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tensor density

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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 ppp, we may define a representation ρpsubscriptρp\rho_{p} of the group G⁢L⁢(ℝk)GLsuperscriptℝkGL(\mathbb{R}^{k}) on the vector space of tensor arrays of rank m,nmnm,n as follows:

(ρp⁢(M)⁢T)j1,…⁢jmi1,…,in=(det(M))p⁢Ml1i1⁢⋯⁢Mlnin⁢(M-1)k1j1⁢⋯⁢(M-1)kmjm⁢Tj1,…⁢jmi1,…,insubscriptsuperscriptsubscriptρpMTsubscripti1normal-…subscriptinsubscriptj1normal-…subscriptjmsuperscriptdetMpsubscriptsuperscriptMsubscripti1subscriptl1normal-⋯subscriptsuperscriptMsubscriptinsubscriptlnsuperscriptsubscriptsuperscriptM1subscriptk1subscriptj1normal-⋯superscriptsubscriptsuperscriptM1subscriptkmsubscriptjmsubscriptsuperscriptTsubscripti1normal-…subscriptinsubscriptj1normal-…subscriptjm(\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 TTT of rank m,nmnm,n and weight ppp 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 yi=Mji⁢xjsuperscriptyisubscriptsuperscriptMijsuperscriptxjy^{i}=M^{i}_{j}x^{j}, a scalar density transforms as follows:

ρp⁢(S)=(det(M))p⁢SsubscriptρpSsuperscriptdetMpS\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 111 because, under a change of coordinates,

(ρ1⁢ϵ)j1,…⁢jm=(det(M))⁢(M-1)k1j1⁢⋯⁢(M-1)kmjm⁢ϵj1,…⁢jmi1,…,in=ϵk1,…⁢kmsubscriptsubscriptρ1ϵsubscriptj1normal-…subscriptjmdetMsuperscriptsubscriptsuperscriptM1subscriptk1subscriptj1normal-⋯superscriptsubscriptsuperscriptM1subscriptkmsubscriptjmsubscriptsuperscriptϵsubscripti1normal-…subscriptinsubscriptj1normal-…subscriptjmsubscriptϵsubscriptk1normal-…subscriptkm(\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

Tj1,…⁢jmi1,…,in⁢(x)=(det(M))p⁢Ml1i1⁢⋯⁢Mlnin⁢(M-1)k1j1⁢⋯⁢(M-1)kmjm⁢Tj1,…⁢jmi1,…,in⁢(y)subscriptsuperscriptTsubscripti1normal-…subscriptinsubscriptj1normal-…subscriptjmxsuperscriptdetMpsubscriptsuperscriptMsubscripti1subscriptl1normal-⋯subscriptsuperscriptMsubscriptinsubscriptlnsuperscriptsubscriptsuperscriptM1subscriptk1subscriptj1normal-⋯superscriptsubscriptsuperscriptM1subscriptkmsubscriptjmsubscriptsuperscriptTsubscripti1normal-…subscriptinsubscriptj1normal-…subscriptjmyT^{{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 MMM is the Jacobian matrix of the change of variables.

Related: 
tensor
Synonym: 
density
Type of Math Object: 
Definition
Major Section: 
Reference
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Mathematics Subject Classification

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Owner: rspuzio
Added: 2005-01-02 - 00:52
Author(s): rspuzio

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(v12) by rspuzio 2013-03-22
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