  It is well known that nonrelativistic quantum mechanics presents a clear
asymmetry between space and time. Much of this asymmetry is attributed to the
lack of Lorentz invariance of the theory. Nonetheless, a recent work [Phys.
Rev. A \textbf{95}, 032133 (2017)] showed that even though this is partially
true, there is a broader physical scenario in which space and time can be
handled in nonrelativistic quantum theory in a more symmetric way. In this
space-time-symmetric formalism, an additional Hilbert space is defined so that
time is raised to the status of operator and position becomes a parameter. As a
consequence, the Hilbert space now requires a space-conditional quantum state
governed by a new quantum dynamics. In this manuscript, we reveal some physical
and mathematical properties of the space-time-symmetric formalism such as:
symmetries between the Hamilton-Jacobi and the space-conditional equation; the
general solution for a time-independent potential; and a new Lagrangian for a
spinless particle in one dimensional. Finally, we present the space-conditional
equation for a particle under the effect of an electromagnetic field, and the
gauge invariance of this equation is proved.
