Dear Robert,

    There is a very simple form to understand it: the equations are coupled and have degrees of freedom. For example in Electrodynamics the conservation of sources is a new equation which introduces a degree of freedom for Maxwell's equations.

Thanks, but as my question states, I am looking for an understanding that is "completely conceptual and pictorial" in nature.

Therefore the answer must be dominated by a space-time framework in which real physical objects move in specific ways because of causal forces or geometrical necessities.

Quite possibly such an explanation is impossible, since much of the supposed action in gauge symmetry/invariance takes place in abstract "phase space(s)".

But I am hoping some sort of translation into a real, causal, space-time modeling is possible. If it is not, then that is an important piece of evidence.

RLO

(I, not-too-secretly, prefer global discrete conformal geometry/invariance since I think it can do more with less in the way of smoke and mirrors)

  If you think in something pictorial (geometrical?) the proper representation of the gauges is a principal fiber bundle where the fiber is the associated gauge group. The trivial topological representation of Electrodynamics can be U(1)xSO(3,1). But this picture is purely mathematical and very far of a simple trajectory in spacetime or something similar due that the gauge relates with the degrees of freedom of equations. In fact is more related with conservation laws than with dynamics.

Yes I think things are getting quite clear.

Another physicist at another cite answered the same question by saying that the most pictorial thing about gauge invarianve he knew of is to be found in Feynman's comment that gauge invariance in the simplest form can be pictured in terms of a rotating vector attached to a photon as it propagates.

Of course no physical thing like a vector is physically attached to a photon.  I readily admit that gauge invariance has given us the standard model which reproduces observed phenomena quite well, albeit with at least 6 fundamental problems that identify the SM as provisional. 

I think we can understand nature in a MUCH less abstract and much more natural manner by extending (or in Einstein's words: "deepening") General Relativity by expanding its symmetries in a sophisticated manner that removes absolute scale but not in the simplest naive way, which does not work.

Dear Robert L. Oldershaw:

     I suggest you to read this article, I hope which would be help you more! you are ask good question. I always ask my self, such as Einstein's words'deepening'. But i could not go further. I think we must have collaboration each other. Found different idea and philosophy which is 'deepening' our understanding for mother nature. Gauge invariance, gauge symmetry .......this kind of theoretical and practical concept must have proper explanation!