A map from lists to list-elements is called "alternating" if any list with repeated elements is mapped to zero.
This has statistical significance: regressions on collinear data are bad, dependent columns add no information, etc.
But alternating maps like determinants and wedge products show up in physics as well. Is there an intuition to go along with the skew symmetry and alternatingness in that domain?
On
The oriented volume spanned by three vectors is a multilinear alternating map of the vectors themselves. That's the starting point.
See, e.g., the book by S.Winitzki.
The alternating property often implies anti-symmetry (=the symmetry of "odd" functions = the opposite of equal).
Anti-symmetry has an intuitive physical explanation.
Consider the axle of a car driving forward. From the passenger’s side, the wheels are spinning "to the right" (clockwise). From the driver’s side, the wheels are spinning "to the left" (anticlockwise). Since they are on the same stiff axle, the wheels are moving the same direction.
Likewise if you draw a ↺ on a glass pane, then it only looks like ↺ from the ink side. If you look through the glass from the non-ink side, the same figure would read “↻”.
electron-scale (picometer) physical interpretation: "spin"
David Hestenes spent his career arguing that the wedge product (which is alternating) would greatly clarify the common understanding of physics—in particular, that quantum mysticism would vanish. Hestenes says the Pauli matrices make spin harder to see whereas an alternating product ∧ make it obvious. Am I looking at a spinning top from above or from below?
ℂ comes with a handedness (the choice of ±i) which becomes the opposite choice after conjugation. Some properties you would learn in an undergraduate ℂ analysis class
Since physics teachers don't teach an alternating product ∧ for a rotating shaft, and don't explain ℂ in very concrete terms either, the world had to suffer What the Bleep Do We Know?!. What the bleep we don't know is ∧, even though it's visible in every spinning object or can be drawn with a dry erase marker on glass.
see page 26 of his Oersted Medal lecture, "Reforming the Language of Physics".
It's not just that movie, though. Raise your hand if you have read at least one quantum book and felt confused by ψ, ψ*, or ψψ* (or ψ*ψ!). Raise your hand if you weren't sure why we needed ℂ or probability or both together, or if you thought a collapsing wave sounded super sci-fi awesome. That was my reaction to many physics books written by full-time academic philosophers and physicists, including textbooks.
This scale of matter (.001 nanometers = one millionth of a micron) is far smaller than what might count as a "physical example" if you don't have access to the lab equipment to perceive electrons, but apparently the human scale of metres has some examples of the alternating property that are easily within reach. Simply find a rotating shaft or draw ↺ on a pane of glass.