3.7. Label indices

Sometimes we want to have indices with no tensorial meaning. For instance the labels l, m of the spherical harmonics can be defined as label indices. This is done using a new head LI, following similar ideas to those bringing us to the head Dir. There are no definitions associated to LI; the user can use LI for any purpose he likes.

LI                        Head denoting label slots in tensors
Labels                    A ficticious vbundle to which all label indices belong

Label indices.

We can define the spherical harmonics as

In[181]:=

DefTensor[Y[LI @ l, LI @ m], S2]

** DefTensor: Defining tensor Y[LI[l], LI[m]] .

In[182]:=

? Y

Global`Y

Dagger[Y]^=Y
DependenciesOfTensor[Y]^={S2}
Info[Y]^={tensor,}
PrintAs[Y]^=Y
SlotsOfTensor[Y]^={Labels,Labels}
SymmetryGroupOfTensor[Y]^=StrongGenSet[{},GenSet[]]
TensorID[Y]^={}
xTensorQ[Y]^=True

The up/down character of the labels only indicates the output position of the indices. Note that the signs inside LI are part of the label, and do not indicate the character.

In[183]:=

Y[LI @ L, LI @ M]

Out[183]=

Y_  ^LM

In[184]:=

{Y[-LI[L], -LI[2]], Y[LI[-L], LI[-2]]}

Out[184]=

{Y_L2^  , Y_  ^(-L - 2)}

In[185]:=

UndefTensor[Y]

** UndefTensor: Undefined tensor Y

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