  In this paper we investigate the consequences of imposing an SO(4) symmetry
both on the scalar field and on spacetime prior to any spontaneous symmetry
breakdown in the electroweak model. First we argue that the local electroweak
gauge symmetry is implied by the initial SO(4) symmetry of the scalar field.
Then we parameterize the vacuum, in a slightly different way with respect to
the conventional assignment in the electroweak model, around the ground state
mainly in two different ways: Once over the local electroweak gauge
transformations and once over the local $\gamma$ basis which generates the
SO(4) symmetry of the scalar field. Through comparing the two, we conclude that
a consistent parametrization, requires the complexification of the scalar
field, which simultaneously requires spacetime to obey local Lorentz invariance
in the broken phase of the electroweak vacuum. After spontaneous symmetry
breakdown and complexification, the SO(3,1) invariant quantity of the scalar
field is identified to be the Higgs mass, which yields $m_H=v/2 \cong 123$ GeV.
Also the vacuum state is normalized to a unit state by rescaling the local
$\gamma$ basis, so that the metric tensor inherently contains the speed of
light in the Higgs vacuum. We shortly consider the validity of the SO(3,1)
invariance property of the vacuum state in terms of the Goldstone modes, in
case the Higgs field departs from the ground state. The main goal of the paper
is to treat spacetime and vacuum as the two faces of the same medallion, and to
relate the local spacetime symmetry with the electroweak gauge symmetry, both
in the broken and unbroken phases of the vacuum via complexification and
spontaneous symmetry breaking.
