Theory of Geometric Unity: Difference between revisions

Theory of Geometric Unity (edit)

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 {| class="wikitable"
-| '''1.''' The Arena (\( X, g_{\mu\nu}\))
+| '''1.''' The Arena (<math> X, g_{\mu\nu}</math>)
-| \(R_{\mu\nu} - \frac{1}{2} Rg_{\mu\nu} + \Lambda g_{\mu\nu} =  \left( \dfrac{8 \pi G}{c^4} T_{\mu\nu}\right)\)
+| <math>R_{\mu\nu} - \frac{1}{2} Rg_{\mu\nu} + \Lambda g_{\mu\nu} =  \left( \dfrac{8 \pi G}{c^4} T_{\mu\nu}\right)</math>
 | the Einstein field equations, which describe gravity in the theory of general relativity
 |-
-| '''2.''' \(G\) (non abelian)
+| '''2.''' <math>G</math> (non abelian)
-\( SU(3) \times SU(2) \times U(1)\)
+<math> SU(3) \times SU(2) \times U(1)</math>
-| \(d_A^*F_A=J(\psi)\)
+| <math>d_A^*F_A=J(\psi)</math>
 | the Yang-Mills equation, which governs all other force fields in Yang-Mill-Maxwell theory
 |-
 | '''3.''' Matter
 Antisymmetric, therefore light
-| \((i \hbar \gamma^\mu \partial_\mu - m) \psi = 0\)
+| <math>(i \hbar \gamma^\mu \partial_\mu - m) \psi = 0</math>
 | the Dirac equation, the equation of motion describing matter particles, or fermions
 |}
@@ Line 43: / Line 43: @@
 === Problem Nr. 1: Einstein's Theory of General Relativity is not a proper Gauge Theory ===
-* From Einstein's general relativity, we take the Einstein projection of the curvature tensor of the Levi-Civita connection $$\nabla$$ of the metric $$P_E(F_{\nabla})$$
+* From Einstein's general relativity, we take the Einstein projection of the curvature tensor of the Levi-Civita connection <math>\nabla</math> of the metric <math>P_E(F_{\nabla})</math>
-* 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 <math>d^\star_A F_A</math>
-'''Idea:''' What if the $$F$$'s are the same in both contexts?
+'''Idea:''' What if the <math>F</math>'s are the same in both contexts?
-Further, supposing these $$F$$'s are the same, then why apply two different operators?
+Further, supposing these <math>F</math>'s are the same, then why apply 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: $$P_E(F_{\nabla h}) \neq  h^{-1} P_E(F_{\nabla}) h $$. 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.
+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: <math>P_E(F_{\nabla h}) \neq  h^{-1} P_E(F_{\nabla}) h </math>. 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 ===
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 <blockquote>
-"So if we're going to take the spin-2 $$G_{\mu\nu}$$ 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?"
+"So if we're going to take the spin-2 <math>G_{\mu\nu}</math> 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>