  This is the first in a series of papers on an attempt to understand quantum
field theory mathematically. In this paper we shall introduce and study BV QFT
algebra and BV QFT as the proto-algebraic model of quantum field theory by
exploiting Batalin-Vilkovisky quantization scheme. We shall develop a complete
theory of obstruction (anomaly) to quantization of classical observables and
propose that expectation value of quantized observable is certain quantum
homotopy invariant. We shall, then, suggest a new method, bypassing Feynman's
path integrals, of computing quantum correlation functions when there is no
anomaly. An exact solution for all quantum correlation functions shall be
presented provided that the number of equivalence classes of observables is
finite for each ghost numbers. Such a theory shall have its natural family
parametrized by a smooth-formal moduli space in quantum coordinates, which
notion generalize that of flat or special coordinates in topological string
theories and shall be interpreted as an example of quasi-isomorphism of general
QFT algebra.
