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KroneckerDelta is sensitive to assumptions:
The symmetry property of KroneckerDelta is automatically taken into account when the indices have symbolic values
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By default, spacetime indices are represented by greek letters and the dimension of spacetime is 4 - you can query about that via
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Note: since Maple 2019, KroneckerDelta[mu, nu] is not considered a tensor (of type Physics:-Library:-PhysicsType:-Tensor), even if and are letters representing tensor indices. Consequently, in all cases, the following does not return the trace, but the number 1
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Care should be taken, however, if you happen to index KroneckerDelta with one index covariant and the other contravariant (preceded by ~), in which case KroneckerDelta[mu, ~nu] is automatically transformed into the spacetime metric g_[mu, ~nu] (both displayed with the greek letter as is standard in physics textbooks), for which g[mu, ~mu], with the index repeated, represents its trace, equal to the spacetime dimension
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Physics:-g_[mu,~nu] = Physics:-g_[mu,~nu]
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There is one scenario, typical in quantum mechanics, in which KroneckerDelta is however used as a tensor, to represent the metric of an Euclidean space, often of dimension 3, and that representation is used when expressing commutator rules. For example,
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Define now and as tensors respectively representing position and momentum
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Set the standard commutator rules using tensor notation there the right-hand side is expressed using KroneckerDelta[j,k]
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| (19) |
The warning message explains the problem: the free indices are not the same at each side of the algebra rule because KroneckerDelta[j, k] is not representing a tensor, even when in an Euclidean space (only) its components do transform like tensor components and remain equal to the identity matrix. From the three options indicated, if you are adapting a worksheet written before Maple 2019, the simplest is option 2:
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After that, all occurrences of KroneckerDelta[m, m], now representing g_[m, m] with Einstein's sum over the repeated indices, will automatically return the trace of the metric instead of the number 1:
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You can undo this mapping of KroneckerDelta into g_ at any time via
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If you are not adapting worksheets of previous releases, however, the best is option 3, because it gives you access to both the tensor and the non-tensor versions of Kronecker . Directly copying from the message above and pasting on an input line,
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| (21) |
Check now the components and behavior of this Kronecker tensor
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while you still have the non-tensor version
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You can now reenter the algebra rules in tensor notation, with the right-hand side corrected using the tensor version delta[j, k]
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