When solving the problems of V. Rubakov's "Classical Theory of Gauge fields" book, I encountered the following phenomenon:
- For a real scalar fields (spin 0) $\phi$, if we consider the action with source $\rho(\overrightarrow{x},t)$, $$S=\int d^4x \left\{ \frac{1}{2}\partial_\mu\phi\partial^\mu\phi-\frac{m^2}{2}\phi^2+\rho\phi\right\}, $$ and that we take the source associated to two charges $q_1$ and $q_2$ in $\overrightarrow{x}_1$ and $\overrightarrow{x}_2$, $$ \rho(t,\overrightarrow{x}) = -q_1\delta^{(3)}(\overrightarrow{x}-\overrightarrow{x}_1)-q_2\delta^{(3)}(\overrightarrow{x}-\overrightarrow{x}_2),$$ then the energy of interaction of those charges (obtained by extracting the divergent terms of the Hamiltonian) decreases when $|\overrightarrow{x}_1-\overrightarrow{x}_2|$ decreases if $q_1q_2>0$. In other words, charges of the same sign attract each other.
- For a (real) vector field (spin 1) $A_\mu$, if we consider the action with source $j^\mu(\overrightarrow{x},t)$, $$S=\int d^4x \left\{ -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-ej_\mu A^\mu\right\}, $$ with $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$, and that we take the source associated to two charges $q_1$ and $q_2$ in $\overrightarrow{x}_1$ and $\overrightarrow{x}_2$, \begin{align} j_0(t,\overrightarrow{x}) &= -q_1\delta^{(3)}(\overrightarrow{x}-\overrightarrow{x}_1)-q_2\delta^{(3)}(\overrightarrow{x}-\overrightarrow{x}_2), \\ j_i(t,\overrightarrow{x}) &= 0, \end{align} then the energy of interaction of those charges (obtained in the same way) decreases when $|\overrightarrow{x}_1-\overrightarrow{x}_2|$ increases if $q_1q_2>0$. In other words, charges of the same sign repulse each other. As in well-known electromagnetsim.
- For a spin 2 field (the metric for example), the behavior switch again: charges of the same sign attract each other. This is pretty intuitive when when it comes to gravity.
The pattern seems pretty clear.
- What is the fundamental reason for this to happend?
- What about half-integer spin fields?
- Does it continue for higher (than 2) spin fields? I heard that higher spin fields are much more difficult and that is was still not much understood. Why is that?
