nLab density

Idea

A density on a manifold of dimension $n$ is a function that to each point assigns an infinitesimal volume (in general signed, and possibly degenerate), hence a volume of $n$-hypercubes in the tangent space at that point. A positive definite density is equivalently a volume element (or a volume form on an oriented manifold).

Definition

For $X$ a manifold its 1-density bundle is the real line bundle associated to the principal bundle underlying the tangent bundle by the 1-dimensional representation of the general linear group

$$\array{ GL(n) &\to& GL(1) \simeq Aut_{Vect}(\mathbb{R}^1) \\ A &\mapsto& |det(A)|^{-1}. }$$

A section of the $1$-density bundle on $X$ is called a $1$-density on $X$.

This is the general object against which one has integration of functions on $X$.

More generally, for $s \in \mathbb{R} - \{0\}$ an $s$-density is a section of the line bundle which is associated to the principal bundle by the representation

$$\array{ GL(n) &\to& GL(1) \simeq Aut_{Vect}(\mathbb{R}^1) \\ A &\mapsto& |\det(A)|^{-s}. }$$

The parameter $s$ is called the weight of the density. In particular for $s = 1/2$ one speaks of half-densities.

Properties and applications

Physical interpretation

We earlier spoke of a density (of weight $1$) $\rho$ as a measure of volume, but in application to physics a density on spacetime (or space) might as easily be a measure of some other extensive quantity $Q$ (say, mass). We then call $\rho$ the $Q$-density (say, mass density); the integral of $\rho$ over a region $R$ is the amount of $Q$ in $R$.

Relative to a nondegenerate notion of volume given by another density $vol$, the ratio $\rho/vol$ is a scalar field, an intensive quantity which is often also referred to as the density. But $\rho$ itself is more fundamental in the geometry of physics.

Wave functions and canonical Hilbert spaces

In the context of geometric quantization one considers spaces of sections of line bundles (“prequantum line bundles”) and tries to equip these with an inner product given by pointwise pairing followed by integration over the base such as to then complete to a Hilbert space.

One can define the integration against a fixed chosen measure, but more canonical is to instead form the tensor product of the prequantum line bundle with the bundle of half-densities. The compactly supported sections of that tensor bundle can then naturally be integrated. This is sometimes called the “canonical Hilbert space” construction (e.g. (Bates-Weinstein)).

Related concepts

• volume form

• canonical bundle, determinant line bundle

• pseudoscalar field

• canonical Hilbert-space of half-densities

• Verdier duality

The following table lists classes of examples of square roots of line bundles

line bundle	square root	choice corresponds to
canonical bundle	Theta characteristic	over Riemann surface and Hermitian manifold (e.g.Kähler manifold): spin structure
density bundle	half-density bundle
canonical bundle of Lagrangian submanifold	metalinear structure	metaplectic correction
determinant line bundle	Pfaffian line bundle
quadratic secondary intersection pairing	partition function of self-dual higher gauge theory	integral Wu structure

References

A textbook account is for instance on p. 29 of

• Nicole Berline, Ezra Getzler, Michele Vergne, Heat kernels and Dirac operators, Springer (2004)

Discussion of half-densities in the context of geometric quantization is in

• Sean Bates, Alan Weinstein, Lectures on the geometry of quantization, pdf