General nonlocality in quantum fields

Abstract

The natural recognition of quantum nonlocality follows from the fact that a quantum wave is spatially extended. The waves of fermions display nonlocality in low energy limit of quantum fields. In this ab initio paper, we propose a complex-geometry model that reveals the effect of nonlocality on the interaction between material particles of spin 12. 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 almost 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), when the electromagnetic fields E⃗ and B⃗ satisfy E⃗2−B⃗2≠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), the quark-confining property can be understood by carrying out the boundary of physical region. It is also found out that under the conventional form of interaction vertex, γμAμ, only when the color group SU(3) is generalized to U(3) is it possible to understand the strongly bound states of quarks.