I have a Lie Algebra with a basis:
- A_1 = [[0 1 0], [0 0 0], [-1 0 0]]
- A_2 = [[0 0 1], [-1 0 0], [0 0 0]]
- A_3 = [[0 0 0], [0 1 0], [0 0 -1]]
The Lie brackets are [A_1,A_2] = A_3 ; [A_1,A_3] = A_1 and [A_3,A_2] = A_2.
I want to compute log(exp(A_1)exp(A_2)). I tried using the well-know Baker-Campbell-Hausdorff formula until order 5 (with the terms already computed as on wikipedia) and also the Tables 1 and 2 of the paper An efficient algorithm for computing... from Casas and Murua published in 2009. The three terms of order 5 that I get are different.
Could someone confirm that or explain the mistakes that I made in the computation?
Thank you in advance for your help.
Best,
FM.
Of course, no one will, until you show us your computation... At least, tell us what are the 3 terms you get.
The well-known result (from Wikipedia for instance) gives at order 5 (in terms of right-nested commutators):
- ([A_2, A_1, A_2, A_1,A_2] + [A_1, A_2, A_1,A_2,A_1]) / 120 = (A_1 + A_2) / 120.
The result using Table 1 of the paper gives at order 5:
- [[[[A_2, A_1], A_1], A_2], A_2] / 180 - [[[A_2, A_1], A_1], [A_1, A_2]] / 120 - [[[A_2, A_1], A_2], [A_1, A_2]] / 360 = A_2 / 180 - A_1 / 120 - A_2 / 360 = - A_1 / 120 + A_2 / 360.
The result using Table 2 of the paper gives at order 5:
- [A_1, [A_1, [[A_1, A_2], A_2]]] / 180 + [[A_1, A_2], [[A_1, A_2], A_2]] / 120 + [[A_1, [A_1, A_2]], [A_1, A_2]] / 360 = A_1 / 360 + A_2 / 120 + A_1 / 180 = 3 A_1 / 360 + A_2 / 120.
Here is a link to the paper: Article An efficient algorithm for computing the Baker-Campbell-Haus...
Well, the two last results (using Table 1 and Table 2 of Casas and Murua) are different because they are expressed in different bases: Table 1 uses a Hall basis, while Table 2 uses a Lyndon basis. On the other hand, the Wikipedia result you quote uses just nested commutators, which do NOT form a basis (you have to take into account multiple identities arising from the Jacobi identity, which have as a consequence that not all the terms appearing in the nested commutator expansion are independent). Hope this helps.
Even if terms are given in different basis, once computed with the Lie brackets that I gave above, shouldn't the results be identical?
I don't think so. The whole BCH series is independent of the chosen basis, but not its truncation up top a given order.
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