From Ramanujan to renormalization: the art of doing away with divergences and arriving at physical results
FFrom Ramanujan to renormalization: the art of doingaway with divergences and arriving at physical results
Wolfgang BietenholzInstituto de Ciencias NuclearesUniversidad Nacional Aut´onoma de M´exicoA.P. 70-543, C.P. 04510 Ciudad de M´exico, Mexico
A century ago Srinivasa Ramanujan — the great self-taught Indian geniusof mathematics — died, shortly after returning from Cambridge, UK, wherehe had collaborated with Godfrey Hardy. Ramanujan contributed numerousoutstanding results to different branches of mathematics, like analysis andnumber theory, with a focus on special functions and series. Here we referto apparently weird values which he assigned to two simple divergent series, (cid:80) n ≥ n and (cid:80) n ≥ n . These values are sensible, however, as analytic con-tinuations, which correspond to Riemann’s ζ -function. Moreover, they haveapplications in physics: we discuss the vacuum energy of the photon field,from which one can derive the Casimir force, which has been experimentallymeasured. We also discuss its interpretation, which remains controversial.This is a simple way to illustrate the concept of renormalization, which isvital in quantum field theory. Keywords: Ramanujan summation, Casimir effect, renormalization, ζ -function
101 years ago Srinivasa Ramanujan (1887-1920) passed away in Madras,at that time part of the British Empire (since 1996 this state capital inSouth-East India is named Chennai). He was one of the greatest geniuses inthe history of mathematics. One way to measure the impact of his work isthrough the amount of mathematical terms that bear his name: the math-ematical online encyclopedia Wolfram Mathworld [1] documents 27 termsnamed after Ramanujan, and his name appears in a total of 205 items; in Phonetically, his last name could be written in Spanish as “Ram´anuchan”. a r X i v : . [ phy s i c s . h i s t - ph ] F e b oth respects, he is among the leading mathematicians of all times. This isparticularly amazing since Ramanujan started to elaborate stunning equa-tions with hardly any mathematical education, and he died at the age ofonly 32 (younger than Mozart, for example).Figure 1: Srinivasa Ramanujan (1887–1920)
Ramanujan was born 1887 in a town called Erode, but at the age of2 his mother took him to the city of Madras, some 400 km away. In theearly 20th century he lived in extreme poverty, at the edge of starvation, buthe discovered a multitude of important mathematical formulae, based on hisincredible intuition — I tend to interpret it as a kind of “pattern recognition”(although it was not an automated process).In 1912 he began to send letters to British mathematicians, trying toattract attention to his discoveries; for a while without success. In January1913 he finally wrote to Godfrey Hardy, a brilliant young mathematicianat Trinity College of Cambridge University, who — together with his long-term collaborator John Littlewood — turned out to be the most influentialBritish mathematician of the first half of the 20th century. They are credited To be explicit, if we rank mathematicians by the number of mathematical items namedafter them, Ramanujan is at position 6, following Euler (71), Gauss (48), Hilbert (33),Fermat (32) and Riemann (31), and followed by Cauchy (26), Dirichlet, Jacobi, Weierstraß(23 each), Euclid (22) and Poincar´e (21). Regarding the number of mentionings in a
Mathworld entry, Ramanujan is at position 18. Ramanujan only obtained from a friend a library copy of a book by George Carr[2], which he studied intensively. It is a collection of formulae and theorems, with littleexplanation, written as an overview for students who are preparing exams. i.e. mostly during World War I. It was not easy for himto get used to the climate, lifestyle and food. Moreover, he suffered fromserious health problems; they had antecedents in his earlier life in India, andthey lead to his decease one year after his return to Madras.Despite appreciating his brilliance, Hardy urged him to take lectures (forinstance, Ramanujan hardly knew anything about complex analysis), and inparticular he insisted in proofs, not just conjectures. That was not easilycompatible with Ramanujan’s mentality, but he published 32 high-impactpapers during his 5 years in Cambridge, 7 of them together with Hardy [5].In 1918 Ramanujan was elected as a Fellow of the Royal Society, as one ofthe youngest members in its history, and half a year later he also became aFellow of the Cambridge Trinity College.Figure 2:
Godfrey Hardy (1877–1947) Being a devout Hindu, Ramanujan was a strict vegetarian, which was highly unusualin England at that time. partitioning, which(surprisingly) involves the number π [6]. Ramanujan traced this number inall kind of contexts; best known is a series that he postulated in Ref. [7](along with a variety of other π -approximation formulae),1 π = √ (cid:88) n ≥ (4 n )!(4 n n !) n n . (1.1)It converges exponentially (despite the factor (4 n )! in the numerator), thusit provides one of the fastest algorithms to compute π . If we truncate at n max = 0 , ,
2, we obtain the corresponding approximation π n max , whichdiffers from the exact value of π as | π − π | (cid:39) . · − , | π − π | (cid:39) . · − , | π − π | (cid:39) . · − . (1.2) How
Ramanujan arrived at such formulae is hard to know: Hardy laterdescribed it as a “process of mingled argument, intuition, and induction,of which he was entirely unable to give any coherent account” [8].Here we are going to address a relatively simple subject, which Ramanu-jan mentioned in his first letter to Hardy [9], and which he had written downbefore in Chapter VI of his Second Notebook [10]. This letter contains twoapparently weird formulae for divergent series, (R) (cid:88) n ≥ n = 1 + 2 + 3 + 4 + 5 + · · · = − , (1.3) (R) (cid:88) n ≥ n = 1 + 8 + 27 + 64 + 125 + · · · = 1120 , (1.4)where the sums run from n = 1 . . . ∞ , and the superscript (R) indicates“Ramanujan summation” [11]. These strange relations have fascinated gen-erations of people; for instance a discussion of eq. (1.3) in YouTube [12],dated 2016, has over 2.4 million views, and over 5000 quite controversialcomments.Of course, it is provocative to write these relations as straight equations,as Ramanujan did (without any superscript), but it fulfills the purpose ofattracting attention and causing debate. Still, in the following we are going4o replace the symbol = by ∧ = , meaning “corresponds to” or “is associatedwith”. In this sense, we are going to show that the fractional values on theright-hand side do have a meaning, not only as a mathematical peculiarity,but they can even be used to derive physical results.Unlike other addressees of Ramanujan’s letters, Hardy recognized thevalues of the Riemann ζ -function , or p -series. For Re z > ζ ( z ) = (cid:88) n ≥ n z = 11 z + 12 z + 13 z + 14 z + . . . . (1.5)In 1739 Leonhard Euler had computed explicit expressions for ζ (2 n ), n ∈ N + ,and later he also conjectured a ζ -functional relation [13]. More than a centurylater, in 1859, Bernhard Riemann established the analytic continuation of the ζ -function to C − { } [14], see Appendix C. In this sense, Hardy noticed thatRamanujan’s results can be interpreted as ζ ( −
1) and ζ ( −
3) (although thesevalues are not explicitly given in Ref. [14]).Riemann was a leading mathematician of the 19th century, and of alltimes, cf. footnote 2. Like Ramanujan he lived his youth in harsh poverty,until he was appointed to a post in G¨ottingen, on Carl Friedrich Gauss’ rec-ommendation. Another analogy is that he soon suffered from health prob-lems. Hoping that a warmer climate might help against his tuberculosis [15](which was also among Ramanujan’s diseases [3]), he spent extensive periodsin Italy, where he died in 1866, at the age of 39.Differences from Ramanujan’s life are that Riemann had access to educa-tion at leading mathematical institutes, in G¨ottingen and Berlin, and that hepublished his results only after elaborating rigorous proofs. His publicationshad an enormous impact, but only Ref. [14] deals with number theory. Therehe discussed the density of prime numbers, and it was in this context that hepostulated the analytic continuation of the ζ -function; the crucial functionalequation is displayed in Appendix C. In contrast to Ramanujan, Riemannwas an expert on complex analysis. Presumably he had hand-written noteswith many more important results, but after his sudden death his house-cleaner burned part of these notes, until some mathematicians managed tostop her [16].Ramanujan did not provide an actual derivation of formulae (1.3) and(1.4), but in the first case he assigned — in Ref. [10] and also his first letter This is another field of common interest of these two geniuses: later Ramanujanproposed his own formula for the prime number density, which is, however, not as accurateas he had expected.
Bernhard Riemann (1826–66) to Hardy — a value to another divergent series, as an intermediate step toarrive at relation (1.3). That series corresponds to a special case of
Dirichlet’s η -function, or alternating ζ -function, η ( z ) = (cid:88) n ≥ ( − n − n z = 11 z − z + 13 z − z . . . , (1.6)which converges for Re z >
0. At Re z > ζ ( z ) − η ( z ) = 2 (cid:88) n ≥ n ) z = 2 − z ζ ( z ) , ζ ( z ) = 11 − − z η ( z ) . (1.7)The latter defines ζ ( z ) in the domain Re z > ∨ z (cid:54) = 1.In particular, Ramanujan wrote down its continuation to [10] E := η ( −
1) = (cid:88) n ≥ ( − n − n = 1 − − . . . ∧ = 14 . (1.8)We are going to confirm this value, and follow his path to relations (1.3) and(1.4), which we finally apply to physics, in particular to the Casimir effect. Heuristic derivation of (cid:80) n ≥ n ∧ = − / Series have both fascinated and confused mathematicians over and over again,throughout history. The famous “paradox” by Zeno, which describes a racebetween Achilles and a tortoise (and further “paradoxes” of a similar style)caused a deep crisis in the mathematics of Ancient Greece (see e.g.
Ref. [17]),because the concept of convergent series — in this case, a geometrical series— had not yet been understood.Here we just take the familiar geometrical series as the point of departure.For | z | < G ( z ) = (cid:88) n ≥ z n = 1 + z + z + z · · · = 1 + z G ( z ) ⇒ G ( z ) = 11 − z . (2.1)The series converges only for | z | <
1, but the final function G ( z ) is de-fined all over C − { } . Moreover, the complex function G ( z ) is holomorphic (or complex analytic , i.e. complex differentiable) in C − { } , and therefore meromorphic in C , which implies that its analytic continuation from the disk | z | < C − { } is unique, cf. Appendix C.This allows us to define Grandi’s series G = 1 − − − · · · = (cid:88) n ≥ ( − n (2.2)by means of analytic continuation, G ∧ = G ( z ) | z = − = 12 . (2.3)We can readily extend this scheme to the η -function in eqs. (1.6), (1.8). Tothis end, we first return to safe grounds, i.e. to | z | <
1, where G ( z ) = 1 + 2 z + 3 z + 4 z · · · = (cid:88) n ≥ n z n − = G (cid:48) ( z ) = 1(1 − z ) . (2.4) We recall that this is a powerful property, which guarantees that the function hasderivatives of any order in its domain of holomorphy, and that it coincides with its powerseries. Moreover, since G (cid:48) ( z ) (cid:54) = 0 it is also conformal, i.e. angle conserving: if we interpretthe function G ( z ) as a map C → C , and consider two curves γ ( z ), γ ( z ), which intersectin z with a certain angle, then the maps of these curves intersect in G ( z ) with the sameangle. G ( z ) is holomorphic as well, again with a (unique) analyticcontinuation to C − { } . This implies in particular E = (cid:88) n ≥ ( − n − n ∧ = G ( z ) | z = − = G = 14 , (2.5)which coincides with Ramanujan’s result (1.8).However, this is not yet what we need in order to assign a value to thenotorious series, which we denote as R := (cid:88) n ≥ n = 1 + 2 + 3 + 4 + 5 + . . . . (2.6)It would correspond to G ( z ) | z =1 , but z = 1 is just the point where thisfunction has its double pole. Following Ramanujan’s line of thought [10], weproceed by introducing another series R ( z ) = 1 − z + 3 z − z + 5 z · · · = (cid:88) n ≥ n ( − z ) n − , (2.7)which also converges at | z | <
1, and we formally obtain R ∧ = R ( − Againwe refer to the safe region, i.e. to the disk | z | <
1, where we take the difference G ( z ) − R ( z ) = 4 z (1 + 2 z + 3 z + . . . ) = 4 z (cid:88) n ≥ nz n − . (2.8)This operation can only be justified inside the convergence disk, but once itis carried out, taking the limit z → − lim z →− (cid:104) G ( z ) − R ( z ) (cid:105) = 14 − R ∧ = − R ⇒ R ∧ = − . (2.9)Thus Ramanujan removed the divergence in a controlled manner, whichleaves an unambiguous finite value, and Hardy noticed that this assignmentcorresponds to R = ζ ( − C = 1 + 1 + 1 + 1 + 1 . . . . The limit z → C ∧ = ζ (0) ∧ = − η (0) ∧ = −G = − , (2.10) The reason for the notation with an index 1 will become clear in Appendix B. z = − R ∧ = − E ∧ = −
112 = ζ ( − . (2.11)So far this may look like a mathematical playground, but in the nextsection we are going to apply this result to a physical toy model, where itleads to sensible results. In Section 4 we proceed to a setting, which refers tophysical phenomenology; for that purpose we will need relation (1.4), whichcorresponds to ζ ( − In this section and beyond, we are going to deal with quantum field the-ory.
General introductions can be found in a number of textbooks, such asRefs. [18] (and a popular science description is given in Ref. [19]), but inorder to follow the derivations in Sections 3 and 4 only very little knowledgeabout it is required. Our notation implicitly refers to the functional integralformulation, where the fields are functions of the space and time variables,with values which can, for instance, be real numbers (then it is a neutralscalar field , as in this section), or vectors (as in Section 4). In general, allfield configurations — i.e. all possible values in each space-time point — areintegrated to obtain expectation values of observables.Here, however, we are only concerned with the ground state contributionsof free fields. For a neutral scalar field we can imagine an (infinite) set ofcoupled harmonic oscillators, one at each space point. A Fourier transformyields oscillators for all possible frequencies.
A priori these frequencies arenot restricted, so if we sum up their ground state contributions to the vacuumenergy density, the result diverges.We are going to be confronted with these ultraviolet (UV) divergences:they require a regularization, i.e. a mathematical modification which makessuch sums (or integral) finite, enabling calculations. In the end we want toremove the regularization, hence we aim at a cancelation of the UV diver-gences. This can often be achieved by subtracting divergent terms, so-called counterterms , which correspond to some limit; without taking that limit, afinite quantity remains. This procedure is known as a renormalization: it Alternatively, in the canonical formalism the fields are operator-valued.
Hendrik Casimir (1909–2000)
As a toy model, we consider a free, neutral, massless scalar field on a line, φ ( t, x ) ∈ R , x ∈ R . At the points x = 0 and x = d > Dirichletboundaries, which force the field to vanish, i.e. φ ( t,
0) = φ ( t, d ) = 0. Inthis interval the field configurations can be Fourier decomposed into standingwaves with wave numbers k n = nπ/d , n = 1 , , . . . (such that sin( k n d ) = 0).In natural units ( (cid:126) = c = 1), they contribute k n / vacuum energy density in this interval we formally obtain ρ d = 12 d (cid:88) n ≥ k n = π d (cid:88) n ≥ n , (3.1)where we encounter the divergent term R , to which Ramanujan assigned thevalue − /
12. We are now going to illustrate — in a physical framework —why, and in which sense, this value is indeed meaningful.10s usual in quantum field theory, we first need a regularization (as wementioned before), but it doesn’t need to be fully specified. We regularize ρ d by performing a substitution n → f ( n ) = n r ( n/ Λ d ) , with r (0) = 1 , (3.2)where f and r are smooth functions on R +0 (an infinite number of timescontinuously differentiable), and Λ is an energy cutoff. If we remove it,Λ → ∞ , we recover the term before regularization. At finite Λ we require lim x → + ∞ f ( x ) = 0 , lim x → + ∞ f ( k ) ( x ) = 0 , (3.3)where f ( k ) is any odd derivative ( k = 1 , , , . . . ) . A simple example of f ( n ) in eq. (3.2) is the heat kernel regularization ,where the function r is exponential, f ( n ) = n exp( − n/ Λ d ), which leads to ageometrical series, (cid:88) n ≥ ne − n/d Λ = Λ d ∂∂ Λ (cid:88) n ≥ e − n/ Λ d = Λ d ∂∂ Λ 11 − exp( − / Λ d )= (Λ d ) −
112 + O (cid:16) d (cid:17) . (3.4)We already see R = − /
12 popping up, but we want to proceed to a broaderperspective, which generalizes the regularization, and which also clarifies therˆole of the UV divergent term, along the lines of Ref. [21].In an infinite interval, d → ∞ , the formal term (3.1) for the energy densityturns into a momentum integral. We expand the difference ρ d − ρ ∞ , at theregularized level, by means of the Euler-Maclaurin formula, N (cid:88) n =1 f ( n ) − (cid:90) N f ( x ) dx = f ( N ) − f (0)2 + (cid:88) j ≥ B j (2 j )! (cid:16) f (2 j − ( N ) − f (2 j − (0) (cid:17) , (3.5) where N ∈ N + . A finite number N represents another component of the UVregularization: in this case, we sum over k n only up to k n, max = N π/d .The powerful formula (3.5) was independently derived by Euler and byColin Maclaurin around 1735. It is very useful in field theory, in particularwhen dealing with finite temperature or finite-size effects. Since we assumethe function f to be smooth and to fulfill the condition (3.3), this series11onverges both at finite N and in the limit N → ∞ . The coefficients in thelast term are the
Bernoulli numbers, which can be defined in a way relatedto eq. (3.4), x − e − x = (cid:88) k ≥ B k x k k ! . (3.6)This yields B = 1, B = , B = , B = − , B = B = B = · · · = 0, ( B , B . . . do not vanish, but we won’t need them).We insert in eq. (3.5) a function f which fulfills the conditions (3.2) and(3.3), and we take the UV limit in two steps: first we let N → ∞ ; due toeq. (3.3) all contributions vanish in this limit. As for the terms at x = 0, wenote that f (0) = 0, f (cid:48) (0) = r (0) = 1, f ( k ) (0) = O ((Λ d ) − k ), k = 2 , , . . . ,hence the second step of the UV limit, Λ → ∞ , leads to ρ d − ρ ∞ = π d (cid:16) − B (cid:17) = − π d , (3.7)where R = − B / − /
12 is crucial, in agreement with eq. (3.4). Werecover Ramanujan’s assignment of a finite value to the divergent series ineq. (3.1), i.e. Ramanujan summation corresponds to the subtraction of theinfinite-volume limit of the vacuum energy density.
This density divergesboth in a finite and in an infinite interval, but the difference , i.e. its finite-size effect, is finite and well-defined. The elimination, or isolation, of a UVdivergent term ((Λ d ) in eq. (3.4)), in order to deal with finite differences,is the basic idea of renormalization. In field theoretic jargon, we have sub-tracted the counterterm ρ ∞ , which cancels the divergence in the series (3.1).Interestingly, this recipe matches exactly the relation ζ ( −
1) = (R) (cid:88) n ≥ n = − B = − , (3.8) If we truncated this expansion at some odd integer J/
2, such that we deal with (cid:80) J/ j =1 . . . , then there is a remainder term R J on the right-hand side. It can be esti-mated as | R J | ≤ ζ ( J ) (cid:82) N dx | f ( J ) ( x ) | / (2 π ) J [22], hence the above assumptions imply lim J → + ∞ R J = 0. Bernoulli numbers were a particular passion of Ramanujan, who had certainly readabout them in Ref. [2]. His very first paper discussed their properties [23]. For instance,he showed that the denominators of B , B , B , B . . . (in lowest terms) all contain theprime factors 2 and 3 exactly once. This becomes obvious if we define g ( x ) = x/ (1 − e − x ) and compute g ( x ) − g ( − x ) = x . Ramanujansummation ; its general properties are defined and explored in Ref. [11]. Forthe purpose of this article, it is sufficient to point out that for a series ofthe form (cid:80) (R) n ≥ n k , k ∈ N , the Ramanujan summation coincides with the ζ -function ζ ( − k ) (defined by analytic continuation), see Appendix C. Itfurther corresponds to the finite term in the Euler-Maclaurin expansion ofthe difference lim N →∞ [ (cid:80) Nn =1 n k − (cid:82) N x k dx ], which can be read off from eq. (3.5),and which generalizes eq. (3.8) to ζ ( − k ) = (R) (cid:88) n ≥ n k = − B k +1 k + 1 , k ∈ N . (3.9)The question remains: beyond the satisfaction of deducing a finite result,why are we interested in this difference? What is its physical meaning? Oneis tempted to reply: the change of the vacuum energy, as a function of d ,implies a force between the Dirichlet boundaries, and the counterterm canbe subtracted since it does not depend on d , so it does not contribute to thisforce. However, there is still a caveat, which is often ignored: the boundariescould also affect the energy outside the interval [0 , d ]. That could contributeto the force between the boundaries, so we have to be careful.A sound approach introduces three Dirichlet boundaries, at the points 0, d , L , with 0 < d < L , see Figure 5 (left). The idea is to keep the extremeboundaries at 0 and L fixed, while the one at d is a variable “piston”. Inthis way, the energy outside the interval [0 , L ] remains constant, while theenergy inside this interval can be computed explicitly, so everything is undercontrol. From eq. (3.7) we obtain the total vacuum energy E ( d ) = − π d − π L − d ) + E out = − πL d ( L − d ) + E out . (3.10)The term E out , which represents the energy outside the interval [0 , L ], isdivergent, but it does not depend on d . Generally a force is obtained asthe negative gradient of the potential energy. In our 1-dimensional case,this operator reduces to the negative derivative with respect to d (the onlyvariable involved). Therefore the term E out is irrelevant for the force actingon the “piston” at d , which is obtained as F ( d ) = − E (cid:48) ( d ) = − πL L − dd ( L − d ) , (3.11)13 d0 x -40-30-20-10 0 10 20 30 40 0 0.2 0.4 0.6 0.8 1 F ( d ) d Figure 5:
Left: Setting for the Casimir effect on a line: we apply Dirichletboundaries at the positions , d and L , with < d < L . Right: the Casimirforce F ( d ) , which acts on the “piston” at d according to eq. (3.11), in unitssuch that L = 1 . and depicted in Figure 5 (right). It is odd with respect to the center d = L/ attractive towards the nearer fixed boundary, at 0 or L . Hence d = L/ L (cid:29) d we obtain a force,which is attractive towards the boundary at 0, F ( d ) (cid:39) − π d , and whichcoincides with the 2-boundary picture of eq. (3.7). Hence, in that picture,ignoring effects outside the interval [0 , d ] is justified after all (varying d doesnot change the energy in the half-line with x > d ).To summarize this section, we have renormalized the system by discard-ing an additive, infinite constant in the energy density, the counterterm ρ ∞ ,which represents the infinite-volume limit. In order to compute the remain-ing finite term, a finite-size effect in this case, a regularization is needed.Then the Euler-Maclaurin formula can be applied, and by removing bothUV cutoffs, k n, max = N d/π → ∞ and Λ → ∞ , we arrive at the finite result(3.7). It does not depend on the choice of the regularizing function f , as longas the conditions (3.2) and (3.3) are fulfilled. This leads to finite values forthe vacuum energy in the interval [0 , L ], and for the force F ( d ) in eq. (3.11),which acts on the “piston”. 14 The Casimir force in 3-dimensional space
We proceed to a realistic situation, which deals with the vacuum energy ofthe photon field in (3 + 1)-dimensional space-time. The simplest setting isshown Figure 6: it involves two parallel, conducting plates, of the samerectangular shape and area A , separated by a short distance d . d A Figure 6:
Setting for an experimental test of the Casimir effect: one measuresthe force between two parallel, conducting plates of area A , separated by ashort distance d . We rely on our experience from the 1-dimensional toy model to conjecturethat it is sufficient to consider the energy E ( d ) between the plates. This isappropriate when the area A is large ( √ A (cid:29) d ). The photon momentumcomponents parallel to the plates — which we denote as k , k — are treatedas continuous. Hence we perform a discrete sum, in analogy to eq. (3.1), onlyover the vertical component k , and the energy between the plates takes the In theory we assume perfect conductivity, this is what it takes to implement exactDirichlet boundaries. The experiments have been performed with well conducting metalplates, which provide a good approximation, cf. Section 5.2. The generalization withrespect to the dielectric constant was theoretically studied by Evgeny Lifshitz [24]. Throughout this article we refer to the standard scenario with static Dirichlet bound-aries. The two-fold generalization of the Casimir effect with dynamical Robin boundariesis discussed for scalar fields in Refs. [25]. E ( d ) = A d ρ ( d ) = 12 A π ) (cid:90) dk dk (cid:88) n ≥ (cid:114) k + k + (cid:16) πnd (cid:17) = A π (cid:88) n ≥ (cid:90) ∞ dK K (cid:114) K + (cid:16) πnd (cid:17) = A π (cid:88) n ≥ (cid:104) K + (cid:16) πnd (cid:17) (cid:105) / (cid:12)(cid:12)(cid:12)(cid:12) ∞ , (4.1)where we have inserted a factor 2 for the two photon polarization states.Note that we have not regularized so far. If we do so and follow theprocedure of Section 3, we can renormalize by subtracting the energy in thesame volume but without plates, E ∞ = A d ρ ∞ . In this difference, firstthe UV contribution due to K → ∞ cancels (a physical interpretation isthat infinitesimally short wavelengths are not sensitive to the presence ofboundaries at a finite distance). Regarding K = 0, we apply the Euler-Maclaurin expansion (3.5) to (cid:80) n ≥ · · · − (cid:82) ∞ dk . . . . This corresponds tothe Ramanujan summation over n , which we again express as a ζ -function, E ( d ) A ∧ = − π π d ζ ( −
3) = − π d ,F ( d ) A = − E (cid:48) ( d ) A = − π d . (4.2)We have used eq. (3.9) and inserted B = − /
30, in agreement with eq. (1.4).Although the sign of B is opposite to B (which we inserted in eq. (3.7)), weobtain again an attractive force between the Dirichlet boundaries: note thatthere is another sign flip due to the integral over K , where the lower boundcontributes.The question arises what magnitude this force takes for realistic sizes A and d , and if such a force can be measured. The first conclusive experimentwas achieved in 1997 by Steve Lamoreaux, who succeeded in measuring theCasimir force to 5 % accuracy [26]. This was soon followed by Umar Mohideenand Anushree Roy [27], who took into account the corrections due to finitetemperature, finite conductivity and the roughness of the surfaces. In theseexperiments, the geometrical structure was a plate and a sphere, because ofthe difficulty in keeping two plates parallel to a very high precision. To obtain E ∞ we start from the term in the upper line of eq. (4.1), convert (in thelarge- d limit) πn/d to the continuous momentum component k , and (cid:80) n ≥ to d/π (cid:82) ∞ dk ,which leads to E ∞ = A d (2 π ) − (cid:82) d k | (cid:126)k | . A =1 . × . d varied from 0 . µ m to 3 µ m. Inorder to compute the predicted force in Newton (N), we have to insert afactor (cid:126) c in eq. (4.2), which leads to F ( d ) (cid:39) − . · − N (cid:16) µ m d (cid:17) A cm . (4.3)Hence the predicted force in this experiment varied between F (cid:39) − . · − Nand − . · − N. Forces of this range are in fact measurable: for instanceRef. [26] used a torsion pendulum and laser interferometry, and Ref. [27]employed an atomic force microscope. In the experiment reported in Ref. [28],one of the parallel plates was the face of a cantilever beam, free to oscillate.The variation of the force was observed by measuring shifts in its resonatorfrequency, by means of a fiber optic interferometer.The precision in recent experiments is around, or below, 1 %.
We don’t know what exactly Ramanujan had in mind when he introducedhis summation of divergent series, which we now denote as Ramanujan sum-mation, such as relations (1.3) and (1.4). In his letter to Hardy he only doc-umented one intermediate step, relation (2.5). In his Second Notebook [10]he additionally hinted at the continuation (2.9), and he mentioned the dif-ference between summation and integration, which we also reviewed. This isa valid argument, and apparently Ramanujan has re-invented an equivalentform of the Euler-Maclaurin formula (he did not use that term [10], nor doesthis formula appear in Ref. [2]).A reason for the sparse documentation in Ramanujan’s notebooks was— in addition to his intuitive way of thinking — that he mostly worked ona slate and only wrote down final results on paper, which was valuable (inparticular for him, who was living in poverty). It is also conceivable that hewas influenced by Carr’s telegram style [2], cf. footnote 3. In any case, his We recall that we have been using natural units. This factor shows that we are dealingwith a relativistic quantum effect. ζ -function, at least withrespect to negative integer arguments, see eq. (3.9). In fact, he also rediscov-ered the analytically continued Γ-function with values in C − { , − , − . . . } .He highlighted this idea in the introduction of his first letter to Hardy [9],unaware that this had been known before; in particular, Riemann had usedit in Ref. [14].These finite values for divergent series may look like a mathematical game,which is rather disconnected from reality. However, it is possible to establishconsistent rules for the Ramanujan summation of divergent series by care-fully dealing with properties like linearity and translation [11]. Moreover, wereviewed their striking application to physics, where they enable the predic-tion of a force, which has in fact been measured. The meaning of the Casimireffect will be discussed in the following two subsections. Numerous authors infer from the experimental observation of the Casimirforce the existence of the vacuum energy of the photon field, ρ vac , as predictedby Quantum Electrodynamics (QED), e.g. Refs. [26–31]. As a typical quo-tation, Ref. [31] states that “the existence of zero-point vacuum fluctuationshas been spectacularly demonstrated by the Casimir effect.” It does, how-ever, not affect usual experiments, which only depend on energy differences, not on the additive constant ρ vac . Still, such an energy density throughoutthe Universe, known as Dark Energy, is indeed manifest since it affects theexpansion of the Universe.It corresponds to the
Cosmological Constant in General Relativity: in itsabsence — which was generally assumed from the 1930s to the 1990s — theexpansion of the Universe would be decelerated. However, at the very endof the 20th century it was observed that the expansion is accelerated . Thisis best described by a positive Cosmological Constant, which corresponds toa Dark Energy density of about ρ DE ≈ (0 .
002 eV) .Unfortunately this value is totally incompatible with the vacuum energydensity ρ ∞ that we discussed. First, ρ ∞ seems to diverge, as we saw, but onemight impose an UV cutoff in the integral (cid:82) d k | (cid:126)k | , most naturally at thePlanck energy. This leads to a finite value ρ Planck , which is, however, much too large, ρ Planck /ρ DE = O (10 ). This was concluded from the distance and redshift of a set of type Ia supernovea [32]. supersymmetry could argue that in a perfectlysupersymmetric world the Dark Energy vanishes (since bosons and fermionsappear in pairs of the same mass, and the fermionic ground state energy isnegative, with the same absolute value [18]). However, even if supersymmetryexists, it has to be badly broken in our low-energy world (otherwise particleslike the “selectron” would have been observed), and the required extent ofbreaking still implies a Dark Energy density, which exceeds ρ DE at least bya factor O (10 ) [30].Hence any evidence for the existence of the QED photon field vacuumenergy ρ vac would be puzzling. Albeit, Julian Schwinger et al. computedthe Casimir force by means of a (complicated) source field technique, in theframework of S-matrix theory, without need to refer to ρ vac [33]. Part ofthe literature concludes from that work that the Casimir experiments do notnecessarily imply the reality of ρ vac , which could be welcome as a remedyagainst the disastrous discrepancy by 121 order of magnitude.One might object that the S-matrix approach has not been shown to beequivalent to full QED. Thus the question remains whether or not relativisticquantum physics could be formulated without ρ vac .If ρ vac exists, in the field theoretic sense, one might wonder whether thefrequency of a photon is affected when it passes through regions of different ρ vac , e.g. when it transversally passes through a Casimir cavity, similar toBernoulli’s Principle in fluid dynamics. Regarding its vertical motion, thereis even a prediction that the speed of light could be affected [34].If we wanted to construct a cavity between two conducting plates with ρ DM = | ρ d | = π / (720 d ), we would need a separation of d ≈ . µ m, whichhappens to be close to the minimal separation in the Padua experiment. One could question what kind of force this really is. It does not seem toappear in the famous list of four forces, which can be described by gaugefields, nor does it match further interactions in the Standard Model of particlephysics (Yukawa couplings and the Higgs field self-coupling). However, beingan effect of the photon field, this force must ultimately be electromagnetic,although this is not explicit in the above discussion. From this perspective,19t can be best described as a van der Waals force between the metal plates.This is the picture that Casimir and Polder originally had in mind [20].So, do we have two equivalent descriptions? This seems puzzling again:in the van der Waals picture, the force depends on the value of the elec-tromagnetic coupling constant α = e / π (cid:39) / i.e. a retarded vander Waals force, which does not require ρ vac . In Jaffe’s own words, “Casimireffects can be formulated and Casimir forces can be computed without refer-ence to zero point energies.” Thus they contradict the paradigm in this field,but this issue remains controversial.Ref. [36] obtains a Casimir force, which does depend on α , such that F ( α = 0) = 0, and F ( α (cid:39) / α → ∞ leads to a finite result: F ( α → ∞ ) just matchesthe force obtained from ρ vac , which is often close to F ( α (cid:39) / . µ m (which is experimentallyrealistic), the consideration with ρ vac is a good approximation if α (cid:29) − [36], which is easily accomplished by the phenomenological value.A wide-spread objection against that point of view refers to examples,where the consideration based on ρ vac leads to a repulsive Casimir force [37], e.g. for specific parallelepipeds [38]. That feature is not easily encompassedby van der Waals forces. For instance Lamoreaux [26] writes: “the Casimirand van der Waals forces are quite different; the van der Waals force is alwaysattractive, whereas the sign of the Casimir force is geometry dependent.”Ref. [40] disagrees and assigns the repulsive result to the negligence of cutoffeffects. In fact, an approach by Ricardo Cavalcanti [41], which is manifestlyfree of any cutoff dependence, only obtains attractive Casimir forces.Jaffe and his collaborators insist that the physical Casimir force is alwaysattractive [36, 40], and therefore compatible with the van der Waals picture.If this alternative to the paradigm — as expressed in Refs. [26–31] — iscorrect, then the approach that we reviewed is not the most precise one, butit is still in agreement with the experimental results. We refer to the van der Waals force in the narrow sense, also known as London–vander Waals force, i.e. the attractive multipole interaction between molecules [35]. On the other hand, Ref. [35] predicted a repulsive Casimir–van der Waals–type force,which agrees with an experiment with interacting materials immersed in a fluid [39]. ρ vac as encoded in QED. There is a consensus, however,that we do not know how to theoretically derive the Dark Matter density ρ DE ,and that we do not understand the enormous discrepancy from the vacuumenergy predicted by quantum field theory.So far our discussion in this subsection focused on the Casimir effect dueto QED, which was described in Section 4. In principle, such an effect alsoexists for other gauge fields, but only for QED it is simple and instructive,in particular because the photon field does not self-interact. This is also theonly case where the Casimir force is experimentally confirmed.For instance in Quantum Chromodynamics (QCD) — the gauge theoryof the strong interaction — this effect is much less transparent because ofthe complicated self-interaction of the gluon field [18], which occurs sincethe QCD gauge group SU(3) is non-Abelian. At low energy its behavior isdominated by non-perturbative effects, which are hard to compute, and whichinduce an intricate vacuum structure. Studies in Euclidean space often focuson the rˆole of instantons [42]. For a static quark–anti-quark pair (which is anidealization), a multipole expansion has been applied to estimate the Casimirforce [43]. Another study [44] deals with the (restricted) Gribov–Zwanzigeraction.Furthermore, there are numerous attempts to theoretically investigate the gravitational
Casimir force, although this is a quantum effect and we do nothave any (fully satisfactory) theory of quantum gravity. Numerous papersrefer to unusual gravitation theories; studies which are (roughly speaking)close to the framework of General Relativity include Refs. [45]. The questionwhether an experimental demonstration of such an effect, with gravitationalwave mirrors, would prove the existence of gravitons is discussed (and nega-tively answered) in Ref. [46].
There are further applications of Ramanujan summation in the perturbativeexpansions of quantum field theory, which can be treated by the ζ -functionregularization [47]. It is an alternative to dimensional regularization, which ismost popular in perturbation theory. The ζ -function regularization removesfrom the beginning the UV divergent terms in the Laurent series by inserting21 ( − k ), thus preventing the necessity of counterterms. Stephen Hawkingadvocated its application in curved space-time [49].Applications of the ζ -function in bosonic string theory are reviewed inRefs. [50]. Ref. [51] summarizes a key point as follows: a particle mass m isobtained as m = 1 σ (cid:104) j + D − (R) (cid:88) k ≥ k (cid:105) , (5.1)where σ is the string tension, D is the space-time dimension (such that aworldsheet lives in D − j is the string excitation num-ber (here also the Planck scale is set to 1). The term with the sum overthe modes k represents the ground state energy E , where one applies Ra-manujan summation, E = − ( D − / σ . The case j = 1 describes spin-1particles with only two polarization states, which must therefore be massless.This condition yields the space-time dimension D = 26, where bosonic stringtheory is formulated [50]. The deeper reason is the requirement to cancel theconformal anomaly. A detailed pedagogical description is given in Ref. [52]. Acknowledgement:
This work was supported by UNAM-DGAPA-PAPIIT,grant number IG100219.
A The failure of partial sums
Many controversial discussions about relations like (1.3) — for instance nu-merous comments on Ref. [12] — refer to partial sums of a few summands.In the framework of divergent series, separating them is conceptually wrongand leads to contradictions. It is entertaining to look at some examples, tosee what one should beware of, e.g. R = 1+(2+3+4)+(5+6+7)+(8+9+10)+ . . . ? = 1+9 R → R ? = − / , (A.1)which deviates from Ramanujan’s value. One might even feel tempted to payattention to this alternative value, since it is consistent with blocks of any Its mathematical equivalence to the heat kernel regularization, see Section 3, wasdemonstrated by Hardy and Littlewood [48]. u ≥ R = 1 + 2 + · · · + u −
12 + (cid:16) u + 12 + · · · + 3 u − (cid:17) + (cid:16) u + 12 + · · · + 5 u − (cid:17) + . . . ? = u −
18 + R u → R ? = − . (A.2) However, we can show that this approach is even intrinsically inconsistent bychoosing blocks of an even number g summands (where the boundary termsare equally divided between the blocks), R = 1 + 2 . . . ( g −
1) + g (cid:16) g g · · · + ( 3 g −
1) + 3 g (cid:17) + (cid:16) g g . . . (cid:17) + . . . ? = g R g → R ? = − g g − , (A.3)which only coincides with the claim (A.2) in the limit g → ∞ .Of course, the applicability of partial sums can be disproved more easily, e.g. in Grandi’s series or Dirichlet’s η -function, if we write them as G = (1 −
1) + (1 −
1) + (1 −
1) + · · · = 1 + ( − − . . . E = (1 −
2) + (3 −
4) + (5 −
6) + · · · = 1 + ( − − . . . which seems to suggest the contradictory values G ? = 0 or G ? = 1, and E ? = ∓ (1+1 + 1 + 1 . . . ) . = ∓C . Again we encounter the series C , which also appearedin Ramanujan’s Second Notebook [10], as we anticipated in eq. (2.10). Wewill come back to it in Appendix B. In that case, a division into blocks of n summands seems to suggest C = (1 + · · · + 1) + (1 + · · · + 1) + . . . ? = n C , C ? = 0or ∞ (while separating k summands C = k + C , C = ∞ ). B Analytic continuation from the unit disk
We now follow the scheme of Section 2 by writing the series under consider-ation in terms of a variable z ∈ C , such that they converge for | z | <
1, andthe divergent series of interest corresponds to the limit z → −
1. Working inthe convergence region | z | < η -function, G ( z ) = 1 + z + z + z + · · · = 11 − z z →− −→ G = 1 − − . . . ∧ = 12 .η ( z ) = 1 + 2 z + 3 z + 4 z + · · · = G ( z ) = G (cid:48) ( z ) = 1(1 − z ) z →− −→ E = 1 − − . . . ∧ = 14 . (B.1)As long as this limit is finite, taking the analytic continuation is trivial. Thisallows us to perform operations, which remain valid in the limit z → −
1, like η ( z ) + G ( z ) = 1 z [ η ( z ) −
1] = 2 + 3 z + 4 z + . . . z →− −→ η ( z ) − G ( z ) = zη ( z ) = z + 2 z + 3 z + . . . z →− −→ − . (B.2)In this framework, the limit z → − z = 1, in particular when we refer to the series C = 1 + 1 + 1 + 1 + 1 . . . (B.3)which worried us before in Section 2 and Appendix A. Here we considerthree different regularizing functions (at | z | < C ( z ) = 1 − z + z − z + z · · · = 11 + z ,C ( z ) = 1 + z + z + z · · · = 11 − z ,C ( z ) = − ( z + z + z + . . . ) = − z − z . (B.4)We can involve the functions C i ( z ) in a variety of relations, such as G ( z ) =(1 + z ) C ( z ) = 1 − (1 + z ) C ( z ). Of course they work both in the form ofseries and of functions, in agreement with C i ( z → − → ∞ . We can alsobuild linear combinations of the functions C i ( z ), for instance G ( z ) = C ( z ) − C ( z ) = 11 − z , (B.5)where the singularity at z = − removed. When we now insert C = C ( z → −
1) = C ( z → − ζ (0), as we saw in eq. (2.10), and whichRamanujan had reported [10], C ∧ = −G ∧ = − = ζ (0) . (B.6)It was (apparently) a step of this kind that Ramanujan performed tocompute the famous series (2.6), R = 1 + 2 + 3 + 4 + 5 . . . . Here we considerthe regularizations R ( z ) = 1 − z + 3 z − z + 5 z · · · = C ( z ) = − C (cid:48) ( z ) = 1(1 + z ) ,R ( z ) = 1 + 2 z + 3 z + 4 z · · · = η ( z ) = 1(1 − z ) . (B.7)We introduced R ( z ) before in eq. (2.7), and we used it, at the regularizedlevel, in identity (2.8), which we can now write in the compact form η ( z ) = R ( z ) + 4 zR ( z ) . (B.8)Again the singularity at z = − ε = z + 1 leads to different Laurent series for R ( ε ) = 1 /ε and R ( ε ) = (1 + ε + 3 ε / / ε + O ( ε ), such that the right-hand side of eq. (B.8)takes the expected form 1 / O ( ε ).If we insert R = R ( −
1) = R ( − R as a finite constant, weretrieve Ramanujan’s famous result14 ∧ = − R ⇒ R ∧ = −
112 = ζ ( − . (B.9)However, this procedure only works when the terms are arranged suchthat the limit of interest ( z → − regular, otherwise this stepis not controlled. Consider for instance the identity R ( z ) + C ( z ) = 2 − z + 4 z − z · · · = 1 z [1 − R ( z )] . (B.10)If we now insert C = C ( −
1) and R = R ( −
1) = − z R ( z ) | z = − , we end upwith C ? = −
1, which contradicts eq. (B.6).Another example, which refers to R , is the identity R ( z ) = R ( z ) η ( z ) = 1(1 + z ) (1 − z ) . (B.11)25arelessly inserting R = R ( −
1) = R ( −
1) purports R ? = R , R ? = 0. Thereason for this fiasco is that eqs. (B.10) and (B.11) are singular at z = − ζ -function, but he rediscovered correct valuesof its analytic continuation. In particular, he must have observed [10] theagreement of the results that he obtained in this way with the finite term inthe series that we call Euler-Maclaurin expansion, cf. Section 3. C The Riemann ζ -function We have seen that a na¨ıve ansatz for the continuation can be plagued bysubtleties when we hit a pole. An unambiguous approach to evaluate serieslike C and R , as well as the cubic series (1.4), combines the terms such that— in the limit of interest — the singularity is removed, as it is done in theapproaches of eqs. (2.8), (B.8), and of eqs. (2.10), (2.11), or by subtracting thecorresponding integral: the result, given in eq. (3.9), coincides with ζ ( − k ), k ∈ N .The underlying concept is analytic continuation: if a complex function f ( z ) ∈ C , z ∈ C , is holomorphic in some region, then its analytic continuationis unique. Hence the existence of a complex derivative is a powerful property:a plausibility argument is that such a map f ( z ), with f (cid:48) ( z ) (cid:54) = 0, is angle-preserving, cf. footnote 6, which constrains the analytic continuation to asingle possibility.Being a pioneer in this field, Riemann extended the ζ -function from theregion with Re z > C −{ } by means of relations [14], which can be condensed into the functionalequation ζ ( z ) = (2 π ) z π sin( πz − z ) ζ (1 − z ) , (C.1)which is valid all over C . We read off ζ ( −
1) = (2 π ) − ( − π / − / n ) = ( n − n ∈ N + , as well as Euler’s Basel formula (cid:80) n ≥ /n = π /
6. Similarly we obtain ζ ( −
3) = 6 ζ (4) / (8 π ) = 1 / ζ (4) = π /
90. This confirms again the Ramanujansummations (1.3) and (1.4). We further see that ζ ( − n ) = 0, ∀ n ∈ N (due to26he sin-function), in agreement with eq. (3.9). Finally we observe a simplepole at z = 1, with the residue lim z → ( z − ζ ( z ) = 1, which is consistent with ζ (0) = − = C .The validity of a series representation of ζ ( z ), which converges all over C − { } , was demonstrated by Helmut Hasse [53]. This formula uses eq. (1.7)and a double sum for η ( z ), ζ ( z ) = 11 − − z (cid:88) n ≥ n +1 n (cid:88) k =0 ( − k (cid:18) nk (cid:19) k + 1) z . (C.2)For z = 0 the sum over k corresponds to (1 − n = δ n, , and we confirm ζ (0) = − /
2. Similarly, for z = − δ n, − δ n, , and weobtain once more ζ ( −
1) = − / ζ ( − n ), n ∈ N . This picture justifiestheir application to the Casimir effect as a basic example of renormalization. References [1] https://mathworld.wolfram.com/[2] G. S. Carr, A Synopsis of Elementary Results in Pure and Applied Math-ematics: Containing Propositions, Formulae, And Methods Of Analysis,With Abridged Demonstrations, C. F. Hodgson and Son, London, 1886.[3] R. Kanigel, The Man Who Knew Infinity: A Life of the Genius Ra-manujan, Charles Scribner’s Sons, 1991.K. Srinivas Rao, Srinivasa Ramanujan: A Mathematical Genius, EastWest Book Pvt Ltd, 1998.[4] G. S. Hardy, S. Ramanujan F.R.S., Nature 105 (1920) 494–495.[5] http://ramanujan.sirinudi.org/html/published papers.html These are the “trivial zeros” of ζ ( z ). According to the famous Riemann Conjecture,all other (“non-trivial”) zeros have Re z = 1 / π { authentically reproduced in Ref. [11], Chapter 8 } in Monatsberichte der Berliner Akademie, November 1859.
A detailed discussion is included in:
H. M. Edwards, Riemann’s Zeta Function, Dover, 2001.[15] J. J. Gray, Bernhard Riemann, in µ m Range, Phys. Rev. Lett. 78 (1997) 5–8.[27] U. Mohideen and A. Roy, Precision Measurement of the Casimir Forcefrom 0.1 to 0.9 µ m, Phys. Rev. Lett. 81 (1998) 4549–4552.[28] G. Bressi, G. Carugno, R. Onofrio and G. Ruoso, Measurement of theCasimir Force between Parallel Metallic Surfaces, Phys. Rev. Lett. 88(2002) 041804.[29] S. Weinberg, The cosmological constant problem, Rev. Mod. Phys. 61(1989) 1–23.K. A. Milton, The Casimir effect: Physical manifestatios of zero-pointenergy, World Scientific, 2001.L. Feng, J. March-Russell, S. Sethi and F. Wilczek, Saltatory Relaxationof the Cosmological Constant, Nucl. Phys. B 602 (2001) 307–328.29. J. E. Peebles and B. Ratra, The Cosmological Constant and DarkEnergy, Rev. Mod. Phys. 75 (2003) 559–606.[30] S. M. Carroll, The Cosmological Constant, Living Rev. Rel. 4:1 (2001)1–56.[31] V. Sahni and A. Starobinsky, The Case for a Positive Cosmological Λ-term, Int. J. Mod. Phys. D 9 (2000) 373–443.[32] A. Riess et al. , Observational Evidence from Supernovae for an Accel-erating Universe and a Cosmological Constant, Astron. J. 116 (1998)1009–1038.S. Perlmutter et al. ∼∼