From Unified Field Theory to Quantum Gravity and Back
Table of Contents
Starting in 1984 it became common to hear from leaders of the theoretical physics community that theoretical or fundamental physics was not about traditional unification of the kind sought for by the like of Albert Einstein.
Terms like “String Theory”, “Theory of Everything” and “Quantum Gravity” replaced unification as the driving force behind the field.
The shift was profound. Students went from working mostly in physical Lorentzian Signature with hyberbolic equations to Euclidean signature to take advantage of the Atiyah-Singer index theorem.
The number of dimensions considered typically dropped from the physical 4 dimensions to the toy dimensions of 2 and 3 to take advantage of complex methods and so-called Chern-Simons like theories respectively, or to ad hoc choice of 10, 11, 12 or 26 dimensions to access ‘Calabi-Yau’ manifolds, Super-Strings and Super-gravity and other exotic mechanisms.
The physically relevant reductive or even semi-simple symmetries related to SU(3) × SU(2) × U(1) were generally replaced with simple groups like a single SU(2) or even U(1) in isolation.
The observed family structure of 3 generations of Fermions increasingly faded from interest or was pushed onto the index theory of Calabi-Yau three-folds, while investigations began to assume the presence of space-time Super-symmetry despite the existence of zero experimental observation for the phenomena.
In short, interest in the direct investigation of the physical world went from the core of physics research to a quaint backwater, as what might be termed the “Toy-Physics era” of String Theory inspired geometric physics began.
To understand how profound the shift truly was, it is helpful to understand what a major keynote address sounded like in theoretical physics in 1983, just before the anomaly cancellation and its embrace by Edward Witten and other String Theorists changed the entire nature of what it meant to be a theoretician working on fundamental physics. Today, it reads almost as an epitaph for the views of a bygone era.
Here is Murray Gell-Mann addressing assembled leaders of the Theoretical Physics community at the 1983 Shelter Island II conference just before the split in the community mediated by the anomaly cancellation:
From Renormalizability to Calculability?
- Why this particular structure for the families?
Why flavor chiral with the left- and right-handed particles are treated differently, rather than, say, vectorlike, in which left and right are transformable into being treated the same?
- Why 3 families?
That’s a generalization of Rabi’s famous question about the muon, which I’ll never forget: ”Who ordered that?”)
The astrophysicists don’t want us to have more than 3 families. Maybe they would tolerate a 4th, but no more, with massless or nearly massless neutrinos; it would upset them in their calculations of the hydrogen and helium isotope abundances.
If the neutrinos suddenly jumped to some huge mass in going from known families to a new one, then they would be less upset.
• How many sets of Higgs bosons are there in the standard theory?
Well, the Peccei-Quinn symmetry, which I’ll mention later, requires at least 2, if you believe in that approach.
If there’s a family symmetry group, there may be more, because we may want a representation of the family symmetry group: maybe there are 6 sets of 4 Higgs bosons; nobody knows.
• Why SU(3) x SU(2) x U(1) in the first place?
The trace of the charge is zero in each family, and that suggests unification with a simple Yang-Mills group at some high energy, or at least a product of simple groups with no arbitrary U(1) factors. If the group is simple or a product of identical simple factors, then we can have a single Yang-Mills coupling constant.”
Murray Gell-Mann, 1983
Sadly, the questions raised in this keynote have not really been answered.
1.4 The Twin Origins Problem
A revolution in geometric physics happened with Simons and C.N. Yang at Stony Brook in the mid 1970s where it was discovered that classical Ehresmannian bundle theoretic geometry was playing the same role beneath classical and Quantum Field Theory that Riemannian geometry was playing in under-girding General Relativity.
The difference in geometric frameworks of Einstein and Bohr are more important than the issue of quantization.
Oddly, perhaps the most succinct synopsis of the main ideas in fundamental physics was given in 1986 by Edward Witten in a way that laid bare that it has been the geometry of physical law rather than the quantum which has constituted our three greatest insights:
Figure 2: Edward Witten Synopsis.
Seen from this perspective:
- (i) corresponds to the Einstein Field Equations of Semi-Riemannian geometry
- (ii) corresponds to the Yang-Mills generalization of Maxwell’s equations to Non-Abelian Ehresmannian gauge theory
- (iii) to the Dirac equation which mixes the bundle structures of both frameworks
The KleinGordon equation for the Higgs Fields with its iconic quartic potential is not mentioned and the quantum is clearly featured not as a rival insight, but as a method of viewing the three main discoveries.
Why did we become focused on quantizing gravity when the underlying data given in (i) and (ii) are themselves of geometrically different origins?
So long as auxiliary principal G-bundles are invoked without compelling justification, there is no sense in which theoretical physics will have a satisfying origin story for the universe. Why then are we not more focused on the ‘Twin Origins Problem’ of why we have separate inexplicable origins for
M1,3 and SU(3) × SU(2) × U(1)?
Ostensibly this is likely to be due to the pessimism that grew up around the lack of progress in Kaluza Klein type theories that sought a single origin.
What we need is not necessarily ‘quantum gravity’ but rather a means of harmonizing General Relativity with the Standard Model in a theory free of paradox.
To our mind, this suggests an emphasis on tying the auxiliary data in (ii) to the fundamental substrate of (i) at the level not of the quantum but at the level of geometry to resolve the tensions between intrinsic and auxiliary bundle geometries.