Tuesday, June 09, 2020

Taylor vs Fourier, smooth vs \(L^2\): battle of beauties and surprises

Simplicity of spectrum vs locality

As a kid, I liked smooth functions early on. You may add, subtract, multiply, and divide numbers and there is also the exponentiation, \(x^y\). At some moment, I realized that all those regular functions may be obtained from the addition, \(\exp(x)\), and its inverse, \(\log(x)\); the infinitely many branches of the latter (as a function of a complex variable) are extremely interesting and subtle but won't be discussed here. Nice smooth (or "holomorphic", informed people could say) functions may be obtained by various compositions of these procedures.

Note that \(x\cdot y = \exp(\log x + \log y)\) and \(x^y = \exp(y \log x)\).

This is the graph of \(1/\Gamma(x)\) on the real axis, a function that must surely be included in the category of nice smooth functions.

OK, for 40 years, I have singled out a subset of the mathematical concepts, the fertile and beautiful ones – those were relevant for the the understanding of the true and really deep truths about mathematics and physics.

Suddenly, there came some surprises. I was much older, maybe even a high school student. But suddenly, one engineering book I was reading was explaining the Fourier series – some years after I understood the Taylor and Laurent series. You know, my attitude was that the infinite sum of nice functions such as the Taylor and Laurent series have to be nice, too.

However, the book showed that you may sum nice functions and obtain self-evidently ugly functions such as piecewise linear functions and... even discontinuous functions! Sodom and Gomorrah, I thought (kids aren't allowed to think even that today so they can't properly meditate about the Taylor and Fourier series, either).

Well, let's be more extreme. You may really write an infinite sum whose result is the array of the delta-functions. Take\[

f(x) = \sum_{n\in \ZZ} \exp(inx)

\] The function \(f(x)\) is periodic with the period \(2\pi\) because every term is. Also, the phases clearly average to zero for any \(x\neq 0\) modulo \(2\pi\). Let's jump over some arguments: we know that the function must be a sum of delta-functions: \[

f(x) = 2\pi \sum_{m\in \ZZ} \delta(x-2\pi m)

\] The prefactor \(2\pi\) may be determined by integrating the function \(f(x)\) over one period, from \(-\pi\) to \(+\pi\), for example. The first expression for the function only gets a contribution from the constant \(n=0\) term and it yields the length of the interval, \(2\pi\). On the other hand, the second form of \(2\pi\) integrates to the prefactor itself because \(\delta(x)\) integrates to one. So the prefactor must be \(2\pi\).

Great. Infinite sums may add up to "totally ugly and unacceptable" functions according to my current world view of that time. (Note that discontinuities and piecewise linear functions may be Fourier-transformed by simply considering the indefinite integral of the delta-function.) How is that possible? What should I do? Well, clearly, after futile attempts to find a mistake in the books, I had to abandon or adapt my world view. ;-)

It is simply true. The sum of infinitely many nice things may be ugly. It means that one must either ban the infinite sums as a legitimate way to extend the set of pretty mathematical functions; or admit that the separation between beautiful and ugly functions is not so clear and sharp. Because I really loved so many Taylor series for \(\exp(x)\) and many other functions, I was really unwilling to ban them as a tool to produce new beautiful functions. Whether I liked it or not, I had to admit the fact that the black and white weren't as segregated as previously thought.

Well, a similar lesson has occurred in many other observations.

The unified conclusion that is valid for all of them says: You may consider pretty and ugly functions and mathematical structures but you just cannot divide the pretty from the ugly without specifying a context. That's why the infinitely smooth functions should better get a particular name, like "smooth functions" or \(C^\infty\), instead of the big word "pretty". The actual point is that there are many kinds of beauty and one should better distinguish them. In various contexts, different types of beauty are relevant – and only when you combine their roles into difficult symphonies, you can get a type of beauty that is universally valid (like the beauty of string theory).

In this particular situation as well as in many others, the "beautiful mathematical structures" have split into two. Functions that are \(C^\infty\)-smooth are surely a special, more pretty subset of "functions in the sense of maps from the real numbers to real numbers" (mathematics and physics undoubtedly use and have to use many adjectives whose broad emotional content may be summarized as "special in a particular nice sense"). And functions obtained by Fourier series are another nicer subset. But these two subsets aren't the same although they have some nontrivial intersection.

Some of the Taylor series may producely nicely smooth, infinitely many times differentiable functions. On the other hand, Fourier series produce periodic functions. In both cases, the "uncountable number of values" \(f(x)\) for every \(x\in\RR\) is replaced by a more manageable, countable package of coefficients \(c_n\) which define the Taylor or Fourier series, respectively. And there may be a one-to-one link between Laurent and Fourier series.

However, mathematics naturally invites you to generalize these infinite sums into two qualitatively different directions. One of them is the analysis of smooth and holomorphic functions, it's what you get if you focus on the "Taylor series" which typically sum over \(n\geq 0\) only. On the other hand, the other one, the "Fourier series", invites you to quickly sacrifice the conditions of differentiability. What you gain is the ability to describe a greater number of functions, including the piecewise linear functions, the functions with discontinuities, and even the distributions starting with the Dirac delta-function.

Physically, the need to describe piecewise linear functions and other functions that are "glued from regional functions" by an "ugly gluing operation" is required by the locality of the laws of physics. You simply need to include wave functions (and even functions describing not only classical waves) that can be modified in two or more regions independently of each other – because this independence is required by locality. In the infinitely many times differentiable smooth functions, an infinitesimal change of the function at one place unavoidably changes what the function does everywhere. But that's not the case for "wave functions" – and this relaxed rule is equivalent to the sacrifice of the infinite smoothness.

Needless to say, both of these representations of "special functions" in terms of a countable number of coefficients have a huge importance in physics. The holomorphic (or meromorphic) functions are extremely important, not only because they appear at many places already in classical physics but especially because the quantum probability amplitudes are naturally meromorphic functions of energies and momenta.

On the other hand, the Fourier series naturally produce the Hilbert spaces of wave functions. Here, another "cut" of the excessively big space of "all functions as maps from reals to reals" takes place: if you change the value of a function \(f(x)\) in a countable number of points \(x\), you don't "really" change the function (and it's "ugly" to work in a scheme where this change matters – it would mean to return to the ugly picture of "functions as maps" that need uncountably many parameters). Several functions that only differ at a countable number of points should be identified; they are expressed by the same Fourier coefficients. On the other hand, we normally demand Fourier series to produce "wave functions from the Hilbert space" which must be \(L^2\)-integrable, a special condition that we wouldn't dare to demand in the context of smooth functions.

Whenever some function in physics may be expressed by similar series, you may usually say which of these two "branches of beautification of functions" you are at. Either the functions that you consider are meromorphic for some physical reasons; or they are supposed to model the "whole Hilbert space" in some way which means that you must be capable of including functions that aren't \(C^\infty\)-smooth, too.

However, I believe that the precise transition between these two "philosophies about pretty functions" must be ultimately blurred and explained in some deeper way. In other words, I believe that to understand the true origin of the wave functions and scattering amplitudes, physicists must understand how to deal with many wave functions that are not \(L^2\)-integrable; and with general dynamical laws that produce amplitudes that are not infinitely smooth. The conditions for the \(L^2\)-integrability (physically inseparable from locality and the physical interpretation of amplitudes as probability amplitudes) or smoothness (physically linked to a sufficient simplicity of the mass spectrum etc.) must be afterthoughts, they shouldn't be placed as assumptions at the very beginning.

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