nLab pseudoform

Idea

A pseudoform is a differential form that has been twisted by the pseudoscalar bundle.

Concretely, this means that when written in two (local) systems of coordinates, the form transforms by an additional factor of $-1$ when the coordinate systems have opposite orientations. For a top-dimension pseudoform, this has the effect that at any given point, the form has a consistent sign regardless of the orientation of the coordinates.

An $n$-pseudoform is the fundamental object of integration on an $n$-manifold. In particular the volume (or area, or length) is a pseudoform.

Definition

Definition

A pseudoform on a manifold $X$ is a section of the tensor product $\Lambda \otimes \Psi$, where

• $\Lambda$ is the exterior algebra of the cotangent bundle on $X$, and

• $\Psi$ is the pseudoscalar bundle on $X$.

By comparison, an untwisted differential form is a section of the exterior algebra $\Lambda$ itself.

Properties

See also discussion under “Twisted and vector-valued forms” at differential form.

Correspondence to untwisted forms given orientation

For a pseudoform $\omega$ on an oriented manifold with orientation $o$, multiplying $\omega$ by the pseudoscalar field $1_o$ whose value at orientation $o$ is everywhere $1$ gives an untwisted form $\omega_o = 1_o \wedge \omega$. Similarly an untwisted form $\alpha$ corresponds to a pseudoform $1_o \wedge \alpha$.

As a result, on an orientable manifold, any pseudoform $\omega$ corresponds to two untwisted forms, one for each orientation, which differ only in sign. Conversely, on an orientable manifold any untwisted form corresponds to two pseudoforms differing only in sign. Even when the manifold is nonorientable, this correspondence holds locally because each point has an orientable neighborhood.

When working on a manifold with a chosen orientation, many authors conflate a pseudoform $\omega$ with the corresponding untwisted form $\omega_o$ and vice versa.

Integration

When integrating differential forms on an $n$-manifold, the most fundamental is the integration of $n$-pseudoforms.

This corresponds to how in multivariable calculus, the formula for change of variables has a factor of the absolute value of the determinant (just like for an $n$-pseudoform), and not of the determinant itself (as one would have for an untwisted $n$-form).

With an orientation, one can of course also integrate (untwisted) $n$-forms; this can be seen as using the orientation to convert the $n$-form to an $n$-pseudoform and integrating that.

More generally one can integrate a $p$-pseudoform on an $n$-manifold $X$ if one has a $p$-submanifold $U$ to integrate over and a pseudoorientation of $U$ in $X$. See integration of differential forms.

Examples

Top-dimensional examples

The volume (or area, or length) on an $n$-manifold is an $n$-pseudoform. On an oriented manifold one often speaks of a volume form as an untwisted $n$-form.

The absolute value of an untwisted $n$-form (on an $n$-manifold) is an $n$-pseudoform, as is the absolute value of an $n$-pseudoform.

Alternatively, both of these examples can be thought of as absolute differential forms. On an $n$-manifold, an absolute $n$-form is equivalent to an $n$-pseudoform.

Electromagnetism

In electromagnetism, the current form $j$, measuring flux of charge, is a 2-pseudoform. This can be seen by considering how one integrates it: you need a 2-surface together with a notion of which way through the surface is forward and which is backward, or in other words a pseudooriented surface.

By contrast the magnetic field strength $B$ is an untwisted 2-form. To integrate $B$ over a surface $U$, one needs not a pseudoorientation but an orientation of $U$ itself, identifying a direction of circulation within the surface rather than a transverse direction through the surface.

Traditionally in electromagnetism these look very similar because one freely converts an orientation to a pseudoorientation and vice versa. The tell when one does so is that it relies on an arbitrary choice of orientation of the ambient 3-dimensional space, namely the right-hand rule?. Comparing the right-hand rule to a left-hand rule, one gets the same physical meaning by integrating $j$ over the same surface with the same pseudoorientation (hence opposite orientations), confirming that the pseudoorientation is primary and $j$ is properly a pseudoform. With $B$, one needs the same orientation on the surface (hence opposite pseudoorientations), confirming that the orientation is primary and $B$ is an untwisted form.

References

Many useful explanations by Toby Bartels and John Baez in this long Usenet thread. In particular:

• Toby on absolute values, and getting a volume from a metric, 2001-12-23

• Toby on examples of pseudo- and untwisted forms in electromagnetism, 2002-02-08