Self-Reduction Rate of a Microtubule

Abstract

We formulate and study a quantum field theory of a microtubule, a basic element of living cells. Following the quantum theory of consciousness by Hameroff and Penrose, we let the system to reduce to one of the classical states without measurement if certain conditions are satisfied(self-reductions), and calculate the self-reduction time $τ_N$ (the mean interval between two successive self-reductions) of a cluster consisting of more than $N$ neighboring tubulins (basic units composing a microtubule). $τ_N$ is interpreted there as an instance of the stream of consciousness. We analyze the dependence of $τ_N$ upon $N$ and the initial conditions, etc. For relatively large electron hopping amplitude, $τ_N$ obeys a power law $τ_N \sim N^b$, which can be explained by the percolation theory. For sufficiently small values of the electron hopping amplitude, $τ_N$ obeys an exponential law, $τ_N \sim \exp(c' N)$. By using this law, we estimate the condition for $τ_N $ to take realistic values $τ_N$ \raisebox{-0.5ex}{$\stackrel{>}{\sim}$} $10^{-1}$ sec as $N$ \raisebox{-0.5ex} {$\stackrel{>}{\sim}$} 1000.