Thanks Frankie, for posting this link to my 2022 paper and its abstract. I think this needs some context if it isn't to leave most (or all) readers nonplussed, so here goes. I'm aware that I'm catering here for a variety of audiences, from the general public to seasoned researchers in general relativity and particle physics. I'll therefore try to keep the highly technical terms to limited parts of this post, and flag up where these are.
Programme of work this is part of
This paper forms part of a hypothesis I've been running with since 2005, and it's looking really promising. I'm now calling it 'Covariant Compactification'. The basic premise is about as stripped down and pure as you can get. It starts with asking what everything is fundamentally made of. Einstein told us that the gravitational field does not act within spacetime, it simply is spacetime - specifically, the curvature of spacetime. Meanwhile, quantum field theories tell us that all fundamental matter and the interactions of this matter can be described as fields. Why have all these extra fields? Why do they exist? If the gravitational field is simply the curvature of spacetime, why should these extra fields not also simply be the curvature of spacetime? Then we wouldn't need to postulate the existence of other fields sitting in this spacetime, in order to describe our universe.
It's easiest to start to examine this idea by looking at the non-gravitational interactions between matter fields (e.g. between quarks, electrons and neutrinos). There are basically two of these present at high energies: the strong and electroweak interactions. The 'gauge principle' sets out how these correspond to transformations acting on matter fields. For these interactions to be described by curvature, we need more than four dimensions of spacetime.
The idea of describing non-gravitational interactions as curvature of a higher-dimensional spacetime goes back around a hundred years. Kaluza-Klein theory described how curvature on a five-dimensional spacetime can be manifested as both gravity and electromagnetism. The extra spatial dimension is assumed at the outset (at least in Klein's model) to be circular - that is, if you go a very short distance along it, you get back to where you started. Indeed, in Klein's model, this distance is far, far smaller than the radius of an atomic nucleus. This idea was later extended to include the whole electroweak interaction and the strong force as well.
I first read of these ideas as a teenager in books such as Superforce, written by FQxI member Paul Davies. I was utterly captivated by them.
However, when I knew enough about physics to read the papers describing them, I started to see deficiencies in the ideas. Klein gave no explanation of why one spatial dimension should be such a small circle, while the rest are so large that we can't see to the ends of them, and they look flat on our everyday scales. A mechanism for arriving at this geometry, called spontaneous compactification, was proposed in 1976 by Cremmer and Scherk. Their paper considers a six-dimensional universe. They explain a mechanism (based on the Higgs mechanism) which takes one from a universe which looks the same in all directions, to one in which two dimensions are tightly rolled up. When I looked into this model, though, I realised that it relied on a Higgs-like field which has symmetries that are not those of the higher-dimensional spacetime. Consequently, it can't really been seen as a 'unification theory', in the way that original the Kaluza-Klein is.
Furthermore, for variants of Klein's model or Cremmer and Scherk's model which included the strong interaction and/or the full electroweak interaction, it seemed to be assumed that the corresponding symmetry transformations acted directly on the extra dimensions. This seemed odd to me, as these transformations usually act on a set of fields whose values at each point are complex numbers; they map one set of complex numbers to another.
My idea was to let the transformations of the non-gravitational interactions act directly only on the complex fields they usually act on. However, they would, essentially, induce rotations in the extra dimensions - this would be an indirect action. In the limit that the extra dimensions 'decompactify' (that is, uncurl), everything would look the same in all spatial dimensions. In this limit, all fields present would only have the symmetries of this full higher-dimensional spacetime.
Pursuing this notion has taken 18 years so far. In the last few years, I've started to get it published, bit by bit. The first bit I needed to publish was a paper which thrashed out the relation between rotations and boosts on the one hand, and curvature on the other. This can be found here or here. (This is written for four-dimensional spacetime, but everything in it is also valid for higher-dimensional spacetime.)
The link that Frankie has provided is to the second paper in this series. I have now written a third, which can be found here. (I submitted this to a journal recently, under the impression that I would not be charged for this particular submission. But it has emerged that I would be charged, so it has now been withdrawn. I'm currently reviewing options for an alternative journal to submit it to.) I have further work in the pipeline.
Easy ways in
These papers go through all the detail of the theory. If you want an easy-to-grasp overview, I have two options.
The first is to go to my website, https://warpedandbroken.com/. The text on this site is aimed at the intelligent layperson. It doesn't assume too much understanding of the physics, but guides you through the background concepts, and then summarises the ideas in my trilogy of papers. It does this over two pages: Warping guides the reader through the the concepts of vectors, scalars and fields, followed by the fundamentals of relativity, including touching on its 'teleparallel' formulation. Breaking does the same for spacetime symmetries and internal symmetries, complex numbers, unification and symmetry breaking. The final section of this page is 'Spontaneous compactification', in which I explain the idea that was developed by Cremmer and Scherk and others in the late 1970s and early 1980s, then summarise my new version of it.
The second is a presentation on my ResearchGate pages. This is designed to be accessible to theoretical physics graduates. Again, it summarises the background necessary to grasp Covariant Compactification, before moving on to summarise my research. This presentation goes beyond what is in the three papers and covers topics I've looked into recently, such as the incorporation of first-generation fermions.
A few points to note on this presentation:
1) You may need to download it for the slides to display properly. You then have the option of playing it as a slideshow in Microsoft Powerpoint, with my commentary.
2) If you do want to download it, please be aware that it's a large file: (129MB).
3) I am intending to re-record this soon, probably breaking it into two parts: one covering background and the other covering my own work. The new version will a) have a punchier delivery and b) be revised in light of recent progress in my research.
What's in this paper and what's in the preprint
Let me now explain what's in the paper Frankie has provided the link for, and what's in my new preprint. For this, I'll need to get a bit more technical, but most or all of terms I use here are explained in the presentation. (I could provide simple explanations of these here, but it would further lengthen this already long comment! I'm happy to post a glossary of them in a further comment if requested.)
Field theories usually have a field equation. This is often seen as the pinnacle of the theory and is usually derivable from the action integral. General relativity, for example, has a field equation which is derivable from the Einstein-Hilbert action. However, you don't need the field equation or the action to understand the basic principles of the theory. You can get a long way through explaining the theory before you need them. Similarly, I have found that it is possible to explain a lot of the theory I have developed without needing a field equation. This is the part covered in the paper.
In Kaluza-Klein theories, spacetime has a form known as a product manifold. I look at the difference between Cartesian product manifolds, which look like higher-dimensional generalisations of cylinders, and more general product manifolds, which look like higher-dimensional generalisations of tubes of varying radius. (To visualise the transition from the former to the latter, imagine taking a sausage-shaped balloon - the cylinder - and squeezing it, or a cartoon of somebody standing on a hose which is being used, causing it to swell up, or a cartoon of a snake swallowing a basketball.) Such 'products' have 'factors' - for example, the factors of a cylinder or tube are a line and a circle (this is explained more thoroughly in the presentation).
I look at the components of the Levi-Civita connection in these spacetimes, which is a key quantity in determining the curvature of the spacetime. In the Cartesian product space, these components fall into two sets. One relates to the factor space describing our familiar four-dimensional spacetime. The other relates to the compact factor space formed by the additional dimensions. However, for the more general product spacetime, there are extra components of the connection. In the appropriate coordinates, these are gauge fields. If there are two extra dimensions, I find that these are U(1) gauge fields. If there are three extra dimensions, they are SU(2) gauge fields. (This should make physicists' ears perk up, because U(1) and SU(2) are groups of symmetries that appear in the non-gravitational interactions. For example, the electromagnetic potential is a U(1) gauge field.)
Now, the curvature of spacetime is described by the Riemann curvature tensor, which is the field strength of the Levi-Civita connection. Unsurprisingly, then, the Riemann curvature tensor for the product spacetime (again, in appropriate coordinates) has the field strength of the gauge field amongst its components. (An example of the field strength of a gauge field is the electromagnetic field, whose components are the three components of the electric field and the three components of the magnetic field.)
Now, say there are six dimensions in total. What determines whether two of these dimensions form a compact subspace, leaving our familiar four-dimensional spacetime, or whether, say, four of them compactify, leaving just a two-dimensional macroscopic space or spacetime? What we need is a quantity which distinguishes between these cases. I show that this 'symmetry breaking pattern' is determined by the way a tensor field (with a particular symmetry property) transforms under changes of coordinates on the higher-dimensional spacetime. We can construct from this tensor field a series of scalars (that is, quantities which remain the same under changes of coordinates) which uniquely determine the dimensionalities of the factor spaces. For a Cartesian product of particularly simple factor spaces, which represents the 'classical vacuum' of the theory, this tensor is the Ricci curvature tensor.
The paper therefore explains what determines the dimensionalities of the factor spaces. However, it doesn't explain the detailed shape of the spacetime, as you move through it. To determine the spacetime curvature at any given point, a field equation is needed. This is what's done in the preprint. The preprint also gives a description of the tensor field that determines the dimensionalities of the factor spaces (beyond the classical vacuum).
It postulates a vector field spread across space and time. It proposes that the tensor field which determines the dimensionalities is a symmetrised derivative of this - that is, it's calculated by working out how fast the vector field changes from one point to another. The preprint then suggests an action integral constructed from this vector field and its derivative. This allows a field equation to be derived. I show that this field equation is the simplest possible generalisation of the classical field equation (Poisson's equation) that is consistent with the principles underlying general relativity.
I then look at combining the field equation with the results of the paper on product manifolds. I show that there are always solutions of the field equation in which there is a flat four-dimensional spacetime, with the other dimensions compactified. I study in depth the case where there are six dimensions, with the extra two dimensions forming a sphere. I show how this solution implies that the vector field must have a staggeringly high density, if the extra dimensions are to become subnuclear in scale. On reflection, this isn't surprising - huge densities are needed to cause fairly limited curvature of spacetime. If matter is to cause dimensions to curl up this tightly, its density must be humungous.
The preprint ends with discussions on a whole series of issues which point the way ahead for this research.
Together, the paper and the preprint constitute a significant revision of the Kaluza-Klein framework, with the following features (again, apologies for the technical language, for any readers not familiar with these terms):
- Particular symmetry breaking patterns are determined by the covariant derivative of an ultra-high-density vector field. In the resulting product spacetime, higher-dimensional tensors break into multiplets which have both Lorentz and internal symmetry indices.
- Unlike most post-1960 Kaluza-Klein theories, the additional dimensions are real, physical space dimensions. But unitary gauge transformations do not act directly on them; rather, these transformations act directly on spinor fields, inducing transformation of the tensor representations contained in the outer product of a spinor and its conjugate.
- Symmetry restoration doesn’t take place at high energies; rather it occurs at zero curvature.
- In this 'decompactification limit', all the space dimensions appear on the same footing.
- Unlike the best-known 1980s theories of spontaneous compactification, the action and field equations are fully covariant under changes of coordinates across all the spacetime dimensions.
- Gravity only propagates in four dimensions; it doesn’t get diluted by extra dimensions as is sometimes claimed of Kaluza-Klein theories.
- The field equation for the vector field is an eigenvalue equation, where the operator is a second order differential tensor, which carries information about the system’s geometry.
- In the appropriate coordinate systems, the Levi-Civita connection components include SO(N) gauge fields. These span some or all of the space of the unitary gauge fields.
- There are no extra spectra of quantum numbers associated with extra-dimensional translations, as feared by O’Raifeartaigh in his “no go theorem”.
I'd be particularly interested to hear any feedback anyone has on the preprint, as I now have a chance to tighten it up before submitting it to another journal.