The Aharonov–Bohm effect teaches a striking lesson: a three-dimensional shield cannot keep a magnetic field away from an electron. Seal the field inside a superconductor, route the electron through field-free space, and it still registers the field as a phase shift. In IDWT the reason is that the electron is a six-dimensional object and the shield is a three-dimensional wall — the field reaches the electron through the dimensions the wall was never built in.
Once that lesson lands, it is tempting to expect the hidden dimensions to help everywhere. If a six-dimensional electron can meet a field a wall was built to block, surely it can also slip around a wall — tunnel through a barrier by detouring through the dimensions the barrier does not occupy. Quantum tunnelling even has a long-standing puzzle that would seem to invite the idea: the time a particle spends crossing a barrier comes out strangely short, and for a thick barrier it stops growing with thickness at all (the Hartman effect). A detour through hidden dimensions looks like an explanation waiting to happen.
It is not. The same principle that makes the shield fail makes the detour fail. Working out why draws a clean line around what the extra dimensions can and cannot do — and it is worth drawing, because the line marks where IDWT says something new and where it simply agrees with ordinary physics.
Uniform in the hidden directions
Start from the principle both cases share. A particle has structure only in its own sector dimensions; in every dimension beyond them it is uniform — present everywhere, at constant amplitude, pinned to no point. A lower-dimensional structure is therefore spread evenly across the hidden directions of every higher sector. The photon, a two-dimensional object, is uniformly present throughout the electron's six dimensions, not confined to the two it nominally occupies.
This is "bound within, free without," and it is the engine of the Aharonov–Bohm result. But it is even-handed: it applies just as much to a barrier as to a field. So before deciding the hidden dimensions offer a way around a wall, we have to ask what, exactly, is uniform in them — and what that uniformity does.
Why a shield fails
In the Aharonov–Bohm setup the electron moves freely; nothing is blocking its path. What the apparatus tries to do is keep a field's influence away from it. The shield removes the field in our three dimensions — that part succeeds, and the measured field along the path really is zero. What the shield cannot do is confine the field's influence to those three dimensions, because the influence was never a three-dimensional thing. It reaches the electron through the hidden coordinates the electron and the field hold in common.
So the hidden dimensions here carry an influence around the shield. A three-dimensional operation acts on the three-dimensional slice; it can erase the field there, but it cannot fence off a structure that extends past the slice. Three dimensions are enough to wipe a field locally, never enough to contain one. The Aharonov–Casher effect is the same story with charge and magnetic moment exchanged: a field-free region in three dimensions does not mean an influence-free path through six.
Why a wall does not
Tunnelling asks the hidden dimensions to do something different, and the difference is everything. A barrier is a region along one observable direction — call it \(z\) — where the electron's energy falls short of the potential. To get past it, the electron must advance in \(z\) through the forbidden stretch. The question is whether moving in the hidden directions can substitute for that.
It cannot, and IDWT's own geometry says so. A barrier is built from ordinary three-dimensional matter, and the potential it presents is set by three-dimensional sources. By the projection theorem that governs static interactions in IDWT, the potential an electron feels is a function of the observable position alone — it takes the same value at every hidden-coordinate value. The barrier is, in effect, the same wall at every depth into the hidden dimensions. Sliding along a hidden direction does not lower it and does not carry the electron forward in \(z\). The hidden directions are transverse to the wall: they run alongside it, not through it.
Write the problem out and the hidden coordinates simply separate off as free directions, decoupled from the barrier entirely. What remains is the ordinary one-dimensional tunnelling problem in \(z\) — the same one standard quantum mechanics solves. IDWT inherits its answer unchanged, including the short, thickness-independent crossing time of the Hartman effect. There is no extra-dimensional shortcut, and IDWT predicts no tunnelling time different from the standard one. On this experiment the framework simply agrees with what is already known.
The rule: hidden dimensions bypass shields, not walls
The two cases look similar — a three-dimensional apparatus, a six-dimensional electron, a hidden remainder — but they ask opposite questions. A shield tries to keep something out; a wall tries to keep something from getting across. The hidden dimensions defeat the first and are powerless against the second, for one reason: they are transverse. A transverse dimension can carry an influence around an obstacle that sits in the slice, because the influence does not need to travel along the blocked direction. It cannot carry a particle across an obstacle that stands in the direction the particle actually has to go, because moving sideways is not moving forward.
So the rule is short. The extra dimensions bypass shields, not walls. Whenever an experiment depends on confining a field or an influence inside a three-dimensional region, IDWT expects the confinement to leak — the influence reaches through the hidden coordinates, and something measurable survives. Whenever an experiment depends on crossing a barrier laid out along an observable direction, the hidden coordinates offer no help, and IDWT returns the ordinary answer.
What the line marks off
This is worth stating plainly because it tells you which puzzles the dimensional picture genuinely speaks to. The Aharonov–Bohm and Aharonov–Casher phases, the electron cloud as a three-dimensional shadow of a six-dimensional path, and entanglement as shared sector coordinates are all containment failures of one kind or another — cases where a three-dimensional view cannot fence in, localize, or separate a structure that extends further, and the leftover reaches us anyway. There the extra dimensions carry the explanation.
Tunnelling times are not on that list. The barrier is a wall along an observable direction, the hidden dimensions are transverse to it, and the framework reduces to the standard problem with nothing to add. Marking that boundary is not a concession; it is what keeps the rest honest. A picture that explained everything equally would explain nothing in particular. This one explains shields and declines walls, and the reason it declines is the same reason it explains — the hidden dimensions are real, the particle truly occupies them, and they run transverse to the directions we measure in. That single fact, read in two directions, is one wave seen from a slice that is sometimes the whole story and sometimes not.
Related reading
The Aharonov–Bohm Effect — the shield that cannot contain a field, the case this principle generalizes.
The Aharonov–Casher Effect — the same containment failure with charge and moment exchanged.
One Space, Six Depths — the nesting of sectors and the "free without" uniformity the rule rests on.
The Single-Electron Double Slit — a containment failure of a different kind: the cloud as a slice of a sharp path.
One Wave — the single-field picture underneath both the shields and the walls.