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.)

  • $\begingroup$ 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. $\endgroup$ Jan 23 at 20:01
  • $\begingroup$ @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? $\endgroup$
    – Astrolabe
    Jan 25 at 17:50

1 Answer 1


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:

\begin{equation} 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} \end{equation} \begin{equation} + 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 \end{equation} \begin{equation} + (\mathrm{terms \; for \; heavy \;light \; quark \; local \; interaction}) \end{equation}

$\quad$ $\sigma$ 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.

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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.