  The possibility of extending the canonical formulation of quantum mechanics
(QM) to a space-time symmetric form has recently attracted wide interest. In
this context, a recent proposal has shown that a spacetime symmetric many-body
extension of the Page and Wootters mechanism naturally leads to the so-called
Quantum Action (QA) operator, a quantum version of the action of classical
mechanics. In this work, we focus on connecting the QA with the
well-established Feynman's Path Integral (PI). In particular, we present a
novel formalism which allows one to identify the "sum over histories" with a
quantum trace, where the role of the classical action is replaced by the
corresponding QA. The trace is defined in the extended Hilbert space resulting
from assigning a conventional Hilbert space to each time slice and then taking
their tensor product. The formalism opens the way to the application of quantum
computation protocols to the evaluation of PIs and general correlation
functions, and reveals that different representations of the PI arise from
distinct choices of basis in the evaluation of the same trace expression. The
Hilbert space embedding of the PIs also discloses a new approach to their
continuum time limit. Finally, we discuss how the ensuing canonical-like
version of QM inherits many properties from the PI formulation, thus allowing
an explicitly covariant treatment of spacetime symmetries.
