  We introduce a new quantum mechanical theory called Spin Matrix theory (SMT).
The theory is interacting with a single coupling constant g and is based on a
Hilbert space of harmonic oscillators with a spin index taking values in a Lie
(super)algebra representation as well as matrix indices for the adjoint
representation of U(N). We show that SMT describes N=4 super-Yang-Mills theory
(SYM) near zero-temperature critical points in the grand canonical phase
diagram. Equivalently, SMT arises from non-relativistic limits of N=4 SYM. Even
though SMT is a non-relativistic quantum mechanical theory it contains a
variety of phases mimicking the AdS/CFT correspondence. Moreover, the infinite
g limit of SMT can be mapped to the supersymmetric sector of string theory on
AdS_5 x S^5. We study SU(2) SMT in detail. At large N and low temperatures it
is a theory of spin chains that for small g resembles planar gauge theory and
for large g a non-relativistic string theory. When raising the temperature a
partial deconfinement transition occurs due to finite-N effects. For
sufficiently high temperatures the partially deconfined phase has a classical
regime. We find a matrix model description of this regime at any coupling g.
Setting g=0 it is a theory of N^2+1 harmonic oscillators while for large g it
becomes 2N harmonic oscillators.
