Quantum Field Theory I

1.2. MANY-BODY QUANTUM MECHANICS 25
Remark: Even if it is not necessary for our purposes, it is hard to resist
the temptation to discuss perhaps the most important pair interaction in the
non-relativistic quantum physics, namely the Coulomb interaction. This is of
general interest not only because of the vast amount of applications, but also
due to the fact that the standard way of dealing with the Coulomb potential in
the p-representation involves a mathematical inconsistency. The way in which
this inconsistency is treated is in a sense generic, and therefore quite instructive.
1
|⃗x−⃗y|
In the x-representation V Coulomb (⃗x,⃗y) = e2
4π
(which is relevant in the case with no external field)
V Coulomb (⃗p 1 ,⃗p 2 ,⃗p 3 ,⃗p 4 ) = e2
4π
∫
, i.e. in the p-representation
d 3 x
(2π) 3 d 3 y
(2π) 3 1
|⃗x−⃗y| ei(⃗p4−⃗p1).⃗x e i(⃗p3−⃗p2).⃗y
(for the sake of brevity, we use the Heaviside-Lorentz convention in electrodynamics
and = 1 units in QM). This integral, however, is badly divergent. The
integrand simply does not drop out fast enough for |⃗x| → ∞ and |⃗y| → ∞.
Instead of giving up the use of the p-representation for the Coulomb potential
energy, it is a common habit to use a dirty trick. It starts by considering the
Yukawa (or Debey) potential energy V Debey (⃗x,⃗y) = e2 1
4π |⃗x−⃗y| e−µ|⃗x−⃗y| , for which
the p-representation is well defined and can be evaluated readily 34
V Debey (⃗p 1 ,⃗p 2 ,⃗p 3 ,⃗p 4 ) = e2 1 1
4π 2π 2 µ 2 +(⃗p 4 −⃗p 1 ) 2δ(⃗p 1 +⃗p 2 −⃗p 3 −⃗p 4 )
Now comes the dirty part. It is based on two simple (almost trivial) observations:
the first is that V Coulomb (⃗x,⃗y) = lim µ→0 V Debey (⃗x,⃗y), and the second is that the
limit lim µ→0 V Debey (⃗p 1 ,⃗p 2 ,⃗p 3 ,⃗p 4 ) is well defined. From this, a brave heart can
easily conclude that V Coulomb (⃗p 1 ,⃗p 2 ,⃗p 3 ,⃗p 4 ) = e2 1
8π
δ(⃗p 3 (⃗p 4−⃗p 1) 2 1 +⃗p 2 −⃗p 3 −⃗p 4 ).
And, believe it or not, this is indeed what is commonly used as the Coulomb
potential energy in the p-representation.
Needless to say, from the mathematical point of view, this is an awful heresy
(called illegal change of order of a limit and an integral). How does it come
about that physicists are working happily with something so unlawful
The most popular answer is that the Debey is nothing else butascreenedCoulomb,
and that in most systems this is more realistic than the pure Coulomb. This is
a reasonable answer, with a slight off-taste of a cheat (the limit µ → 0 convicts
us that we are nevertheless interested in the pure Coulomb).
Perhaps a bit more fair answer is this one: For µ small enough, one cannot say
experimentally the difference between the Debey and Coulomb. And the common
plausible belief is that measurable outputs should not depend on immeasurable
inputs (if this was not typically true, the whole science would hardly be possible).
If the mathematics nevertheless reveals inconsistencies for some values of
a immeasurable parameter, one should feel free to choose another value, which
allows for mathematically sound treatment.
34 ∫
For⃗r = ⃗x−⃗y, V Yukawa (⃗p 1 ,⃗p 2 ,⃗p 3 ,⃗p 4 ) = e2 d
3 r d 3 y 1
4π (2π) 3 (2π) 3 r e−µr e i(⃗p 4−⃗p 1 ).⃗r e i(⃗p 3−⃗p 2 +⃗p 4 −⃗p 1 ).⃗y
= e2
4π δ(⃗p 1 +⃗p 2 − ⃗p 3 −⃗p 4 ) ∫ d 3 r e −µr
(2π) 3 e r i⃗q.⃗r , where ⃗q = ⃗p 4 − ⃗p 1 . The remaining integral in
∫ 1
the spherical coordinates: ∞
(2π) 2 0 dr r ∫ e−µr 1
−1 dcosϑ eiqrcosϑ = 1 ∫ 1 ∞
2π 2 q 0 dr e−µr sinqr.
For the integral I = ∫ ∞
0 dr e−µr sinqr one obtains by double partial integration the relation
I = q
µ 2 − q2
µ 2 I, and putting everything together, one comes to the quoted result.