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:

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

The Cosmological Quantum State from Deformation Quantization

I will show that the Weyl-Wigner-Groenewold formalism of the cosmological quantum state is described by the Moyal-Wheeler-DeWitt deformation quantization equation with symplectic solutions in the Moyal-Wigner phase space: this is philosophically of foundational importance since quantum gravity necessitates that spacetime is quantized in a way described by the Moyal-Wheeler-DeWitt equation:

    \[\tilde H\left( {x + \frac{i}{2}{{\overrightarrow \partial }_x},{\Pi _x} - \frac{i}{2}{{\overrightarrow \partial }_x}} \right)W\left( {x,{\Pi _x}} \right) = 0\]

where in the deformation quantization Hilbert space formalism, the deformed operator in the scalar product relative to \tilde H is given by:

    \[\begin{array}{*{20}{l}}{\left\langle {\Psi ,{{\left( {d{X^\mu }d{X^\mu }} \right)}_\Theta }\Psi } \right\rangle = \left\langle {\Psi ,{{\left( {d{X^\mu }} \right)}^2}_\Theta \Psi } \right\rangle }\\{ = {{\left( {2\pi } \right)}^{ - d}}\mathop {\lim }\limits_{\varepsilon \to 0} \int {\int {dx{\mkern 1mu} dy{\mkern 1mu} {e^{ - ixy}}} } \chi \left( {\varepsilon x,\varepsilon y} \right) \cdot }\\{\left\langle {\Psi ,U(y)\alpha {\mkern 1mu} {\Theta _x}\left( {d{X^\mu }} \right)} \right\rangle }\\{ = {{\left( {2\pi } \right)}^{ - d}}\mathop {\lim }\limits_{\varepsilon \to 0} \int {\int {dx{\mkern 1mu} dy{\mkern 1mu} {e^{ - ixy}}} } \chi \left( {\varepsilon x,\varepsilon y} \right){b^\mu }\left( {x,y} \right)}\end{array}\]

with \Theta a skew-symmetric matrix on {\mathbb{R}^d}, and \chi \in f_{DQ}^{ps}\left( {{\mathbb{R}^d} \times {\mathbb{R}^d}} \right), \chi \left( {0,0} \right) = 1, and the deformation quantization differential operator is given by:

    \[{\left( {d{X^\mu }} \right)^2}_\Theta = \int {{e^{ - 2{a_\mu }{{\left( {\Theta X} \right)}_\mu }}}} {\left( {d{X^\mu }} \right)^2}\]

The central property of the cosmological quantum state is that it must entail the emergence of a classical universe satisfying all of the observable properties induced by the Friedmann-Robertson-Walker space-time, where such a FRW-space-time flat line-element, curved by quantum deformation, is:

    \[{\left( {d{s^2}} \right)_\Theta } = d{\hat t^2} - {e^{H\hat t}}d{\hat x_ \bot }^2\]

with curved space-time metric:

    \[{\left( {{\eta _{\mu \nu }}} \right)_\Theta } = {e^{ - 2{a_\mu }{{\left( {\Theta \hat x} \right)}_\mu }}}{\eta _{\mu \nu }}\]