Nonlocality III: General Nonlocality in Quantum Fields
  The waves of fermions display nonlocality in low energy limit of quantum fields. In this \QTR{it}{ab initio} paper we propose a complex-geometry model that reveals the affection of nonlocality on the interaction between material particles of spin-1/2. To make nonlocal properties appropriately involved in a quantum theory, the special unitary group SU(n) and spinor representation $D^{(1/2,1/2)}$ of Lorentz group are generalized by making complex spaces--which are spanned by wave functions of quantum particles--curved. The curved spaces are described by the geometry used in General Relativity by replacing the real space with complex space and additionally imposing the analytic condition on the space. The field equations for fermions and for bosons are respectively associated with geodesic motion equations and with local curvature of the considered space. The equation for fermions can restore all the terms of quadratic form of Dirac equation. According to the field equation it is found that, for the U(1) field [generalized Quantum Electrodynamics (QED)], when the electromagnetic fields $\vec E$ and $\vec B$ satisfy $\vec E^2-\vec B^2\neq 0$, the bosons will gain masses. In this model, a physical region is empirically defined, which can be characterized by a determinant occurring in boson field equation. Applying the field equation to U(3) field [generalized Quantum Chromodynamics (QCD)], the quark-confining property can be understood by carrying out the boundary of physical region. 
Hai-Jhun Wanng