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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.
Physical Principles
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.
Special Relativity
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.
Mathematical Methods
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.
Gravity
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
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.
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 Friedman models are described.
Relativistic Quantum Theory
Quantum electrodynamics is perhaps the most empirically accurate theory known to established science, but it is fraught with mathematical difficulties and divergence problems. The approach here is based heavily on the Dirac-Von Neumann interpretation and focusses on showing that the general considerations of relativy and quantum mechanics lead to quantum and classical electrodynamics. This approach gives a simpler view of underlying physics than is generally presented from quantum field theory. The resulting calculations and predictions using Feynman rules are, of course, the same. Attention will be drawn to mathematical issues raised by the approach. These will be addressed in Relational Quantum Gravity.
Foundations of Quantum Theory
Quantum theory 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 to physical reality. All that is required 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.
Time Evolution
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.
Quantum Electrodynamics
With the exception of special relativity,
Quantum electrodynamics» is the most
empirically accurate» theory known to established science, but it is fraught with mathematical difficulties and divergence problems. I follow a straightforward approach, and draw attention to issues which this raises. I believe that this approach gives a much truer view of underlying physics than is generally presented from quantum field theory. It is shown that the simple interaction in which a Dirac particle emits or absorbs a photon explains the form of Maxwell’s equations of electromagnetism.
- (under construction)
Dealing with Paradox - (to be written).
Relational Quantum Gravity
Empirical Evidence
Future Directions
Outline ↑ Philosophical Foundations →
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