| Summary: | We describe non-equilibrium <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>ϕ</mi><mn>4</mn></msup></semantics></math></inline-formula> 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>ϕ</mi><mn>4</mn></msup></semantics></math></inline-formula> 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.
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