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.”
