The sum of all natural numbers is not \(-1/12\)It's not hard to see that the title is in some "tension" with at least one of the titles of older TRF blog posts about this issue,
Zeta-function regularization (2007)These texts of mine were actually linked to by some Pharyngula's readers who are my semi-fans – folks who would say how brilliant I am but who would never avoid mentioning that "thee shall not read any Lumo's articles on the global warming" and "he is a [right-wing] as*hole" of a sort. These untrue libels are apparently mandatory in the current Academia.
Why is the sum of integers equal to \(-1/12\) (2011)
At any rate, Pharyngula's "contribution" doesn't say much more than the title. He also copies a segment of a text by Mark Chu-Carroll,
Bad Math from the Bad Astronomer,who didn't like the regulated value of the sum, either. The only person among these notorious over-the-edge left-wing activists who liked the regulated sum was the Bad Astronomer Phil Plait, according to his enthusiastic article
When Infinity Is Actually a Small, Negative Fraction (Slate),which was supplemented with an extra disclaimer after an avalanche of dissatisfied reactions from some other randomly ejaculating liberals.
Plait's text has also embedded an enthusiastic video about the regulated sum:
It was posted on Brady Haran's Numberphile YouTube channel a week ago. It describes/copies the Euler method that I presented both in the 2007 article and the 2011 article.
I have already discussed about four different ways to calculate the natural value of the sum, \[
1+2+3+4+5+\dots = - \frac{1}{12},
\] in the blog posts from 2007 and 2011. In this entry, I would like to focus on some sociological aspects of this issue – and the different scientific disciplines' viewpoint on the sum.
First, let me begin with the high-school or freshman undergraduate perspective on the sum – and folks like P.Z. Myers haven't managed to go beyond this level. What do we mean by the sum of the infinite number of terms? Well, we mean the limit of the partial sums:\[
E_0 = \sum_{n=1}^\infty n = \lim_{N_{\rm max}\to\infty} \sum_{n=1}^{N_{\rm max}} n
\] I will use the symbol \(E_0\) for the sum to remind you that in some units, it is the energy of the ground state of some physical system. To fully appreciate this simple straightforward definition of the infinite sum, you must know what the "limit" means. In this case, the limit exists and is equal to \(L\) if and only if\[
\forall \epsilon\gt 0, \,\, \exists N_\epsilon:\,\, \forall N \gt N_\epsilon:\,\, \abs{ L - \sum_{n=1}^N n } \lt \epsilon.
\] In plain mathematical English, the partial sums have to converge to \(L\) which means that for an arbitrarily narrow neighborhood of \(L\), i.e. for an arbitrarily short interval \((L-\epsilon,L+\epsilon)\), one can find some minimum length of the partial sums \(N_\epsilon\) such that all the partial sums that are this long or longer produce results that belong to this interval.
This is pretty much the only meaning of the infinite sum – the only method to deal with the bizarre symbol \(\infty\) used as the upper limit of the sum – that an undergraduate freshman student who has no extra interests in maths and physics knows. This student may easily prove that the sum of positive integers cannot be negative because all the partial sums are positive and the limit of a sequence of positive numbers cannot be negative. By the same argument, he may prove that the sum cannot be fractional because all the partial sums are integers.
The real problem is that the definition of the sum involving the limit of partial sums – limits that way too often "diverge" or "refuse to exist" – isn't the only definition or the best definition or the most natural definition that may be connected to the sum. There exist better definitions of the infinite sum – numerous definitions that turn out to be more natural in physics applications – and they generally produce the result \(-1/12\). It is no trick or sleight-of-hand. The value \(-1/12\) is really the right one and the rightness may be experimentally verified (using the Casimir effect).
These physics applications of the sum of positive integers are pretty much derived from the master example, the ground state energy of a string. The energy of a relativistic string is given by the Hamiltonian which looks like\[
H = \int\dd \sigma \,\zav{ \frac 12 p(\sigma)^2 + \frac 12 x'(\sigma)^2 }
\] Let's not be picky about the limits of the integral and coefficients, among other things (such as the extra spatial indices that \(x\) and \(p\) usually carry). The point is that once the theory is quantized, the operator \(H\) above – the Hamiltonian – is equivalent to the sum of infinitely many quantum harmonic oscillators' Hamiltonians. Each of these Hamiltonians is linked to one Fourier mode for which the \(x\) and \(p\) are multiplied by functions like \(\cos n\sigma\) (or sine or the complex exponential) whose frequency scales with \(n\). And because the zero-point (ground state) energy of the harmonic oscillator is \(\hbar\omega/2\) and \(\omega\sim n\), we get the contribution scaling like \(n\) from each of these harmonic oscillator which is why the overall zero-point energy of \(H\) will be proportional to \[
E_0 = \sum_{n=1}^\infty n.
\] But an important detail we must realize is that when we play with these Fourier expansions and harmonic oscillator, mathematics and Nature don't tell us\[
E_0 = \sum_{n=1}^\infty n,\\
\text{...please compute the sum...}\\
\text{...using the partial sums, ...}\\
\text{...your loving Nature...}
\] Nature just doesn't give us any similar hints how the sum should be evaluated. The whole concept of "partial sums" is just one possible man-made concept. It implicitly includes the assumption that the low-frequency Fourier modes have a higher priority and the effect of modes with excessively high frequencies may be quantified by simply forgetting about them.
But none of these assumptions – assumptions which are really just man-made social conventions – really follows from the mathematics that encodes the laws of Nature. When we switch between different forms of the formula for the Hamiltonian, some of the manipulations are formal which means that some forms may be more well-defined than others. But again, Nature really doesn't tell us which of the forms is superior. We must find out.
Experiments always measure finite values of quantities such as the energy so from a physics viewpoint, all prescriptions and methods of calculation that produce answers such as \(\infty\) or "the limit doesn't exist" are just immediately excluded. A physics theory isn't allowed to work like that. Getting rid of the "spurious infinities" that just mask the important finite answers isn't just an option for a physicist; it is his unquestionable duty.
So physicists have learned to deal with similar sums in smarter ways, ways that remove the "spurious infinities" that just mask the important finite results. The symbol \(\infty\) is always the same thing and pretty much carries no information. Instead, the physical predictions must carry lots of information. Only finite numbers may convey such information, the dependence on various physical variables, and so on. The infinities in the results are anonymous scum that has to be uprooted.
When the procedures are done properly, we sometimes find out that the correct finite answer is somewhat undetermined and an experimentally measured parameter (e.g. the value of the elementary electric charge) has to be substituted. (In renormalizable theories, the number of infinite sums that are replaced by a finite number is finite – we learn about the values of a finite number of "patterns of diverging integrals" and all others may be expressed as their functions which is what keeps the theory predictive: infinitely many predictions may be made once you measure a finite number of parameters.)
In the case of the sum of the positive integers, the correct finite result that should be assigned to the result is unique. You may reconcile this finite result with your low-brow, straightforward, common-sense freshman undergraduate intuition if you redefine the Hamiltonian and include some terms that subtract the infinite part. The formula for the Hamiltionian without these extra terms is just a "heuristic template" or an "inspiration" and you may say that the "right Hamiltonian" is more complicated – less directly expressible in different forms. Alternatively, you may keep the simple-minded form of the Hamiltonian and redefine the meaning of the infinite sums etc.
Quantum physicists really need similar procedures in their everyday work but they don't have a monopoly over these methods and the finite results acquired with their help. In fact, some of the best mathematicians have determined the right values of these superficially divergent or ill-defined sums long before these sums became important in physics. Leonhard Euler (1707-1783) was able to calculate the value \(-1/12\) for the sum of integers using one of the "old-fashioned" methods that are at risk of being similar to some other methods producing wrong results if the process is performed by a less ingenious mathematician than Euler. The right methods generally preserve the "analytic continuation" of some holomorphic functions of some complex variables but I don't want to turn this text into another technical essay explaining all the possible "wrong regularizations" and how they differ from the right ones.
Many other top mathematicians have spent quite some time by calculating similar sums – sometimes sums that are still in the "ballpark of the physically relevant ones" but that are more complex than anything that physicists have actually needed.
Other mathematicians have developed alternative definitions of the summation that agree for convergent sums but are more likely to produce finite answers if the sum diverges using the trivial limit-of-partial-sums definition. These definitions of the sums that improve the convergence are known under many names borrowed from their inventors. For almost every letter in the alphabet, you find a summation, for example:
Abel summation, Borel summation, Cesàro summation, ..., Lambert summation, Lindelöf summation, ..., Ramanujan summation...and so on. I couldn't forget about Srinivasa Ramanujan who has really gained some profound (according to a modern physicist's opinion) insights on the superficially divergent and ill-defined sums. His prescription is the most far-reaching one in the list – it produces finite answers in many cases (including the result \(-1/12\) for the sum of positive integers) and it is so general that according to some straightforward definitions of the "sum", it can't even be called "a sum". Nevertheless, Ramanujan's summation has the key algebraic properties expected from a sum and this is what really matters for Nature's decision whether some quantity deserves to be called a sum.
Borel summation is helpful for a clarification of the character of the divergence of the perturbative series in most quantum field theories (and perturbative string theory) – and why this divergence doesn't imply an inconsistency.
Especially some of these summation methods are much smarter than the naive man-made prescription using the partial sums. But we must understand that just like the partial-sum definition, they're still man-made inventions. A mathematician may simply define concepts and symbols, including \(\sum_{n=1}^\infty a_n\), in many ways. But what's important to realize is that none of the man-made inventions and none of the man-recommended procedures is automatically relevant for calculations in physics – or natural sciences.
So it's not true that physics is forever constrained to use a particular "Mann summation" named after a particular man (I chose "Mann" for the sake of wittiness – because everyone knows that almost no person boasting this name is able to sum anything). Instead, the truth in physics is determined by Nature's wisdom that is only accessible to us through the experiments and observations (and whatever deeper we may induce from them).
In particular quantum theories, like in the two-dimensional conformal field theory that is used as a world sheet description of string theory, there exist "more natural, less man-dependent" rules that eliminate many wrong ways to sum expressions. Conformal field theories underlying the string world sheet have to obey the axioms – mutual locality of operators, some world sheet duality, and especially modular invariance (two/many ways of visualizing the torus). These conditions restrict the possible value of the true observables (which may be more or less directly extracted from experiments) – such as scattering amplitudes, correlation functions of operators, and so on. In particular, modular invariance is the main physical principle that implies that the result \(-1/12\) for the sum of positive integers is the right one; the constant \(-1/12\) itself is inherited from some identities obeyed by the \(\theta\) and especially \(\eta\) functions.
All the other concepts and quantities that appear in the middle of the calculations as intermediate results are "questionable". In physics, your goal is really to predict particular measurable quantities such as cross sections – and to design algorithms that allow one to determine many such predictions at the same moment. Pretty much any algorithm to produce lots of predictions in physics includes lots of "intermediate calculations" but all conceivable intermediate calculations, not just some randomly selected ones "sponsored" by particular people, are allowed to compete.
That is the reason why it's always naive and ultimately wrong to assume that some particular ways to calculate, like the low-brow limit-of-partial-sums definition of the infinite sums, is the right component of the calculations. It almost never is. It doesn't matter that a stringent instructor was beating you for you to memorize some particular rules or definitions (e.g. the limits and the partial sums); your pain isn't enough to make these algorithms and definitions relevant and right in a quantum field theory, for example. All these seemingly divergent or ill-defined sums and integrals have to interpreted and calculated using the best knowledge and intuition that is rooted in previously successful theories and ways of thinking (those that may produce self-consistent predictions that agree with the empirical data).
So physics is never a permanent slave of some limited axiomatic system in mathematics. In mathematics, people invent concepts which is why these concepts may be rigorous. As Einstein and others liked to emphasize, what is rigorous can't be automatically applied to the real world; what is applicable to the real world is necessarily at least partially non-rigorous. In spite of this independence of mathematics and physics, it is clear that physics is revealing a particularly natural and important way to look at many mathematical structures (not only the infinite sums), a way that good mathematicians who don't want to be stuck at the freshman undergraduate level should get familiar with.
I want to end this essay with a few words on self-confidence.
It's often right to be stubborn about some insights and principles that one is sure about, that have worked many times. But P.Z. Myers' fight against the finite value of the sum of positive integers is a random ejaculation outside his domain of expertise. How can one determine that his effective rules – that have apparently worked pretty well in the past – are no longer good in a given situation and that he should lower his self-confidence?
Well, it is a difficult question, especially because the answer depends on what kind of questions we discuss. However, I still want to give you something like two answers to this general question – or two recipes that should replace the answer.
One point I want to make is that in many cases, you should be able to see that the "simple" answer you are tempted to defend (like in Myers' case) is just too simple and the people who disagree with you can't possibly be stupid enough to misunderstand things like the partial sums. I am convinced that Myers must know that the physicists and mathematicians who still regulate the seemingly divergent sums must be aware of the partial sums and the fact that the partial sums of positive integers are neither fractional nor negative. He must know that they're smarter than himself! He must know that they are calculating many things that he doesn't know and these things have been successfully tested in numerous experiments. That should have been a reason for him to avoid writing his low-brow naive eight-sentence attack claiming that the "sum cannot be \(-1/12\)".
The other point I want to make is that the "limits of our knowledge" (and "limits of validity of our theories" as well!) is something we should be interested in, too; these limits should be a part of our knowledge and our theories, too! This thesis is closely analogous to the recommendation to young experimental physicists to never forget about the error margins of the quantities that they measure. The error margins are just parts of the information. If you don't have a clue how large the error is or might be, the mean value becomes sort of meaningless, especially if you only talk about one such mean value (or a small number of them). Your mean value will almost always disagree (at least a little bit) with someone else's mean value and without any idea about the error margins, you won't know whether you should be worried about the disagreement!
It's similar with the limits of your knowledge and limits of validity of your favorite theories. Unless you are a much better string theory (the theory of everything) expert than Edward Witten, all your theories you use and believe ultimately break down at some point or in some "fuzzy region" and it is your duty to have a clue where these boundaries are located. If you "know" a theory but you have no clue in what class of phenomena it may be trusted, you really know just an "uncertain part" of a theory. P.Z. Myers' knowledge about the infinite sums could be OK to get a passing grade in a freshman undergraduate mathematics exam; but he should know that his body of knowledge is not good enough to evaluate sums and integrals that appear in quantum field theories and string theory because he has never calculated a damn thing in these fields!
To summarize, there are numerous reasons that should have convinced him not to write a naive negative text about a topic he has no clue about – he could have avoided a controllable example showing that he has no clue pretty much about anything that he has assertive opinions about. P.Z. Myers included his scream "the sum cannot be what it is" into his Skepticism category which is bizarre – the incorporation implicitly says that the regulated sum is a supernatural or religious phenomenon. This fact unfortunately shows that the scientific skepticism of these atheists is nothing else than the opposition to everything that these average people don't understand with the help of their common sense combined with their mediocre high-school and freshman undergraduate knowledge. These opinions may sometimes be right but they may sometimes be wrong, too, and they're more likely to be wrong if they talk about topics in which the "skeptics'" opinions have been never tested or verified (that's the case of P.Z. Myers and all of physics or mathematics – and beyond).
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