Matchete`

RelabelIndices

relabels all repeated indices in expr in a canonical way.

Details and Options

The following options can be given:

	Expand	True	expand out products and positive integer powers in expr before the relabeling of indices. Use this option set to False only with caution, since this can lead to aninconsistent relabeling of indices.
	Unique	False	specifies whether to use unique or canonical index labels.

Examples

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Basic Examples (1)

Define a fermionic theory with an SU(3) gauge group and a flavor index:

RelabelIndices identifies all dummy indices in an expression and relabels them to their canonic form:

The NiceForm printing uses the symbols form IndexAlphabet, FundAlphabet, AdjAlphabet for printing the canonic index labels.

The canonic index labels used are d$$i where i=1,2,... for every type of index present separately:

Generalizations & Extensions (1)

RelabelIndices relabels all dummy indices, that is, all indices appearing exactly twice in a term.

The exception to this rule are flavor indices. Since heavy fields must be in their mass basis for the matching, their mass matrices must be diagonal and therefore only carry a single flavor index:

RelabelIndices is aware which flavor indices should be interpreted as belonging to diagonal couplings and which not. Therefore, it can be applied to these expressions:

To manually define a coupling with a flavor index that should be interpreted as diagonal one can use:

Similarly the flavor indices of FlavorSum objects are treated properly:

Options (2)

Expand (1)

The option Expand determines whether the expression should be expanded out before the index relabeling:

Warning: use this function with Expand->False with caution as this can lead to an inconsistent relabeling of indices.

Unique (1)

The option Unique->True uses unique index labels instead of the canonic ones by using the Unique function.

Using unique index labels can be required, for example, when applying replacement rules to enure non overlapping dummy indices.

Applications (2)

Simplification of expressions (1)

By introducing canonical index labels different equivalent structures can be combined:

Replacement rules (1)

RelabelIndices[expr,Unique->True] can be used to ensure a proper index relabeling when working with replacement rules:

Possible Issues (2)

Triple indices (1)

Error messages are generated if more than two indices of the same type with the same label are found:

Relabeling without expanding (1)

When using the option Expand->False some expressions might generate error messages when relabeling, since the individual contractions of the indices in each term can no longer be clear identified.

One should always use Expand->True as long as the opposite is not strictly necessaty.