String-Theory, Witten-Reduction, Einsteinian Space-Time(s), and 4-D-Analysis

By George Shiber Thursday, February 11, 2016Saturday, August 17, 2019 In Cosmology, Grand Unification Physics, M Theory, Mathematical Physics, Mathematics, Philosophy Of Physics, Philosophy Of Science, Physics, Super String Theory, Supersymmetry, Time Einsteinian Space-Time, M-Theory

Let no man who is not a Mathematician read the elements of my work ~ Leonardo da Vinci

The aim of this series of posts is to show that string-theory can be a direct and fundamental dynamical theory of four-dimensional symmetric space-times and derive Witten’s identification of

${S_{grav}} = \int\limits_{{M_1}} {dt\left\{ { - \frac{3}{k}a{{\left( {\frac{{da}}{{dt}}} \right)}^2}} \right\}}$

with dilaton coupling, leading directly to four-dimensional space-times via

$\begin{array}{c}Z = \sum\limits_{{M_1} - top} {\int\limits_{{M_1}} {\left[ {dh} \right]} } \left[ {dx} \right]\left[ \phi \right] \cdot \\{e^{ - \,\left( {\frac{i}{h}{S_{RFW}}} \right)}}\end{array}$

where ${S_{RFW}}$ is

${S_{RFW}} = {S_{grav}} + {S_{mat}} + {S_{geo}}$

First (1), let’s take an axiomatic look at string theory. Where the ‘ $a$ ‘s are the Witten-Wiegmann Kac-Moody quantization terms. There is a target space, including space-time, evolving on a two-dimensional Riemannian world-sheet ${\Sigma _g}$ with genus $g$ and metric ${h_{\alpha \beta }}$ with the generalized sigma model action being

$S = \sum\limits_{g = 0}^\infty {\int\limits_{{\Sigma _g}} {{d^2}} } \sigma {A_i}\left( {Z,\not \partial Z} \right){\Phi _i}\left( Z \right)$

with ${A_i}\left( {Z,\not \partial Z} \right)$ being a complete set of target coordinates and theirs derivations with respect to the world-sheet coordinates ${\sigma ^\alpha }\left( {\alpha = 1,2} \right)$ and noting that ${A_i}\left( {Z,\not \partial Z} \right)$ are in a one-to-one correspondence with states of string theory. ${\Phi _i}\left( Z \right)$ are string 2-D target-sections fields-coupling constants. The renormalisable bosonic part of

$S = \sum\limits_{g = 0}^\infty {\int\limits_{{\Sigma _g}} {{d^2}} } \sigma {A_i}\left( {Z,\not \partial Z} \right){\Phi _i}\left( Z \right)$

$\begin{array}{l}S = \frac{1}{{4\pi \alpha '}}\int\limits_{{\Sigma _g}} {{d^2}} \sigma \left\{ {\sqrt h {h^{\alpha \beta }}{{\not \partial }_\alpha }{X^\mu }{G_{\mu \nu }} + {\varepsilon ^{\alpha \beta }}{{\not \partial }_\alpha }{X^\mu }{{\not \partial }_\beta }{X^\nu }{B_{\mu \nu }}} \right\} + \\\alpha '\int\limits_{{\Sigma _g}} {{d^2}} \sigma \sqrt h \,\phi \left( X \right)R + {S_{{\mathop{\rm int}} }}\end{array}$

with $\alpha '$ the inverse string tension, ${X^\mu }\left( {\mu = 1,...,D} \right)$ the space time coordinates. Backgrounds ${G_{\mu \nu }}$ , ${B_{\mu \nu }}$ and $\phi$ are respectively the space-time metric, axion-field and the dilaton-field and ${S_{{\mathop{\rm int}} }}$ is the internal part of of the above sigma model action. Moreover (2), the partition function $Z\left( {{\phi _i}} \right)$ , which is closed with respect to the space-time effective action $\Gamma \left( {{\phi _i}} \right)$ , is

$\begin{array}{c}Z\left( {{\phi _i}} \right) = \sum\limits_{g = 0}^\infty {{e^{\rho \chi }}} \int\limits_{{\Sigma _g}} {\left[ {d{h_{\alpha \beta }}} \right]} \left[ {d{X^i}} \right]\\ \cdot \left[ {dZ} \right]{e^{\frac{i}{h}S}}\end{array}$

where ${e^\rho }$ is the string coupling constant, $\rho$ is the constant part of the dilaton field, $\chi$ is the ${\Sigma _g}$ -Euler characteristic and $\chi \left( {{\Sigma _g}} \right) = 2\left( {g + 1} \right)$ . Also (3) the N-points correlation functions, keeping

${e^{\frac{i}{h}S}}\prod\limits_{i = 1}^N {{A_i}}$

in mind for later, is

$\begin{array}{l}\frac{{{\delta ^N}\Gamma \left( {{\Phi _1},...,{\Phi _N}} \right)}}{{\delta {\phi _1},...,\delta {\phi _N}}} = \left\langle {\prod\limits_{i = 1}^N {{A_i}} } \right\rangle = \\\sum\limits_{g = 0}^\infty {\left[ {{e^{\left( {N + 1} \right)\rho \chi }}\int\limits_{{\Sigma _g}} {\left[ {d{h_{\alpha \beta }}} \right]\left[ {d{X^i}} \right]\left[ {dZ} \right]\left\{ {{e^{\frac{i}{h}S}}\prod\limits_{i = 1}^N {{A_i}} } \right\}} } \right]} \end{array}$

with background satisfying the quantum equation of motion

$\frac{{\delta \,\Gamma \left( {{\Phi _i}} \right)}}{{\delta \,{\Phi _i}}} = 0$

Given all the mathematics in this post, I have reasons, as should we all, to suspect a need for string-theory to be re-interpreted, and here are some:

– The degeneracy of string vacua (solvable as I will show).

– No known mechanism for supersymmetry breaking (solvable).

– The ‘time independent’ backgrounds problem (solvable).

– The perturbative interpretation of the Polyakov series seems to be incompatible with the connection between loops corrections and the Planck constant.

– In the light-cone gauge, the world-sheet time-variable coincides with the space-time variable ${x^0}$ . The Fredholm evolution of spatial coordinates on the world-sheet is given by ${x^i} = {x^i}\left( {\sigma ,{x^0}} \right)$ , $\sigma$ is the spatial world-sheet coordinate. Note when the evolution of strings in space is concerned, this dependence seems natural. But if we are interested about locality in the spatial part of the universe the $\sigma$ -dependence of ${x^i}$ necessitates us to consider $\sigma$ as a scale of the universe on equal footing with the cosmic time ${x^0}$ , which indicates that the natural interpretation of the world sheet is actually a two dimensional parameter base-space upon which the universe and matter evolve.

Development of M-theory due to Witten, duality and D/p-branes solutions in string theory, and the Holographic principle, the AdS/CFT correspondence, and the Hawking-’t Hooft proposal on the construction of black holes, are clearly deep advances in our understanding of string theory in a technical and conceptual context. But they are not sufficient by themselves to solve the above listed problems. It seems to me that a solution hinges on a Witten-reductional framework for low dimensional effective description of initially higher dimensional symmetric dynamical systems, with matter evolving on homogeneous isotropic four-dimensional space-time ${M_4}$ , and the gravitational equations coupled to matter reduce to a dynamical system with a one dimensional base space ${M_1}$ . This isomorphically reflects the existence of Picard-differentiable fibration

$\left\{ {\begin{array}{*{20}{c}}{{f_1}:E \to {M_4}}\\{{f_2}:{M_4} \to {M_1}}\end{array}} \right.$

thus getting a ${G_1}$ –gauge invariance and general covariant action. The analogous action on ${M_1}$ corresponding to ${f_i}$ with

$G = {G_1} \times \frac{{{G_{ext}}}}{{SO\left( 3 \right)}}$

${G_1}$ and ${G_{ext}}$ are respectively the ${M_4}$ -internal symmetry and the geometric one. This action describes evolution upon ${M_1}$ and so we get a fundamental law governing $G$ -orbital evolution upon the orbit space $\frac{E}{G} = {M_d}$ ,

$d = \dim \frac{E}{G} = \dim E - \dim G$

according to a $G$ -gauge invariant and generally covariant action

${S_d} = \int\limits_{{M_d}} {\left\{ {\sqrt g \phi R + {L_{mat}} + {L_{gauge}} + {L_{top}}} \right\}}$

$\phi$ is the dilaton,

$\int\limits_{{M_d}} {\left\{ {{L_{mat}} + {L_{gauge}}} \right\}}$

the matter and gauge field action, and

$\int\limits_{{M_d}} {\left\{ {{L_{top}}} \right\}}$

the topological term which picks the vacuum. Hence the partition function corresponding to

${S_d} = \int\limits_{{M_d}} {\left\{ {\sqrt g \phi R + {L_{mat}} + {L_{gauge}} + {L_{top}}} \right\}}$

is given by the polyakov-looking series

$Z\left( {{\phi _i}} \right) = \sum\limits_{M - {\rm{topologies}}}^\infty {\left\{ {\int\limits_{{M_d}} {\left[ {d{g_{\alpha \beta }}} \right]\left[ {d\Psi } \right]\left[ {dZ} \right]{e^{\frac{i}{h}{S_d}}}} } \right\}}$