The second point concerns unification of tensor, there are some issues on
contracted indices, with some boilerplate
from sympy.tensor.tensor import *
L = TensorIndexType('L')
i0, i1, i2 = tensor_indices('i0:3', L)
A = tensorhead('A', [L]*2, [[1]*2])
The arguments
>>> A(i0, i1).args
(1, (A(L,L),), (i0, i1))
>>> A(i0, -i0).args
(1, (A(L,L),), (dummy_index_0, -dummy_index_0))
That is, the third argument is the list of indices, and in the contracted
form the index i0 gets replaced by dummy_index_0, which makes unification
hard:
>>> from sympy.unify import *
>>> list(unify(A(i0, i1), A(i0, i2), {}, variables=[i2]))
[{i2: i1}]
>>> list(unify(A(i0, -i0), A(i1, -i1), {}, variables=[i1]))
[{}]
The second unify expression does not return anything because the arguments
i0 and i1 have been substituted. A possible solution is to keep the
arguments of a TensMul as they have been created, and store the
dummy_index_0 inside the corresponding TIDS object.
A possible implementation of TensorDerivative (by which I mean the
derivatives usually employed to get the equation of motions from the
Lagrangian in QFT) can simply rely on rewriterule contained in the unify
module.
The ability to define new objects such as TensorDerivative,
PartialDerivative, CovariantDerivative is the primary reason why I want to
make the tensor expression objects just containers, and possible transfer
all index manipulation algorithms to either TIDS or a new object.
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