• The Lindblad Master Equation, Feynman-Kac Formula, and the Measurement Problem

    Any adequate account of how micro-causality and quantum coherence can explain the emergent-property of spacetime and how the Wheeler-DeWitt problem of time can be solved must incorporate a theory of how the Lindblad master equation solves the Markov quantum fluctuation problem as well as showing how the quantum Jarzynski-Hatano-Sasa relation can be homologically defined globally for both, Minkowski space and Friedmann-Robertson-Walker generalized Cartan space-times. This is a step towards those goals. Consider a wave-function \left| {{\psi _t}^{S,m,c}} \right\rangle and the entropic quantum entanglement relation of the total system consisting of ‘S’, ‘m’ and the quantum-time measuring clock ‘c’ subject to Heisenberg’s UP. It follows then that the probability that any given initial state \left| {\psi _t^{S,m,c}} \right\rangle evolves for a time T that undergoes N jumps during intervals \delta t centered at times {t_1},...,{t_N} is given by:

        \[\begin{array}{l}{\left( {2\delta t{\kappa ^2}/G} \right)^N}{\rm{Tr}}\left\{ {{e^{ - i{{\tilde H}_{eff}}\left( {T - {t_N}} \right)}}} \right. \cdot \\\hat a{e^{ - i{{\hat H}_{eff}}}}\left( {{t_N} - {t_{N - 1}}} \right)\hat a...\,\hat a{e^{ - i{{\hat H}_{eff}}t}}\\ \times \left| {\psi _t^{S,m,c}} \right\rangle \left\langle {\psi _t^{S,m,c}} \right|{e^{i{{\tilde H}^\dagger }_{eff}{t_1}}}{{\hat a}^\dagger }...\,\left. {{{\hat a}^\dagger }{e^{i{{\tilde H}^\dagger }_{eff}\left( {T - {t_N}} \right)}}} \right\}\end{array}\]

    So, the master equation:

        \[\begin{array}{l}{{\dot \rho }_{00}} = - i\left[ {{{\hat H}_0},{\rho _{00}}} \right] + \frac{{2{\kappa ^2}}}{G}\hat a{\rho _{00}}{{\hat a}^\dagger }\\ - \frac{{{\kappa ^2}}}{G}{{\hat a}^\dagger }\hat a{\rho _{00}} - \frac{{{\kappa ^2}}}{G}{\rho _{00}}{{\hat a}^\dagger }\hat a\end{array}\]

    is valid iff the Markovian approximation is faithful and valid only on time-scales longer than 1/{\Gamma _1}, hence the jump occurs during an interval \delta t \sim 1/{\Gamma _1} centered on {t_i}. Therefore, with the Hamiltonian:

        \[{\hat H_I} = \kappa \left( {{{\hat a}^\dagger } \otimes \hat b + \hat a \otimes {{\hat b}^\dagger }} \right)\]

    where \left( {\hat a,\hat b} \right);\left( {{{\hat a}^\dagger },{{\hat b}^\dagger }} \right) are the lowering/raising operators for the system and output mode respectively, it follows that the total system satisfies the master equation:

        \[\begin{array}{c}\dot \rho = - i\left[ {\hat H,\rho } \right] + {\Gamma _1}\hat b\rho {{\hat b}^\dagger } - \frac{{{\Gamma _1}}}{2}{{\hat b}^\dagger }\hat b\rho \\ - \frac{{{\Gamma _1}}}{2}\rho {{\hat b}^\dagger }\hat b + {\Gamma _2}{\sigma _z}\rho {\sigma _z} - {\Gamma _2}\rho \\ \equiv L_s^L\rho \end{array}\]

    where the Pauli operator {\sigma _z} acts on the output mode and L_s^L is the Liouville superoperator. Given that it is a linear equation, it has a solution given as:

        \[\rho ({t_2}) = \exp \left\{ {L_s^L\left( {{t_2} - {t_1}} \right)} \right\}\rho ({t_1})\]

    and so the evolution of the density matrix {\rho _t} is given by the Lindblad master equation:

        \[\begin{array}{l}{\partial _t}{\rho _t} = - i\left[ {{H_t},{\rho _t}} \right] + \sum\limits_{i = 1}^I {\left( {{V_i}{\rho _t}V_i^\dagger } \right.} \\\left. { - \frac{1}{2}V_i^\dagger {V_i}{\rho _t} - \frac{1}{2}{\rho _t}V_i^\dagger {V_i}} \right)\end{array}\]

  • 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:

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


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

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