# tensor density

## 0.1 Heuristic definition

A tensor density is a quantity whose transformation law under change of basis involves the determinant of the transformation matrix (as opposed to a tensor, whose transformation law does not involve the determinant).

## 0.2 Linear Theory

For any real number $p$, we may define a representation $\rho_{p}$ of the group $GL(\mathbb{R}^{k})$ on the vector space of tensor arrays of rank $m,n$ as follows:

 $(\rho_{p}(M)T)^{i_{1},\ldots,i_{n}}_{j_{1},\ldots j_{m}}=(\mathop{\rm det}(M))% ^{p}M^{i_{1}}_{l_{1}}\cdots M^{i_{n}}_{l_{n}}(M^{-1})_{k_{1}}^{j_{1}}\cdots(M^% {-1})_{k_{m}}^{j_{m}}T^{i_{1},\ldots,i_{n}}_{j_{1},\ldots j_{m}}$

A tensor density $T$ of rank $m,n$ and weight $p$ is an element of the vector space on which this representation acts.

Note that if the weight equals zero, the concept of tensor density reduces to that of a tensor.

## 0.3 Examples

The simplest example of such a quantity is a scalar density. Under a change of basis $y^{i}=M^{i}_{j}x^{j}$, a scalar density transforms as follows:

 $\rho_{p}(S)=(\mathop{\rm det}(M))^{p}S$

An important example of a tensor density is the Levi-Civita permutation symbol. It is a density of weight $1$ because, under a change of coordinates,

 $(\rho_{1}\epsilon)_{j_{1},\ldots j_{m}}=(\mathop{\rm det}(M))(M^{-1})_{k_{1}}^% {j_{1}}\cdots(M^{-1})_{k_{m}}^{j_{m}}\epsilon^{i_{1},\ldots,i_{n}}_{j_{1},% \ldots j_{m}}=\epsilon_{k_{1},\ldots k_{m}}$

## 0.4 Tensor Densities on Manifolds

As with tensors, it is possible to define tensor density fields on manifolds. On each coordinate neighborhood, the density field is given by a tensor array of functions. When two neighborhoods overlap, the tensor arrays are related by the change of variable formula

 $T^{i_{1},\ldots,i_{n}}_{j_{1},\ldots j_{m}}(x)=(\mathop{\rm det}(M))^{p}M^{i_{% 1}}_{l_{1}}\cdots M^{i_{n}}_{l_{n}}(M^{-1})_{k_{1}}^{j_{1}}\cdots(M^{-1})_{k_{% m}}^{j_{m}}T^{i_{1},\ldots,i_{n}}_{j_{1},\ldots j_{m}}(y)$

where $M$ is the Jacobian matrix of the change of variables.

Title tensor density TensorDensity 2013-03-22 14:55:18 2013-03-22 14:55:18 rspuzio (6075) rspuzio (6075) 12 rspuzio (6075) Definition msc 15A72 density tensor