All high mathematics serves to do is to beget higher mathematics. ~ Ashim Shanker!
Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere. W. S. Anglin, in Mathematics and History
In my last post I showed via AdS/CFT analysis, that gravity is an emergent holographic notion: namely, that one can holographically derive (logically deduce) gravity from conformal field theoretic entropic properties of quantum entanglement, and that such a property is a necessary condition for the ‘bundle’ existence of the gravitonic field. To do so, I had to deform CFT by source-fields via the addition of , which is a dual AdS theory with a bundle field and a boundary condition
with the conformal dimension, a local operator and equals the number of indices of substracting the contravariant ones to get the AdS/CFT quasi-isomorphic Maldacena correspondence ( = AdS/CFT correspondence), thus the identity
with the left-hand side being the vacuum expectation value of the time-ordered exponential of the operator over CFT, the right-hand side being the quantum gravity functional with topological-conformal boundary condition, thus leading to holographic emergence, and in a sense, elimination, of gravity. Recall that the Heisenberg Uncertainty Relation holds for energy and time, leading to many anomalies for the Green‘s function of string-propagation:
with the Lagragian, due to the fact the superpositionality with respect to energy makes Feynman path-summation:
incoherent since some topologies will degenerate and violate existence conditions for tangent bundles over Minkowski spacetime and some will not correspond to the categorical CFT-manifold, and hence we need to replace the Green’s function with the Källén–Lehmann spectral representation. This is where the GKP-Witten Relation enters with all its glory:
with background deficit angle and the curvature localized on the boundary with an angular deficit:
hence solving the ‘Ricci/dilaton’ problem I discussed in my last post, since now the holographic formula is
with the ‘magical’ expression ( being the string lenght):
and with that, the GKP-Witten relation solves the ‘Ricci/dilaton’ problem for the action of supergravity theory.
Now let me set up the mathematical context needed to show, in a forthcoming post, that even in M-Theory, or for that matter: any quantum-gravity theory, one cannot coherently quantize gravity in a way that satisfies General Relativistic ‘necessity-criteria’ – as I will show that this would imply, via gravitonic quantum entanglement, the point-‘instantaneous’ collapse of spacetime to a zero-dimensional point like singularity. Not a pretty picture! To do that I have to show that boundary AdS/CFT admits of a ‘local’ symmetry in the bulk theory that is dual to a ‘global’ symmetry corresponding to the boundary and that the (Gubser-Klebanov-Polyakov)-Witten relation deduces the Green correlation functions and that they must have negative Källén–Lehmann spectral representation
with being the gauge-theoretic positive-definite spectral density function.
In the AdS/CFT duality, one must note that the second derivative of the on-shell action principle with respect to the bulk second-quantized field, must, by unitarity, be identical to the Green function of the current
with being the Euclidean time-ordering, and the Green function.
For equation 1. to be true, the connected Green function should provably reduce to the static …