coupling constant ( which is proportional to the asymptotic field) and therefore is the analog of our renormalization for problems with static sources. In this case the equation of motion is $$D\phi^\prime=-j^\prime
\label {neom}$$ where $$D\equiv
(1+f)^2 (-\nabla^2+m^2) -2(1+f)
\partial_\mu f \partial^\mu -(1+f)\nabla^2f$$ and $$j^\prime\equiv
j(1+f).$$ Clearly the Green’s function $G^\prime$ ( the inverse of $D$) and current $j^\prime$ are both fairly complicated. The use of the new Hamiltonian density ${\cal
H}^\prime $ gives $$\begin{aligned}
E^\prime=\int {\rm d}^3 r{1\over 2}[ (1+f)^2
\vec\nabla\phi^\prime\cdot\vec\nabla\phi^\prime \nonumber \\
+(\phi^\prime)^2 \vec\nabla f\cdot\vec\nabla f
+2\phi^\prime(1+f)\vec\nabla f\cdot\vec\nabla\phi^\prime+
(m\phi^\prime)^2(1+f)^2+2j\phi^\prime(1+f)].\end{aligned}$$ Integration by parts and the equation of motion Eq.(\[neom\]) allows one to obtain $$E^\prime \, = \, {1\over 2}\int {\rm d}^3 r
j(1+f)\phi^\prime \, =\,{1\over 2}\int {\rm d}^3 r j^\prime(\vec
r)G^\prime(\vec
r,\vec r^\prime)j^\prime(\vec r^\prime),$$ which is just the original energy since $(1+f)\phi^\prime=\phi$.

Thus we have seen two explicit examples in which transformations of field variables change the equation of motion, the Green’s functions and the currents without changing, the physical observable, the energy of the system.

These same arguments of Eqns. (5-10) can be used to show that the various n-point vertex functions of the also depend on the specific choice of field. The generalization of the argument to 