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Relational Quantum Gravity incorporates the special and general theories of relativity, as well as quantum mechanics and quantum electrodynamics. To show how it reconciles these theories using the teleconnection, I first review these theories from first principles. The treatments here are kept as simple as possible and are aimed at an introductory level.
A brief view of the history of relationist ideas, leading up to a description of the underlying physical principles which Einstein built into relativity and using a minimal amount of mathematics.
Philosophical Foundations
An overview of fundamental philosophical ideas of space and time which underlie the development of physical theory, leading up to Einstein’s general principle of relativity. I aim here to provide some historical and philosophical context for relational ideas. The remainder of the site is dedicated to the verification of these ideas, by means of the full development of a mathematical model, and the description of empirical tests.
Quantum Weirdness
Quantum theory is famous for the paradoxical way in which it describes the universe. More strictly the paradoxes lie in the “standard”
Copenhagen interpretaion of quantum mechanics, not in its mathematical structure. Some physicists adopt
other interpretations, but these appear inadequate to properly describe properties of matter in a complete, coherent, and consistent manner. Many declare that interpretation is not the business of physics. This section contains a general description of quantum theory and of paradox in the Copenhagen interpretation, and describes how the paradoxes are addressed in relational quantum gravity.
In spite of the mystique surrounding it, special relativity is remarkably simple and straightforward. If you have understood the philosophical ideas leading to the general principle of relativity, and can remember a bit of school level algebra, then you can prove for yourself results like Lorentz contraction, time dilation and Einstein’s most famous formula, E = mc2. This page leads you through it. All you need to do is rearrange a few equations.
Basics of Curvature
A gentle introduction to ideas encapsulated in the mathematics of Riemannian Geometry.
The Equivalence Principle
Newton’s laws are restated after replacing Newtonian absolute space with spacetime, leading to Einstein’s
equivalence principle, that gravity is an
inertial force, from which it follows that gravity is a manifestation of spacetime curvature.
This is the maths which I regard as essential to an understanding of modern physics. I have kept it to a minimum and made the treatments as simple as possible. Without it, I don’t think it is possible to have a full understanding of either quantum theory or general relativity, but if the mathematics is understood then neither of those theories hold any great problems.
Miscellaneous Methods
Before moving on to describe general relativity and quantum theory, I need to briefly mention a few really useful bits of mathematics which you may not have done in school, or if you did do them you may need a reminder because they are not the sort of thing most people have to think about too much in daily life.
Introduction to Vector Space
I often think
vector space is the most useful abstract structure in mathematics, almost that it is the only useful structure. Vector space shows how almost everything in mathematics can be reduced to multiplication and adding up. This makes it really easy. Vector space is the foundation from which the mathematics of both quantum theory and general relativity are built, it is vital to much of mathematics, and has applications in statistics, computing, and in all areas of science. In this treatment, I try to leave out most of the jargon and illustrate that, in good mathematics, elegance and simplicity go hand in hand.
A quick trip through some essential concepts and language.
Introduction to Tensors
Tensors are built from vectors. They provide the mathematical structure used to describe states of many particles in quantum mechanics, as well as the structure to express general physical law in general relativity. The idea in their construction is remarkably straightforward. Tensors simplify physical laws and show that the magnetic force is just the electrostatic force after Lorentz transformation.
General relativity expresses Einstein’s physical ideas on the nature of time and space in the mathematical language of tensors and Riemannian geometry.
Concepts of General Relativity
In general relativity, Einstein put his physical ideas on the nature of time and space, into the mathematical language of
tensors and
Riemannian geometry.
Riemann Curvature
If index gymnastics were a physical sport, this page would be a training session for the fit and athletic. In it, the covariant derivative is established from local parallelism, the
Riemann curvature tensor is found, properties are analysed, and the
Einstein curvature tensor is found and shown to obey the
contracted Bianchi identity, which has importance in
Einstein’s law of gravitation. From a philosophical perspective, the important aspect is that manipulations in mathematics introduce no new physical principles, and merely express relationships which necessarily hold in a universe obeying the general principle of relativity and in which we can translate objects through small distances. If you are prepared to take that on faith, you can skip the calculations and move quickly on to the next section. That is quite reasonable. You may reflect that these calculations have been checked and rechecked by tens of thousands of mathematicians since their original formulation in the 19th century. Such is the requirement of reproducibility in a strict approach to science. If you take the strict approach, that nothing should be taken on faith, and require that logic, rather than authority, should be the final arbiter, there is no help for it; you have to do the training session. No sympathy can be afforded to those who decry authority and yet are too idle or unfit to do the training.
Einstein’s Law of Gravitation
It is shown that, for weak gravitational fields, the effect of gravitational redshift on geodesic motion gives an identical acceleration to that of a classical gravitational field. Einstein combined Newton’s law of gravity with his three laws of motion into a single tensor law. The Schwarzschild metric, describing gravity in the region of a star or a planet, is calculated. Black holes are introduced.
Large Scale Structure of the Universe
“Einstein’s biggest blunder”, the
cosmological constant, is introduced. Weyl’s postulate is described, which treats the motions of galaxies as a “cosmic fluid” and allows us to talk of “cosmic time” and the large scale structure of the universe. Spaces of constant curvature are treated and the meaning of cosmological expansion is described. The cosmological principle, which essentially states that the universe is everywhere the same at any cosmic time, is used to derive Friedmann’s equation for the expansion of the universe. The equation is solved and the Friedmann models are described.
Quantum logic is introduced as a natural language for discussing measurement results in a universe consisting of only particles and relationships between them, and in which spacetime is not fundamental but emerges from those relationships. Relativistic considerations impose severe constraints, leading to the necessity for spin and antimatter, and to a precise description of the electron.
Foundations of Quantum Theory
Quantum theory is often thought to be conceptually incomprehensible as physical theory, but it is as much a theory of language as it is a theory of physics. Properties like wave function collapse apply to statements about what we know of a situation, not directly to physical reality. All that is required to understand it is a bit of mathematical trickery applied to a language describing general principles of measurement.
Observable Quantities
It has been said that measurement of time and position sufficient for the study of all physical quantities. For example, a classical measurement of velocity may be reduced to a time trial over a measured distance, and a typical measurement of momentum of a particle involves plotting its path in a bubble chamber, being a set of positions over a time interval. In the most general case, measurement of position might refer to the position of particles other than the one under study, such as the position of a pointer. The language of quantum logic is here found sufficient to discuss probabilities for the results of any measurement where this is so.
Evolution of Quantum States
The inner product allows us to calculate probabilities for the outcome of a measurement provided that we know the ket describing hypothetical measurement at the time of measurement. This is only useful if we can calculate the ket at any time, t, from a known previous measurement result. The probability interpretation requires that time evolution is determined from a first order wave equation, the Schrödinger equation. Relativistic considerations dictate that Newton’s first law is obeyed for non-interacting particles.
The Dirac Equation
There is no relativistic Schrödinger equation for a spinless particle. A physical model requires the inclusion of spin. The simplest solution is the Dirac equation, which describes the electron and predicts the existence of antimatter. Thus, the general relational principles incorporated into relativity and quantum logic are found to be sufficient to determine physical properties of fundamental particles.
States of Many Particles
The underlying idea is simple. Formal clauses are combined using the tensor product to model logical conjunction,
AND. The resulting structure,
Fock space, contains clauses about the hypothetical measurement of all the particles under consideration, and allows that particles of the same type are indistinguishable from each other.
Interactions are modelled as a perturbation to the motion of free particles, using quantum logical
OR to write the statement that, at each instant, either a particle interacts with another particle, or it does not, in which case its wave function evolves as a free particle. Relativistic considerations are used to derive the locality condition, showing that particles must meet in order to interact and which gives meaning to the claim that particles are point-like. Conservation of 3-momentum is demonstrated, showing that classical Newtonian mechanics is a consequence of the relational principles described in relativity and quantum logic.
With the exception of special relativity,
Quantum electrodynamics is the most
empirically accurate theory known to science, but it is fraught with mathematical difficulties and divergence problems. I follow a straightforward approach, based heavily on the Dirac-Von Neumann interpretation. The focus is on showing that the general considerations of relativity and quantum mechanics lead to both quantum and classical electrodynamics.
The Dirac Field Operator
I introduce qed by constructing the Dirac field operators, which are used to describe the interactions of Dirac particles, demonstrate that they obey locality and I define the current density observable.
The Photon Field Operator
The possibilities for interactions between Dirac particles and other matter are limited by covariance. The most straightforward interaction, known as the minimal interaction, is with a vector particle, the photon. I introduce its properties and extend quantum logic to describe the behaviour of a particle whose position cannot be determined directly because it is only created or annihilated in interaction.
Classical Electromagnetism
In keeping with the idea that particles are the fundamental building blocks of matter, and have behaviour constrained by quantum theory and relativity, classical electromagnetism has not been assumed in this account. To see that classical electromagnetism is the consequence of particle interactions we need to show that
is a conserved current, and that the Lorentz force law and Maxwell’s equations follow from the minimal interaction in which a photon is emitted from, or absorbed by, a Dirac particle.
The accepted wisdom is that Feynman diagrams should not be taken literally, but should merely be treated as aids to calculation. That is not how Feynman regarded them, as described in, for example,
QED: The Strange Theory of Light and Matter. In the interpretation of quantum mechanics used here, they are understood as statements in quantum logic. Since we cannot say what combination of particle interactions takes place between measurements, we must sum the possibilities using quantum logical
OR. In Feynman diagrams only lines and vertices have meaning. They represent structures of matter as configurations of particle interactions in the absence of background of space or spacetime.
A principle application of quantum theory, and of quantum electrodynamics in particular, is in the interpretation of the results of scattering experiments. I describe the scattering cross section, show how it may be calculated from Feynman rules. The prototypical scattering experiment, on which much research into the properties of elementary particles has been based, was the
Geiger-Marsden gold foil experiment carried out under the supervision of
Ernest Rutherford. I obtain the Rutherford formula for non-relativistic Coulomb scattering, verified by the experiment, and from which we deduce the structure of the atom containing a dense charged nucleus.
Quantum electrodynamics has had considerable success experimentally, but is renowned for difficulties with infinite quantities. Certain methods used to remove divergences have been severely criticised, even by some of the founders of the field. I describe here how divergences can be legitimately avoided. When correct methods are followed there are no remaining issues in qed.
The construction of qed as a model of particle interactions using finite dimensional Hilbert space is straightforward, but sweeps thorny issues under the carpet. The problems are deep enough that most physicists believe that this approach is fundamentally flawed. I move on from the description of standard theory from a relational viewpoint, and describe my own research into a resolution.
Quantum Covariance deals with the issue that manifest covariance requires a continuum model.
The Teleconnection shows how quantum theory can be consistently defined when classical spacetime is curved. I conclude that quantum field theory should be interpreted strictly as a theory of discrete
particles. A consequence is that the arguments which led to Minkowski metric in
special relativity must be adapted and lead to
Einstein’s Field Equation in the case for a single gravitating particle.
Quantum Covariance
In order to analyse redshift effects between remote initial and final quantum states, we need to study the relationship between inertial reference frames defined locally with speed of light equal to unity and
τ−ρ coordinates used in a Penrose diagram of the universe with a non-physical metric,
. The factors of two in the non-physical metric are equivalent to a two way stretch of the wave function, and yields surprising results in conjunction with local fluctuations in geometry. Gravitational lenses in deep space have four times greater effect for given mass. The interpretation of redshifts by standard formulae leads to an illusory component in the velocities of astronomical objects.
To introduce the idea of the teleconnection I will here describe it diagramatically for a Friedmann Cosmology. In this case the analysis is particularly simple. The result is that the cosmological redshift of light from a distant galaxy is proportional to the square of the expansion parameter, not linear with it as predicted by the
Levi-Civita connection. A more general, and more mathematical, treatment will be given on the
next page.
A
connection defines the notion of parallel in vector spaces defined at nearby points of a manifold. The teleconnection defines the parallel displacement of momentum in quantum mechanics from an initial state to a final state when the reference matter used to describe the initial state is remote from that used to describe the final state. When the initial and final states are determined with respect to nearby reference matter, the teleconnection is equivalent to the
Levi-Civita connection.
In order to analyse redshift effects between remote initial and final quantum states, we need to study the relationship between inertial reference frames defined locally with speed of light equal to unity and
τ−ρ coordinates used in a Penrose diagram of the universe with a non-physical metric,
. The factors of two in the non-physical metric are equivalent to a two way stretch of the wave function, and yields surprising results in conjunction with local fluctuations in geometry. Gravitational lenses in deep space have four times greater effect for given mass. The interpretation of redshifts by standard formulae leads to an illusory component in the velocities of astronomical objects.
Usually quantum electrodynamics is approached from the viewpoint that the underlying (meta)physical structures are fields. I do not think there is any justification for this beyond the undeniable mathematical similarity between relativistic quantum field theory and the treatment of quasi-particles in
condensed matter physics. Field theorists argue that, in like manner, the fundamental particles of Nature are really quasi-particles. In my view, argument by analogy is not a valid scientific methodology; it is often the case that identical mathematics can be applied to quite distinct physical situations. We have no means to conclude from similar mathematics that the vacuum has a physical structure analogous to that of a
superfluid. Despite considerable effort over the years, field theory has not been able to resolve fundamental mathematical problems in qed. Meanwhile, particle qed has largely been neglected. This section describes the issues and summarises the steps taken in relational quantum gravity to address the problems with the particle view.
Relational quantum gravity modifies the standard, continuum, form of qed and introduces an effective minimal interval of proper time between interactions of an elementary particle. The result of this modification is that instead of Minkowski spacetime, we find a curved spacetime obeying Einstein’s field equation.
The Emergence of Spacetime Structure
In relational quantum gravity, the most fundamental description of matter is illustrated by
Feynman diagrams. In Feynman diagrams, space, and hence curvature, have no meaning, and emerge only in the classical correspondence from the mean behaviour of large populations of particles.
In classical general relativity, a
singularity may be described as a point at which the known laws of physics necessarily break down. The singularities of interest to physics are the big bang (and the big crunch) and black holes. We may expect that known laws of physics break down not just at a singularity, but also close to it.
The predicted redshift relation under the teleconnection, together with observational evidence from supernovae, lead to a closed Friedmann model with no
cosmological constant or
cold dark matter, and to the resolution of other problems in astrophysics.
Supernovae Redshifts and Cosmological Parameters
The Teleconnection predicts a different redshift relation from that of standard general relativity. The redshift relation has a number of testable implications. A revised relationship between redshift and age offers the prospect of a reconciliation between observation and galaxy evolution models. The magnitude-redshift relation has been analysed using the
Union compilation compilation of data from the
Supernova Cosmology Project. The quality of the fits is such that any improvement of the teleconnection no-
Λ model over the standard
concordance model is wholly insignificant.
Pioneer 10 and
Pioneer 11 were the first spacecraft to investigate
Jupiter and explore the
outer solar system. For some years they sent back Doppler information interpreted as an anomalous acceleration toward the Sun (
Anderson et al.). This cannot be accounted for by classical physics, but is predicted in relational quantum gravity.
Galaxy Rotation Curves, CDM and MOND
Cold dark matter (CDM) has been hypothesised to make sense of a number of observations in Cosmology, but the theory is not without problems.
MOND has offered an alternative, but also does not work in all cases. The
teleconnection offers a no-CDM alternative in which illusory velocities account for the phenomenology of MOND without modifying Newtonian dynamics
The Local Slope of the Rotation Curve
CDM and
MOND allow flat
galaxy rotation curves, but it is observed that the local gradient of the rotation curve is not flat. It appears that CDM defies general relativity as well as elementary particle physics and earth based laboratory experiments, and MOND fails in its aim of replacing Newtonian gravity with a law predicting the slope of the rotation curve from visible matter.
Radial Velocity Test
Solar Motion Relative to the Metal-poor Halo
In an
analysis of solar motion relative to halo stars, we found that the motion of the Sun with respect to halo stars within a cone with axis in the direction of Galactic rotation is significantly faster than its motion with respect to stars outside of that cone. This result can be explained by an
illusory component of radial velocity in accordance with the prediction of
relational quantum gravity. On account of the small population of halo stars this test does not demonstrate an illusory component of radial velocity at the 3σ level, or lead to a precise calculation of the orbital velocity of the Sun, but it does offer independent supporting evidence for the results of the regression test.
This section is entirely Newtonian, and has no bearing on relational quantum gravity. I studied the kinematics of local stars to look for evidence of the spectral shift prediction, but, once familiar with the structure of the local velocity distribution, I realised that it fitted a simple model of galactic spiral structure. Much to my surprise, it turned out that apparently no one had seen the model before, and that the field is usually studied using an analysis based on Ptolemy’s epicycles, and fraught with simple mathematical mistakes.
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