-------------------------------------------------
S9d. Perturbation theory and instantaneous forces
-------------------------------------------------
In classical relativity theory, causality demands that all forces
are retarded. In relativistic quantum theory, this principle is
somewhat obscured, due to the approximations needed to get a
dynamical picture. The general practice is to expand in powers
of v/c, where v is a velocity and c is the speed of light.
When doing this, the resulting formulas look instantaneous at
each order of perturbation theory, which might invite unfounded
conclusions.
However, the same already happens at the classical level, where
the situation is easy to understand. The retarded terms must
reappear when summing terms to all order.
This is most easily seen by noting that a retarded differential
equation (for simplicity 1D, but the 4D case is similar)
dx(t)/dt = f(x(t-tau)),
when expanded in powers of the small parameter tau, becomes a
higher order ordinary differential equation at fixed order.
To see this, differentiate the original equation k times and
introduce new functions
x_0=x, x_1=dx/dt, ..., x_k=d^kx/dt^k
to get a system of retarded differential equations.
Then expand the equation for dx_k/dt up to order n-k.
Then substitute terms on the right hand side.
The approximate equation is manifestly instantaneous, but it
describes the perturbative behavior of the retarded equation.
Thus perturbation theory in v/c cannot be used to decide about the
instantaneous or retarded nature of quantum dynamics.