the Kronecker delta symmetric tensor - Maple Help

Physics[KroneckerDelta] - the Kronecker delta symmetric tensor

 Calling Sequence KroneckerDelta[$\mathrm{mu},\mathrm{nu}$]

Parameters

 mu, nu - the indices, as names representing integer numbers or the numbers themselves

Description

 • KroneckerDelta[mu, nu], displayed as ${\mathrm{\delta }}_{\mathrm{\mu ,\nu }}$, is a computational representation for the KroneckerDelta tensor. When its indices assume integer values it returns 1 or 0, according to whether the indices are equal or different. The values 0 and 4, or for the case any dimension set for the spacetime, represent the same object. When its indices have a symbolic value, say mu, and are equal, it returns the dimension of spacetime if the index is a spacetime index, or the dimension of space if the index is a space index, or 1 otherwise.
 • Computations performed with the Physics package commands take into account Einstein's sum rule for repeated indices - see . and Simplify. The distinction between covariant and contravariant indices in the input of tensors is done by prefixing contravariant ones with ~, say as in ~mu; in the output, contravariant indices are displayed as superscripts. For contracted indices, you can enter them one covariant and one contravariant. Note however that - provided that the spacetime metric is galilean (Euclidean or Minkowski), or the object is a tensor also in curvilinear coordinates - this distinction in the input is not relevant, and so contracted indices can be entered as both covariant or both contravariant, in which case they will be automatically rewritten as one covariant and one contravariant. Tensors can have spacetime and space indices at the same time. To change the type of letter used to represent spacetime or space indices see Setup.
 • The %KroneckerDelta command is the inert form of KroneckerDelta, so it represents the same mathematical operation but without performing it. To perform the operation, use value.
 • Some automatic checking and normalization are carried out each time you enter KroneckerDelta[...]. The checking is concerned with possible syntax errors. The automatic normalization takes into account the symmetry of KroneckerDelta[mu,nu] with respect to interchanging the positions of the indices mu and nu.

Examples

 > $\mathrm{with}\left(\mathrm{Physics}\right):$
 > $\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true}\right)$
 $\left[{\mathrm{mathematicalnotation}}{=}{\mathrm{true}}\right]$ (1)

The default spacetime of the Physics package is a Minkowski spacetime with dimension 4, so the values 0 and 4 represent the same object

 > ${\mathrm{KroneckerDelta}}_{0,0}={\mathrm{KroneckerDelta}}_{4,4}$
 ${1}{=}{1}$ (2)
 > ${\mathrm{KroneckerDelta}}_{0,3}$
 ${0}$ (3)

KroneckerDelta can also be used with values of its indices that are beyond the spacetime dimension - and are different KroneckerDelta returns zero.

 > ${\mathrm{KroneckerDelta}}_{5,6}$
 ${0}$ (4)
 > ${\mathrm{KroneckerDelta}}_{6,6}$
 ${1}$ (5)

The symmetry property of KroneckerDelta is automatically taken into account when the indices have symbolic values

 > ${\mathrm{KroneckerDelta}}_{\mathrm{μ},\mathrm{ν}}-{\mathrm{KroneckerDelta}}_{\mathrm{ν},\mathrm{μ}}$
 ${0}$ (6)

By default, spacetime indices are represented by greek letters and the dimension of spacetime is 4 - you can query about that via

 > $\mathrm{Setup}\left(\mathrm{spacetimeindices},\mathrm{dimension}\right)$
 $\left[{\mathrm{dimension}}{=}{4}{,}{\mathrm{spacetimeindices}}{=}{\mathrm{greek}}\right]$ (7)

Set the space indices to be represented by lowercaselatin letters till h

 > $\mathrm{Setup}\left(\mathrm{spaceindices}=\mathrm{lowercaselatin_ah}\right)$
 $\left[{\mathrm{spaceindices}}{=}{\mathrm{lowercaselatin_ah}}\right]$ (8)

The trace of KroneckerDelta is equal to the dimension; note the use of the sum rule for repeated indices, both for spacetime and space only indices

 > ${\mathrm{KroneckerDelta}}_{\mathrm{μ},\mathrm{ν}}$
 ${{\mathrm{\delta }}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}$ (9)
 > $\genfrac{}{}{0}{}{\phantom{\mathrm{μ}=\mathrm{ν}}}{}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{}}{\mathrm{μ}=\mathrm{ν}}$
 ${4}$ (10)
 > ${\mathrm{KroneckerDelta}}_{a,b}$
 ${{\mathrm{\delta }}}_{{a}{,}{b}}$ (11)
 > $\genfrac{}{}{0}{}{\phantom{a=b}}{}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{}}{a=b}$
 ${3}$ (12)

When the indices are not of spacetime or space kind (typical use in quantum mechanics Bra/Ket notation)

 > ${\mathrm{KroneckerDelta}}_{i,i}$
 ${1}$ (13)
 > ${\mathrm{KroneckerDelta}}_{A,B}$
 ${{\mathrm{\delta }}}_{{A}{,}{B}}$ (14)
 > $\genfrac{}{}{0}{}{\phantom{A=B}}{}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{}}{A=B}$
 ${1}$ (15)
 >