  We give a general $SU(2)_L\times SU(2)_R$ $\times U(1)_{EM}$ sigma model with
external sources, dynamical breaking and spontaneous vacuum symmetry breaking,
and present the general formulation of the model. It is found that $\sigma $
and $\pi ^0$ without electric charges have electromagnetic interaction effects
coming from their internal structure. A general Lorentz transformation relative
to external sources $J_{gauge}$ $=(J_{A_\mu},J_{A_\mu ^\kappa})$ is derived,
using the general Lorentz transformation and the four-dimensional current of
nuclear matter of the ground state with $J_{gauge}$ = 0, we give the
four-dimensional general relations between the different currents of nuclear
matter systems with $J_{gauge}\neq 0$ and those with $J_{gauge}=0$. The
relation of the density's coupling with external magnetic field is derived,
which conforms well to dense nuclear matter in a strong magnetic field. We show
different condensed effects in strong interaction about fermions and
antifermions, and give the concrete scalar and pseudoscalar condensed
expressions of $\sigma_0$ and $\pi_0$ bosons. About different dynamical
breaking and spontaneous vacuum symmetry breaking, the concrete expressions of
different mass spectra are obtained in field theory. This paper acquires the
running spontaneous vacuum breaking value $\sigma_0^{\prime},$ and obtains the
spontaneous vacuum breaking in terms of the running $\sigma_0^{\prime}$, which
make nucleon, $\sigma $ and $\pi $ particles gain effective masses. We achieve
both the effect of external sources and nonvanishing value of the condensed
scalar and pseudoscalar paticles. It is deduced that the masses of nucleons,
$\sigma $ and $\pi $ generally depend on different external sources.
