Summability methods rFrom Wikipedia, the free encyclopediaContents1 Ramanujan summation 11.1 Summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Sum of divergent series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Regularization (physics) 42.1 Realistic regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.1 Conceptual problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.2 Paulis conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.3 Opinions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.4 Minimal realistic regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Transport theoretic approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Resummation 83.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Riesz mean 94.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.2 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.4 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 114.4.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.4.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.4.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11iChapter 1Ramanujan summationNot to be confused with Ramanujans sum.Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value toinnite divergent series. Although the Ramanujan summation of a divergent series is not a sumin the traditional sense,it has properties which make it mathematically useful in the study of divergent innite series, for which conventionalsummation is undened.1.1 SummationRamanujan summation essentially is a property of the partial sums, rather than a property of the entire sum, as thatdoesn't exist.If we take the EulerMaclaurin summation formula together with the correction rule using Bernoullinumbers, we see that:12f (0) + f (1) + + f (n 1) +12f (n)=12 [f (0) + f (n)] +n1k=1f (k)=n0f(x) dx +pk=1Bk+1(k + 1)![f(k)(n) f(k)(0)]+ RpRamanujan[1] wrote it for the case p going to innity:xk=1f(k) = C +x0f(t) dt +12f(x) +k=1B2k(2k)!f(2k1)(x)where C is a constant specic to the series and its analytic continuation and the limits on the integral were not speciedby Ramanujan, but presumably they were as given above. Comparing both formulae and assuming that R tends to 0as x tends to innity, we see that, in a general case, for functions f(x) with no divergence at x = 0:C(a) =a0f(t) dt 12f(0) k=1B2k(2k)!f(2k1)(0)where Ramanujan assumeda =0 . By takinga = we normally recover the usual summation for convergent series.For functions f(x) with no divergence at x = 1, we obtain:C(a) =a1f(t) dt +12f(1) k=1B2k(2k)!f(2k1)(1)12 CHAPTER 1. RAMANUJAN SUMMATIONC(0) was then proposed to use as the sum of the divergent sequence. It is like a bridge between summation andintegration.1.2 Sum of divergent seriesIn the following text, () indicates Ramanujan summation. This formula originally appeared in one of Ramanujansnotebooks, without any notation to indicate that it exemplied a novel method of summation.For example, the() of 1 - 1 + 1 - is:1 1 + 1 =12()Ramanujan had calculated sums of known divergent series. It is important to mention that the Ramanujan sums arenot the sums of the series in the usual sense,[2][3] i.e. the partial sums do not converge to this value, which is denotedby the symbol() . In particular, the() sum of 1 + 2 + 3 + 4 + was calculated as:1 + 2 + 3 + = 112()Extending to positive even powers, this gave:1 + 22k+ 32k+ = 0 ()and for odd powers the approach suggested a relation with the Bernoulli numbers:1 + 22k1+ 32k1+ = B2k2k()It has been proposed to use of C(1) rather than C(0) as the result of Ramanujans summation, since then it can beassured that one series k=1 f(k) admits one and only one Ramanujans summation, dened as the value in 1 of theonly solution of the dierence equationR(x) R(x +1) =f(x) that veries the condition 21R(t) dt =0 .[4]This denition of Ramanujans summation (denoted as n1 f(n) ) does not coincide with the earlier dened Ra-manujans summation, C(0), nor with the summation of convergent series, but it has interesting properties, such as:If R(x) tends to a nite limit when x +1, then the series n1 f(n) is convergent, and we haven1f(n) = limN[Nn=1f(n) N1f(t) dt]In particular we have:n11n= where is the EulerMascheroni constant.Ramanujan resummation can be extended to integrals; for example, using the Euler-Maclaurin summation formula,one can writeaxmsdx =ms2axm1sdx + (s m) ai=1ims+ amsr=1B2r(ms+1)(2r)!(m2r+2s)(m2r + 1 s)axm2rsdx1.3. SEE ALSO 3which is the natural extension to integrals of the Zeta regularization algorithm.This recurrence equation is nite, since for m2r < 1 ,adxxm2r= am2r+1m2r + 1whereI(n,) =0dxxn(see zeta function regularization).With , the application of this Ramanujan resummation lends to nite results in the renormalization ofquantum eld theories.1.3 See alsoBorel summationCesro summationDivergent seriesRamanujans sum1.4 References[1] Bruce C. Berndt, Ramanujans Notebooks,Ramanujans Theory of Divergent Series, Chapter 6, Springer-Verlag (ed.),(1939), pp. 133-149.[2] The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation.Retrieved20 January 2014.[3] Innite series are weird. Retrieved 20 January 2014.[4] ric Delabaere, Ramanujans Summation, Algorithms Seminar 20012002, F. Chyzak (ed.), INRIA, (2003), pp. 8388.Chapter 2Regularization (physics)In physics, especially quantum eld theory, regularization is a method of dealing with innite, divergent, and non-sensical expressions by introducing an auxiliary concept of a regulator (for example, the minimal distance in spacewhich is useful if the divergences arise from short-distance physical eects). The correct physical result is obtainedin the limit in which the regulator goes away (in our example, 0 ), but the virtue of the regulator is that for itsnite value, the result is nite.However, the result usually includes terms proportional to expressions like 1/ which are not well-dened in the limit 0 . Regularization is the rst step towards obtaining a completely nite and meaningful result; in quantum eldtheory it must be usually followed by a related, but independent technique called renormalization. Renormalization isbased on the requirement that some physical quantities expressed by seemingly divergent expressions such as 1/ are equal to the observed values. Such a constraint allows one to calculate a nite value for many other quantitiesthat looked divergent.The existence of a limit as goes to zero and the independence of the nal result from the regulator are nontrivialfacts. The underlying reason for them lies in universality as shown by Kenneth Wilson and Leo Kadano and theexistence of a second order phase transition. Sometimes, taking the limit as goes to zero is not possible. This is thecase when we have a Landau pole and for nonrenormalizable couplings like the Fermi interaction. However, even forthese two examples, if the regulator only gives reasonable results for 1/ and we are working with scales of theorder of 1/ , regulators with 1/ 1/ still give pretty accurate approximations. The physical reason whywe can't take the limit of going to zero is the existence of new physics below .It is not always possible to dene a regularization such that the limit of going to zero is independent of the regu-larization. In this case, one says that the theory contains an anomaly. Anomalous theories have been studied in greatdetail and are often founded on the celebrated AtiyahSinger index theorem or variations thereof (see, for example,the chiral anomaly).Specic types of regularization includeDimensional regularization[1]PauliVillars regularizationLattice regularizationZeta function regularizationCausal regularization[2]Hadamard regularization2.1 Realistic regularization42.1. REALISTIC REGULARIZATION 52.1.1 Conceptual problemPerturbative predictions by quantum eld theory about quantum scattering of elementary particles, implied by acorresponding Lagrangian densities, are computed using the Feynman rules, a regularization method to circumventultraviolet divergences so as to obtain nite results for Feynman diagrams containing loops, and a renormalizationscheme. Regularization method results in regularized n-point Greens functions (propagators), and a suitable limitingprocedure (a renormalization scheme) then leads to perturbative S-matrix elements. These are independent of theparticular regularization method used, and enable one to model perturbatively the measurable physical processes(cross sections, probability amplitudes, decay widths and lifetimes of excited states). However, so far no knownregularized n-point Greens functions can be regarded as being based on a physically realistic theory of quantum-scattering since the derivation of each disregards some of the basic tenets of conventional physics (e.g., by not beingLorentz-invariant, by introducing either unphysical particles with negative metric or wrong statistic, or discrete space-time, or lowering the dimensionality of space-time, or some combination thereof. . . ). So the available regularizationmethods are understood as formalistic technical devices, devoid of any direct physical meaning. In addition, thereare qualms about renormalization. For a history and comments on this more than half-a-century old open conceptualproblem, see e.g.[3][4] .[5]2.1.2 Paulis conjectureAs it seems that the vertices of non-regularized Feynman series adequately describe interactions in quantum scat-tering, it is taken that their ultraviolet divergences are due to the asymptotic, high-energy behavior of the Feynmanpropagators. So it is a prudent, conservative approach to retain the vertices in Feynman series, and modify only theFeynman propagators to create a regularized Feynman series. This is the reasoning behind the formal Pauli-Villarscovariant regularization by modication of Feynman propagators through auxiliary unphysical particles, cf [6] andrepresentation of physical reality by Feynman diagrams.In 1949 Pauli conjectured there is a realistic regularization, which is implied by a theory that respects all the establishedprinciples of contemporary physics.[6][7] So its propagators (i) do not need to be regularized, and (ii) can be regardedas such a regularization of the propagators used in quantum eld theories that might reect the underlying physics.The additional parameters of such a theory do not need to be removed (i.e. the theory needs no renormalization) andmay provide some new information about the physics of quantum scattering, though they may turn out experimentallyto be negligible. By contrast, any present regularization method introduces formal coecients that must eventuallybe disposed of by renormalization.2.1.3 OpinionsDirac was persistently, extremely critical about procedures of renormalization. So in 1963 :in the renormalizationtheory we have a theory that has deed all the attempts of the mathematician to make it sound. I aminclined to suspectthat the renormalization theory is something that will not survive in the future, [8] So he was expecting a realisticregularization.About the skepticism that there is a realistic regularization Salam's remark[9] in 1972 is still relevant: Field-theoreticinnities rst encountered in Lorentzs computation of electron have persisted in classical electrodynamics for seventyand in quantum electrodynamics for some thirty-ve years. These long years of frustration have left in the subjecta curious aection for the innities and a passionate belief that they are an inevitable part of nature; so much sothat even the suggestion of a hope that they may after all be circumvented - and nite values for the renormalizationconstants computed - is considered irrational.However, in t Hoofts opinion:[10] History tells us that if we hit upon some obstacle, even if it looks like a pureformality or just a technical complication, it should be carefully scrutinized. Nature might be telling us something,and we should nd out what it is.By Dirac:[11] One can distinguish between two main procedures for a theoretical physicist. One of them is to workfrom the experimental basis ...The other procedure is to work from the mathematical basis.One examines andcriticizes the existing theory. One tries to pin-point the faults in it and then tries to remove them. The diculty hereis to remove the faults without destroying the very great successes of the existing theory. The diculty with a realisticregularization is that so far there is none, although nothing could be destroyed by its bottom-up approach; and there isno experimental basis for it.6 CHAPTER 2. REGULARIZATION (PHYSICS)2.1.4 Minimal realistic regularizationConsidering distinct theoretical problems, Dirac[8] in 1963 suggested: I believe separate ideas will be needed to solvethese distinct problems and that they will be solved one at a time through successive stages in the future evolution ofphysics. At this point I nd myself in disagreement with most physicists. They are inclined to think one master ideawill be discovered that will solve all these problems together. I think it is asking too much to hope that anyone willbe able to solve all these problems together. One should separate them one from another as much as possible and tryto tackle them separately. And I believe the future development of physics will consist of solving them one at a time,and that after any one of them has been solved there will still be a great mystery about how to attack further ones.According to Dirac:[11] Quantumelectrodynamics is the domain of physics that we knowmost about, and presumablyit will have to be put in order before we can hope to make any fundamental progress with other eld theories, althoughthese will continue to develop on the experimental basis.Diracs two preceding remarks[8][11] suggest that we should start searching for a realistic regularization in the caseof quantum electrodynamics (QED) in the four-dimensional Minkowski spacetime, starting with the original QEDLagrangian density.The path-integral formulation provides the most direct way from the Lagrangian density to the corresponding Feyn-man series in its Lorentz-invariant form.[5] The free-eld part of the Lagrangian density determines the Feynmanpropagators, whereas the rest determines the vertices. As the QED vertices are considered to adequately describeinteractions in QED scattering, it makes sense to modify only the free-eld part of the Lagrangian density so asto obtain such regularized Feynman series that the Lehmann-Symanzik-Zimmermann reduction formula provides aperturbative S-matrix that: (i) is Lorentz invariant and unitary; (ii) involves only the QED particles; (iii) dependssolely on QED parameters and those introduced by the modication of the Feynman propagatorsfor particularvalues of these parameters it is equal to the QED perturbative S-matrix; and (iv) exhibits the same symmetries as theQED perturbative S-matrix. Let us refer to such a regularization as the minimal realistic regularization, and startsearching for the corresponding, modied free-eld parts of the QED Lagrangian density.2.2 Transport theoretic approachAccording to Bjorken and Drell, it would make physical sense to sidestep ultraviolet divergences by using moredetailed description than can be provided by dierential eld equations. And Feynman noted about the use of dier-ential equations: . . . for neutron diusion it is only an approximation that is good when the distance over which weare looking is large compared with the mean free path. If we looked more closely, we would see individual neutronsrunning around. And then he wondered, Could it be that the real world consists of little X-ons which can be seenonly at very tiny distances? And that in our measurements we are always observing on such a large scale that we cantsee these little X-ons, and that is why we get the dierential equations? . . . Are they [therefore] also correct only asa smoothed-out imitation of a really much more complicated microscopic world?[12]Already in 1938, Heisenberg[13] proposed that a quantum eld theory can provide only an idealized, large-scaledescription of quantum dynamics, valid for distances larger than some fundamental length, expected also by Bjorkenand Drell in 1965. Feynmans preceding remark provides a possible physical reason for its existence.2.3 References[1] 't Hooft, Veltman M.: Regularization and renormalization of gauge elds, Nucl. Phys. B44 (1972), p.189213.[2] Scharf, G.: Finite Quantum Electrodynamics: The Causal Approach, Springer 1995.[3] T. Y. Cao and S. S. Schweber, The Conceptual Foundations and Philosophical Aspects of Renormalization Theory,Synthese 97: 1 (1993) 33108.[4] L. M.Brown, editor, Renormalization (Springer-Verlag, New York 1993).[5] S. Weinberg. The Quantum Theory of Fields. Cambridge University Press, Cambridge 1995.: Vol. I, Sec. 1.3 and Ch.9.[6] F. Villars. Regularization and Non-Singular Interactions in Quantum Field Theory. Theoretical Physics in the TwentiethCentury, edited by M. Fierz and V. F. Weiskopf, Interscience Publishers, New York 1960.: 78106.2.3. REFERENCES 7[7] Pauli, W., Villars, F. On the Invariant Regularization in Relativistic Quantum Theory, Rev. Mod. Phys, 21, 434-444 (1949)[8] P.A.M. Dirac. The Evolution of the Physicists Picture of Nature. Scientic American, May 1963: 4553.[9] C.J.Isham, A.Salam and J.Strathdee, `Innity Suppression Gravity Modied Quantum Electrodynamics, Phys. Rev. D5,2548 (1972).[10] G. t Hooft, In Search of the Ultimate Building Blocks (Cambridge University Press, Cambridge 1997).[11] P.A.M. Dirac (1968). Methods in theoretical physics. Unication of Fundamental Forces by A. Salam (CambridgeUniversity Press, Cambridge 1990): 125143.[12] R. P. Feynman, R. B. Leighton and M. Sands: The Feynman Lectures on Physics, Vol. II (Addison-Wesley, Reading, Mass.,1965), Sec.127.[13] W. Heisenberg (1938). Uber die in der Thorie der Elementarteilchen auftretende universelle Lange. Annalen der Physik32: 2033. Bibcode:1938AnP...424...20H. doi:10.1002/andp.19384240105.Chapter 3ResummationIn mathematics and theoretical physics, resummation is a procedure to obtain a nite result from a divergent sum(series) of functions. Resummation involves a denition of another (convergent) function in which the individualterms dening the original function are re-scaled, and an integral transformation of this new function to obtain theoriginal function. Borel resummation is probably the most well-known example. The simplest method is an extensionof a variational approach to higher order based on a paper by R.P. Feynman and H. Kleinert.[1] In quantummechanicsit was extended to any order here,[2] and in quantum eld theory here.[3] See also Chapters 1620 in the textbookcited below.3.1 References[1] Feynman R.P., Kleinert H. (1986). Eective classical partition functions. Physical ReviewA34 (6): 50805084.Bibcode:1986PhRvA..34.5080F. doi:10.1103/PhysRevA.34.5080. PMID 9897894.[2] Janke W., Kleinert H. (1995). Convergent Strong-Coupling Expansions fromDivergent Weak-Coupling Perturbation The-ory. Physical ReviewLetters 75 (6): 287. arXiv:quant-ph/9502019. Bibcode:1995PhRvL..75.2787J. doi:10.1103/physrevlett.75.2787.[3] Kleinert, H., Critical exponents from seven-loop strong-coupling 4 theory in three dimensions. Physical Review D 60,085001 (1999)3.2 BooksHagen Kleinert, Critical Properties of 4-Theories, World Scientic (Singapore, 2001); Paperback ISBN 981-02-4658-7 (also available online) (together with V. Schulte-Frohlinde).8Chapter 4Riesz meanIn mathematics, the Riesz mean is a certain mean of the terms in a series. They were introduced by Marcel Rieszin 1911 as an improvement over the Cesro mean. The Riesz mean should not be confused with the BochnerRieszmean or the StrongRiesz mean.4.1 DenitionGiven a series {sn} , the Riesz mean of the series is dened bys() =n(1 n)snSometimes, a generalized Riesz mean is dened asRn=1nnk=0(k k1)skHere, the n are sequence with n and with n+1/n 1 as n . Other than this, the n are otherwisetaken as arbitrary.Riesz means are often used to explore the summability of sequences; typical summability theorems discuss the caseof sn=nk=0 an for some sequence {an} . Typically, a sequence is summable when the limit limnRn exists,or the limit lim1,s() exists, although the precise summability theorems in question often impose additionalconditions.4.2 Special casesLet an= 1 for all n . Thenn(1 n)=12ic+ici(1 + )(s)(1 + + s) (s)sds =1 + +nbnn.Here, one must take c > 1 ; (s) is the Gamma function and (s) is the Riemann zeta function. The power seriesnbnn910 CHAPTER 4. RIESZ MEANcan be shown to be convergent for > 1 . Note that the integral is of the form of an inverse Mellin transform.Another interesting case connected with number theory arises by taking an= (n) where (n) is the Von Mangoldtfunction. Thenn(1 n)(n) = 12ic+ici(1 + )(s)(1 + + s)(s)(s) sds =1 + +(1 + )()(1 + + )+ncnn.Again, one must take c > 1. The sum over is the sum over the zeroes of the Riemann zeta function, andncnnis convergent for > 1.The integrals that occur here are similar to the NrlundRice integral; very roughly, they can be connected to thatintegral via Perrons formula.4.3 References^ M. Riesz, Comptes Rendus, 12 June 1911^ Hardy, G. H. & Littlewood, J. E. (1916). Contributions to the Theory of the Riemann Zeta-Function andthe Theory of the Distribution of Primes (PDF). Acta Mathematica 41: 119196. doi:10.1007/BF02422942.Volkov, I.I. (2001), Riesz summation method, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-44.4. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 114.4 Text and image sources, contributors, and licenses4.4.1 Text Ramanujansummation Source: https://en.wikipedia.org/wiki/Ramanujan_summation?oldid=661533646 Contributors: The Anome,Giftlite, Dratman, Histrion, Crislax, EmilJ, Wavelength, SmackBot, Melchoir, Silly rabbit, Berland, Sammy1339, Atulsvasu, CBM, Cy-debot, JamesBWatson, Albmont, David Eppstein, NorthAce, Addbot, Raulshc, LucienBOT, FoxBot, Ybab321, Brad7777, Permafrost46,Kasuga and Anonymous: 24 Regularization(physics) Source: https://en.wikipedia.org/wiki/Regularization_(physics)?oldid=674735506 Contributors: Michael Hardy,Ancheta Wis, Fropu, David Schaich, Jrme, Linas, Rjwilmsi, Chobot, KasugaHuang, SmackBot, Silly rabbit, Xxanthippe, Maliz,Racepacket, VolkovBot, Yartsa, Phe-bot, Eebster the Great, Addbot, Yobot, Niout, Jim1138, Ulric1313, Materialscientist, Citation bot,Xqbot, Omnipaedista, , Astiburg, Naviguessor, Maschen, Greggp42, Bibcode Bot, AvocatoBot, Mark viking and Anonymous: 14 Resummation Source: https://en.wikipedia.org/wiki/Resummation?oldid=654380815 Contributors: Phys, Lumidek, Count Iblis, Rjwilmsi,Conscious, Dialectric, SmackBot, Rrburke, Yobot, Bibcode Bot, AHusain314 and Anonymous: 3 Riesz meanSource: https://en.wikipedia.org/wiki/Riesz_mean?oldid=674665920 Contributors: Michael Hardy, Charles Matthews, Giftlite,Rich Farmbrough, Bender235, Linas, RussBot, SmackBot, Silly rabbit, Addbot, Uncia, Luckas-bot, Constructive editor, Citation bot 1,Onkekabonke, EmausBot, K9re11 and Anonymous: 14.4.2 Images File:Ambox_important.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/b4/Ambox_important.svg License: Public do-main Contributors: Own work, based o of Image:Ambox scales.svg Original artist: Dsmurat (talk contribs)4.4.3 Content license Creative Commons Attribution-Share Alike 3.0