Saturday, August 22, 2015

Trump: a Perturbative Approach

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

(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.


(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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