AdS/CFT Duality, GKP-Witten Relation, and U(1)-Symmetric Holography

By George Shiber Saturday, May 2, 2015Friday, October 26, 2018 In Grand Unification Physics, M Theory, Mathematical Physics, Philosophy Of Physics, Philosophy Of Science, Super String Theory, Supersymmetry AdS/CFT, entanglement

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 $\int {{d^4}} xJ(x)\vartheta (x)$ , which is a dual AdS theory with a bundle field $J$ and a boundary condition

${J^{\Delta - d + k}} = {J_{CFT}}$

with $\Delta$ the conformal dimension, a local operator $\vartheta$ and $\kappa$ equals the number of indices of $J$ substracting the contravariant ones to get the AdS/CFT quasi-isomorphic Maldacena correspondence ( = AdS/CFT correspondence), thus the identity

$\begin{array}{c}{\left\langle {\Im \{ \exp (\int {{d^4}x{J_{4D}}(X)\vartheta (x))\} } } \right\rangle _{CFT}} = \\{Z_{AdS}}\left[ {\frac{{\lim }}{{boundary}}J{\omega ^{\Delta - d + k}} = {J_{4D}}} \right]\end{array}$

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:

$G({X_{{\sigma _t}(a)}},{X_{{\sigma _t}(b)}}) = \int {{}^sD\exp \left( { - \int_{{\sigma _t}(b)}^{{\sigma _t}(a)} {d{\sigma _t}\int_0^\pi {d{\sigma _t}} } } \right)} \,L$

with $L$ the Lagragian, due to the fact the ${\sigma _t}$ superpositionality with respect to energy makes Feynman path-summation:

$P = \frac{\sum }{{{\rm{Topologies}}}}d\mu {\prod _{\mu {\sigma _s}\,{\sigma _t}}}d{X_\mu }({\sigma _s},{\sigma _t})$

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:

${Z_{CFT}} = {e^{ - S{\,_{GRAVITY}}}}({\phi _i})$

with background deficit angle $\Delta \varphi = 2\pi (1 - n)$ and the curvature localized on the boundary with an angular deficit:

$R = 4\pi (n - 1) \cdot \delta ({\gamma _A}) + ...\,{\rm{regular terms}}$

with action

$\begin{array}{c}{S_A} = \frac{1}{{16\pi {G_N}}}\int {d{x^{d + 2}}} \sqrt g R + \to \\{\rm{Area}}\frac{{({\gamma _A})}}{{4{G_N}}} \cdot (n - 1)\end{array}$

giving us

$\begin{array}{*{20}{c}}{{S_A} = \frac{\partial }{{{\partial _n}}}{\rm{log}}{\mkern 1mu} {\rm{T}}{{\rm{r}}_A}{\mkern 1mu} {\rho ^n} = - \frac{\partial }{{{\partial _n}}}{\rm{log}}\left( {\frac{{{Z_n}}}{{{{\left( {{Z_1}} \right)}^n}}}} \right)}\\\begin{array}{c} = {\rm{Area}}\frac{{\left( {{\gamma _A}} \right)}}{{4{G_N}}} \cdot \\\delta {S_{GRAVITY}} = 0 \to {\gamma _A} = {\rm{ minimal surface!}}\end{array}\end{array}$

with

${S_{GRAVITY}} = \frac{1}{{16\pi {G_N}}}\int {d{x^{d + 2}}} \sqrt {{g_{\mu \nu }}} R + ... \to \frac{{{\rm{Area(}}{\gamma _A})}}{{4{G_N}}} \cdot (n - 1)$

hence solving the ‘Ricci/dilaton’ problem I discussed in my last post, since now the holographic formula is

${S_A} = \frac{{\left| {{\gamma _A}} \right|}}{{4G_N^3}} = \frac{c}{3}{\rm{log}}\left( {\frac{{{l_s}}}{a}} \right)$

with the ‘magical’ expression ( ${l_s}$ being the string lenght):

$c = \frac{{3R}}{{2G_N^{(3)}}}$

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

$\Delta (p) = \int_0^\infty {d{\mu ^2}} \rho ({\mu ^2})\frac{1}{{{p^2} - {\mu ^2} + i\varepsilon }}$

with $\rho ({\mu ^2})$ 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 $U(1)$ second-quantized field, must, by unitarity, be identical to the Green function of the $U(1)$ current

$\begin{array}{c}1.\quad {\left. {\frac{{{\delta ^2}{S_{CFT}}}}{{\delta {A_\mu }(x)\delta A\nu (x')}}} \right|_{u = 0}}\\ \to {G^{\mu \nu }}(x - x') = - {\left\langle {{T_E}{J^\mu }(x){J^\nu }(x')} \right\rangle _G}\end{array}$

with ${T_E}$ being the Euclidean time-ordering, and $G$ the Green function.

For equation 1. to be true, the connected Green function ${G^{ij}}(x)$ should provably reduce to the static response function ${K^{ij}}(x)$ in the stationary limit of the following ‘identity’ 2.

$2.\quad \mathop {{K^{ij}}}\limits^\_ (k) \equiv {(2\pi )^{p + 1}}{\left. {\frac{{{\delta ^2}_{CFT}}}{{\delta A_i^\dagger ( - k)\delta A_i^\dagger (k)}}} \right|_{u = 0}}$

thus, from the limit, one gets the conjectural equation

$3.\quad \mathop {{K^{ij}}}\limits^ - (k)\mathop = \limits^? \mathop {{G^{ij}}}\limits^ - (\omega = 0,k)$

But this cannot be true since the holographic Källén–Lehmann spectral representation implies that

$\mathop {{K^{ij}}}\limits^ - (k) > 0$

whereas $\mathop {{G^{ij}}}\limits^ - (\omega = 0,k) < 0$

Now, the definite-negativeness of $\mathop {{G^{ij}}}\limits^ - (0,k)$ can be derived from the Källén–Lehmann spectral representation of the ‘connected’ Green function:

$4.\quad \mathop {{G^{\mu \nu }}}\limits^ - ({\omega _n},k) = \int_{ - \infty }^\infty {\frac{{d\omega }}{{2\pi }}} \frac{{\mathop {{\rho ^{\mu \nu }}(\omega ,k)}\limits^ - }}{{i{\omega _n} - \omega }}$

where ${\omega _n} = 2\pi n/\beta$ must be the Matsubara frequencies and $\mathop {{\rho ^{\mu \nu }}}\limits^ -$ is a Fourier spectral functional transform of:

$5.\quad {\rho ^{\mu \nu }}(t,x) \equiv \left\langle {\left[ {{J^\mu }(t,x),{J^\nu }(0,0)} \right]} \right\rangle$

However, this spectral function satisfies

$\mathop {{\rho ^{\mu \mu }}}\limits^ - (\omega ,k)/\omega > 0$

thus leading to the definite-negativeness of:

$6.\quad \mathop {{G^{ii}}}\limits^ - (o,k) = - \int_{ - \infty }^\infty {\frac{{d\omega }}{{2\pi }}} \frac{{\mathop {{\rho ^{ii}}(\omega ,k)}\limits^ - }}{\omega } < 0$

no ‘summing’ over ‘i’. Because the ‘connected’ Green function and the holographic Källén–Lehmann spectral representational functional differ by a sign,

$3.\quad \mathop {{K^{ij}}}\limits^ - (k)\mathop = \limits^? \mathop {{G^{ij}}}\limits^ - (\omega = 0,k)$

must be false!

A resolution to this contradiction is obtained by noting that the AdS/CFT bulk theory has gauge symmetry and the boundary theory has background-local $U(1)$ symmetry: hence the $U(1)$ current ${J^\mu }$ does contain an external source field ${A_\mu }$ . In such a case, the Källén–Lehmann spectral representational functional can differ from the Green function, and the GKP-Witten relation yields the holographic Källén–Lehmann spectral function instead of the Green function. To show how this works, take a complex scalar field $\phi$ coupled to the the electromagnetic field ${A_\mu }$ :

$\begin{array}{c}{S_A}\left[ \phi \right] = \int {{d^{p + 1}}} x \cdot \\\left( {{{\left| {\left( {{{\not \partial }_\mu } - ie{A_\mu }} \right)} \right|}^2} + V\left( {\left| \phi \right|} \right)} \right)\end{array}$

Now, the current

$7.\quad {J^\mu } = - \frac{{\delta {S_A}}}{{\delta {A_\mu }}} = - ie{\phi ^\dagger }\mathop {{\partial ^\mu }}\limits^ \leftrightarrow \phi - 2{e^2}{A^\mu }{\left| \phi \right|^2}$

contains the electromagnetic field ${A_\mu }$ by the background local $U(1)$ symmetry. Now, one can generate the functional

$8.\quad Z{\rm{[}}A{\rm{]}} = {e^{W{\rm{[}}A{\rm{]}}}} = \int {D{\phi ^\dagger }} D{\phi ^{ - {S_A}{\rm{[}}\phi {\rm{]}}}}$

thus deriving the current expectation value as

$9.\quad \left\langle {{J^\mu }} \right\rangle = - \left\langle {\frac{{\delta {S_A}}}{{\delta {A_\mu }}}} \right\rangle = \frac{{\delta W{\rm{[}}A{\rm{]}}}}{{\delta {A_\mu }}}$

with a ‘response’ functional

${K^{\mu \nu }}$

given by

$\begin{array}{c}10.\quad {K^{\mu \nu }}(x): = - \frac{{\delta \left\langle {{J^\mu }(x)} \right\rangle }}{{\delta {A_\nu }(0)}} = - \frac{{{\delta ^2}W{\rm{[}}A{\rm{]}}}}{{\delta {A_\nu }(0)\delta {A_\mu }(x)}}\\ = {G^{\mu \nu }}(x) - \left\langle {\frac{{\delta {J^\mu }(x)}}{{\delta {A_\nu }(0)}}} \right\rangle \\ = {G^{\mu \nu }}(x) + 2{e^2}{\delta ^{\mu \nu }}\delta (x)\left\langle {{{\left| {\phi (x)} \right|}^2}} \right\rangle \end{array}$

where ${G^{\mu \nu }}$ is the ‘connected’ Green function for the current ${J^\mu }$

$\begin{array}{c}11.\quad {G^{\mu \nu }}(x) = \\ - {\left\langle {{T_E}{J^\mu }(x){J^\nu }(0)} \right\rangle _G}\end{array}$

Therefore, the Källén–Lehmann spectral representation functional differs from the ‘connected’ Green function by the second term of

${G^{\mu \nu }}(x) + 2{e^2}{\delta ^{\mu \nu }}\delta (x)\left\langle {{{\left| {\phi (x)} \right|}^2}} \right\rangle$

Hence, the ‘negativity’ is not reflected in the Källén–Lehmann spectral representation functional and from

${G^{\mu \nu }}(x) + 2{e^2}{\delta ^{\mu \nu }}\delta (x)\left\langle {{{\left| {\phi (x)} \right|}^2}} \right\rangle$

one gets the GKP-Witten relational implication:

${A_\mu } = 0$

Applied in a forthcoming post to the quantum gravitational field

$Q_\mu ^{{G_{ST}}}(R_{{S_{({\sigma _s},{\sigma _t})}}}^{2d})$

I will derive, via quantum entanglement and gravitonic vacuua analysis, the collapse of spacetime to a zero-dimensional point-like singularity that also violates the Theory of Special Relativity (in fact, what will it not violate), and the GKP-Witten Relation will be central in that analysis. Note: the AdS/CFT holography principle entropically implies the ’emergence’-property, and thus quantum-field-theoretic elimination, of the gravitational field.