatus. Such notion is at the basis of the operational definitions of the scalar product and of the adjoint of a transformation.

In this paper we approach the question of the existence of a (x, p) phase space in a new way. Rather than abandoning all hope of constructing such a phase-space for quantum phenomena, we take aspects from both the Wigner-Moyal and Bohm approaches and show that although there is no unique phase space, we can form ‘shadow’ phase spaces. We then argue that this is a consequence of the non-commutative geometry defined by the operator algebra.

In previous publications we constructed a kind of theory with hidden variables – Pre-quantum Classical Statistical Field Theory (PCSFT). The role of hidden variables was played by classical fields. Since the corresponding phase space of classical fields has the infinite dimension, the model was quite complicated from the mathematical viewpoint. It was based on integration over the infinite dimensional Hilbert space. In this note we consider a finite dimensional illustration for PCSFT. Now phase space has a finite dimension, all integrals are usual Gaussian integrals. It becomes completely evident that quantum mechanics can be considered as just a simple algorithm for approximation of Gaussian integrals.

Dyson published in 1990 a proof due to Feynman of the Maxwell equations. This proof is based on the assumption of simple commutation relations between position and velocity. We first study a nonrelativistic particle using Feynman formalism. We show that Poincaré's magnetic angular momentum and Dirac magnetic monopole are the direct consequences of the structure of the SO(3) Lie algebra in Feynman formalism. Then we show how to extend this formalism to the dual momentum space with the aim of introducing Non-commutative quantum mechanics which was recently the subject of a wide range of works from particle physics to condensed matter physics.

Snyder Space is a modified formulation of quantum mechanics, proposed by Snyder in Phys. Rev. 71, 38-41 (1947) to circumvent infinities in high-energy physics. Snyder space proposes discrete space, continuous time, continuous momentum and continuous energy. It also leads to modified commutation relations for position and momentum operators. Snyder's algebra is therefore related to current theories beyond the standard model, such as noncommutative quantum mechanics, minimal length uncertainty relations, and dynamical quantization. Snyder obtains his algebra by postulating an underlying space on which both position and momentum operators are defined and by requiring Lorentz invariance of the theory. When solving problems in Snyder space one can in principle study the equations as they appear in coordinate space, momentum space, or in the underlying space. Alternatively one can transform Snyder space problems into modified problems in standard quantum mechanics using operator methods. In the limit that the spatial lattice parameter a vanishes, Snyder space reduces to standard quantum mechanics, thereby giving a quantitative meaning to the preposition “beyond” in “beyond the quantum”. We give a brief introduction to Snyder space and quote results for some simple systems. Finally we speculate on the connection with other approaches including some that assume the existence of a structure underlying quantum mechanics.

Stochastic electrodynamics (SED), an alternative theory to quantum phenomena based on laws of classical physics is shortly reviewed and compared with quantum electrodynamics. Experiments supporting the existence of zero-point fluctuating radiation field, the key concept of SED, are discussed. Relation between measurements of the black-body radiation spectrum and noise is analysed to define conditions under which the zero-point component of radiation or noise can be observed. Further, it is shown that stability of weakly localized orbits, measured in disordered solid state systems, can be explained by the presence of zero-point fluctuations of vacuum.

We study anew the behaviour of an otherwise classical bound particle immersed in a radiation field that includes the zero-point field component of average energy (1/2)ħω per mode. The presence of this field introduces an essential stochasticity into the dynamics of the particle, characterized by Planck's constant ħ; this has been the basis for stochastic electrodynamics. Both the near field and the particle are affected substantially by their continuous interaction. Stationary solutions are in principle possible when a balance is achieved between the mean powers emitted and absorbed by the particle. By demanding that the ensuing approximate stationary solutions satisfy an ergodic principle, we are led to a resonant response that is linear in the Fourier amplitudes of the field; this is the essence of linear stochastic electrodynamics. The connection with the matrix formulation of quantum mechanics can be readily made, with the resonance frequencies of the ergodic solutions corresponding to the quantum mechanical transition frequencies. Some implications of these results for the understanding of quantum phenomena are briefly discussed.

The work-in-progress on the conjectured origin of the inertia reaction force (Newton's Second Law) in quantum vacuum fields is discussed and reviewed. It is first pointed out that the inertia reaction force is not a fundamental effect at the particle level, but an emergent macroscopic phenomenon that appears in large condensed aggregates. A brief sketch of the analysis that leads to the derivation of the electromagnetic vacuum contribution to the inertia reaction force is presented, in several complementary ways and also in a fully covariant way. All derivations were initially done within Stochastic Electrodynamics and more recently, we briefly report here for the first time, they have been reformulated within ordinary Quantum Electrodynamics. Analysis leading to an expression for, what we can call, the vacuum electromagnetic field contribution to the inertia reaction force, is briefly reviewed. As an example, the case of an ordinary electromagnetic (microwave) cavity is briefly mentioned with its associated very small but nonnegligible inertial mass of the interior of the microwave cavity case (i.e., the cavity alone not considering its walls). Next, it is briefly mentioned that the results for inertial mass can be passed to passive gravitational mass. Thus some light is thrown on the origin of the Weak Equivalence Principle, which equates inertial mass to passive gravitational mass. Finally we mention the derivation of Newton's gravitational force expression that easily follows from this analysis. Unfortunately, all this has been accomplished just for the electromagnetic vacuum case, as contribution by the other quantum vacuum fields have not been calculated. This specially refers to the gluonic vacuum, which presumably contributes the lion's share of the inertia reaction force in ordinary objects. Furthermore, the origin of what constitutes active gravitational mass has still not been considered within this approach. I.e., why a massive object “bends” space-time still remains unexplained.

A specific form of the Hopf map S3 → S2 is introduced. The map is represented by the fields θ and φ describing a (infinitely) extended, topologically stable field structure in three dimensional space performing an internal rotation. That ‘bunched’ field shows properties that can be identified with properties of a particle: discrete mass and charge, spin, left or right-handed chirality, and a far-reaching field oscillating in phase with the internal rotation of the structure. In this model the particle is not introduced as a basic object but appears as a specific field concentration of the extended field, which is considered to be the fundamental entity.

The Kerr-Newman (KN) solution to Einstein's equation shows a gyromagnetic factor g = 2, typical of a Dirac spinor. This fact has prompted many attempts to consider this solution as the exterior metric for a fundamental spin 1/2 particle. In the present work, the KN solution is proposed as the exterior and interior solution for a fundamental particle, leading to a redefinition of the particle concept. By considering the extended interpretation of Hawking and Ellis, other properties like the spacetime spinorial structure, mass and charge follow from its non- trivial geometry. A crucial point of the model is the excision of the ring singularity present in the original KN solution. This excision removes non-causal regions of the solution, and is consistent with its metric structure. Although the spacetime dimension of the singularity is of the order of the particles's Compton wavelength, which for the electron is λ = 10⁻¹¹cm, the space dimension of the ring is found to vanish. In the three-dimensional space, therefore, it is a point-like object, a property that validates the concept of “fundamental particle” of the model.

Dirac electron theory and QED do not take into account gravitational field, while the corresponding Kerr-Newman solution with parameters of electron has very strong stringy, topological and non-local action on the Compton distances, polarizing space-time and deforming the Coulomb field. We discuss the relation of the electron to the Kerr's microgeon model and argue that the Kerr geometry may be hidden beyond the Quantum Theory. In particular, we show that the Foldi-Wouthuysen ‘mean-position’ operator of the Dirac electron is related to a complex representation of the Kerr geometry, and to a complex stringy source. Therefore, the complex Kerr geometry may be hidden beyond the Dirac equation.

In classical electrodynamics, extended with gradients of the electric and magnetic fields, a linear soliton is presented which bears features of the Kerr-Newman electron of electro-gravity. This is considered as a model for the electron, having a ring shape, with diameter equal to the Compton length ħ/mc and thickness smaller by the fine structure constant. The soliton has a finite mass, a spin-½, a g = 2 factor, and an electric quadrupole moment that is also “twice too large”. From this setup, all relativistic corrections to the classical version of the Pauli Hamiltonian are derived. There appears an additional, spin-dependent quadrupolar force that may vanish on the average. Particle-antiparticle annihilation may become explained on the basis of electromagnetic attraction.

Non-collapse interpretations of quantum mechanics set themselves the task of developing a self-consistent and empirically adequate version of quantum mechanics that does not make use of the projection postulate (or collapse of the wavefunction). Only unitary evolution is allowed in these interpretations, so that superpositions are always maintained during evolution—even in measurements. In this paper we discuss how this deterministic mathematical scheme can be brought into accordance with the usual statistical predictions of quantum mechanics. In particular, we investigate how the Born probability rule fits in.

A seldom recognized fundamental difficulty undermines the concept of individual “state” in the present formulations of quantum statistical mechanics and quantum information theory. The difficulty is an unavoidable consequence of an almost forgotten corollary proved by Schrödinger in 1936 and perused by Park in 1968. To resolve it, we must either reject as unsound the concept of state, or else undertake a serious reformulation of quantum theory and the role of statistics. We restate the difficulty and discuss alternatives towards its resolution.

It is argued that local realism is a fundamental principle, which might be rejected only if experiments clearly show that it is untenable. Forty years after Bell's work no experiment has provided a valid, loophole-free, violation of local realism which, in my opinion, is thus reinforced. I study a simple, but wide, family of local realistic models and derive new inequalities almost insensitive to the detection loophole. I argue that quantum mechanics, with some change in the theory of measurement, might be compatible with local realism.

This is a transcript of the round table discussion, moderated by G. 't Hooft, that took place at the end of the workshop “Beyond The Quantum” in the Lorentz Center of the University of Leiden, the Netherlands, 29 May - 2 June 2006.

It displays current views on foundations of quantum mechanics.

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