The Orchestrated Objective Reduction (Orch OR) theory, proposed by physicist Roger Penrose and anesthesiologist Stuart Hameroff, relies on a complex mathematical framework that combines quantum mechanics, neurobiology, and theories of consciousness. This page explores the key mathematical concepts underlying Orch OR.
At the heart of Orch OR is the concept of quantum superposition, described by the wave function:
\[ \Psi = c_1\psi_1 + c_2\psi_2 + ... + c_n\psi_n \]
Where \(\Psi\) represents the quantum state, and \(c_i\) are complex coefficients.
Penrose proposed a modification to the Schrödinger equation that incorporates gravitational effects:
\[ i\hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m}\nabla^2\Psi + V\Psi - Gm^2\int \frac{|\Psi(\mathbf{r'})|^2}{|\mathbf{r} - \mathbf{r'}|} d^3r' \Psi \]
This equation combines quantum mechanics with Newtonian gravity, forming a basis for gravitational effects on quantum states.
The time for objective reduction is given by:
\[ \tau \approx \frac{\hbar}{E_G} \]
Where \(\tau\) is the coherence time, \(\hbar\) is the reduced Planck constant, and \(E_G\) is the gravitational self-energy of the superposed state.
Hameroff proposed that microtubules in neurons could perform quantum computations. This involves modeling microtubules as cellular automata capable of quantum operations.
Entanglement between quantum states in different neurons is described by the density matrix:
\[ \rho = \sum_i p_i |\psi_i\rangle \langle\psi_i| \]
Where \(p_i\) are probabilities and \(|\psi_i\rangle\) are pure quantum states.