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:

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

Quantum Non-Locality and Wave Function Collapse

The foundational crises caused by the wave function collapse need no introduction. Here I will address the obvious need for a collapse theory that accounts simultaneously for quantum entanglement, time-symmetry and eliminability of any incorporation of a ‘decohering-observer’. The reason is clear, and not simply due to the universal wave functional collapse or the meta-entanglement Nakajima–Zwanzig-Wigner regress paradoxes. Here’s how to appreciate the acuteness of the problem. Look at the following collapse equation:

    \[\begin{array}{*{20}{l}}{d\left| {{\psi _t}} \right\rangle = \left[ { - \frac{i}{\hbar }} \right.Hdt + \sqrt \lambda \int {{d^3}} x\left( {N(\bar x) - {{\left\langle {N(\bar x)} \right\rangle }_t}} \right)}\\{ \cdot d{W_t}(\bar x) - \frac{\lambda }{2}\int {{d^3}} x\left. {{{\left( {N(\bar x) - {{\left\langle {N(\bar x)} \right\rangle }_t}} \right)}^2}dt} \right]\left| {{\psi _t}} \right\rangle }\end{array}\]

ensuring the positivity of the reduced density matrix:

    \[\forall t\left\langle {\left. \psi \right|{{\hat \rho }_S}(t)\left| \psi \right.} \right\rangle \ge 0\]

entails that the Lindblad quantum-jump equation:

    \[\begin{array}{l}{\rm{d}}\hat \rho _S^C = - i\left[ {{{\hat H}_S},\hat \rho _S^C} \right]{\rm{d}}t - \\\frac{1}{2}\sum\limits_\mu {{\kappa _\mu }} \left[ {{{\hat L}_\mu },\left[ {{{\hat L}_\mu },\hat \rho _S^C} \right]} \right]{\rm{d}}t + \\\sum\limits_\mu {\sqrt {{\kappa _\mu }} } W\left[ {{{\hat L}_\mu }} \right]\hat \rho _S^C{\rm{d}}{W_\mu }\end{array}\]

is solvable iff {L_k} commutes with the position-operator, with:

    \[\begin{array}{l}W\left[ {\hat L} \right]\hat \rho \equiv \hat L\hat \rho + \hat \rho {{\hat L}^\dagger } - \\\hat \rho {\rm{Tr}}\left\{ {\hat L\hat \rho + \hat \rho {{\hat L}^\dagger }} \right\}\end{array}\]

and {\rm{d}}{W_\mu } the Weiner-quantum-increments.

However:

    \[\tilde U: = - \frac{\lambda }{2}\int {{d^3}} x{\left( {N(\bar x) - {{\left\langle {N(\bar x)} \right\rangle }_t}} \right)^2}dt(...)\]

commutes with the energy operator. By the Heisenberg energy-time uncertainty principle, the collapse equation cannot be integrated to get the collapse-localization double-integral in 4-D:

    \[{\int {\int {\left| {\Psi _{{t_4}}^I\left( {x,X} \right) + \Psi _{{t_4}}^{II}\left( {x,X} \right)} \right|} } ^2}dxdX = 1\]

Now applying the wave-particle duality to:

    \[\int { - \frac{\lambda }{2}} {d^3}xd{W_t}(\bar x) + \int {\frac{i}{\hbar }Hdt + \sqrt \lambda \int {{d^3}} xd{W_t}(\bar x)} \]

we face the problem of decoherence-induced space-time localization, since the time-local non-Markovian master equation:

    \[\frac{{\rm{d}}}{{{\rm{d}}t}}{\hat \rho _S}(t) = \hat K(t){\hat \rho _S}(t)\]

cannot be solvable consistent with the Lindblad quantum-jump equation also being solvable.

And here’s the paradox: the wave-particle duality is equivalent to the generalized Heisenberg uncertainty principle, and that would imply that the above collapse-localization double-integral in 4-D cannot have a solution and by entanglement entropy relation, that would contradict the Heisenberg time-energy uncertainty principle, thus the wave-particle duality ceases to make sense in the time-symmetry group representation of the corresponding Hilbert space of the quantum system.