Jason, what one means by "function" is that operation which transforms a set from one value into another. So there is a set of coordinate points describing dimension -- like the 6 points of 3 dimension discrete space and the 10 non-redundant points of 4 dimension continuous spacetime -- making the 16 points of the Minkowski space matrix which includes the redundant set. A continuous wave function in 4 dimensions accounts for the discretely measured result in 3 dimensions, evolving in time.
As even Einstein averred, there is nothing to prevent the extension of physical reality to higher dimensions than 4, " ... so long as there are good physical reasons to do so." That's what quantum field theory and its extradimensional extension, supersymmetric string theory, is all about.
The wave function of conventional quantum theory is a mathematical, probabilistic, function. Not physically real.
The continuous functions of classical physics assume the 4 dimension limit, such that all measurement results are described within that matrix as physically real results (Einstein ~ "All physics is local."). The probabilistic function of quantum mechanics assumes the 3 dimension limit, in which the classical observation is dependent on at least one result orthogonal to the observer -- therefore, for every local physically real measure, there is a "nonlocal" result that isn't real, and continuous time evolution drops out of the equations in favor of the state vector evolution.
So it doesn't matter whether one chooses a continuous model or a probabilistic model, the wave function cannot be physically real if the upper limit of all measurement functions is 4 dimensions.
To ask whether the wave function is physically real in dimensions > 4, however, still begs the question of locality; i.e., because a local observer is always positioned at the origin, or singularity -- therefore, only if the measure space is simply connected can we guarantee that all measurement results are local and the function is continuous. This is because the probabilistic functions of an n-dimension Hilbert space are all discrete rolls of the dice that beg a nonlocal result for every roll regardless of the dimensionality of the space, and the wave function collapses -- converges on -- the local result. On the other hand, a continuous wave function in any number of dimensions does not collapse, and so we get the quantum interpretation of Hugh Everett III -- wherein classical probability (event bifurcations) predicts an infinite set of "verses" independent of our own. This saves quantum theory from having to abandon the infinite-dimension Hilbert space formalism, in order to explain why all our results are local (classical) yet discontinuous with the classical measurement schema. It makes the quantum solution equal to the analytical solution; boundary conditions are randomly generated by event bifurcation, yet not continuous with the spacetime of our measure space.
If you want your ghosts to be both local and real, you should prefer the continuous functions of classical physics, in which all fields of the measure space affect, and are affected by, the states of all other fields -- continuously. This is what general relativity teaches us. Unfortunately, general relativity only applies "up to diffeomorphism," because boundary conditions for continuous physical functions have to be arbitrarily assigned. Therefore, one cannot be sure of a point of origin that satisfies locality because every origin satisfies locality -- there is no privileged observer frame, no dependence on coordinate geometry.
A model of locally real ghostly phenomena, therefore, is not differentiable from any other physics, i.e., phenomena that we can measure and for which we can deterministically record a position and describe an effect. So "ghostly action at a distance" is ruled out, meaning that no field is discretely disconnected from the simply connected classical field influences.
Conventional quantum theory rules out any classically real -- that is, local -- effects from ghosts, meaning disembodied spirits with causal abilities. What the conventional theory does allow, though, is far more problematic than interfering spirits; "action at a distance" avers that disconnected fields assert causal influence on local phenomena in a much more mystical way than can be imagined by the existence of ghosts.