• 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

  • A String-Theoretic Derivation of Canonical General Relativity

    Before discussing the canonical formulation of Einstein’s TGR and the relation it bears to string-dynamics and the critical relation between the total string-theory action and the Nieh–Yan-Barbero-Immirzi action, note that the Hilbert action is a functional of the metric tensor, given by:

    {S_D} = {\int {\left( {^{\_\left( 4 \right)}g} \right)} ^{1/2}}{\,^{\left( 4 \right)}}R{d^4}x

    also note a crucial relation to the D-p-brane partition function for closed strings, which is:

    P_{{\rm{int}}}^{Dp} \equiv \not Z = \sum\limits_{\gamma = 0}^\infty {\underbrace {\int {{{\not D}^{SuSy}}\gamma {{\not D'}^{SuSy}}X{e^{S_{cld}^s}}} }_{{\rm{Topologies}}}}

    where {\not D^{SuSy}} is the supersymmetry group covariant derivative. Since the closed string action satisfies the variational equation:

    \begin{array}{c}\delta S_{cld}^s = - \frac{1}{{2\pi \alpha '}}\int_{\partial E_S^5} {{d^2}} \sigma d\,\Omega {\left( {{\phi _{INST}}} \right)^2}{\varepsilon ^{\alpha \beta }}{{\not \partial }_\alpha }{X^\mu }{{\not \partial }_\mu }{\lambda _\nu }\\ = - \frac{1}{{2\pi \alpha '}}\int_{\partial E_S^5} {{d^2}} d\,\Omega {\left( {{\phi _{INST}}} \right)^{ - 1/2}}\sigma \,{{\not \partial }_\mu }X\nu {\left( {{\varepsilon ^{\alpha \beta }}{{\not \partial }_\beta }{X^{^\nu }}{\lambda _\mu }} \right)^{{e^{ - H_3^b}}}}\end{array}

    thus no topology in the sum is degenerate, and hence the closed string has a solvable action in 4-D curved space-time described by {S_D} that needs no renormalization, where the closed string action coupled to the instanton field is:

    \begin{array}{*{20}{c}}{S_{cld}^s = - \frac{1}{{4\pi \alpha '}}\int_{\partial E_{{S_D}}^5} {{d^2}} \sigma d{\mkern 1mu} \Omega {\mkern 1mu} {{\left( {{\phi _{INST}}} \right)}^2}\sigma \sqrt { - \gamma } \left( {\phi \left( {\bar X} \right)} \right.{R_{icci}} + {\gamma ^{\alpha \beta }}{\partial _\alpha }{X^\mu }{g_{\mu \nu }}\left( {\bar X} \right)}\\{ + \frac{1}{{\sqrt { - \gamma } }}{\varepsilon ^{ - H_3^b}}{\partial _\alpha }{X^\mu }{\varepsilon ^{\alpha \beta }}{\partial _\beta }{X^\nu }{b_{\mu \nu }}{{\left( {\bar X} \right)}^2}}\end{array}

    Recall that in the canonical formalism for Einstein’s TGR as developed by Dirac and Arnowitt, Deser and Misner