xTras Package Symbol

AllContractions

AllContractions[expr]
returns a sorted list of all possible full contractions of expr over its free indices.

AllContractions[expr, frees]
returns all possible contractions of expr that have frees as free indices.

AllContractions[expr, frees, sym]
returns all possible contractions of expr with the symmetry sym imposed on the free indices frees.

MORE INFORMATION

The first argument can also be a list, in which case AllContractions returns the union of all contractions for each of the members of the list.

AllContractions returns {} if it is not possible to take contractions, or throws an exception if it encounters an error.

The indices in the second argument have to be given as an IndexList or as a List.

The input expression expr cannot have dummy indices.

The number of free indices of expr plus those in frees has to be even.

The free indices of the input have to belong to the same tangent bundle.

The metric of that tangent bundle has to be symmetric.

The first argument can also be given in pseudo index-free notation.

AllContractions does not take multi-term symmetries into account, only mono-term symmetries.

Specifying free indices with the second argument is equivalent to adding an auxiliary tensor with indices frees to expr, computing all contractions, and varying with respect to the auxiliary tensor afterwards.

The following options can be given:

Verbose	False	display information while computing
Parallelization	True	whether or not to distribute the computation over all available kernels
SymmetrizeMethod	ImposeSymmetry	how to symmetrize the free indices
UncontractedIndices	None	how many indices should not be contracted
FreeMetrics	All	contractions with this many of free metrics will be included
AuxiliaryTensor	Default	name of the auxiliary tensor used internally

Possible settings for Verbose include:

	False	don't display any information while computing
	True	display status information while computing

Possible settings for SymmetrizeMethod include:

	ImposeSymmetry	explicity symmetrize the free indices
	ImposeSym	implicitly symmetrize the free indices
	None	do not symmetrize the free indices, but instead keep the auxiliary tensor

Possible settings for UncontractedIndices include:

	None	contract over all indices (i.e. all indices of the input expression and additionally specified free indices)
	n	do not contract n of the indices, where is an integer between 0 and the total number of indices.

Parallelization only works for Mathematica version 7 and higher.

When UncontractedIndices is greater than 0, the uncontracted indices are treated equal, and AllContractions returns only one item for all contractions whose uncontracted index configurations are equal up to permutations.

Possible settings for FreeMetrics include:

	All	contractions with any number of free metrics will be included
	None	only contractions without free metrics will be included
	n	only contractions with 0 to n number of free metrics will be included
	{m,n}	only contractions with m to n number of free metrics will be included

Possible settings for AuxiliaryTensor include:

	Default	automatically generate a name for the internally used auxiliary tensor
	head	the symbol head will be used as the auxiliary tensor

If the auxiliary tensor is specified via the option AuxiliaryTensor, it must either be non-existing beforehand (in which case it will be defined afterwards), or it must have the exact same index structure and symmetry of the free indices specified in the second and third argument of AllContractions.

AllContractions uses a semi brute-force algorithm. In the worst case scenario (i.e. when the underlying expression has no symmetry) its efficiency is exponential in the number of free indices.

EXAMPLES

CLOSE ALL

Basic Examples (2)

Typing all the indices can become cumbersome, so instead we can use the pseudo index-free notation:

Typing all the indices can become cumbersome, so instead we can use the pseudo index-free notation:

In[1]:=

Out[1]=

Scope (2)

AllContractions doesn't take multi-term symmetries like the Bianchi identities into account.

The last two contractions in the above are related by a factor of 2:

The efficiency of AllContractions is exponential in the number of free indices:

Options (4)

The symmetrization of the free indices is by default explicit:

The symmetrization can be made implicit by specifying SymmetrizeMethod → ImposeSym.

Alternatively, the auxiliary tensor may be kept with SymmetrizeMethod → None.

The name of the auxiliary tensor, if kept by specifying SymmetrizeMethod → None, can be changed as follows:

By specifying UncontractedIndices all indices are no longer contracted:

This is in fact different than giving free indices in the second argument, because the command above doesn't take the different ordering of the free indices into account.

Only contractions with a given number of free metrics are returned when FreeMetrics is specified: