Yeti: Я не знаю, что там делают Шаталин с Абалкиным, а мне бы научиться огонь добывать! ......................................................
(Критика чистого безумия)................................................
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problem is simple. The author considers first a $\delta$-like
potential $V_0 (x)=c_0\cdot\delta (x)$ and calculates scattering
amplitudes (reflection/transmission amplitudes). In particular, he
obtains the "low-energy" formula: $T=-i p/c_0$. It is just a
regular calculation. Everything is physically reasonable and no
renormalization is necessary so far. In particular, when the potential
coefficient $c_0$ tends to infinity, the transmission amplitude
vanishes and the incident wave is completely reflected. It is
comprehensible in case of a positive $c_0$, but it holds as well for a negative value of $c_0$ in the "wave" mechanics.
that, the author considers another interaction potential $V_1
(x)=c_1\cdot\delta' (x)$. This potential gives "undesirable" results.
Replacing $V_1 (x)$ with a "regularized" version of this kind $V_1(x) = c_1\cdot[\delta (x+a)-\delta(x-a)]/2a $, the author obtains a "regularized" amplitude $T=-i a p/c_1 ^2$. Again, so far
so good. When $a\to 0$, the transmission amplitude tends to zero
too. It is qualitatively comprehensible because each $c_1\delta
(x)/a$ grows in its absolute value as in case of $|c_0| \to
\infty$ in the problem considered just above. $V_1$ is a highly reflecting potential. Factually, it separates the region $x\lt 0$ from the region $x\gt 0$, like an infinite barrier of a finite width.
But the author
does not like this result. He wants fulfilling a "low-energy theorem"
for this potential too. He wants a non zero transmission amplitude! I do
not know why he wants this, but I suspect that in "realistic" cases we
use interactions like $V_1$ because we do not know how to write
down something like $V_0$. As well, it is possible that in
experiment one observes a non zero transmission amplitude and
renormalization of $V_1$ "works". So, his desire to obtain a
physical result from an unphysical potential is the main "human
phenomenon" happening in this domain. We want right results from a wrong
theory. We require them from it! (The second theory is wrong because of wrong guess of potential.)
course, a wrong theory does not give you the right results whatever
spells one pronounces over it. So he takes an initiative and replaces
a wrong result with a right one: $T=-i p/c_0$. Comparing it with
$T=-i a p/c_1 ^2$, he concludes that they are equivalent if $c_0=c_1^2/a$. The author denotes $c_0$ as $c_R$ and calls
it a "phenomenological parameter" to compare with experiment via the
famous formula $T=-i p/c_0$. After finding it from experimental
data, the author says that a theory with $V_1$ describes the
Is his reasoning convincing to you as a way of doing physics?
If one is obliged to manipulate the calculation results with saying that
$c_1$ is "not observable", I wonder why and for what reason one
then proposes such a $V_1$ and insists on correctness and
uniqueness of it? Because it is "relativistic and gauge invariant"?
Because after renormalization it "works"? And how about physics?
the renormalized result belongs to another theory (to a theory with
another potential). Then why not to find it out from physical reasoning
and use instead of $V_1$? This is what I call a theory reformulation. Am I not reasonable?