Don't be mislead by the great stock put into the 'left generalized
eigenstates' of section 3.5.2. They're really not that fantastic,
and with 20/20 hindsight, could have been guessed, 'trivially'.
Here's why. In (undergrad) textbook physics, things happen in a
Hilbert space where eigenfunctions are in a sense 'trivially
self-dual' because they are self-orthogonal. For example,
the sines and cosines of teh fourrier series are orthogonal to
each other; Lagendre, Laguerre, Hermite, Chebysheff polynomials
are all 'self-dual' with respect to some measure (weight). The
textbooks end there: I, at least, forgot that the universe is
bigger than that.
The Bernoulli polynomials have a dual that is quite completely
different: some generalized functions. However, that dual has
all the right properties that we expect:
$ 1 = \sum_{n=0}^\infty |B_n>
is the unit operator, as always, and
$ \delta_{m,n} = $
is the orthogonality relationship, as always. What is so unusual
here is that $ = B_n(x) $ is a Bernoulli polynomial,
whereas its dual is not a polynomial at all, and not even an
ordinary function, but is the generalized function
$ = (-)^n / n! [\delta^{(n-1)} (x-1) - \delta^{(n-1)} (x) ]
or equivalently
$ = (1/n!) d^n/dx^n $
The proof for teh first equation is that its a fancy way of
expressing the Euler-Maclaurin expansion: for function f(x),
we recognize that
$ f(x) = \sum_{n=0}^\infty $
is just Euler-Maclaurin. See, for example, Abramowitz & Stegun
equation 23.1.32, and set m=1 and $ p=\infty $ (i.e. we assumed
f(x) was infinitly differentiable.)
So, once we found that the right eigenstates in 3.5.1 were
Bernoulli polynomials, then we should have 'known' the result
of 3.5.2: the 'dual' of a Bernoulli polynomial is given
by the generalized functions that make the Euler-Maclaurin series
possible. At first, there seemed to be something 'magical'
happeinging with the derivation of the left eigenstates. But
with this hindsight, this now seems rather mundane.
Questions/Work items:
- Build a table of (generalized function) duals to other
well-known non-self-dual polynomials. For example, we
start with the Appell (Scheffer A-type zero) polynomials
as given by R.P. Boas, Jr. and R.C. Buck,
Polynomial Expansions of Analytic Functions
Springer-Verlag, 1964.
- Are there any text-book level, garden-variety problems
that have Bernoulli or Appell polynomial eigenstates,
something simple enough for a first-year-grad-level
textbook on e.g. quantum mechanics? How many quantum
problems go unsolved because the above non-self-dual
Hilbert space is not common knowledge?
Divergent Series
Another bit that seems 'magic' at first is the manipulations
of the formally divergent series of section 3.5.2. This only
seems magic because of the general lack of familiarity of
physics students with formally divergent series. For example,
the Euler series
w(z) = \sum_{q = 0}^{+\infty} q! z^{q+1}
is formally divergent; but for Re(z) < 0, its just the nice,
analytic exponential integral in hiding. Keywords: Borel
resummation, Gevrey development. There is a theorm
(Gevrey's theorm ???) that the analytic function is uniquely
determined by the formally divergent series, and thus it is
'safe' to confuse the formally divergent series and its
finite, analytic resummation.
Questions/Work items:
- Quantium mechanics is rife with formally divergent series
(that's what renormalization is about). Just how
general can one make this 'gevrey theorm'? General enough
to prove that the finite, renormalized solutions of
formally divergent series are unique?
- Many physicists (e.g. Roger Penrose) express a certain
distaste for things like QED because of the occurance of
formally divergent expressions, and conclude that there
must be some other theory (e.q. quantum gravity) that
will be finite, with no need for renormalization. But
in fact, is this a chimera, due to the lack of familiarity
of physicists with the machinery of formally divergent sums
and thier role as asymptotic expansions of analytic functions?
Is the fact that QED works so well, and gives 12 decimal places
with only 5th or sixth order corrections just a manifestation
of the 'hyperconvergence' generally seen in formally divergent
sums?