Simple Foldy-Wouthuysen expansion of the dirac hamiltonian

[milksushi]

Hi, I am new to this code, but I believe what I am trying to do should be very simple to implement. Starting with the dirac hamiltonian with an external potential, I just want to compute a Foldy-Wouthuysen block diagonalization to some order 1/m^n. For now I'm just trying to compute it to order 1/m. In other words, let:

$i\partial_t \Psi =H\Psi=(-i\gamma^0\gamma^jD_j + \gamma^0 m - \gamma^0 V)\Psi$
where j ranges across three cartesian indices.

Then define $S=\frac{1}{2m-2V}\gamma^jD_j$ and simplify $H'=e^{iS}He^{-iS}=H+i[S,H]-\frac{1}{2}[S,[S,H]]+\cdots$ keeping terms up to order $1/m^n$. It may be necessary to define subsequent transformations $S_1,S_2\ldots$ but the idea is to find some $H'$ that, to order $1/m^n$, commutes with $\gamma^0$.

For example, the simplest case is $n=1$ and $V=0$, which results in the minimal coupling hamiltonian and is not too hard to compute by hand. I am interested in higher order terms like spin-orbit coupling. My hope is to write a program where I can plug in some $S$ and FeynCalc will simplify the algebra using dirac identities and simple user-defined commutators, like $[D_j,V]=\nabla V$ or $D_jD_j=(\partial_j-ieA_j)(\partial_j-ieA_j)=\nabla^2-2ieA\cdot\nabla-ie\nabla A-eA^2$ etc. In other words, just something that understands how derivative operators commute through quantum fields and can help simplify that big expression that comes out.

I'm not certain this is the best program to use for this purpose, but its what ai suggested. Unfortunately it doesn't seem like ai really understands how to use this program yet, and so far I haven't managed to figure out how to implement this. At first I was trying to define a momentum 4-vector and explicitly specify how it commutes with V and A, but maybe it would be better to use FCPartialD[*, i] wherever there's a derivative and explicitly include the quantum field $\Psi$? In that case I'm not sure if the program would simplify the expression nicely, and I'm not sure how I would define $V$. I've already tried a few things which were probably misguided, so I figured I would ask here before trying anything else. Any advice is appreciated.

[vsht]

Sorry for the late reply. I know how FWT works and I think that one could implement it in FeynCalc, but that would require lots of hand-written code to cover all combinations of commutators and make DotSimplify move different noncom objects past each other in the right way.

It's definitely not something you can get with one line of code using built-in routines.