  In this paper, we introduce a framework of new mathematical representation of
Game Theory, including static classical game and static quantum game. The idea
is to find a set of base vectors in every single-player strategy space and to
define their inner product so as to form them as a Hilbert space, and then form
a Hilbert space of system state. Basic ideas, concepts and formulas in Game
Theory have been reexpressed in such a space of system state. This space
provides more possible strategies than traditional classical game and
traditional quantum game. So besides those two games, more games have been
defined in different strategy spaces. All the games have been unified in the
new representation and their relation has been discussed. General Nash
Equilibrium for all the games has been proposed but without a general proof of
the existence. Besides the theoretical description, ideas and technics from
Statistical Physics, such as Kinetics Equation and Thermal Equilibrium can be
easily incorporated into Game Theory through such a representation. This
incorporation gives an endogenous method for refinement of Equilibrium State
and some hits to simplify the calculation of Equilibrium State. The more
privileges of this new representation depends on further application on more
theoretical and real games. Here, almost all ideas and conclusions are shown by
examples and argument, while, we wish, lately, we can give mathematical proof
for most results.
