Nope: Precursor Field With a Background State Has to be Discontinuous

In the last post, I had come to the conclusion that the proposed R3 + I precursor field that would give rise to the particle zoo and EM and other fields could have twists and not have a discontinuity. This posed a problem, however, since quantization of a unitary twist field depends on the twist not being able to dissipate–that the discontinuity provides a “lock” that ensures particle stability over time. Further study has revealed that the extra I dimension does NOT topologically enable a continuous field that could contain twists.

The proof is simple. If the two ends of the twist are bound to the background state, but there is a field twist in between, it must be possible to create some other path connected to the endpoints that does not have a twist, since the background state must completely surround the twist path–see the diagram below. But this is impossible, because in a continuous system it must be topologically possible to move the paths close to each other such that an epsilon volume contains both paths yet has no discontinuities. Since this field is unitary and orientable (I like to use the car seat cover analogy, which is a plane of twistable balls for infinitesimal field elements), there is no “zero” magnitude possible. Somewhere in the epsilon volume there must be a region where the field orientations show a cut analogous to a contour integral cut.

It doesn’t matter how many dimensions the field has, if I’ve thought this through correctly, twists always require a discontinuity in a unitary orientable vector field.

This is a relief in most ways–otherwise this whole scheme falls apart if twists can dissipate. The only way a twist can unravel is in a collision with another twist of the opposite spin or some other similar geometrical construct.




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