What does over determination mean?

[pellman]

What does "over determination" mean?

I have come across instances of the phrase "over determination" in discussing Einstein's attempt at a unified field theory.

For example:

"Again and again, Einstein returned to the question of whether one could obtain the quantum of action from an over determination of the field variables."
-- Jagdish Mehra, "Einstein, Hilbert and the Theory of Gravitation"

 

[Halliday]

pellman:

I'm not certain what the quote you gave necessarily means, exactly. However, in general, "over determination", or an "over determined" system is where one has more equations to satisfy than unknowns to be determined. (From a linear perspective, at least. Of course, non-linear systems can have cases where one has exactly the same number of equations and unknowns, while still not having solutions---at least not solutions within the prescribed space, such as real numbers, etc.)

In this case, one has the very real possibility that the system is not solvable (no solutions exist). Another possibility is that the system of equations is redundant, in the sense that the system can actually be reduced to a simpler system that more closely matches the number of unknowns to be determined.

However, in some cases, the system of equations may not be reducible to be commensurate with the number of unknowns, and, yet, solutions do exist. (Incidentally, this can only occur in non-linear systems.) This is where one ends up having to look to prove the existence of solutions, before wasting time trying to solve a potentially unsolvable system. (Einstein was faced with this issue when he developed General Relativity.)

One potential application of such a concept would be to nail down the mass spectrum of fundamental particles with an over determined system. Since the present Standard Model can accommodate arbitrary masses, because the particle masses are simply parameters of the model, one could envision having some kind of overdetermined system that would only yield solutions for a specific mass spectrum. (Hopefully one that matches our observations, of course.)

 

[pellman]

Thanks, Halliday.