In general, quantum field theories (QFT) require regularizations and infinite
renormalizations due to ultraviolet divergences in their loop calculations.
Furthermore, perturbation series in theories like QED are not convergent
series, but are asymptotic series. We apply neutrix calculus, developed in
connection with asymptotic series and divergent integrals, to QFT,obtaining
finite renormalizations. While none of the physically measurable results in
renormalizable QFT is changed, quantum gravity is rendered more manageable in
the neutrix framework.Comment: 10 pages; LaTeX; version to appear in J. Phys. A: Math. Gen. as a
  Letter to the Edito

Bender C M

Cheng T P

H van Dam

Jauch J M

Ng Y J van Dam H

Peskin M E

Schwinger J

van der Corput J G

Veltman M J G

Weinberg S

Y Jack Ng

English

arXiv

Crossref

Neutrix calculus and finite quantum field theory

In general, quantum field theories (QFT) require regularizations and infinite renormalizations due to ultraviolet divergences in their loop calculations. Furthermore, perturbation series in theories like QED are not convergent series, but are asymptotic series. We apply neutrix calculus, developed in connection with asymptotic series and divergent integrals, to QFT, obtaining finite renormalizations. While none of the physically measurable results in renormalizable QFT is changed, quantum gravity is rendered more manageable in the neutrix framework

Ng, Y Jack

van Dam, H

Carolina Digital Repository

arXiv:hep-th/0410285v4  4 Apr 2005Neutrix Calculus and Finite Quantum Field TheoryY. Jack Ng∗ and H. van DamInstitute of Field Physics,Department of Physics and Astronomy,University of North Carolina,Chapel Hill, NC 27599-3255AbstractIn general, quantum field theories (QFT) require regularizations and infinite renormalizationsdue to ultraviolet divergences in their loop calculations. Furthermore, perturbation series in the-ories like QED are not convergent series, but are asymptotic series. We apply neutrix calculus,developed in connection with asymptotic series and divergent integrals, to QFT, obtaining fi-nite renormalizations. While none of the physically measurable results in renormalizable QFT ischanged, quantum gravity is rendered more manageable in the neutrix framework.PACS numbers: 03.70.+k, 11.10.Gh, 11.10.-z∗ E-mail: yjng@physics.unc.edu1The procedure of regularization and renormalization is a big step forward in making senseof the infinities that one encounters in calculating perturbative series in quantum field the-ories. The result is a phenomenal success. For example, Quantum Electrodynamics (QED),the paradigm of relativistic quantum field theories, suitably regularized and renormalized, isarguably the most accurate theory ever devised by mankind. Yet in spite of the impressivephenomenological successes, the specter of infinite renormalizations has convinced many,including such eminent physicists as Dirac and Schwinger, that we should seek a bettermathematical and/or physical foundation for quantum field theory, without simultaneouslytearing down the towering edifice we have built on the existing one. In another development,Dyson [1] has shown that the series as defined by the Feynman rules in QED is not a con-vergent series and has suggested that it is instead an asymptotic series in the fine structureconstant α, i.e., in the number of internal integrals (for given outside lines). In this paper,we propose to apply neutrix calculus, in conjunction with Hadamard integrals, developedby J.G. van der Corput [2] in connection with asymptotic series and divergent integrals, toquantum field theories in general, and QED in particular, to obtain finite results for thecoefficients in the perturbation series. (A more detailed discussion[3] will appear elsewhere.)The replacement of regular integrals by Hadamard integrals in quantum field theory appearsto make good mathematical sense, as van der Corput observed that Hadamard integrals arethe proper tool to calculate the coefficients of an asymptotic series. (Actually Hadamardintegrals work equally well for convergent series.)We begin by recalling the definition of asymptotic series[4]. The series f(x) = a0+a1(x−b) + a2(x− b)2 + ... for finite b is an asymptotic series if and only if there exists an n0 > 0,such that for n > n0,limx→b1(x− b)n |f(x)− a0 − a1(x− b)− . . .− an(x− b)n| = 0, (1)with the remnant being at most ∼ (x− b)n+1.Next, following van der Corput [2], we define a neutrix as a class of negligible functionsdefined in a domain, which satisfy the following two conditions: (1) the neutrix is an additivegroup; (2) it does not contain any constant except 0. Let us illustrate the concept with thefollowing example considered by Hadamard: for s real,∫ 2ξxs−1 dx =s−12s − s−1ξs for s 6= 0log 2− log ξ for s = 0(2)2For s > 0, the integral converges even as ξ → 0. For s ≤ 0, Hadamard neglects ξs/s andlog ξ as ξ → 0. Here we have a neutrix which we will call N(0), consisting of functionsν(ξ) = ǫ(ξ) + c1ξs + c2 log ξ, where ǫ(ξ) → 0 as ξ → 0, and where c1, c2, and s are arbitraryconstants. This results in writing∫ 2N(0)xs−1 dx =s−12s for s 6= 0log 2 for s = 0(3)Note the analytic extension in the complex s plane of the answer for Re s > 0 to the entirecomplex plane with the exclusion of s = 0.Before applying neutrices to QED, we need to consider the generalized Hadamard neutrixHa defined to contain the negligible functionsν(ξ) = U(ξ) + ǫ(ξ), (4)where ǫ(ξ) → 0 as ξ → a. Each of the functions U(ξ) is defined by an asymptotic seriesbased on a:U(ξ) ∼ Σ∞h=0χh (ξ − a)Ψh logkh(ξ − a). (5)Here χh, Ψh and integers kh ≥ 0 are independent of ξ, Re Ψh → ∞ as h→ ∞, and logkh(ξ)stands for (log(ξ))kh. An example is provided by∫ bHa(z − a)−1 logk(z − a) dz = (k + 1)−1 logk+1(b− a). (6)Similarly, we can define the Hadamard neutrix H∞ by Eq. (4) where now ǫ(ξ) → 0 as ξ → ∞and the function U(ξ) has a Hadamard development in powers of ξ−1 in its asymptotic series:U(ξ) ∼ Σ∞h=0χh ξΨh logkh ξ, (7)where Re Ψh → −∞ as h→ ∞.We can now demonstrate a very valuable property of the Hadamard neutrix. Recallthat in the theory of distributions developed by Schwartz, generalized functions usuallycannot be multiplied. Consider, for example, the one-dimensional Dirac delta functionmultiplying itself δ(x) × δ(x). This product is not mathematically meaningful because itsFourier transform diverges:∫∞−∞dk2π1× 1 −→ ∞, (8)3where we have used the convolution rule in Fourier transform and have noted that theFourier transform of the Dirac delta function is 1. In contrast, the Hadamard method doesallow multiplication for a wide class of distributions. For the example of δ(x) × δ(x), theHadamard-neutralized Fourier transform of the product∫ H∞H−∞dk2π1× 1 = 0, (9)yieldsδ(x)× δ(x) = 0, (10)a mathematically meaningful (though somewhat counter-intuitive) result!In doing quantum field theory in configuration space, we multiply operator-valued dis-tributions of quantum fields. Or, in a slightly different interpretation, we multiply singularfunctions such as Feynman propagators. As we will see, the use of neutrix calculus allowsone to put these products on a mathematically sound basis. Let us now generalize theabove discussion for 1-dimensional Dirac delta functions to the case of (3 + 1)-dimensionalFeynman propagators∆+(x) =∫ d4p(2π)4eip·xp2 +m2 − iǫ (11)=14πδ(x2)− m8π√−x2 − iǫH(2)1 (m√−x2 − iǫ), (12)where H(2) is the Hankel function of the second kind and we use the (+++-) metric. TheFourier transform of ∆+(x)×∆+(x) (which appears in certain quantum loop calculations)is given by∫d4x e−ip·x∆+(x)∆+(x) =∫d4k(2π)41k2 +m2 − iǫ1(p− k)2 +m2 − iǫ (13)=i4(2π)2D − i4(2π)2∫ 10dz log(1 +p2m2z(1− z)), (14)whereD =1iπ2∫d4k(k2 +m2)2=∫∞0k2dk2(k2 +m2)2, (15)with the second expression of D obtained after a Wick rotation. But D is logarithmicallydivergent. Hence ∆+(x)×∆+(x) is not mathematically well defined. Let us now see how the4Hadamard-van der Corput method gives mathematical meaning to this product. Obviously,it is in the calculation of the logarithmically divergent D where we apply neutrix calculus.Introducing dimensionless variable q = k2/m2, we bring in H∞ to write D asD =∫ H∞0qdq(q + 1)2= −1, (16)where we have recalled that, for q → ∞, log q is negligible in the Hadamard neutrix H∞.It follows that, in the neutralized version, ∆+(x) × ∆+(x) ∼ δ(4)(x)+ regular part (whereδ(4)(x) is the 4-dimensional Dirac delta function), a much more mathematically palatableobject.As the first example in the application of neutrices to QED, let us consider the one-loopcontribution to the electron’s self energyΣ(p) = −ie2∫ d4k(2π)4γµ[−γ · (p− k) +m]γµ[k2 + λ2][(p− k)2 +m2] , (17)where m is the electron bare mass and we have given the photon a fictitious mass λ toregularize infrared divergences. Expanding Σ(p) about γ · p = −m,Σ(p) = A+ B(γ · p+m) +R, (18)one finds (cf. results found in Ref.[5])A = − α2πm(32D +94), (19)B = − α4π(D − 4∫ 1λmdxx+112), (20)where α = e2/4π is the fine structure constant and D is given by Eq. (15) and Eq. (16)in the pre-neutralized and neutralized forms respectively. We note that R, the last pieceof Σ(p) in Eq. (18), is finite. Mass renormalization and wavefunction renormalization aregiven by mren = m − A and ψren = Z−1/22 ψ respectively with Z−12 = 1 − B. Now, sinceD = −1 is finite, it is abundantly clear that the renormalizations are finite in the frameworkof neutrix calculus. There is no need for a separate discussion of the electron vertex functionrenormalization constant Z1 due to the Ward identity Z1 = Z2.The one-loop contribution to vacuum polarization is given byΠµν(k) = ie2∫d4p(2π)4Tr(γµ1γ · (p+ k2) +mγν1γ · (p− k2) +m). (21)5A standard calculation[5] shows that Πµν takes on the formΠµν = δm2ηµν + (k2ηµν − kµkν)Π(k2), (22)where ηµν is the flat metric (+++-),δm2 =α2π(m2D +D′), (23)andΠ(k2) = − α3π(D +56) +2απ∫ 10dx x(1− x) log(1 +k2m2x(1− x)), (24)withD′ =1iπ2∫d4pp2 +m2, (25)and D given by Eq. (15). Just as D is rendered finite upon invoking neutrix calculus (seeEq. (16)), so is D′:D′ = m2∫ H∞0qdqq + 1= 0, (26)since both q and log q, for q → ∞, are negligible in H∞. Thus neutrix calculus yields a finiterenormalization for both the photon mass and the photon wavefunction Aµren = Z−1/23 Aµ(and consequently also for charge eren = Z1/23 e) where Z−13 = 1−Π(0). In electron-electronscattering by the exchange of a photon with energy-momentum k, vacuum polarizationeffects effectively replace e2 by e2/(1−Π(k2)), i.e.,e2 → e2eff =e21−Π(k2)=e2renZ3(1−Π(k2))=e2ren1− (Π(k2)− Π(0)) . (27)Eq. (24) can be used, for k2 ≫ m2, to show thatαeff(k2) =α1− α3πlog(k2exp(5/3)m2) . (28)Thus we have obtained the correct running of the coupling[6] with energy-momentum inthe framework of neutrices. In fact, the only effect of neutrix calculus, when applied toQED (and other renormalizable theories), is to convert infinite renormalizations (obtainedwithout using neutrix calculus) to mathematically well-defined finite renormalizations. As6far as we can tell, all (finite) physically observable results of QED are recovered. In passingwe mention that the use of neutralized integrals does not affect the results of axial triangleanomalies.As shown by the appearance of photon mass in the above discussion of vacuum polar-ization, the application of neutrix calculus to the energy-momentum cutoff regularization,though straightforward and natural, is ill suited for more complicated theories like thoseinvolving Yang-Mills fields. For those theories, one should use other more convenient regu-larization schemes. It is amusing to note that already in 1961 van der Corput suggested that,instead of finding the appropriate neutrices, one can continue analytically in any variable(presumably including the dimension of integrations) contained in the problem of tacklingapparent divergences to calculate the coefficients of the corresponding asymptotic series. Inhind sight, one recognizes that this was the approach taken by ’t Hooft and Veltman whospearheaded the use of dimensional regularizations[6]. Let us now explore using neutrixcalculus in conjunction with the dimensional regularization scheme. In that case, negligiblefunctions will include 1/ǫ where ǫ = 4−n is the deviation of spacetime dimensions from 4. Inthe one-loop calculations for QED, the internal energy-momentum integration is now over ndimensions. The forms of Σ(p) and Πµν remain the same as given by Eqs. (18) and (22), butnow with δm2 = 0. Using the approximation for the gamma function, Γ(ǫ) = ǫ−1−γ+O(ǫ),where γ ≃ .577 is the Euler-Mascheroni constant, and the approximation f ǫ ≃ 1 + ǫ log f ,for ǫ≪ 1, one findsA = αm4π[3(γ − log 4π) + 1] + α2πm∫ 10dx(1 + x) logD0,B = α4π[1 + γ − log 4π] + αm2π∫ 10dxx(1− x2)m2x2 + λ2(1− x) +α2π∫ 10dx(1− x) logD0, (29)where D0 = m2x2 + λ2(1− x), andΠ(k2) =α3π[γ − log 4π] + 2απ∫ 10dx x(1− x) log[m2 + x(1− x)k2],1Z3= 1− α3π[γ − log 4π]− α3πlogm2. (30)By design, the generalized neutrix calculus renders all the renormalizations finite. Again,all physically measurable results of QED appear to be recovered. In this article we haveexplicitly considered QED to one-loop only. But we expect that higher-loop calculationscan be handled in the same way according to neutrix calculus. It will be interesting to7see explicitly whether neutrix calculus, applied to higher-loop calculations, can provide newinsights in the issue of overlapping divergences.In the framework of quantum field theory for the four fundamental forces, the divergenceproblem is particularly severe for quantum gravity. Using dimensional regularization, ’tHooftand Veltman[7] found that pure gravity is one-loop renormalizable, but in the presence of ascalar field, renormalization was lost. For the latter case, they found that the countertermevaluated on the mass shell is given by ∼ ǫ−1√gR2 with R being the Ricci scalar. Similarresults for the cases of Maxwell fields and Dirac fields etc (supplementing the Einstien field)were obtained[8]. It is natural to inquire whether the application of neutrix calculus couldimprove the situation. The result is that now essentially the divergent ǫ−1 factor is replacedby −γ+ constant.It has not escaped our notice that neutrix calculus may ameliorate the hierarchy problemin particle physics. The hierachy problem is due to the fact that the Higgs scalar self-energiesdiverge quadratically, leading to a stability problem in the standard model of particle physics.But neutrix calculus treats quadratic divergences no different from logarithmic divergences,since both divergences belong to (the negligible functions of) the neutrix. Neutrix calculusmay also ameliorate the cosmological constant problem in quantum gravity. The cosmologi-cal constant problem can be traced to the quartic divergences in zero-point fluctuations fromall quantum fields. But again, neutrix calculus treats quartic divergences no different fromlogarithmic divergences. Indeed, for a theory of gravitation with a cosmological constantterm, the cosmological constant receives at most a finite renormalization from the quantumloops in the framework of neutrix calculus.We conclude with a comment on what neutrix calculus means to the general questionof renormalizability of a theory. We recall that a theory is renormalizable if, in loop cal-culations, the counterterms vanish or if they are proportional to terms in the original La-grangian (the usual renormalization through rescaling). It is still renormalizable if, to allloops, the counterterms are of a new form, but only a finite number of such terms exist. Bythis standard, neutrix calculus does not change the renormalizability of a theory, since itmerely changes potentially infinite renormalizations to finite renormalizations. On the otherhand, non-renormalizable terms, i.e., terms with positive superficial degree of divergence, aretolerated in neutralized quantum field theory. In a sense it is a pity that we have lost renor-malizability as a physical restrictive criterion in the choice of sensible theories. However, we8believe that this is actually not as big a loss as it may first appear. Quite likely, all realistictheories now in our possession are actually effective field theories.[9, 10] They appear tobe renormalizable field theories because, at energies now accessible, or more correctly, atsufficiently low energies, all the non-renormalizable interactions are highly suppressed. Bytolerating non-renormalizable terms, neutrix calculus has freed us from the past dogmaticand rigid requirement of renormalizability. (Having said that, given a choice between renor-malizable field theories and effective field theories, we still prefer the former to the latterbecause of the former’s compactness and predictive power. But the point is that both typesof theories can be accommodated in the framework of neutrix calculus.) Furthermore, ifthe application of neutrix calculus to loop calculations results in a term of a new form (likethe Pauli term in QED) that is finite, then we have a prediction which, in principle, can bechecked against experiments to confirm or invalidate the theory in question. For the lattercase, we will have to modify the theory by including a term of that form in the Lagrangian,making the parameter associated with the new term an adjustable parameter rather thanone that is predicted by the theory. This loss of predictive power is again not as big a lossas one may dread.Lastly we should emphasize that, for renormalizable theories as well as non-renormalizabletheories (like quantum gravity?), neutrix calculus is a useful tool to the extent that it isrelevant for asymptotic series and lessens the divergence of the theories. Based on our studyso far, we tentatively conclude that neutrix calculus has banished infinities from quantumfield theory, rendering perturbative quantum field theory mathematically meaningful.We thank J.J. Duistermaat, E. M. de Jager, T. Levelt, and T. W. Ruijgrok for encour-agement and for kindly providing us with relevant references of the work by J.G. van derCorput. We thank C. Bender and K. A. Milton for useful discussions. We also thank L. Ngand X. Calmet for their help in the preparation of this manuscript. We are grateful to thelate Paul Dirac and Julian Schwinger for inspiring us to look for a better mathematicalfoundation for quantum field theory. This work was supported in part by DOE and by theBahnson Fund of University of North Carolina at Chapel Hill.[1] Dyson F J 1952 Phys. Rev. 85 6319[2] van der Corput J G 1959 J. Analyse Math. 7 281; 1960 MRC Technical Reports # 128 -130 (University of Wisconsin at Madison); 1961 Voordrachten over neutrixrekening (StichtingMathematisch Centrum, Amsterdam). For more references, see MathSciNet under “neutrix”.[3] Ng Y J and van Dam H 2005 Preprint hep-th/0502163[4] Bender C M and Orszag S A 1978 Advanced Mathematical Methods for Scientists and Engi-neers (New York: McGraw-Hill)[5] Jauch J M and Rohrlich F 1955 The Theory of Photons and Electrons (Cambridge, MA:Addison-Wesley)[6] Peskin M E and Schroeder D V 1995 An Introduction to Quantum Field Theory (Reading,MA: Addison-Wesley); Cheng T P and Li L F 1984 Gauge Theory of Elementary ParticlePhysics (Oxford: Claredon Press)[7] ’t Hooft G 1973 Nucl Phys. B62 44; Veltman M J G 1976 in Quantum Theory of Gravitation(Amsterdam: North Holland) (Les Houches session XXVIII)[8] Deser S and van Nieuwenhuizen P 1974 Phys. Rev. Lett. 32 245; Deser S, Tsao H S and vanNieuwenhuizen P 1974 Phys. Lett. 50B 491[9] Schwinger J 1970, 1973, 1989 Particles, Sources, and Fields Vol I, II, III (Reading, MA:Addison-Wesley)[10] Weinberg S 1995 The Quantum Theory of Fields, Volume I (Cambridge, UK: CambridgeUniversity Press)10

Neutrix Calculus and Finite Quantum Field Theory

Neutrix Calculus and Finite Quantum Field Theory

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