  We develop a non-perturbative method for calculating partition functions of
strongly coupled quantum mechanical systems with interactions between
subsystems described by a path integral of a dual system. The dual path
integral is derived starting from non-interacting subsystems at zeroth order
and then by introducing couplings of increasing complexity at each order of an
iterative procedure. These orders of interactions play the role of a dual time
and the full quantum partition function is expressed as a transition amplitude
in the dual system. More precisely, it is expressed as a path integral from a
deformation-operators dependent initial state at zero time/order to the
inverse-temperature dependent final state at later time/order. We provide three
examples of strongly coupled systems with first-order, second-order and
higher-order interactions and discuss a possible emergence of space-time,
quantum field theories and general relativity in context of the dual path
integral.
