Well-posedness and decoherence transition for nonlinear p-adic Dirac networks

Prove global well-posedness for the nonlinear four-spinor p-adic Dirac equation obtained by adding a non-local weight kernel and componentwise nonlinear activation, and characterize the transition between its unitary regime and its Lindblad-type decohering regime.

Background

The paper proposes extending the free p-adic Dirac continuous-time quantum walk by adding a non-local interaction kernel W(x,y), a componentwise nonlinear activation function, and a bias term. This would produce a four-spinor analogue of the scalar p-adic quantum neural networks discussed in earlier work.

The proposed extension is expected to interpolate between the unitary free dynamics when W=0 and a decohering, Lindblad-type regime when W is nonzero. The paper does not establish existence and uniqueness theory for the resulting nonlinear equation or explain the dynamical transition between these regimes.