Non-Equilibrium φ4 Theory in a Hierarchy: Towards Manipulating Holograms in Quantum Brain Dynamics

We describe non-equilibrium phi(4) theory in a hierarchical manner to develop a method for manipulating coherent fields as a toy model of introducing control into Quantum Field Theory (QFT) of the brain, which is called Quantum Brain Dynamics (QBD). We begin with the Lagrangian density of phi(4) model, where we adopt 2-Particle-Irreducible (2PI) effective action, and derive the Klein-Gordon equation of coherent fields with a damping term as an input-output equation proposed in areas of morphological computation or reservoir computing. Our analysis is extended to QFT in a hierarchy representing multiple layers covering cortex in a brain. We find that the desired target function is achieved via time-evolution in the Klein-Gordon equations in a hierarchy of numerical simulations when a signal in both the input and output prevails over noise in the intermediate layers. Our approach will be applied to control coherent fields in the systems (in a hierarchy) described in the QFT framework, with potential applications allowing the manipulation of quantum fields, especially holograms in QBD. We could then provide realistic physical degrees of freedom of a light-matter system in the contexts of quantum cognition and the associated free-energy principle.