How the String Theory Worldsheet 'Knows' All About Spacetime Physics

This is how the worldsheet quantum theory knows all about spacetime physics. Setting the stage first. Since branes are 'generalizations', and are BPS, the supergravity solution in the multi-brane harmonic function form is:

H_p^{{\rm{array}}} = 1 + \sum\limits_{n = - \infty }^{ + \infty } {\frac{{r_p^{7 - p}}}{{{{\left| {{{\widetilde r}^2} + {{\left( {{X^{p + 1}} - 2\pi nR} \right)}^2}} \right|}^{(7 - p)/2}}}}}

with:

{r^2} = {\left( {{X^{p + 1}}} \right)^2} + {\left( {{X^{p + 2}}} \right)^2} + ... + {\left( {{X^{p + 9}}} \right)^2} = {\widetilde r^2} + {\left( {{X^{p + 1}}} \right)^2}

Thus, I can now derive:

H_p^{{\rm{array}}} \sim 1 + \frac{{r_p^{7 - p}}}{{2\pi R}}\frac{1}{{{{\widetilde r}^{6 - p}}}}\int\limits_{ - \infty }^\infty {\frac{{du}}{{{{\left( {1 + {u^2}} \right)}^{\left( {7 - p} \right)/2}}}}}

Hence, the integral is:

\int\limits_{n = - \infty }^{ + \infty } {\frac{{du}}{{{{\left( {1 + {u^2}} \right)}^{\left( {7 - {p^n}} \right)/2}}}}} = \frac{{\sqrt {2\pi n{R^n}} {\mkern 1mu} \Gamma \left[ {\frac{1}{2}\left( {6 - {p^n}} \right)} \right]}}{{\Gamma \left[ {\frac{1}{2}\left( {7 - {p^n}} \right)} \right]}}

After checking renormalization, one gets:

H_p^{{\rm{array}}} \sim H_{p + 1}^{{\rm{array}}} = 1 + \frac{{\sqrt {\alpha '} r_{p + 1}^{7 - \,\left( {p + 1} \right)}}}{R}\frac{1}{{{{\widetilde r}^{7 - \,\left( {p + 1} \right)}}}}

which is the correct harmonic function for a D(p+1)-brane. The relevance of H_{p + 1}^{{\rm{array}}} is that via Green's functional analysis, it yields the string coupling of the dual 25-D theory:

{e^{{\Phi _{bos}}}} = {e^{\Phi _{bos}^{{e^{{\phi _{si}}}}}}}\frac{{{{\alpha '}^{1/2}}}}{{2\pi nR}}

which is key to the T-duality transformation properties of propagating background matter fields in 4-dimensional space-time, with {\Phi _{bos}} the bosonic field configuration corresponding to the string world-sheet, whose variable is {\phi _{si}}, yielding the two following key relations:

\begin{array}{c}({T_p}\left( {2\pi \sqrt {\alpha '} } \right){e^{ - {\Phi _{bos}}}}\prod\limits_{i = 1}^{p - 1} {\left( {2\pi {R_p}} \right)} = \\{T_{p - 1}}{e^{ - \Phi _{bos}^{1/2}}}\prod\limits_{i = 1}^{p - 1} {\left( {2\pi {R_{p - 1}}} \right)} \end{array}

and

\frac{d}{{{d_{{\sigma _p}}}}}\int\limits_{{\rm{worldvolumes}}}^p {{e^{H_{p + 1}^{{\rm{array}}}}}} + \underbrace {\sum\limits_{{\sigma _p}}^D {{{\left( {S_p^D} \right)}^{ - H_{p + 1}^{{\rm{array}}}}}} }_{{\rm{topologies}}}

Can a Dirac-Schrödinger 'Quantum Space Analysis' Solve the Problem of Time Scale-Invariantly?!

We saw how the omega-simplified Wheeler-DeWitt equation:

\left( {{\hbar ^2}\left( {\frac{1}{{24}}\frac{{{{\not \partial }^2}}}{{\not \partial {\Omega ^2}}} - \frac{1}{2}\frac{{{{\not \partial }^2}}}{{\not \partial {\phi ^2}}}} \right) - 6\kappa {e^{4\Omega }} + {e^{6\Omega }}V(\phi )} \right)\psi \left( {\Omega ,\phi } \right) = 0

with the corresponding scalar product

\left\langle {\psi ,\phi } \right\rangle : = i\int_{\Omega = {\rm{const}}} {d\phi } \left( {{\psi ^ * }\frac{{\not \partial \phi }}{{\not \partial \Omega }} - \phi \frac{{\not \partial {\psi ^ * }}}{{\not \partial \Omega }}} \right)

which is conserved in \Omega -time by virtue of:

\left( {{\hbar ^2}\left( {\frac{1}{{24}}\frac{{{{\not \partial }^2}}}{{\not \partial {\Omega ^2}}} - \frac{1}{2}\frac{{{{\not \partial }^2}}}{{\not \partial {\phi ^2}}}} \right) - 6\kappa {e^{4\Omega }} + {e^{6\Omega }}V(\phi )} \right)\psi \left( {\Omega ,\phi } \right) = 0

allowed us to derive what looks like a Hilbert space inner-product, until one looks at the right-hand-side of:

there can be no selection of an intrinsic time functional hence, the non-existence of a suitable Killing-vector follows: this is the 'Hilbert space problem' for time. Here, I shall try and recover a notion of time via a Dirac-Schrödinger analysis.

On the Klein-Gordon 'Hilbert-Space-Problem' of Time

Given, as I showed, that the Eulerian fluid action, after the ADM splitting, is

{S_F}\int_\mathbb{R} {dt} \int_\sigma {{d^3}} x\sqrt q N{\rho _0}\left( {\sqrt {\left( {{v_\mu }{n^\mu }} \right) - {v_a}{v^a}} - TS} \right)

with

\left\{ {\begin{array}{*{20}{c}}{{\chi _1} = {p_\alpha } = 0}\\{{\chi _2} = {p_\beta } - \alpha \pi = 0}\\{{\chi _3} = {p_\theta } = 0}\\{{\chi _4} = {p_S} - \theta \pi }\end{array}} \right.

the entropy S, given the Mandelstam-Tamm formulation of a time-energy uncertainty relation, entails the undefinability of inner products of functional differential operators for this form of the Wheeler-DeWitt equation:

and at face value, that is a no-go theorem for the 'existence' of time outside of string-theory's AdS/CFT duality, but not to be addressed here. Let me
ignore the operator-ordering problem to reduce the WdW-equation to

\left( {{\hbar ^2}{e^{ - 3\Omega }}\left( {\frac{1}{{24}}\frac{{{{\not \partial }^2}}}{{\not \partial {\Omega ^2}}} - \frac{1}{2}\frac{{{{\not \partial }^2}}}{{\not \partial {\phi ^2}}}} \right) - 6k{e^\Omega } + {e^{3\Omega }}V(\phi )} \right)\psi \left( {\Omega ,\phi } \right) = 0

and try and define an inner product via the Klein-Gordon interpretation of quantum gravitational geometrodynamics and proceed from there.