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tensor density (Definition)

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).

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:

$\displaystyle (\rho_p (M) T)^{i_1, \ldots, i_n}_{j_1, \ldots j_m} = (\mathop{\r... ..._{k_1}^{j_1} \cdots (M^{-1})_{k_m}^{j_m} T^{i_1, \ldots, i_n}_{j_1, \ldots j_m}$

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.

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:

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

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,

$\displaystyle (\rho_1 \epsilon)_{j_1, \ldots j_m} = (\mathop{\rm det}(M)) (M^{-... ...j_m} \epsilon^{i_1, \ldots, i_n}_{j_1, \ldots j_m} = \epsilon_{k_1, \ldots k_m}$

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

$\displaystyle T^{i_1, \ldots, i_n}_{j_1, \ldots j_m} (x) = (\mathop{\rm det}(M)... ...1}^{j_1} \cdots (M^{-1})_{k_m}^{j_m} T^{i_1, \ldots, i_n}_{j_1, \ldots j_m} (y)$
where $ M$ is the Jacobian matrix of the change of variables.



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See Also: tensor

Other names:  density
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Cross-references: Jacobian matrix, variable, functions, neighborhood, coordinate, manifolds, fields, Levi-Civita permutation symbol, Transforms, scalar, weight, rank, tensor arrays, vector space, group, representation, real number, tensor, matrix, determinant, change of basis, transformation
There are 7 references to this entry.

This is version 9 of tensor density, born on 2005-01-01, modified 2005-02-18.
Object id is 6608, canonical name is TensorDensity.
Accessed 4364 times total.

Classification:
AMS MSC: 15A72 (Linear and multilinear algebra; matrix theory :: Vector and tensor algebra, theory of invariants)
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