# nLab density

Contents

### Context

#### Bundles

bundles

fiber bundles in physics

## Constructions

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## 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 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 given by the determinant homomorphism (sign representation):

$det \;\colon\; GL(n) \to GL(1) \simeq Aut_{Vect}(\mathbb{R}^1) \,.$

A section of the density bundle on $X$ is called a density on $X$ (in physics also: a pseudoscalar).

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 via the determinant to the power of $s$:

$det^{s} \;\colon \; GL(n) \to GL(1) \simeq Aut_{Vect}(\mathbb{R}^1) \,.$

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)).

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

## References

A textbook account is for instance on p. 29 of

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

Last revised on March 20, 2020 at 13:27:50. See the history of this page for a list of all contributions to it.