I'm a bit embarrassed to post this, as it might be trivial but what the hey? I'm stuck!
The problem is to reduce this:
into this:
There is going to be a factor of -2i in the process.
If it matters, there is an integration over all k in the unreduced form. These (minus a couple of numerical factors) are the intregrands.
More information upon request.
-Dan
Just for completeness, the Minkowski four-vector for k is . In case you need it.
Okay, I think I know where the problem is in my explanation. I know this is what you are doing Hartlw and I think that this is what you are saying also, Plato. You're actually moving backward, if I'm right. I'm going to give you the whole mess, like I should have in the beginning.
So. I'm trying to find the commutator
where
and thus
Some definitions. k, a, and b are 4-vectors in Minkowski space, and likewise for operators a and b. The 4-vector product is defined as .
We also have the commutators
So on to the commutator:
Using the a and b commutators, integrating over k', and using the 4-vector product to separate the components, for example, we get:
I understand all of this and this is where I took the integrand out. We didn't use because we needed to separate out the 3-vector k to do the last integration over k. The next line should read
But I don't know how to get from the penultimate line to the final.
-Dan
Well that took a while to verify. (Which means I tried to keep each step verifiable, ie. I actually made sure each term was Mathematically correct, not necessarily Physically correct.)
Hartlw: Thank you. It worked out nicely. I had thought that there may be a way to extract an essentially 0 integral but I couldn't figure out how to do it.
Thanks to you and Plato. Couldn't have done it without both of you. And thanks for putting up with slow to understand student that I occasionally am.
-Dan