  This paper presents a comprehensive study of linear-quadratic (LQ) mean field
games (MFGs) in Hilbert spaces, generalizing the classic LQ MFG theory to
scenarios involving $N$ agents with dynamics governed by infinite-dimensional
stochastic equations. In this framework, both state and control processes of
each agent take values in separable Hilbert spaces. All agents are coupled
through the average state of the population which appears in their linear
dynamics and quadratic cost functional. Specifically, the dynamics of each
agent incorporates an infinite-dimensional noise, namely a $Q$-Wiener process,
and an unbounded operator. The diffusion coefficient of each agent is
stochastic involving the state, control, and average state processes. We first
study the well-posedness of a system of $N$ coupled semilinear
infinite-dimensional stochastic evolution equations establishing the foundation
of MFGs in Hilbert spaces. We then specialize to $N$-player LQ games described
above and study the asymptotic behaviour as the number of agents, $N$,
approaches infinity. We develop an infinite-dimensional variant of the Nash
Certainty Equivalence principle and characterize a unique Nash equilibrium for
the limiting MFG. Finally, we study the connections between the $N$-player game
and the limiting MFG, demonstrating that the empirical average state converges
to the mean field and that the resulting limiting best-response strategies form
an $\epsilon$-Nash equilibrium for the $N$-player game in Hilbert spaces.
