  We develop a new background independent Moyal star formalism in bosonic open
string field theory. The new star product is formulated in a half-phase-space,
and because phase space is independent of any background fields, the
interactions are background independent. In this basis there is a large amount
of symmetry, including a supersymmetry OSp(d|2) that acts on matter and ghost
degrees of freedom, and simplifies computations. The BRST operator that defines
the quadratic kinetic term of string field theory may be regarded as the
solution of the equation of motion A*A=0 of a purely cubic background
independent string field theory. We find an infinite number of non-perturbative
solutions to this equation, and are able to associate them to the BRST operator
of conformal field theories on the worldsheet. Thus, the background emerges
from a spontaneous-type breaking of a purely cubic highly symmetric theory. The
form of the BRST field breaks the symmetry in a tractable way such that the
symmetry continues to be useful in practical perturbative computations as an
expansion around some background. The new Moyal basis is called the $\sigma
$-basis, where $\sigma$ is the worldsheet parameter of an open string. A vital
part of the new star product is a natural and crucially needed mid-point
regulator in this continuous basis, so that all computations are finite. The
regulator is removed after renormalization and then the theory is finite only
in the critical dimension. Boundary conditions for D-branes at the endpoints of
the string are naturally introduced and made part of the theory as simple rules
in algebraic computations. A byproduct of our approach is an astonishing
suggestion of the formalism: the roots of ordinary quantum mechanics may
originate in the rules of non-commutative interactions in string theory.
