This one was rejected as too speculative for sci.physics.research.
Hi, gang!
For two particle systems, the application of quantum mechanics and a change of variable allow the separation of the problem into "one concerning only the centre of mass of the system, and another which describes the behavior of a particle of mass mu under a potential V(r)." (Alistair I. M. Rae, Quantum Mechanics, John Wiley and Sons, New York, 1981, p. 189.
If you have a small machine shop with two lathes, two mills, two surface grinders, two cylindrical grinders, and two of every other machine tool needed, and duplicate tooling, than taken as a system of 2v machine tools, the system is capable of self-replication. (The foundry is a separate thing. Don't worry about it.)
This does not contradict the finding of Wigner in "On the impossibility of self-replication" in "The Logic of Personal Knowledge" because the machinist, an agent not included in Wigner's analysis of structures growing in a nutrient "sea", is self-replicating (alive).
I assert that a properly trained machinist inherently knows how to operate such an array to self-replicate, given time, because the machinist is a living, self-replicating being, but special training in the theory of self-replication may help. It may take generations to acheive it if it is done one machine part at a time, but a theoretical solution might be achieved in one machinist's lifetime, and a computer calculation might be a matrix operation that would complete in seconds, or days. Once stated, the theoretical basis can be taught, in context, to students at the appropriate level of instruction in mere minutes.
v is finite and may be 2, for a small shop, or up to around 7.
If n is 1, we have a pair of self-replicating machine tools and then can consider a growing population of them. This idea of growth doesn't work in an array very well because it's constrained to pairs of machine tools. Multiple pairs of machines. It's rather over constrained. In particular, cross pairings start to get all, well, complicated.
If we start with an large enough array of pairs of machine tools ( a fully equipped shop) then the array is "universal", able to construct any product of industry, and in theory, can be reduced to a single pair of identical, universal self-replicating machine tools: the Holy Grail of Mechanical Engineering.
Goncz's Postulate is : "You Need Two of Everything"
If and only if you start with a pair of universal self-replicating machine tools, then each tool in the growing population is indistinguishible from (functionally identical to) its fellow, so every possible pairing in a population is a valid pairing in which one machine may reproduce a part of the other and there are no cross pairings to get in the way. In other words, the population gets busy, starts growing faster, and we get more and more of the little devils. And then exclusion principles, entanglement, and other interesting properties will probably start showing up.
If we can accomplish this, the cost of guns, if not butter, should fall, producing new wealth for all to share.
For a system of two particles with position vectors r1 and r2, and with mass m1= m2, we form the center of mass of the system, bold R, and the relative position bold r:
bold R = ( m1*r1 + m2*r2 ) / ( m1 + m2 ) and bold r = r1 - r2
The center of mass of a circular machine tool array in full assembly is fixed, the position vector magnitudes are constant, but the mass of each machine tool is distinct, and it may vary as one only of each pair is disassembled to relase an internal part for replication by the array.
So the wave function of this system will in general be a function of the masses of the particles. That is, if a machine tool's current mass is m.r, and its fully assembled mass is m.t, then m.r