I wish I could connect these n-transports to something having a concrete physical meaning! For years ago I tried to understand n-parallel transport (or perhaps it was something related;-)) in terms of simple geometric mental images. I try to formulate my mis-understandings using the ancient terminology still used by physicists like me and these mental images. No arrows nor commuting diagrams which make me mad!

n=1: One starts with a parallel transport from point a to b along curve C_1(a,b). 1-parallel transport defines a map between fibers.

n=2: 1-parallel transport along C_1(a,b) is parallelly transported to a 1-parallel transport along curve C_1(c,d). One can say that one parallelly transports curve instead of point. 2-connection would define this parallel transport of parallel transport. One obtains a kind of square like structure C_2(a,b|c,d).

n>2: One can continue this and obtains at n:th level parallel transport of parallel transport of…..

Some comments.

a) The ordered exponential representation for parallel transport suggests that n=1 parallel transport could define n-parallel transport. Probably something trivial and un-interesting.

b) If the n-connection is non-flat, the n-parallel transport depends on how the curve evolves from the initial state to the final state.

c) A physically highly attractive possibility is generalized general coordinate invariance stating that the parallel transport depends only on the n-surface spanned by the curve. Is n-parallel transport induced by 1-parallel transport the only solution to this requirement?

d) One can wonder about the counterparts of geodesic lines. 1-parallel transport leaves the tangent vector field of geodesic line invariant. n-parallel transport should leave invariant the n-form defining tangent spaces of a geodesic n-surface? For n-parallel transport induced by 1-parallel transport geodesic sub-manifolds would probably result. What is the n-counterpart for the equations of geodesic line? Could one model the behaviour of extended objects in gravitational fields using these kind of equations?

e) One could also generalize the notion of holonomy group. 2-holonomy group would be associated with cylinder-like surfaces C_2(a,b|a,b) with topology DxS^1. At higher levels you would have topology DxS^1xS^1 and so on. You could also consider closed curve at n=1 level and get hierarchy of n-holonomy groups associated with n-tori. Of course also other topologies can result if the parallel transport is such that the surface develops pinches. Could one generalize the notion so that one cold assign say 2-parallel transport to a 2-torus. What to do when the curve for 1-parallel transport decomposes into two separate pieces? Just hop? Why not?

For years ago I assigned this kind of hierarchical structure of parallel transports to a hierarchical structure defined by infinite primes. I believe that this kind of abstractions about abstractions about…, thoughts about thoughts about… , statements about statements about… and repeated second quantization, represent fundamental new physics especially relevant for quantum consciousness theories.

Cheers,

Matti

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