Observed Field Content

12.5 The Failure of Unification May Be Solved by Dirac Square Roots

If we accept the colloquial description of the Dirac equation as the square root of the Klein-Gordon equation, we see that solutions of a first order operator can guarantee solutions of a more general second order equation.

This was oddly at the fore when the so-called ‘Self-Dual’ Yang-Mills equation burst onto the scene in that

.. (12.7)

indicating that an equation linear in the curvature was powerful enough to guarantee the solution of a differential equation in the curvature via the Bianchi identity. This suggested to the author in the early 1980s in a seminar taught at the University of Pennsylvania that the Self-Dual equations were actually not so much meant to be Instanton equations, but were somehow more accurately the Einstein Field Equations in disguise as the square root of the YangMills-Maxwell equations.

Confusing this picture was the fact that the Einstein equations are usually viewed as equations for a metric rather than a connection, and the fact that the self-duality operator does not work for signatures other than (4, 0),(2, 2) and (0, 4), all of which are non-physical. However, we have now attacked both of these issues in the construction of the Observerse so as to be able to address the viability of the idea that the Einstein and Yang-Mills curvature equations are so related.

Thus, in a Dirac pair, the Yang-Mills and Klein-Gordon equations would be assigned to a second order strata and the Einstein and Dirac equations to a first order strata, with a relationship between the two understood as above.

In our case of fundamental physics, there are so far four basic equations for each of the known fundamental fields,

Spin Name Field Order

0 Klein-Gordon Higgs Field φ 2 1 2 Dirac Lepton and Hadron Fields ψ 1 1 Yang-Mills Gauge Bosons A 2 2 Einstein Gravitons g 2 (12.8) In some sense, this can be replaced in GU by Naive Spin Name Field Order 0 Klein-Gordon w Potential Yang-Mills-Higgs Field φ 2

‘Dirac-Rarita-Schwinger’ Lepton and Hadron Fields ν, ζ 1 ‘10 Yang-Mills Gauge Bosons $ 2 ‘10 ‘Chern-Simons-Einstein’ Tω 1 (12.9)

suggests a Dirac Square Root Unification. That is, the two first order equations live inside a square root structure of a different equation that contains the two second order equations. In an extreme abuse of notation we might write

Einstein-Dirac = p Yang-Mills-Higgs-Klein-Gordon (12.10)

to be maximally suggestive of the kind of Dirac Square Root unification we have in mind.

12.6 Metric Data Transfer under Pull Back Operation is Engine of Observation.

The metric tensor has traditionally been seen as an instrument of measurement of length and angle. This of course, is purely classical, arising as it does in both Special and General Relativity. The puzzle of Quantum measurement is, however, rather different, as it involves the application of Hermitian operators on Hilbert spaces to find eigenvectors as the possible post-measurement states, with their corresponding Eigenvalues as the experimental results.

Figure 6: Observation and The Observerse.

But in GU a different picture is possible. Consider X as if it were an old fashioned Victrola and the Metric as analogous to an old fashioned stylus with Y being a phonograph. What appears to be happening on the Victrola is largely a function of where the stylus alights on the phonograph. From the point of view of the listener, each track or location on the phonograph is a different world, while from the perspective of the record manufacturer the album is a single unified release. In this way, the world of states of Y is merely being sampled and displayed as if it were the only thing happening on X.

12.7 Spinors are Taken Chimeric and Topological to Allow Pre-metric Considerations

It has been very difficult to get upstream from Einstein’s concept of space-time for a variety of reasons. In particular, the dependence of Fermions on the choice of a metric in fact appears to doom us to beginning with the assumption of a metric if we wish to consider leptonic or hadronic matter. Yet this dependence must be partially broken if we are to harmonize metric-generated gravity from within metric-dependent Quantum Field Theory.

Many years ago, Nigel Hitchin demonstrated that, while the elliptic index of a Dirac operator in Euclidean signature was an invariant by the Atiyah-Singer index theorem, the dimension of the Kernel and Co-Kernel could jump under metric variation. Since that time Jean-Pierre Bourguignon and others have expended a great deal of work tracking Spinors under continuous variation of the metric.

Given the odd way in which Spinors appear to be both intrinsically topological (e.g. the topological Aˆ-genus) but confoundingly tied to the metric, we have sought to search for the natural space over which the topological nature of spinors is most clearly manifest. In essence, this has lead us to attempt to absorb the metric structure into a new base space made of pure measuring devices but constructed from the purely topological representation of GL( f r + s, R) on the homogeneous spaces GL( f r + s, R)/Spin(r, s) 59

12.8 Affine Space Emphasis Should Shift to A from Minkowski Space

There appear to be many difficulties when attempting to do Quantum Field Theory in curved space. Thus there has always been a question in the author’s mind as to whether the emphasis on affine Minkowski space M1,3 is a linearized crutch to make the theorist’s model building easier, or whether there is something actually fundamental about affine space analysis. In some sense, GU attempts to split the difference here. We find the emphasis on Minkowski space misplaced, but not the focus on affine theory, as no matter how curved Space-time may be, there is always an affine space that is natural and available with a powerful dictionary of analogies to relate it to ordinary and super-symmetric Quantum Field Theory:

Special Relativity/QFT to GU Relativity, QFT GU Analog Affine Space M1,3 A Model Space R 1,3 N = Ω1 (ad)

Core Symmetries Spin(1, 3) H = Γ∞(PH ×Ad H) Inhomogeneous Poincare Group G Extension = Spin(1, 3) n R1,3 = H n N Fermionic Extension Space-Time SUSY (ν, ζ) ∈ Ω

… (12.11)

This also makes more sense from the so-called super-symmetric perspective. If, historically, supercharges are to be thought of as square roots of translations, then in the context of a ‘superspace’ built not on M1,3 but on A, supercharges would have an honest affine space to act and translate where they would appear as square roots of operators or gauge potentials. This would also allow a framework where Supersymmetry12 could be formally active without the introduction of artificial superpartners which have been remarkable in their failure to materialize at expected energies. In this framework, the supercharges may already be here in the form of the ν and ζ fields as this would not be space-time supersymmetry.