One learns early-on in calculus
(or, if you are well brought-up, at your mother’s knee), to be wary of conditionally convergent series. If the series is not absolutely convergent, then you may not
arbitrarily rearrange the order of terms, as that could well alter the
limit. Indeed, Riemann proved a
surprisingly strong result:
that with an appropriate permutation, you can come up with any ‘limit’
you like:
I was reminded of that caveat,
reading the repeated strictures of Roger Penrose (in Road to Reality) on
the ‘renormalization’ of divergent series (in QED and QFT), where many infinite
bumps may be thus swept under the rug.
More generally, he
questions many uses of perturbation
theory, which turns a problem
into that of summing an infinite perturbation series. These are admittedly tricky (“There is absolutely no
guarantee that perturbative methods result in a convergent solution. In fact,
asymptotic series are the norm.” -- Wiki), but widely used (and perhaps abused)
in physics.
To illustrate the power of an
unbridled perturbative approach, we now shall prove, by its methods, a
surprising result:
Donald Trump is a feminazi
Proof:
(Step zero) Begin with Trump in his ground-state.
(Step one) Perturb this by addition of a couple of
boobs.
(Step two) Add a Hitler-style
toothbrush-moustache.
(Step three) Within the limits of experimental
error, higher orders of perturbation can be ignored.
Voilà!
∎
(The technique used in that proof is known technically as the "Mr Potato Head algorithm".)
(The technique used in that proof is known technically as the "Mr Potato Head algorithm".)
Donald Trump, looking mildly perturbed |
Thus we see that, using the
powerful techniques of Perturbation Theory, we can achieve results scarcely
obtainable otherwise.
~
[Update 28 August 2015] The above was a sotie -- a learnèd jest.
But there is a deeper sense in which the notion of “perturbation” does
apply. Namely, in quantum theory, where the Schroedinger
wave-function blithely evolves in smooth linear splendor, entirely
non-committed as to the values of such parameters as location or momentum: when suddenly, it is ‘perturbed’ -- probed,
interfered with -- in the form of an ‘observation’. At which point, “the wave-function collapses”, and out
rolls a value for the parameter measured (orientation of the polarization, or
whatever), like a silver dollar rolling out of a slot-machine.
In the case of an amateur running
for President (and this would apply to retired neurosurgeons, pizza moguls, generals,
whomever, as well as to casino magnates or Miss Universe impresarios), the “perturbation”
comes not from a physicist (‘observer’), but from a reporter or
interviewer; but the effect is the
same. Within the mists
thitherto swirling within the novice’s head, there is no conception of a
position on: the Armenian
genocide (who did they kill?),
quantitative easing, the independence of the continuum hypothesis, the
Protocols of the Elders of Zion, the role of the Illuminati in the moon-landing
hoax, or whether Jeb Bush was really born in Florida as he claims or rather in
Communist China as the short form of his birth-certificate clearly states (the
long form claims that he was never born at all). To keep the
conversation flowing, an opinion -- a position -- is required: and voilà ! Suddenly our neophyte, without having given the matter
a moment’s thought, has an actual stance on, say, the nuclear deal with Iran
(the product of years of work by both diplomats and boffins, from several
leading nations).
.
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