Trauma of Modern Physics: The Wave Function

The root of the present crisis of modern physics can be traced back to Schrödinger's equation for a system of $N$ electrons formulated nearly 100 years ago in terms of a wave function 

• $\Psi (x,t)$ with $x=(x_1,x_2,...x_N)$ 

as a complex number depending on $N$ three dimensional (3d) coordinates $x_i$ for $i=1,..,N$ plus a time coordinate $t$, in total $3N$ spatial coordinates plus time. The Schrödinger equation specifies how $\Psi (x,t)$ changes over time starting from some given initial configuration $\Psi (x,0)$. 

This may look harmless from purely notational point of view, but the wave function is a monster from both conceptual and computational point of view, since it depends on so many spatial coordinates.  

The conceptual difficulty is to give the wave function a physical (ontic) meaning in 3d physical space, and there is no resolution in sight. Physicists have simply given up resorting to "Shut up and calculate".

The computational difficulty is that even a coarse discretisation of each coordinate into say 10 different positions makes $\Psi$ to depend on $10^{3N}$ numbers, which already for $N=10$ is beyond the capacity of any thinkable computer. In particular, the specification of the initial configuration $\Psi (x,0)$ is a daunting task. 

Since modern physics is based on Schrödinger's equation (+ relativity theory), the above difficulties have remained as the deep trauma behind the present crisis, witnessed in many books including:

• The Wave Function (eds Ney and Albert).
• Philosophy of Physics: Quantum Mechanics (Tim Maudlin)  

To get perspective on the nature of $\Psi (x,t)$ let us compare with the Monadology of Leibniz with the world seen as a collection of monads or simple substances each one with its own mind capable of forming its own (blurred) conception of the world in interaction with all other monads. 

The simple particles without mind of early atomism are here given minds capable of perception and so connect to the wave-function. The trouble with such a many-mind theory is that there is no common 3d space as basis for some ontology, only a collection of separate individual views impossible for an external observer to make sense of. 

To make progress, the standard Schrödinger equation with its hopeless wave-function must be replaced by another atomistic model which has an ontic meaning and is computable. Real Quantum Mechanics offers such a model. It may be the model Schrödinger was searching for when realising that his equation did not make sense and so in despair giving up and leaving quantum mechanics to Bohr and Heisenberg.