Let two particles P1 and P2 of masses M1 and M2 be racing towards each other along a connecting radius, under the effect of their mutual gravitational interaction. Take the center of gravity of the system as its rest frame.
Theoretically , if the particles have no extension (Translation: electro-magnetic or nuclear forces that oppose the collision by an effective radius around each particle) their velocities at the moment of collision will be infinite.
What this implies is that one can define a massive particle as a singularity in the gravitational potential at some point of space-time. This is rendered explicit in the mathematical theory of Black Holes. The phenomenon of space-time curvature in General Relativity then becomes the study of the interactions generated by singularities in the global gravitational field of the universe, the so-called "Large Scale Structure of Space-Time". This approach is in line with Roger Penrose's claim that one can derive the universal metric entirely from the topological character of the singular points on the boundary of space-time. (cf: The Road to Reality and other writings).
To give an example: physicists working at the LHC hope to find evidence for the Higgs Boson by smashing together beams of protons accelerating at incredible speeds, then examining the fall-out of this man-made catastrophe. One presumes that they will restrict their searches to a tiny region of Geneva, Switzerland. For some reason one can be confident that they will not feel it necessary to examine the contents of the can of ginger ale that I have stored in my refrigerator. This is because the expectation of finding the Boson locally is so much higher in Geneva than that of finding it anywhere else!
According to the Schrödinger- Neumann hypothesis of the "collapse of the wave packet", the probability, or norm of the Schrödinger wave function, of finding a particle in a given location, jumps to 1 the moment it is found, and shrinks to 0 everywhere else in the universe.
However, the probability of locating a particle before it is found is distributed around the entire universe. Given that the Schrödinger norm is an everywhere positive function over all of space, it has an absolute maximum at one or more locations. That maximum is a critical point of the probability distribution. Therefore one can define a "particle" as this critical point, (or, in the presence of entanglement, a collection of critical points).
One does not need to worry about the absolute minima. Because the global probability is 1, the value of the norm must rapidly go to 0 as one travels out to the far reaches of the universe.
Seen from this vantage, the barriers separating General Relativity from Quantum Theory can be reduced to the way one defines an particle (more generally any system in isolation, invariant under local forces.) The question then becomes: should one picture a particle as a singularity (gravitational field) or as a critical point of the Schrödinger probability distribution? This apposition may lie at the root of the gulf separating the two great theories of physics. If my ideas do indeed open up a new way to look at Quantum Gravity, the Nobel Committee in Stockholm can send the check to my stock-broker.
Apropos: at a critical point the derivative dy/dx = 0. At a singularity dy/dx = ∞. Does this point to some deep principle of dualism? Who knows? Only the phantom (wave) knows! Enough said...