AdS/CFT Duality, String Ontology, and the ‘Illusion’ of Gravity

By George Shiber Wednesday, April 29, 2015Friday, October 26, 2018 In Grand Unification Physics, M Theory, Mathematical Physics, Philosophy Of Physics, Philosophy Of Science, Super String Theory, Supersymmetry AdS/CFT, philosophy of science

You will find truth more quickly through delight than gravity. Let out a little more string on your kite. ~ Alan Cohen!

General Relativity and quantum field theory are by far the most successful scientific theories. They are however, if not impossible, extremely hard to reconcile and the main obstacle has to do with renormalizability. In a nutshell, unlike all other particles, whether in a first or second-quantizational context, the graviton does not respond to charge directly: it is massless, because the gravitational force theoretically has unlimited range and must be a spin-2 boson, and so, responds directly to mass and energy, and since they carry energy, they respond to themselves: hence, they self-gravitate. The problem with that is quantum physics ‘tells’ us that gravitons are particles as well as waves: but, and that will be a central issue, particles are theoretically construed as 0-dimensional pointlike entities; so a pointlike graviton gravitates more strongly as you get closer and closer to it. But its gravitational field can only be interpreted as the emission of other gravitons (call those ‘children’-gravitons and the original one the ‘parent’). The gravitational field very close to the ‘parent’ graviton is extremely strong: that implies that the ‘children’ gravitons have enormous energy and momenta, by the uncertainty relation. The trouble now is that ‘children’ gravitons will also themselves emit gravitons (‘grand-children’), and the whole process, by non-renormalizability, runs away and one cannot keep track of the effects of such hyper-exponential gravitonic emissions, and one has to deal with actual infinities that cannot be eliminated. There is good news: enter String Theory.

The key philosophical point is that ontologically, particles are not interpreted as 0-dimensional manifoidal ‘points’ in an abstract mathematical structure, but as vibrational modes of strings that have non-zero dimension in actual physical ‘space’ and ‘time’. As strings dynamically ‘move’ – propagate – through space and time, they define a world-sheet which is isomorphic to a 2-dimensional Riemannian manifold, denoted by $R_{{S_{({\sigma _s},{\sigma _t})}}}^{2d}$ , where the ${\sigma _s}$ and ${\sigma _t}$ represent the string space and time coordinates that define the world-sheet, which is a two-dimensional manifold describing the embedding of a string in spacetime. Now, ${\sigma _s}$ and ${\sigma _t}$ are subject to quantum fluctuation and the only way these fluctuations can be consistent with a quantization of $R_{{S_{({\sigma _s},{\sigma _t})}}}^{2d}$ is if spacetime satisfies the equations of General Relativity – hence, the existence of $R_{{S_{({\sigma _s},{\sigma _t})}}}^{2d} \cong \,\,{W_{St}}$ , where ${W_{St}}$ is the string world sheet, plus quantum physics implies a finite theory of quantum gravity: QP + $R_{{S_{({\sigma _s},{\sigma _t})}}}^{2d}$ = Quantum Theory of Gravity. This seems magical and it is. Let us dig deeper though.

Recall that energy and time are related by the Heisenberg uncertainty principle: let ${H_{E,t}}$ denote that relation. On the tangent bundle of $R_{{S_{({\sigma _s},{\sigma _t})}}}^{2d}$ , in 2nd-quantization form, vectors ${V_s}:\, = \frac{{\partial {X_\mu }}}{{\partial {\sigma _s}}}$ and ${V_t}:\, = \frac{{\partial {X_\mu }}}{{{\partial _{{\sigma _t}}}}}$ where ${X_\mu } \equiv {X_\mu }({\sigma _s},{\sigma _t})$ represent the spacetime coordinates of $R_{{S_{({\sigma _s},{\sigma _t})}}}^{2d}$ have, by ${H_{E,t}}$ , an ambiguation problem: ‘t’ and ‘s’ as they occur in $'\mu '$ do not satisfy the quantum fluctuation of $R_{{S_{({\sigma _s},{\sigma _t})}}}^{2d}$ since $'\mu '$ only makes sense in a classical GR ontological framework of 0-dimensional point-particality, whereas ‘t’ and ‘s’ as they occur in ${V_s}:\, = \frac{{\partial {X_\mu }}}{{\partial {\sigma _s}}}$ and ${V_t}:\, = \frac{{\partial {X_\mu }}}{{{\partial _{{\sigma _t}}}}}$ are non-classical and imply non-zero-dimensionality of time and space. What gives? A deeper look always repays. In the QP + $R_{{S_{({\sigma _s},{\sigma _t})}}}^{2d}$ setting, the metric is ${g_{ab}} = {\partial _a}{X_\mu }{\partial _b}{X^\mu }$ , deriving the area gives us

$d\,{\rm{Area}} \sim \sqrt {{\rm{det}}\left| {{g_{ab}}} \right|} d{\sigma _s}d{\sigma _t}$

with action

${S_{ws}} = \int {d{\sigma _s}d{\sigma _t}\,L({\sigma _s},{\sigma _t})}$

where the Lagrangian $L$ is given by

$L = \frac{1}{{2\pi {\alpha ^ * }}}\sqrt {\mathop {X_\mu ^2}\limits^ \cdot {X^ * }^{{\mu ^2}} - {{\left( {\mathop {X_\mu ^2}\limits^ \cdot {X^ * }^\mu } \right)}^2}}$

Now note that by ${H_{E,t}}$ , after disambiguation of the spatial coordinates as they occur in ${X_\mu }({\sigma _s},{\sigma _t})$ , by substituting in $'\mu '$ and ${\sigma _t}$ , as one must, in order for the supersymmetric covariant derivative(s)

${}^sD = \frac{\partial }{{\partial \theta }} - i\theta \frac{\partial }{{\partial {\sigma _t}}}$ and ${}^sD = \frac{\partial }{{\partial \theta }} - i\theta \frac{\partial }{{\partial {\sigma _s}}}$

and

${}^s{D^\dagger } = \frac{\partial }{{\partial {\theta ^ * }}} - i\theta \frac{\partial }{{\partial {\sigma _t}}}$

and

${}^s{D^\dagger } = \frac{\partial }{{\partial {\theta ^ * }}} - i\theta \frac{\partial }{{\partial {\sigma _s}}}$

to determine the dynamics of string propagation, one must be able to differentiate under the action integral:

$\frac{d}{{d{\sigma _t}}}\int {d{\sigma _s}d{\sigma _t}L({\sigma _s},{\sigma _t})}$ ; but, by ${H_{E,t}}$ , ‘time‘ as it occurs in $'{\sigma _t}'$ is in superpositionality with energy, and therefore would have no solution. One has to keep in mind that integration is a smoothing process, and ${\sigma _t}$ superpositionality and ${H_{E,t}}$ both imply that one cannot integrate over a quantized spacetime in a way that is consistent with General Relativity. Hence, string propagation in spacetime would be uninterpretable physically: to see this, one has to look at the Green’s function for string propagation, say from ${X_{{\sigma _t}(a)}}$ at ‘time’ ${\sigma _t}(a)$ to ${X_{{\sigma _t}(b)}}$ at ‘time’ ${\sigma _t}(b)$

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

The problem now is clear: ${H_{E,t}}$ implies that the Lagrangian has no solution, and so the path ‘integral’

$P = \sum\limits_{{\rm{Topologies}}} {d\mu \,DX{\,^{ - {\rm{Area}}}}}$

with

$DX = \prod\nolimits_{\mu ,{\sigma _s},{\sigma _t}} {d{X_\mu }} ({\sigma _t},{\sigma _s})$

being the ‘outer’ measure of integration, also would not make mathematical nor physical sense. Since the Lagragian

$L = \frac{1}{{2\pi {\alpha ^ * }}}\sqrt {\mathop {{X_\mu }^2}\limits^ \cdot {X^{ * {\mu ^2}}} - {{\left( {\mathop {{X_\mu }{X^{ * \mu }}}\limits^ \cdot } \right)}^2}}$

has no solution as it occurs in Green’s function for string propagation, one cannot ‘sum’ over all topologies as is demanded by unitarity for

$P = \sum\limits_{{\rm{Topologies}}} {d\mu \,DX{\,^{ - {\rm{Area}}}}}$

since some will be degenerate and will violate Special Relativity’s sacred cow:

$E = m{c^2}$

and that, by quantum fluctuations of the cross-sectional Riemannian manifold $R_{{S_{({\sigma _s},{\sigma _t})}}}^{2d}$ , falsifies General Relativity. Back to square one?! After all it was the quantum fluctuations of $R_{{S_{({\sigma _s},{\sigma _t})}}}^{2d}$ crossed-sectioned, that gave us a quantum theory of General Relativity: do we have a contradiction?! Well, yes and no, as is usually the case in science. First, one must ask: why are there 2 types of strings, an open and a closed one? It turns out that this dualism is the key in String-Theoretic SuperSymmetric unification of Quantum Field Theory (QFT) and General Relativity (GR).

Open strings represent the gauge theory of QFT while the closed strings represent Gravity: put another way, the gravitational field equation in a second-quantizational context, would essentially include the graviton in its solution. But, String Theory as a second-quantized field theory, is a theory of supergravity. This seems like magic: the gravitational field must, by gauge constraints, have general covariance, so the gravity is given by

$S = \frac{1}{{16\pi {G_{10}}}}\int {{d^{10}}} {X_{_\mu }}({\sigma _t},{\sigma _s},)\sqrt { - g} \,R$

with

${{G_{10}}}$

the Newtonian constant in 10-Dimensions. The coupling constant in String Theory is the string coupling constant, ${g_s}$ , and in supergravity, the coupling constants are ${G_{10}}$ and ${{\rm{g}}_{YM}}$ : they are mathematically related via the expasion

${g_{\mu \nu }} = {\eta _{\mu \nu }} + {h_{\mu \nu }}$

whereby one can derive the action for gravity in schematic form (‘infinities’ again!)

$(1)\quad S = \frac{1}{{16\pi {G_{10}}}}\int {{d^{10}}} {X_\mu }({\sigma _s},{\sigma _t})\left\{ {\partial h\,\partial h} \right. + h\,\partial h\,\partial h + {h^2}\partial h\,\partial h + ...\}$

so one can see that graviton-emission is in exact proportion to $G_{10}^{1/2}$ , and a closed string’s emission is in exact proportion to ${g_s}$ , hence the crucial derivation:

${G_{10}} \propto g_s^2$

Now, by gauge and unitarity constraints, and in parallel, as demanded by AdS/CFT (Anti-de Sitter/Conformal Field Theory Correspondence – also known as Maldacena Duality and Gauge/Gravity Duality), the schematic gauge action is

$(2)\quad S = \frac{1}{{g_{YM}^2}}\int {{d^{p + 1}}} x\{ \partial A\,\partial A + {A^2}\partial A + {A^4}...\}$

by solving (1) and (2), one gets the scientific ‘miracle’:

${G_{10}} \simeq g_s^2l_s^8$

where ${l_s}$ is the string length, and thus, we get the supergravity relation,

$g_{YM}^2 \simeq g_{_s}^2\,l_s^{p - 3}$

by topological expansion of String Theory, the action becomes in schematic form,

$(3)\quad S = \frac{1}{{16\pi {G_{10}}}}\int {{d^{10}}} x\sqrt { - g} {e^{ - 2\phi }}\{ R + 4{(\nabla \phi )^2}\} + ..........$

where

$\phi$

is the dilaton scalar field: but now, we hit a serious problem – the crucial aspect of (3) is that the Ricci scalar part has the dilaton factor ${e^{ - 2\phi }}$

and the anomaly now is that the Ricci scalar will not be an invariant of General Relativity’s (Pseudo)-Riemannian manifold since the supersymmetric covariant derivative is highly non-linear! That means

$\frac{d}{{{d_{{\sigma _t}}}}}\int {{d^{10}}} {X_\mu }({\sigma _s},{\sigma _t})\sqrt { - g} \,{e^{ - 2\phi }}\{ R + 4{(\nabla \phi )^2} + ...$

has no solution: thus even supersymmetric quantum gravity cannot be consistent with the existence of a Minkowskian metric tensor; thus, in conjuction with

${H_{E,t}}$

one can see that spacetime becomes separable and therefore not consistent with General Relativity.

A dual way to appreciate the severity of the problem is to see that the Ricci scalar $R$ in (3) does not satisfy the supergravity version of the Einstein-Hilbert action in 10-Dimensions as is required by gauge and consistency constraints: that in turn implies the catastrophe, namely, that the Newtonian ‘constant’ ${G_{10}}$ becomes hyper-dynamical. A totally unacceptable result, and will be addressed in forthcoming posts.

Good news (with some bad and surely ugly ones as well: such is the beauty of scientific meta-epistemology and the meta-mathematical analysis of physics): a holographic ‘illusionary’ emergent interpretation of gravity via Anti–de Sitter/Conformal Field Theory (AdS/CFT) Correspondence.

AdS/CFT Duality is a correspondence between string theory on a space ${S_T}$ and a quantum field theory WITHOUT GRAVITY on the conformal boundary

$B\,_{{S_T}}^C( - G)$

of ${S_T}$ :

${S_T} \equiv B\,_{{S_T}}^C( - G)$

where the dimension of

$B\,_{{S_T}}^C( - G)$

is LOWER by at least ONE or more. ${S_T}$ is isomoprphic to an orbifold in a field-theoretic setting, for there to be an exact solution to the problem of the ‘constancy’ of Newton’s constant in 10 dimensions, and if the tangent bundle of

$R_{{S_{({\sigma _s},{\sigma _t})}}}^{2d}$

is to be well-definable: this results in constricting us to conformal field theory. One such Duality is Type II-B String Theory on $Ad{S_5} \times {S^5}$ space where ${S^5}$ is a 5-Dimensional hypersphere, and a supersymmetric N = 4 (Yang-Mills) Conformal Gauge Theory on the 4-Dimensinal boundary of $Ad{S_5}$ :

Question: what guarantees that the $Ad{S_5}$ orbifold has a boundary?! One can answer that by ‘Weyl transformation’ because the supergravity theta functional is invariant under its representation, and the boundary: ${B_{Ad{S_5}}}$ of $Ad{S_5}$ has the desired quantum conformal invariances. To see this, pinch any coordinatization, the half-space one is the best:

${d_s}^2 = {(kz)^{ - 2}}(d{z^2} + {\eta _{\mu \nu }}d{x^\mu }d{x^\nu })$ , a Weyl transformation would then give us, via $w = kz$ , the conformal boundary

$d{s^2} = d{z^2} + {\eta _{\mu \nu }}d{x^\mu }d{x^\nu }$

at $z = 0$ , one gets the desired Minkowski metric. One may ask philosophically how the AdS/CFT correspondence works since it involves a strict isomorphism: the answer is given by deforming CFT by source fields via the addition of

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

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

with $\Delta$ the conformal dimension of the local operator $\vartheta$ and $k$ equals the number of indices of $J$ substracting the contravariant ones: the beauty now is that ONLY gauge-invariant operators are admissible, and so, for every gauge-invariant operator, there is a dual source field expressed via

$\left\langle \Im \right.\{ \exp (\int {{d^4}} x{J_{4D}}(x)\vartheta (x)){\left. \} \right\rangle _{CFT}} = {Z_{AdS}}\left[ {\frac{{{\rm{lim}}}}{{boundary}}J{w^{\Delta - d + k}} = {J_{4D}}} \right]$

Now the left-hand side is the vacuum expectation value of the time-ordered exponential of the operator over CFT; the right-hand side is the quantum gravity functional with the conformal boundary condition: this implies that gravity is an emergent holographic notion: namely, one can holographically deduce gravity from conformal field theoretic entropic properties of quantum entanglement! One immediately notices also that the stress-energy operator on the CFT side is dual to the transverse components of the metric on the AdS side. So by AdS/CFT dimensional duality/reductive elimination of gravity, one hits two birds with one stone: no need to 2nd-quantize the gravitational field in a non-separable spacetime and gravity might turn out to be an ‘illusion’ and thus poses no real physical explanatory or causal problems: that is, the renormalization problem disappears as opposed to having to be solved mathematically.

The problem of matching the conformal SuSy in 4-Dimension with AdS SuSy in 5-Dimensions will be the topic of another post.