Clifford-Kähler Algebras and M-Theory Compactification

Continuing from my work on the relation between Clifford algebraic symmetries and M-theory, here I will initiate an analysis of compactification via the derived Kähler-Atiyah bundle associated with Clifford-Kähler manifolds. Recall that whenever 2 or more D-branes coincide, there is a Clifford algebraic symmetry whose generators allow us to derive the total action:

    \[\begin{array}{*{20}{l}}{{S^{Total}} = \frac{1}{{2\pi {\alpha ^\dagger }12}}\int\limits_{{\rm{world - volumes}}} {{d^{26}}} x{\mkern 1mu} d{\mkern 1mu} \Omega {{\left( {{\phi _{Inst}}} \right)}^2}\sqrt {\frac{{ - {g_{\mu \nu }}}}{{ - \gamma }}} {\mkern 1mu} {e^{ - {c_{2n}}/{\Upsilon _\kappa }(\cos \varphi )}} \cdot }\\{\left( {{R_{icci}} - 4{{\left( {{{\not D}^{SuSy}}\left( {{\phi _{Inst}}} \right)} \right)}^2}} \right) + \frac{1}{{12}}H_{3,\mu \nu \lambda }^bH_3^{b,\mu \nu \lambda }/A_\mu ^H + \sum\limits_{D - p - branes} {S_{Dp}^{WV}} }\end{array}\]

and since D-p-branes are metaplectic solitons in closed string-theory, by the von Neumann boundary condition, there is a natural coupling of the super-Higgs field A_\mu ^H to the world-sheet of a string through its boundary:

    \[{S_{open}} = {S_{cld}} + \int\limits_{{\rm{end - points}}} {d\tau } A_\mu ^H{\bar X^\mu }{e^{ - H_3^b}}d{\mkern 1mu} \Omega {\left( {{\phi _{Inst}}} \right)^{\exp {\kern 1pt} ({c_{2n}}/{\Upsilon _\kappa }(\cos \varphi )}}\]

Hence, A_\mu ^H lives on a p+1 dimensional subspace with a {\Upsilon _\kappa }(\cos \varphi ) contribution, yielding the world-volume action:

    \[S_{Dp}^{WV} = {S_{cld}} + \int\limits_{{\rm{end - points}}} {d\tau } A_\mu ^H{\bar X^\mu }d{\mkern 1mu} \Omega {\left( {{\phi _{Inst}}} \right)^2}{e^{ - \left( {H_3^b} \right)/{\Upsilon _\kappa }(\cos \varphi )}} + {e^{{c_{2n}}/{\Upsilon _\kappa }(\cos \varphi )}}/H_3^b\]

and since world-volumes have conformal invariance, by solving the n-loop level equation of motion

Spontaneous Quantum-to-Classical Cosmological Collapse Dynamics

The cosmological primordial perturbations of the universe, implicitly defined by the Wheeler–DeWitt equation:

    \[\begin{array}{l}\tilde H\Psi = \left( {\frac{{2\pi G{\hbar ^2}}}{3}} \right.\frac{{{\partial ^2}}}{{\partial {\alpha ^2}}} - \frac{{{\hbar ^2}}}{2}\frac{{{\partial ^2}}}{{\partial {\phi ^2}}}\\ + \,{e^{6\alpha }}\left( {V\left( \phi \right) + \frac{\Lambda }{{8\pi G}}} \right) - 3{e^{4\alpha }}\left. {\frac{k}{{8\pi G}}} \right)\Psi \left( {\alpha ,\phi } \right) = 0\end{array}\]

a partial differential equation determining a wave-function not defined in space or time or spacetime, with:

    \[\Psi { \approx _{Heisb}}\exp \left( {i{S_0}\left[ {{h_{ab}}} \right]/\hbar } \right)\psi \left[ {{h_{ab}},\left\{ {{x_n}} \right\}} \right]\]

and \psi satisfies an approximate Schrödinger equation:

are clearly quantum in origin. One of the central foundational philosophically pressing problems in physics is to describe a ‘collapse’ dynamics that explains the classical features consistent with astrophysical data. Given the ‘no-time’-property of the Wheeler–DeWitt equation: namely, that it lacks an external time parameter and it lacks a first derivative with an imaginary Schrödinger time-factor, as well as its linearity and symmetrization, we face a deep conflict with the Lindblad equation:

    \[\begin{array}{l}\frac{{d{{\hat \rho }_S}}}{{dt}} = - \frac{i}{\hbar }\left[ {{{\hat H}_S},\hat \rho } \right] + \\\gamma \sum\limits_j {\left[ {{{\hat L}_j}{{\hat \rho }_S}\hat L_j^\dagger - \frac{1}{2}\left\{ {\hat L_j^\dagger {{\hat L}_j},{{\hat \rho }_S}} \right\}} \right]} \end{array}\]

given that its central properties are time-asymmetry and entanglement-entropic-irreversibility, and whose Lindbladian:

    \[\gamma \left[ {\hat S{{\hat \rho }_S}{{\hat S}^\dagger } - \frac{1}{2}\left\{ {{{\hat S}^\dagger }\hat S,{{\hat \rho }_S}} \right\}} \right]\]

describes the non-unitary evolution of the density operator, with:

    \[\gamma \equiv 2\pi \int_0^\infty {d\omega J\left( \omega \right)\delta \left( \omega \right)} \]

Besides the problem of the undefinability of the Lindbladian system-bath interaction:

    \[\left\{ {\begin{array}{*{20}{c}}{{{\hat H}_{SB}} = \hbar \left( {\hat S{{\hat B}^\dagger } + {{\hat S}^\dagger }\hat B} \right)\quad }\\{{{\hat H}_B} = \hbar \sum\limits_k {{\omega _k}\hat a_k^\dagger } {{\hat a}_k}}\end{array}} \right.\]

and

    \[\left\{ {\begin{array}{*{20}{c}}{\left[ {\hat S,{{\hat H}_S}} \right] = 0}\\{\hat S\left( t \right) = \hat S\quad ;\quad \hat B = \sum\limits_k {g_k^ * } {{\hat a}_k}}\\{\hat B\left( t \right) = {e^{\frac{i}{\hbar }{{\hat H}_B}t}}\hat B{e^{ - \,\frac{i}{\hbar }{{\hat H}_B}t}}}\end{array}} \right.\]

in the quantum gravitational cosmology context: see Derivation of the Lindblad Equation for technical details, we already face the tripartite conflict of time