Mathematically for every 3D pseudovector $x^i$ there is a 2-form $F_{ij}=\epsilon_{ijk}x^k$ such that the 2-form transforms properly under all orthogonal transformations. Therefore I would expect it would be more natural to write physical quantities such as angular momentum $\textbf{L}$ or magnetic field $\textbf{B}$ in terms of their corresponding 2-forms.

Is there any physical insight as to why these quantities behave the way they do apart from experimental verification. If it is simply the way they are, is there any insightful interpretation of their corresponding 2-forms? I seem to be able to get some intuition from looking at the vectors but none at all by analysing the 2-forms.

Is the way these vectors physically behave related to their pseudoness? For example the rather odd direction of magnetic force.


1 Answer 1


Angular momentum is a very instructive example to look at - in particular, to look at how the notion of angular momentum (or, in fact, rotation), changes when you consider more or fewer than the usual three spatial dimensions. The proper notion of angular momentum that generalizes to all dimensions is $L = \vec r\wedge \vec p$, i.e. a 2-form, the wedge product of position and momentum. In three dimensions, the Hodge dual of this form is the ordinary pseudovector of angular momentum, and in fact one might define the cross product in 3d as the Hodge dual of the wedge product. You should think of the 2-form $L$ as describing the plane in which the rotation happens, together with some numbers encoding its direction and speed.

Let's start in one dimension: There is no rotation for an object in one dimension - it can only move forward and backward, and nowhere else. This corresponds to the second exterior power of a one-dimensional vector space being zero - there are no 2-forms, hence no rotation.

In two dimensions, there is clearly rotation, imagine a one-dimensional "rod" spinning in a plane. You might be tempted to describe the rotation as the 3d vector of angular momentum perpendicular to the plane, but this is an extrinstic description. If the world were truly two-dimensional, this description would not be available - but the description by two-forms is available. 2-forms in 2 dimensions are dual to 0-forms, i.e. scalars, so rotation in a fixed plane is fully described not by a vector, but by a number - its magnitude tells you how fast the rotation is and the sign whether it is clockwise or counter-clockwise.

In three dimensions, we get the familiar duality between 2-forms and 1-forms/vectors. But note that there is really nothing about rotation that would force you to describe it as "rotation about an axis" rather than "rotation in a plane" - the two descriptions/interpretations are fully dual, and it is the latter that generalizes to all dimensions.

In four dimensions...well, I get that this is not visual anymore, but think about special relativity, and the Lorentz transformations, which are generalized rotations - they are generated by 2-tensors, not vectors, and their associated conserved quantity is a 2-tensor, the energy-momentum tensor, not a vector.

Note that I have nowhere relied on the "pseudoness" of the angular momentum vector in 3d. It's an artifact of the Hodge dual not commuting with reflections, but it's really not the defining property of a "pseudovector". A "pseudovector" is not a vector at all, it is intrinsically a 2-form, and especially when you generalize to other dimensions you must respect that, as I also pointed out here.

That the magnetic force is a pseudovector and not a vector is something you can only appreciate after switching to the covariant formulation of electromagnetism and recognizing the electric and magnetic fields as certain parts of the electromagnetic field strength tensor - and once again, you will find that going to other dimensions shows that the magnetic field is not fundamentally a "vector" at all - in particular, it is equivalent to a scalar in 2 spatial dimensions, and to a more complicated $d-2$-form in higher dimensions. You might think of the magnetic 2-form as a collection of planes the velocities of charged particles are "dragged along" in the sense that the more parallel their velocity is to these planes, the stronger is the magnetic force that tries to make them describe circles in those planes:

Consider that the vector $\vec B$ is perpendicular to the planes its Hodge dual 2-form describes and that the Lorentz force is $\vec v\times \vec B$. This is maximal when $\vec v$ and $\vec B$ are perpendicular, i.e. when $\vec v$ lies in the plane the dual encodes, and the Lorentz force is also also perpendicular to $\vec B$, so it always lies in that plane.

  • $\begingroup$ Naive question, but how can the exterior product of two vectors $L=\vec{r}\wedge\vec{p}$ be a 2-form? I thought a 2-form had to contains things like (in Cartesian coordinates) $dy\wedge dz$, $dz\wedge dx$ and $dx\wedge dy$. $\endgroup$
    – Peter4075
    Feb 3, 2020 at 8:47
  • 1
    $\begingroup$ @Peter4075 $L = \vec r \wedge \vec p$ was a bad choice of notation on my part. What I really meant was to take the wedge of the duals of the vector fields $\vec r(\vec r)$ and $\vec p(\vec r)$ (i.e. $x\mathrm{d}x + y \mathrm{d}y + z \mathrm{d}z$ and $p_x \mathrm{d}x + p_y \mathrm{d}y + p_z \mathrm{d}z$). $\endgroup$
    – ACuriousMind
    Feb 3, 2020 at 17:46

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