  A nonlocal response theory was developed to describe a many-electron system
within the neighborhood of a nanostructure radiating the longitudinal and
transverse electric fields, which are fundamentally reduced to the scalar and
vector potentials (SP and VP). The coexistence of the SP and VP incidences
distinguishes such a near-field optical system from the ordinary optical
system, in which only the VP (under the Coulomb gauge) incidence survives far
from the light source. This fact is the motivation for equal treatment of the
SP and VP as the cause of the response in the near-field optical system. In the
semiclassical treatment, the linear and nonlinear single susceptibilities are
derived in the form of Heisenberg operators by the functional derivatives of
the action integral of the matter with respect to the SP and VP. These single
susceptibilities relate the SP and VP (as the cause) to the induced charge and
current densities (as the result), and guarantee charge conservation and gauge
invariance; this theory is free from gauge-fixing. It is necessary to consider
the quantum many-electron effect (exchange-correlation effect) to make the
ground state bounded in the non-perturbed system. This is done by employing the
fundamental idea of density functional theory, instead of the ordinary unequal
treatment of the SP and VP, that is, remaking the SP into a Coulomb interaction
between electron charges. Applying the present linear response theory to the
non-metallic material in a limited near-field optical system reveals that the
electric field with the associated permittivity is not suitable quantity to
describe the response, instead, the SP and VP with associate single
susceptibility are essential.
