Geometric Unity

2.3 Higgs Sector Remains Geometrically Unmotivated

For most of the 20th century, fundamental physics was split into two halves, only one of which was geometric. Then, in the mid 1970s, the quantum sector was discovered to have a basis in differential geometry with the advent of the Wu-Yang dictionary of Simons, Wu and Yang. Gauge potentials corresponded to the geometer’s notion of a connection and Fermi fields fit with Atiyah and Singer’s rediscovery of the Dirac operator in a bundle theoretic context.

After that point, fields of fractional spin, spin 1 and spin 2 were all well motivated in either Ehresmannian or Riemannian geometry. This left the curious case of the spin 0 Higgs sector with its so-called Mexican Hat potential. While the Higgs sector could be described bundle theoretically, it was not fully natural for geometers to consider a spinless field valued in a Lie Algebra and governed by a quartic potential, despite the seemingly geometric nature of the Mexican Hat shape.

2.4 Geometric Unity

Mapping out the various reasons that Riemannian and Ehresmannian geometry have continued to progress side by side reveals that there is a trade-off to be had between the two.

In the case of Ehresmannian geometry, the leading advantage has to be the freedom to accommodate any observed field content found through experiment through the use of auxiliary bundles structures. As a compensating secondary advantage, the use of the gauge group to simultaneously transform the field content together with the derivative structures which furnish the differential equations that govern propagation allows us to consider a greatly reduced set of truly distinct configurations without being overwhelmed by unnecessary redundancies that we would face in the group’s absence.

The power of these advantages are so central to the massive edifice that is modern Quantum Field Theory that the advantages of classical Einsteinian gravity based on Riemann’s geometry theory seem highly restrictive and almost provincial or perhaps quaint by comparison.

Geometry \ Advantage Primary Secondary Riemannian Projection Operators Distinguished Connection Ehresmannian Content Freedom Gauge Group

(2.11)

The leading advantage of Einstein’s theory would most likely be thought by most to be the ability to ‘Project’ or contract the full Riemann Curvature tensor back onto a subspace of Symmetric 2-tensors which can be put in correspondence with the tangent space to the parameter space of metrics.

This gives the Einstein theory the flavor of having the matter and energy warp space directly, Analytic

(2.12)

rather than a change in the curvature represented by a differential operator providing the link. This however pays a high price in that it treats the two copies of Λ2 in Λ2 ⊗Λ

2 where the Riemann curvature tensor lives symmetrically while the gauge group acts only on the second factor. This seemingly negates much of the advantage of working within the Riemannian paradigm if one is interested in harmonizing GR with QFT.

There is however a small hope. If one can give up the freedom of being able to choose auxiliary gauge content as needed in the hopes of avoiding the double origin problem, then it may be possible to work within an extremely narrow class of theories which are amenable to the advantages of both geometries. But such geometric frameworks are likely to be extremely restrictive. Thus one must hope that the Standard Model and General Relativity would be of a highly unusual and non-generic geometric variety. And the main hope here is that observed quantum numbers of (ii) and (iii) that have no explanation within Witten’s point (i) seem to us to be highly suggestive that the Standard Model with its three generation and 16 particle per generation structure is of exactly the non-generic theory that carries both attributes.

This brings us to the less scientific and more human point. Let us ask the question whether the incentive structures of physics select for or against the search for a highly specific and restrictive class of geometries on which to focus. Attempts to find such a class are likely to fail, be considered as numerological and quixotic, and be career limiting. As such, leading physicists would likely avoid such theories in favor of flexible frameworks which do not paint the investigator into a research corner.

To a seasoned investigator, trading gauge invariance of the action and the freedom to choose field content to fit the needs of a problem for mere Einstein projection maps and a distinguished connection, is likely to seem akin to a naive Magic Beans trade where something of great value is bartered away for something of obviously lesser or even dubious importance. Here that trade is auxiliary content freedom and gauge covariance for contraction operators and a distinguished choice of connection.

It is the assertion here that this is not only an advantageous trade but likely a necessary one. That is, we do not need auxiliary freedom because of our good fortune in the Standard Model, and we can buy back the gauge invariance after exploiting the riches of projection operators and the choice of a distinguished connection.

2.5 Geometric Harmony vs Quantum Gravity

It is said that Gravity has to be quantized because of the paradox that since every particle creates a gravitational field, a classical localization would ultimately have to be as uncertain as the quantum particle’s location under observation.

In fact, this presupposes that the answer will be found in the simple Einsteinian space-time paradigm where the argument is maximally persuasive.

However, even here, there is a significant issue that is often glossed over.

Because the group Gl(4 f , R) does not carry a finite dimensional copy of the 15 fractional spin representations, if we allow the space time metric to become uncertain, we find a profound puzzle in the fractional spin fields.

While the Spin-1 force particles and Spin-0 Higgs fields may be difficult to measure between metric observations, the bundles in which they live are well defined in the absence of a definite choice of metric.

This however is not true for fractional spin fields.

Should the metric ever successfully be ‘quantized’ in a manner similar to the other fields, there will be no finite dimensional bundle for the hadrons and leptons during the period between observations. Not only do the waves become uncertain but the media in which the waves would live would appear to be uncertain or even said to vanish. One of the goals of GU is to ensure that the bundles for spinors and Rarita-Schwinger matter do not vanish in the absence of a metric.

By the above reasoning, gravity must be somehow harmonized with quantum fields in an as yet unspecified way more persuasive than the argument that metric gravity must be quantized on the same footing as the other fields.

Thus, whether gravity is to be harmonized or quantized, it is the goal of GU to decouple the existence of the fractional spin bundles as the medium for matter waves from the assumption of a metric in the ultimate quantum theory.