Testing a conjecture on the origin of space, gravity and mass

Abstract

A Planck-scale model for the microscopic degrees of freedom of space and gravity, based on a fundamental principle that involves fluctuating one-dimensional strands, is tested. Classical and quantum properties of space and gravitation, from the field equations of general relativity to gravitons, are deduced. Predictions include the lack of any change to general relativity at all sub-galactic scales, the validity of black hole thermodynamics, the lack of singularities and the lack of unknown observable quantum gravity effects. So far, all predictions agree with observations, including the validity of the maximum luminosity or power value $c^5/4G$ for all processes in nature, from microscopic to astronomical. Finally, it is shown that the strand conjecture implies a model for elementary particles that allows deducing ab-initio upper and lower limits for their mass values.

Appendices

Appendix 1: On the circularity of the fundamental principle

On the one hand, the crossing switch of Fig. 1 is assumed to take place in space. On the other hand, space, distances and physical observables are assumed to arise from strands. The apparent circularity can be avoided—to a large degree, but not completely—by increasing the precision of the formulation.

Crossing switches take place in background space. In the strand conjecture, background space is defined by the observer. In contrast, physical space, physical distances and physical observables arise from strands and their crossing switches. When space is flat, background space and physical space coincide. Otherwise, they do not; in that case, background space is (usually) the local tangent space of physical space. A similar situation arises for the concept of time.

In nature, any observation of a change implies the use of (background) time; any observation of difference between objects or systems implies the use of separation in (background) space. Indeed, a local background space—observer-defined and usually observer-dependent—is required to describe any observation, or simply, to talk about nature. In the strand conjecture, it is equally impossible to define crossing switches or any Planck unit without a background. The strand conjecture asserts that a description of nature without a background space and time is impossible.

Every use of the term ‘observation’ or ‘observable’ or ‘physical’ implies and requires the use of a background space and time. All the illustrations of the present work are drawn in background space. In contrast, physical space—an observable in general relativity, dynamical and pseudo-Riemannian—arises through crossing switches of strands. The local background space agrees with physical space only locally, where the crossing switches being explored are taking place. In fact, the need for a background space to describe nature is rooted in a deeper issue.

Background space is what is needed to talk about nature. Physical space is everything that can be measured about space: curvature, vacuum energy, entropy, temperature, etc.

There is a fundamental contrast between nature and its precise description. The properties of nature itself and the properties of a precise description differ and contradict each other. A precise description of nature requires axioms, sets, elements, functions, and in particular continuous background space, continuous background time and points in background space and background time. In contrast, due to the uncertainty relations, at the Planck scale, nature itself does not provide the possibility to define points in physical space or time; physical space and time are not continuous at smallest scale, and in fact, physical)space and time are emergent. In short, observer space, or background space, differs in its properties from physical space.

Any precise description of nature thus requires a limited degree of circularity in its definition of physical time and space with the help of background time and background space. Therefore, an axiomatic description of all of nature is impossible. An axiomatic description is only possible for those parts of nature that avoid the fundamental circularity, such as quantum theory, or special relativity, or quantum field theory, or electromagnetism, or general relativity. Even though Hilbert asked for an axiomatic description of physics in his famous sixth problem, no claim for an axiomatic description of all of nature (all of physics) has ever appeared in the literature. Any unified description of nature must be circular. The strand conjecture, like or any other unified model, can be tested by asking whether it is a consistent, complete and correct description of nature. So far, this appears to be the case for strands.

An important example for the difference between an axiomatic description and a consistent, complete, correct—but somewhat circular—description is the dimensionality of space. The number of dimensions of (background and physical) space is not a consequence of the fundamental principle or of some axiom; the number of dimensions is assumed in the fundamental principle right from the start. Tangles only exist in three dimensions. Only three dimensions allow a description of nature that is consistent, complete and correct: only three dimensions allow crossing switches, particle tangles, spin 1/2, Dirac’s equation and Einstein’s field equations.

Appendix 2: From strands to quantum theory and the standard model Lagrangian

Fig. 10

A configuration of two skew strands, called a strand crossing in the present context, allows defining density, orientation, position and a phase. These are the same properties that characterize a wave function. The freedom in the definition of phase is at the origin of the choice of gauge. For a complete tangle, the density, the phase and the two (spin) orientation angles define, after spatial averaging, the two components of the Dirac wave function $\Psi$ of the particle and, for the mirror tangle, the two components of the antiparticle

Fig. 11

In the strand conjecture, the wave function and the probability density are due, respectively, to crossings and to crossing switches at the Planck scale. The wave function arises as time average of crossings in fluctuating tangled strands. The probability density arises as time average of the crossing switches in a tangle. The tethers—strand segments that continue up to large spatial distances—generate spin 1/2 behaviour under rotation and fermion behaviour under particle exchange. The tangle model also ensures that fermions are massive and move slower than light (see text)

This appendix provides an extremely short summary of references [16] and [23], which explain how quantum theory, quantum field theory and the full Lagrangian of the standard model arise from strands. The tangle model for massive quantum particles is illustrated in Figs. 10 and 11. The figures visualize that crossings have properties similar to those of wave functions, and that time-averaged crossing switches have the same properties as probability densities.

Starting from the fundamental principle and Dirac’s belt trick, tangles of fluctuating strands in flat (physical) space indeed describe matter particles and wave functions: the wave function of a particle is the strand crossing density of its fluctuating tangle. In other words, wave functions arise as local time averages of strand crossings. More specifically, to get the value of the wave function at a certain position in space, the local time average of the strand crossings at that position is taken, averaging over a time scale of (at least) a few Planck times. In this way, a density and a phase can be defined, for each ‘position’ in space. As usual for quantum theory, also in the strand conjecture physical space and time have to be defined before defining the concept of wave function. The probability density for a particle is the local time average of the crossing switch density of its fluctuating tangle. A detailed exploration [16, 23] shows that strands produce a Hilbert space, the quantum phase, interference, contextuality and freedom in the definition of the absolute phase value.

Moving particles are advancing rotating tangles. Antiparticles are mirror tangles rotating in the opposite direction. Fluctuating rational tangles made of two or more strands imply spin 1/2 behaviour under rotation and, above all, Dirac’s equation [14]. For systems of several particles, tangles reproduce fermion behaviour and entanglement. Tangles of strands are fully equivalent to textbook quantum theory and predict the lack of any extension or deviation, up to Planck energy. For example, the principle of least action is the principle of fewest crossing switches. In this way, strands also explain the origin of the principle of least action [23].

No new physics arises in the domain of quantum theory. Strands only visualize quantum theory; they do not modify it. Every quantum effect is due to crossing switches—and vice versa. The visualization of quantum effects with strands requires that strands remain unobservable in principle, whereas their crossing switches are observable.

Fig. 12

An illustration of two Feynman diagrams of quantum electrodynamics in the tangle model

Tangles also allow deducing quantum field theory. Exploring all possible tangles, it appears that rational, i.e. unknotted tangles reproduce the known spectrum of elementary particles and their properties [16, 23]. Every massive elementary particle is represented by an infinite family of rational tangles made of either two or three strands. Quarks are made of two strands; all other massive elementary particles are made of three strands. Three generations for quarks and for leptons arise. The Higgs itself is represented by a braid. The family members for each elementary particle differ among them only by the number of attached braids. The structure of each elementary particle tangle explains the spin value, parity, charge and all other quantum numbers.

Models for the massless bosons also arise. In particular, a photon is a single, twisted strand. Photons are emitted or absorbed by topologically chiral tangles, i.e. by fermion tangles that are electrically charged. Figure 12 illustrates the strand conjecture for quantum electrodynamics. Only three kinds of massless bosons arise, each kind due to one Reidemeister move. The boson generator algebras turn out to be the well-known U(1), broken SU(2) and SU(3) of the three gauge interactions [16, 23]. The violation of parity in the weak interaction and the way that the massless bosons of SU(2) acquire mass are also explained.

A detailed investigation shows that tangles reproduces every propagator and every Feynman vertex observed in nature—and no other ones. Particle mixing appears naturally. The correct couplings between fermions and bosons also arise. As a result, the full Lagrangian of the modern standard model arises, term by term, including PMNS mixing of Dirac neutrinos, without any addition or modification [16, 23].

In short, strands predict the lack of any physics beyond the standard model. Discovering such an effect or any new influence between quantum field theory and gravitation at sub-galactic scales—apart from the cosmological constant, the particle masses and the other constants of the standard model, including their running with energy—would falsify the strand conjecture. This terse summary of the implications of strands for quantum field theory allows proceeding with the exploration of space and gravity.