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Theory of Geometric Unity: Difference between revisions
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== Background == | == Background == | ||
=== Starting point: three observations by Edward Witten === | |||
[[File:GU_triangle.png|500px|right]] | [[File:GU_triangle.png|500px|right]] | ||
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[[File:Geometric unity puzzle pieces.png|500px|right]] | [[File:Geometric unity puzzle pieces.png|500px|right]] | ||
{| class="wikitable" | {| class="wikitable" | ||
| '''1.''' The Arena (<math> Xg_{\mu\nu}</math>) | | '''1.''' The Arena (<math> Xg_{\mu\nu}</math>) | ||
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* From Yang-Mills-Maxwell-Anderson-Higgs theory of gauge fields, we take the adjoint exterior derivative coupled to a connection $$d^\star_A F_A$$ | * From Yang-Mills-Maxwell-Anderson-Higgs theory of gauge fields, we take the adjoint exterior derivative coupled to a connection $$d^\star_A F_A$$ | ||
''' | === Problem Nr. 1: Einstein's Theory of General Relativity is not a proper Gauge Theory === | ||
'''Idea:''' What if the $$F$$'s are the same in both contexts? | |||
But we're applying two different operators. | |||
'''Thus the question becomes:''' Is there any opportunity to combine these two operators? | |||
A problem is that the hallmark of the Yang-Mills theory is the freedom to choose the data, the internal quantum numbers that give all the particles their personalities beyond the mass and the spin. We can allow the gauge group of symmetries to act on both sides of the equation, but the key problem is that: $$P_E(F_{\Delta^{LC} h}) \neq h^{-1} P_E(F_{\Delta^{LC} }) h $$. If we act on connections on the right and then take the Einstein projection, this is not equal to first taking the projection and then conjugating with the gauge action. The gauge rotation is only acting on one of the two factors. Yet the projection is making use of both of them. So there is a fundamental incompatibility in the claim that Einstein's theory is a gauge theory relies more on analogy than an exact mapping between the two theories. | |||
=== Problem Nr. 2: Spinors are sensitive to the metric === | |||
'''Observation:''' Gauge fields not depend on the existence of a metric. One-forms are defined whether or not a metric is present. But for spinors (fermion fields) this is not the case. | |||
'' | <blockquote> | ||
"So if we're going to take the spin-2 Gμν field to be quantum mechanical, if it blinks out and does whatever the quantum does between observations. In the case of the photon, it is saying that the waves may blink out, but the ocean need not blink out. In the case of the Dirac theory, it is the ocean, the medium, in which the waves live that becomes uncertain itself. So even if you're comfortable with the quantum, to me, this becomes a bridge too far. So the question is: "How do we liberate the definition?" | |||
How do we get the metric out from its responsibilities? It's been assigned far too many responsibilities. It is responsible for a volume form; for differential operators; it's responsible for measurement; it's responsible for being a dynamical field, part of the field content of the system." | |||
</blockquote> | |||
=== Problem Nr. 3: === | |||
== Layman Explanation == | == Layman Explanation == | ||