  Quantum games represent the really 21st century branch of game theory,
tightly linked to the modern development of quantum computing and quantum
technologies. The main accent in these developments so far was made on
stationary or repeated games. In this paper we aim at initiating the truly
dynamic theory with strategies chosen by players in real time. Since direct
continuous observations are known to destroy quantum evolutions (so-called
quantum Zeno paradox) the necessary new ingredient for quantum dynamic games
must be the theory of non-direct observations and the corresponding quantum
filtering. Apart from the technical problems in organising feedback quantum
control in real time, the difficulty in applying this theory for obtaining
mathematically amenable control systems is due partially to the fact that it
leads usually to rather nontrivial jump-type Markov processes and/or degenerate
diffusions on manifolds, for which the corresponding control is very difficult
to handle. The starting point for the present research is the remarkable
discovery (quite unexpected, at least to the author) that there exists a very
natural class of homodyne detections such that the diffusion processes on
projective spaces resulting by filtering under such arrangements coincide
exactly with the standard Brownian motions (BM) on these spaces. In some cases
one can even reduce the process to the plain BM on Euclidean spaces or tori.
The theory of such motions is well studied making it possible to develop a
tractable theory of related control and games, which can be at the same time
practically implemented on quantum optical devices.
