*Article* **Non-Equilibrium Quantum Brain Dynamics: Super-Radiance and Equilibration in 2 + 1 Dimensions**

#### **Akihiro Nishiyama 1,\*, Shigenori Tanaka 1,\* and Jack A. Tuszynski 2,3,4,\***


Received: 15 October 2019; Accepted: 26 October 2019; Published: 30 October 2019

**Abstract:** We derive time evolution equations, namely the Schrödinger-like equations and the Klein–Gordon equations for coherent fields and the Kadanoff–Baym (KB) equations for quantum fluctuations, in quantum electrodynamics (QED) with electric dipoles in 2 + 1 dimensions. Next we introduce a kinetic entropy current based on the KB equations in the first order of the gradient expansion. We show the H-theorem for the leading-order self-energy in the coupling expansion (the Hartree–Fock approximation). We show conserved energy in the spatially homogeneous systems in the time evolution. We derive aspects of the super-radiance and the equilibration in our single Lagrangian. Our analysis can be applied to quantum brain dynamics, that is QED, with water electric dipoles. The total energy consumption to maintain super-radiant states in microtubules seems to be within the energy consumption to maintain the ordered systems in a brain.

**Keywords:** non-equilibrium quantum field theory; quantum brain dynamics; Kadanoff–Baym equation; entropy; super-radiance
