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In 4d minkowski spacetime the time dependent wave equation of a free particle is $$\Psi(x,t) = \left({a \over a + i\hbar t/m}\right)^{3/2} e^{- {x^2\over 2(a + i\hbar t/m)} }.$$ I notice that there is the imaginary unit in the equation, and in minkowski spacetime space can be thought of as imaginary time, and time can be thought of as imaginary space. In euclidean spacetime the minus sign in the equation for the spacetime interval $${{\Delta}s}^2={{\Delta}x}^2+{{\Delta}y}^2+{{\Delta}z}^2-(c{\Delta}t)^2$$ would get replaced by a plus sign, so that the spacetime interval between two events would be $${{\Delta}s}^2={{\Delta}x}^2+{{\Delta}y}^2+{{\Delta}z}^2+{{\Delta}w}^2.$$ Is the imaginary unit in the time dependent wave equation of a free particle related to time being imaginary space, and space being imaginary time? The time dependent wave equation is time symmetric, but if I simply replace the imaginary unit with 1, I do not get a time symmetric equation.

In a 4d euclidean spacetime, in which every spacetime coordinate of an event was always a real number, would the wave equation be the same as in 4d minkowski spacetime, and have the imaginary unit?

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