Baker–Campbell–Hausdorff (CBH) Formula question

Answers and Replies

  • #2
Bill_K
Science Advisor
Insights Author
4,155
195
I've been trying to understand the explicit CBH formula

http://en.wikipedia.org/wiki/Baker–...er.E2.80.93Campbell.E2.80.93Hausdorff_formula

However, I don't really understand how to know what the values of [itex]r_i[/itex] and [itex]s_i[/itex] are, where [itex]1\leq i\leq n[/itex].
Like it says,

Where sn and rn are non-negative integers.
The sum runs over all possible values of sn and rn, where sn + rn > 0. They apparently stand for the number of X's and Y's in the multiple commutator. (Glad I don't have to prove this! :wink:)
 
  • #3
I agree, proving it would be awful, to say to least.

This may be a dumb question, but how do I know what the possible values of [itex]r_n[/itex] and [itex]s_n[/itex] are? I feel like I'm over-thinking this.

For example, if I consider [itex][X,Y][/itex], then [itex]r_n=s_n=r_1=s_1=1[/itex]. If I just had the BCH formula as written by Dynkin, how would I know that [itex]r_1=s_1=1[/itex] for [itex]n=1[/itex]?
 

Related Threads on Baker–Campbell–Hausdorff (CBH) Formula question

Replies
2
Views
13K
Replies
5
Views
8K
  • Last Post
Replies
1
Views
5K
  • Last Post
Replies
6
Views
9K
Replies
2
Views
4K
  • Last Post
Replies
2
Views
2K
Replies
2
Views
2K
Top