Non-abelian magnetic field interaction

Consider the magnetic part of a non-Abelian field. I want to know if we can define Hamiltonian for the interaction of this sector with the spin (something such as Hamiltonian of interaction of magnetic moment and electromagnetic field.)

• You asked about magnetic part of non-abelian interaction like strong interactions. Color charge interactions can be decomposed into chromoelectric and chromomagnetic fields like explained in physics.stackexchange.com/questions/426507/…. You later ask about Hamiltonian interaction of magnetic moment and electromagnetic field which is abelian. Please charify this confusion. Commented Jan 23, 2023 at 20:01
• @OrdinarySuperhero I mean if the spin can interact with the non-Abelian magnetic field? For example if we can define Hamiltonian for the interaction of the electron spin with the chromomagnetic field? Commented Jan 25, 2023 at 17:50

As Lorentz invariance is inherently followed in a quantum field theory, we ususally encounter quantities that are 4-vectors. But in some cases like NRQCD, it is more convenient to resolve the Yang-Mills field strength tensor into terms like what we have in classical EM field theory.

When we integrated out the hard scale of mass, $$m$$ from the full QCD lagrangian, we get the effective field theory of NRQCD. The heavy quark part of the NRQCD lagrangian is given as:

$$$$L_\psi = \psi^\dagger \Biggl\{ iD^0 + c_k^{(1)} \frac{{\textbf D}^2}{2m_1} + c_4^{(1)} \frac{{\textbf D}^4}{8m_1^3} + c_F^{(1)} g \frac{\sigma \cdot {\textbf B}}{2m_1}$$$$ $$$$+ c_D^{(1)} g\frac{{\textbf D} \cdot {\textbf E}- {\textbf E} \cdot {\textbf D}}{8m_1^2} + ic_S^{(1)}g\frac{\sigma \cdot({\textbf D} \times {\textbf E}- {\textbf E} \times {\textbf D})}{8m_1^2} \Biggl\} \psi$$$$ $$$$+ (\mathrm{terms \; for \; heavy \;light \; quark \; local \; interaction})$$$$

where,
$$\quad$$ $$\sigma$$ are the Pauli matrices,
$$\quad$$ $$\textbf{D}=\nabla-ig\mathbf{A}$$,
$$\quad$$ $$\textbf{E}^i=F^{i0}$$ is the chromoelectric field
$$\quad$$ $$\textbf{B}^i=-\varepsilon_{ijk} F^{jk}/2$$ is the chromomagnetic field.
$$\quad$$ $$F^{jk}$$ is the spacial part of the full QCD field strength tensor, $$F^{\mu\nu}$$.
$$\quad$$ $$c$$ terms are the Wilson coefficents of the EFT.

The bracket term is the free field part plus the coupling to soft gluons. In this EFT heavy quark-antiquark pairs cannot be created anymore, so it is convenient to use non-relativistic Pauli spinors ($$\psi$$) instead of Dirac spinors for the heavy quark spin states.

The term with $$\sigma \cdot {\textbf B}$$ is remenicent of quark spin interaction with chromomagnetic field. So, it is possible to invoke chromoelectric and chromomagnetic field description for non-abelian gauge theory, but it is suitable only for some non-relativistic systems.

Reference:
page 26, Effective Field Theories of QCD for Heavy Quarkonia at Finite Temperature, Jacopo Ghiglieri,arXiv:1201.2920 [hep-ph]