and the solution to all orders is
ψ = φ+GUφ+GUGUφ+···= φ+∑∞
n=1(GU)nφ (12.27)The interpretation of the first term is free propagation; of the second term is a single interaction
with the potential; of the subsequent terms of higher order exchanges with the potential.
12.5 Short range versus long rangeV and massless particles
The basic assumption in scattering theory is that the potentialV(x) tends to zero as|x|→∞. It
is under this condition that the above description of the scattering process off a target really makes
sense. Namely, in the far past, we have a wave (or more physically, a wave packet) incoming from
spatial infinity where its behavior is that of a free particle. At finite time, and a finite distance
away from the target, the particle interacts with the target, and then again moves out to spatial
infinity at far future times, again behaving as a free particle.
But even ifV(x)→0 asr=|x| → ∞, the potentialV may tend to zero at different possible
rates. One distinguishes the following two behaviors,
- Short ranged: the potentialV vanishes at least exponentiallyV(x)∼e−r/ξwith distancer
outside a bounded region of space; this includes the case whereV vanishes identically outside
a bounded region of space of linear sizeξ. The smallest suchξ >0 is called therange of the
potential. One example is the Yukawa potential, given byV(x)∼e−r/ξ/r. - Long Ranged: the potentialV vanishes like a powerV(x)∼r−α of distance whereα > 0
is called theexponent. One example is the Coulomb potentialV(x)∼ 1 /r, for which the
exponent isα= 1.
The reason for this distinction has a deep physical underpinning. In relativistic quantum theories,
interactions cannot be instantaneous, and must be mediatedby the exchange of signals that travel at
the speed of light or slower. In fact, these signals are particles themselves. In relativistic quantum
field theory, the interaction potential may be obtained from summing up the contributions of
repeated exchanges of particles. If the mediator of the interaction is massless, then the resulting
interaction potential in the non-relativistic limit will always be long-ranged, while if the particle is
massive, the interaction potential will be short ranged. The range of the interaction is related to
the mass via the formula
ξ=̄h
mc(12.28)
namely, it is the Compton wave length.
The four forces of Nature behave as follows,- Gravitational: long-ranged, since it also obeys the Coulomb potential; the mediator of the
gravity is the massless graviton.