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Neural Network for Quantum Brain Dynamics: 4D CP$^1$+U(1) Gauge Theory on Lattice and its Phase Structure

We consider a system of two-level quantum quasi-spins and gauge bosons put on a 3+1D lattice. As a model of neural network of the brain functions, these spins describe neurons quantum-mechanically, and the gauge bosons describes weights of synaptic connections. It is a generalization of the Hopfield model to a quantum network with dynamical synaptic weights. At the microscopic level, this system becomes a model of quantum brain dynamics proposed by Umezawa et al., where spins and gauge field describe water molecules and photons, respectively. We calculate the phase diagram of this system under quantum and thermal fluctuations, and find that there are three phases; confinement, Coulomb, and Higgs phases. Each phase is classified according to the ability to learn patterns and recall them. By comparing the phase diagram with that of classical networks, we discuss the effect of quantum fluctuations and thermal fluctuations (noises in signal propagations) on the brain functions.