Tom Lawrence

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.)

a) Paper

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.

b) Preprint

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.

Summary
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.

Thanks Steve. Your post covers a lot of grounds, mentioning many aspects of modern theoretical physics. I'll respond on just one point: the nature of quantization.

Our current understanding of quantization, which is reflected in phrases in your post such as “how to quantise this quantum gravitation”, is as a mathematical procedure. However, I am increasingly of the opinion that when we have a truly fundamental theory of all the interactions and all matter, the effects of quantization will emerge naturally from physical considerations.

Let me see if I can explain this. For single, non-relativistic particles, (that is, moving at sufficiently low velocities relative to an observer), the transition from classical mechanics to quantum mechanics is a mathematical procedure: introducing a wavefunction and differential operators, or introducing operator matrices, or replacing Poisson brackets by Moyal brackets. The results seem bizarre, but they agree with experiment without further mathematical manipulation. (They also reduce to classical mechanics in appropriate limits of scale - the “correspondence principle”.)

The real problems come when we try to quantise relativistic fields. To start with, we find energy locked up in the vacuum - the “ground state”. Then, we find that if we try to calculate masses and energies, we get infinite values. We need to use a procedure called a “regularisation scheme” to pare off an infinite value; this is then subtracted to give us a finite value – a procedure called “renormalization”. I'm deeply uncomfortable with all of this. And I'm not alone. Dirac said “I think that the present methods which theoretical physicists are using are not the correct methods. They use what they call a renormalization technique, which involves handling infinite quantities and this is not really mathematically a logical process. I would say that it is just a set of working rules, rather than a correct mathematical theory and I don't like this whole development at all. I think some other important discoveries will have to be made before these questions are put into order.” [Interview with Friedrich Hund in Gottingen in 1982 on YouTube, 17 mins 37s from start.] And even Feynman said of quantum electrodynamics: “No, you are not going to be able to understand it… my physics students don’t understand it either. That is because I don’t understand it. Nobody does… It’s a problem that physicists have learned to deal with: They’ve learned to realise that whether they like a theory or they don’t like a theory is not the essential question… The theory of quantum electrodynamics describes Nature as absurd from the point of view of common sense”. [QED: The strange theory of light and matter, p9-10.]

What I've come to realise is that regularisation represents a way of saying “this part of the calculation involves a regime for which we don't yet have a theory”, or essentially, “Here Be Dragons”. Consequently in my view, a fundamental theory shouldn't need quantization and renormalization, because it should cover all regimes, such as the ultraviolet limits and non-perturbative effects that traditional quantum field theories have such difficulty with.

We would still need a correspondence principle, or rather correspondence principles. We would need to show that this theory reduces to the standard model and to general relativity in the appropriate limits. This is where I think the Aharonov-Bohm effect, both for electromagnetism and gravity, may be particularly useful: it could give us insights into these transitions or correspondences.

Indeed, I mentioned in my previous post that any waves could experience the effect. This got me wondering whether anyone had looked into whether it occurs for a fluid, and performed an experiment of this nature. I found that they had. Last night, I found a discussion of this lovely paper by Berry et al. I’ll take some time to digest its contents fully, but I think it’s worth summarising part of it now. In Section 4, the authors consider water waves in a bath, passing either side of a plughole. Water is draining out of the plughole, causing the water around the plughole to swirl around. This is shown in Figure 4 for various different speeds of rotation. They had a dipper off to the right of the photos, causing plane waves to ripple from the right-hand side towards the left. As the wavefronts pass the vortex, they break in two. On one side, they are advanced by the vortex, while on the other side they are retarded. Consequently, you get a dislocation of the wavefront on the left hand side of the photos, which is stronger for some rotation speeds than others. If you were to have a “detector” to the left of the apparatus which records the passing wave amplitudes, you’d find that it records constructive or destructive interference between the two parts of the wave.

It's a beautiful demonstration that the core phenomenon in the Aharanov-Bohm effect is a wave phenomenon, that occurs in classical wave mechanics just as much as it does in particle physics.

Where quantum mechanics comes in is in interpreting these waves as particles. My strong suspicion is that this is where the infinite quantities come from – for example, from assuming that wave packets can be localised to infinitesimal points, rather than to points which are so small that they appear to be infinitesimal.

I greatly enjoyed listening to all three parts of this end-of-year review, presented by Zeeya Merali and Ian Durham. As always, it was fascinating and provided plenty of food for thought.

In this third part, though, there was one particular section which I feel moved to comment on here. This was Ian Durham’s interpretation of the gravitational Aharonov-Bohm effect. He provided a very thought-provoking commentary, while clearly choosing his words with great care. (I have transcribed them below for the aid of readers of this post). Nonetheless, there are aspects of it which make me keen to offer an alternative interpretation. I’d be really interested to hear the thoughts of Ian or anyone else reading this post on this alternative.

In essence, he started by pointing out that this experimental result highlighted similarities between gravity and electromagnetism. I’m not going to challenge this; my alternative interpretation agrees with his narrative on this point.

However, he then implied that gravity is not a “regular force” because it’s a curvature of spacetime, and that its formulation in this way made it less “real” than electromagnetism. This is where my interpretation differs.

Einstein showed that the classical mechanics of Newton needs modification when considering relative velocities that are a significant fraction of that of light. Minkowski showed that this was most elegantly done by treating space and time as parts of a four-dimensional spacetime. Similarly, Maxwell's electromagnetism is most simply expressed in a spacetime formulation. Our current best theories of gravity and electromagnetism, general relativity (GR) and quantum electrodynamics (QED) respectively, both result from further modifying these (special) relativistic versions of the classical theories, but in very different ways. In my recent essay, General Relativity: its beauty, its curves, its rough edges ... and its lessons for gauge fields, I describe how considering relativistic mechanics under a generalised change of coordinates (for example, to a system representing a rotating observer) introduces inertial forces, described by a connection and covariant derivatives. It is then a relatively small and natural step to incorporate curved spacetime within this theory, which is found to describe gravity. The procedure for transitioning from relativistic electromagnetism to QED can be found in any textbook on quantum field theory.

It seems to me that the impetus for developing a quantum field theory of gravity largely comes from the predictive power of QED and the success of the quantum field theory framework for unifying non-gravitational forces. However, GR is also a highly successful theory in its ability to predict and explain observations. Furthermore, infinite values only result in GR in very extreme circumstances, such as the (inaccessible) centre of black holes and the very beginning of the universe – in contrast to QED, where infinities must be removed from almost every calculation in order to get experimentally observed values.

So we have two independent pathways from (special) relativistic versions of classical theories to our best theories of two forces. Both are highly predictive, but of these, GR does not need the process of renormalisation to extract physical values. In light of all this, there is no convincing reason why a unified theory of all the fundamental interactions should be a quantum field theory, rather than a geometric field theory or an as yet unknown type of theory. The development of geometric unified field theories on curved spacetimes began with Kaluza and Klein in the 1920s. It was developed significantly in the late 1970s and early 1980s. The resulting Kaluza-Klein theories and theories of spontaneous compactification have deficiencies, but in my recent paper Product manifolds as realizations of general linear symmetries [eprint here] and my new preprint Covariant compactification: a radical revision of Kaluza-Klein unification, I show how many of these deficiencies can be overcome. Furthermore, this is done much more simply and elegantly than seems possible for overcoming the barriers to quantising gravity.

This whole body of theory relies on similarities between gravity and non-gravitational forces, such as the relation to symmetry groups and the coupling of matter to these forces being described by covariant derivatives. I think Ian has provided a valuable service in highlighting how this result helps to flesh out the similarities still further. But I disagree with any implication that this can be taken as evidence that gravity can be cast into a similar form to the quantum field theories of the electroweak and strong forces. It could equally be that a common description of them is based on geometry and curvature, as described in my papers, or an as yet unknown formulation.

To be fair, such possibilities were to some extent alluded to in the subsequent discussion between Ian and Zeeya, with Zeeya questioning why gravity should have a relationship with the curvature of spacetime while this does not appear to exist for electromagnetism, and the two of them discussing whether electromagnetism may have a corresponding relation to “something else”. The Kaluza-Klein answer to this is that the “something else” might be additional degrees of curvature in a higher dimensional space-time.

To come back to the Aharonov-Bohm effect, an objection to my interpretation might be that this effect, whether for electromagnetism or gravity, is a quantum effect. Indeed, it was described by Zeeya as “quantum weirdness” in this podcast. But I would say that it is more accurately described as a wave effect. It applies whenever a wave interacts with an electromagnetic potential – or, in this case, with a gravitational potential. Going on the description of it given in “Quantum Field Theory” by Ryder, the essential feature of the effect is that the potential changes the wavenumber of a wave, as it passes through the potential. (For a quantum wavefunction, this wavenumber is the momentum divided by h-bar, the reduced Planck constant.) This can happen in a region in which the field strength is zero. If a wave splits before passing through the region with significant values of the potential, the two parts of it may have their wavenumber changed differently along their trajectories, resulting in a phase difference on the far side. Thus if there is a detector on the far side, it will record an interference pattern. This need not be a quantum effect; any waves, including waveforms of a classical field, could in theory experience the same effect. Quantum mechanics only comes into play with the interpretation of these waves as the wavefunction of a particle.

Indeed, for gravity, you don't even need a potential difference between two paths for the potential to affect a body’s momentum in a region of zero field strength. Presumably for the gravitational Aharonov-Bohm effect, there is a minimal coupling of the atom’s momentum to the connection which describes the gravitational potential. This minimal coupling also appears in the covariant derivative for a vector. In particular, it appears in the geodesic equation for a macroscopic body in freefall. In this case, a change of gauge represents a change of coordinate system. The new potential can be experienced by the body as an inertial force (for example, a centrifugal force), even in a region with zero field strength (zero gravity). (This is explained in detail in my paper Tangent space symmetries in general relativity and teleparallelism [eprint here].) Perhaps this parallel, between inertial forces and the momentum shift in the Aharonov-Bohm effect, might provide insights into a common description for gravity and electromagnetism.

Ian Durham’s commentary:
“…according to general relativity, gravity is not really a force per se like the other forces – it's the curvature of space-time. So it sort of has this difference: in the standard model of particle physics, we assume that if eventually, gravity will fit in as one of the forces of nature, that it will behave in a similar way, in that there will be a carrier particle. In this case, it would be called the graviton … and that sort of thing. So you've got these two ideas: you've got the quantum idea that gravity is just like any other force, and then you have the general relativity view that it's not really a force at all, it’s this curvature of space-time. The question is, which one of those is it? The full answer’s that we don't know yet, but what's fascinating about this result is that it suggests that gravity is more like this idea of a regular force: the gravitational field behaves more like a standard electromagnetic field. So the fact that we see this effect with gravitational fields is extremely suggestive that there is a correspondence there, in that, yes, just like electromagnetism and the strong nuclear force and the weak nuclear force, gravity is a force just like those others are, and has similar kind of behaviour.
“Well, is general relativity wrong? Well, no, of course, it's not wrong in the sense that we rely on general relativistic equations every day to ensure that the GPS system works, etc, but it suggests that the interpretation of it as a curvature of spacetime, may either be incorrect or maybe incomplete because then the question becomes, maybe it is the curvature of spacetime, but maybe spacetime now develops a sort of ontological status unto itself that maybe in some interpretations it doesn't have, meaning that it possesses the properties of a field in that sense. Again these are open questions, but it nudges us in one particular direction. It says: yes, gravitational fields do behave very much like electromagnetic fields in this particular case and therefore it suggests that it's like a real thing; it’s not just a curvature of spacetime.”