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renormalisation, regularisation and number theory

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"Shortly after relativistic quantum field theory was discovered, it became clear that it suffered from serious conceptual and technical deficiencies. The only known way of extracting any information from the fundamental equations was to expand in a power series in the coupling constant. Such an expansion is called perturbation theory. Unfortunately, it turns out that each term of perturbation theory, except for the leading order contribution, is singular, and so the power series as it stands is meaningless! The path-breaking work of Feynman, Schwinger, Dyson, and others resulted in a procedure of extracting finite parts of the singular expressions encountered in perturbation theory known as renormalization theory. Renormalization theory removes singularities from perturbation theory at the expense of introducing a number of arbitrary constants whose values should be determined by experiment. That leads one to the requirement that the number of such constants should be finite (otherwise one could "explain" any experiment) and that they should be measurable parameters of the theory. Any quantum field theory satisfying these requirements is called renormalizable. These concepts led to some of the most remarkable developments in physics. The theory of interacting electrons and photons, quantum electrodynamics, turns out to be renormalizable and leads to fantastic agreement with experiment.

The requirement of renormalizability became a paradigm of quantum field theory, and it proved extremely fruitful. Guided by it, particle physicists generalized quantum electrodynamics to include other types of interactions: the Weinberg-Salam model unifying electromagnetic and weak interactions and the standard model unifying electromagnetic, weak, and strong interactions. As of today there is a consensus that the standard model is the correct theory of elementary interactions. One disturbing fact about this theory is that gravity has so far resisted inclusion into the framework of renormalizable quantum field theory.

This renormalizability paradigm underwent a dramatic revision in the seventies as a result of Wilson's renormalization group theory. According to Wilson, we do not need to know the details of the "true" theory of elementary interactions which is valid at all energy scales. It may as well be that the fundamental theory is not a quantum field theory at all. All we have at our disposal is an effective theory [and] a low- energy limit of this fundamental theory. The concept of renormalizability thus acquires a new meaning: renormalizable theories, rather than being "fundamental" are merely those theories which survive the scaling down from the fundamental scale to the "laboratory scale. Non-renormalizable theories get wiped out in the process of taking this limit."

A. Lesniewski, from "Noncommutative Geometry", Notices of the AMS 44 no. 7 (August 1997) 800-805

"Let us briefly remind the reader of the historical road which finally led to the renormalization group concept. The problem of infinities which appeared in electrodynamics when one tried to compute the various physical quantities has been partly solved by a procedure called renormalization, introduced by Tomonaga, Schwinger, Feynman and Dyson. It mainly consists in replacing the calculated infinite values of mass, charge and field by their observed finite values, and then compute again all the other physical quantities. The remarkable result is that once these three fundamental quantities are renormalized, the predictions for other physical quanitites (Lamb shift, anomalous magnetic moments, radiative corrections and so on) become finite and in extraordinary precise agreement with experiments. After renormalizations, QED becomes the second best experimentally verified theory among all physical theories."

L. Notalle, from Fractal Space-Time and Microphysics - Towards a Theory of Scale Relativity (World Scientific, 1993).

K. Shirish, S-duality and chaos" (preprint 11/2020)

[abstract:] "The Renormalization group in field theories happens to resemble dynamical systems in many ways. In this paper, we discuss the unexpected connection between chaos and duality in field theories. In a sense, that various dual field theories can emerge at the end of chaotic RG trajectories, and hence strong-weak duality in quantum field theory is a direct result of the chaotic flow of the renormalization group. This suggests that various properties of field and string theories could come into existence due to chaotic RG flow. We also conjecture the existence of dual quantum field theories in the half strip of the Riemann zeta function."

G. Caginalp and B. Ion, "Probabilistic renormalization and analytic continuation" (preprint 08/2020)

[abstract:] "We introduce a theory of probabilistic renormalization for series, the renormalized values being encoded in the expectation of a certain random variable on the set of natural numbers. We identify a large class of weakly renormalizable series of Dirichlet type, whose analysis depends on the properties of a (infinite order) difference operator that we call Bernoulli operator. For the series in this class, we show that the probabilistic renormalization is compatible with analytic continuation. The general zeta series for $s\neq 1$ is found to be strongly renormalizable and its renormalized value is given by the Riemann zeta function."

G. Sierra, "The Riemann zeros and the cyclic Renormalization Group" (preprint 10/2005)

[abstract:] "We propose a consistent quantization of the Berry-Keating Hamiltonian xp, which is currently discussed in connection with the non trivial zeros of the Riemann zeta function. The smooth part of the Riemann counting formula of the zeros is reproduced exactly. The zeros appear, not as eigenstates, but as missing states in the spectrum, in agreement with Connes adelic approach to the Riemann hypothesis. The model is exactly solvable and renormalizable, with a cyclic Renormalization Group. These results are obtained by mapping the Berry-Keating model into the Russian doll model of superconductivity. Finally, we propose a generalization of these models in an attempt to explain the oscillatory part of the Riemann's formula."

A. LeClair, J.M. Román, G. Sierra, "Log-periodic behaviour of finite size effects in field theories with RG limit cycles" (preprint 12/03)

[abstract:] "We compute the finite size effects in the ground state energy, equivalently the effective central charge c_eff, based on S-matrix theories recently conjectured to describe a cyclic regime of the Kosterlitz-Thouless renormalization group flows. The effective central charge has periodic properties consistent with renormalization group predictions. Whereas c_eff for the massive case has a singularity in the very deep ultra-violet, we argue that the massless version is non-singular and periodic on all length scales."

[from introduction:] "We derive an approximate analytic expression for c_eff in terms of Riemann's zeta function..."

[from conclusion:] "On a broader note, there appears to be a network of deeply interrelated concepts and techniques, namely RG limit cycles, discrete scale invariance, complex exponents, fractals, log-periodicity, quantum groups (with real q), zeta function regularizations, number theory, etc., whose full significance needs to be clarified."

A. Petermann, "The so-called renormalization group method applied to the specific prime number logarithmic decrease"

"A so-called Renormalization Group (RG) analysis is performed in order to shed some light on why the density of prime numbers in N^* decreases like the single power of the inverse naperian logarithm."

"...in this note, our aim is to look for the deep reason why the density of primes decreases with the single power of the natural logarithm. We hope that we have been able to shed some light on this fact: the breaking of a symmetry, namely that of scale invariance...is the very factor responsible for this specific decrease.

The coincidence of the results obtained is striking when compared to the formulas of the first non-trivial approximation of Quantum ChromoDynamics...But a main common feature emerges: in both cases the two fields are afflicted by the same broken symmetry, that of scale invariance."

an idea to be explored - speculative notes on possible phenomenon relating number theory, fractal geometry, Notalle's scale invariance, Renormalisation Group, etc.

Here is an intriguing excerpt from "On Fourier and Zeta(s)" by J.-F. Burnol:

"We are mainly inspired by the large body of ideas associated with the Renormalization Group, the Wilson idea of the statistical continuum limit, and the unification it has allowed of the physics of second-order phase transitions with the concepts of quantum field theory. Our general philosophical outlook had been originally deeply framed through the Niels Bohr idea of complementarity, but this is a topic more distant yet from our immediate goals, so we will leave this aside here.

We believe that the zeta function is analogous to a multiplicative wave-field renormalization. We expect that there exists some kind of a system, in some manner rather alike the Ising models of statistical physics, but much richer in its phase diagram, as each of the L-functions will be associated to a certain universality domain. That is, we do not at all attempt at realizing the zeta function as a partition function. No, the zeta function rather corresponds to some kind of symmetry pattern appearing at low temperature. But the other L-functions too may themselves be the symmetry where the system gets frozen at low temperature.

Renormalization group trajectories flow through the entire space encompassing all universality domains, and perhaps because there are literally fixed points, or another more subtle mechanism, this gives rise to sets of critical exponents associated with each domain: the (non-trivial) zeros of the L-functions. So there could be some underlying quantum dynamics, but the zeros arise at a more classical level, at the level of the renormalization group flow."

[expand this excerpt]

M.D. Missarov, "Random fields on the adele ring and Wilson's renormalization group", Ann. Inst. H. Poincaré 49 (1989) 357-367

R. Padma and H. Gopalkrishna Gadiyar, "Renormalisation and the density of prime pairs"

"Ideas from physics are used to show that the prime pairs have the density conjectured by Hardy and Littlewood. The proof involves dealing with infinities like in quantum field theory."

B. Fauser, "On a relation between Bogoliubov's renormalization operator and number theory" - talk given at Annual DPG Spring Conference, Hannover, March 24-28, 2003

B. Fauser and P.D. Jarvis, "The Dirichlet Hopf algebra of arithmetics" (preprint, 11/05)

[abstract:] "In this work, we develop systematically the 'Dirichlet Hopf algebra of arithmetics' by dualizing addition and multiplication maps. We study the additive and multiplicative antipodal convolutions which fail to give rise to Hopf algebra structures, obeying only a weakened (multiplicative) homomorphism axiom. The consequences of the weakened structure, called a Hopf gebra, e.g. on cohomology are explored. This features multiplicativity versus complete multiplicativity of number theoretic arithmetic functions. The deficiency of not being a Hopf algebra is then cured by introducing an 'unrenormalized' coproduct and an 'unrenormalized' pairing. It is then argued that exactly the failure of the homomorphism property (complete multiplicativity) for non-coprime integers is a blueprint for the problems in quantum field theory (QFT) leading to the need for renormalization. Renormalization turns out to be the morphism from the algebraically sound Hopf algebra to the physical and number theoretically meaningful Hopf gebra. This can be modelled alternatively by employing Rota-Baxter operators. We stress the need for a characteristic-free development where possible, to have a sound starting point for generalizations of the algebraic structures. The last section provides three key applications: symmetric function theory, quantum (matrix) mechanics, and the combinatorics of renormalization in QFT which can be discerned as functorially inherited from the development at the number-theoretic level as outlined here. Hence the occurrence of number theoretic functions in QFT becomes natural."

[abstract:] "Renormdynamic equations of motion and their solutions are given. New equation for NBD distribution and Riemann zeta function invented. Explicit forms of the z-Scaling functions are constructed."

[abstract:] "In this article we provide the complete proof of the result announced in arXiv:1210.7717 about the construction of scale invariant non-Gaussian generalized stochastic processes over three dimensional $p$-adic space. The construction includes that of the associated squared field and our result shows this squared field has a dynamically generated anomalous dimension which rigorously confirms a prediction made more than forty years ago, in an essentially identical situation, by K. G. Wilson. We also prove a mild form of universality for the model under consideration. Our main innovation is that our rigourous renormalization group formalism allows for space dependent couplings. We derive the relationship between mixed correlations and the dynamical systems features of our extended renormalization group transformation at a nontrivial fixed point. The key to our control of the composite field is a partial linearization theorem which is an infinite-dimensional version of the Koenigs Theorem in holomorphic dynamics. This is akin to a nonperturbative construction of a nonlinear scaling field in the sense of F. J. Wegner infinitesimally near the critical surface. Our presentation is essentially self-contained and geared towards a wider audience. While primarily concerning the areas of probability and mathematical physics we believe this article will be of interest to researchers in dynamical systems theory, harmonic analysis and number theory. It can also be profitably read by graduate students in theoretical physics with a craving for mathematical precision while struggling to learn the renormalization group."

[abstract:] "Two methods to prove the Riemann Hypothesis are presented. One is based on the modular properties of $\Theta$ (theta) functions and the other on the Hilbert–Pólya proposal to find an operator whose spectrum reproduces the ordinates $\rho_n$ (imaginary parts) of the zeta zeros in the critical line: $s_n=1/2+i\rho n$. A detailed analysis of a one-dimensional Dirac-like operator with a potential $V(x)$ is given that reproduces the spectrum of energy levels $E_n = \rho_n$, when the boundary conditions $\Psi_E(x=-\infty)=\pm\Psi_E(x=+\infty) are imposed. Such potential $V(x)$ is derived implicitly from the relation $x=x(V)=\frac{\pi}{2}2 (dN(V)/dV), where the functional form of $N(V)$ is given by the full-fledged Riemann–von Mangoldt counting function of the zeta zeros, including the fluctuating as well as the $O(E^{-n})$ terms. The construction is also extended to self-adjoint Schroedinger operators. Crucial is the introduction of an energy-dependent cut-off function $\Lambda(E)$. Finally, the natural quantization of the phase space areas (associated to nonperiodic crystal-like structures) in integer multiples of $\pi$ follows from the Bohr–Sommerfeld quantization conditions of Quantum Mechanics. It allows to find a physical reasoning why the average density of the primes distribution for very large $x(O(\frac{1}{\log x}))$ has a one-to-one correspondence with the asymptotic limit of the $inverse$ average density of the zeta zeros in the critical line suggesting intriguing connections to the renormalization group program."

[abstract:] "The heat trace asymptotics on the noncommutative torus, where generalized Laplacians are made out of left and right regular representations, is fully determined. It turns out that this question is very sensitive to the number-theoretical aspect of the deformation parameters. The central condition we use is of a Diophantine type. More generally, the importance of number theory is made explicit on a few examples. We apply the results to the spectral action computation and revisit the UV/IR mixing phenomenon for a scalar theory. Although we find non-local counterterms in the NC $\phi^4$ theory on $\T^4$, we show that this theory can be made renormalizable at least at one loop, and may be even beyond."

M. Marcolli and A. Connes, "From physics to number theory via noncommutative geometry. Part II: Renormalization, the Riemann-Hilbert correspondence, and motivic Galois theory", from Frontiers in Number Theory, Physics, and Geometry: On Random Matrices, Zeta Functions, and Dynamical Systems (Springer, 2006)

The following is a good introduction, with historical background, etc.:

N. Robles, "Zeta Function Regularization" (MSc thesis, Imperial College London, 2009)
 

J. S. Dowker and R. Critchley , "Effective Lagrangian and energy-momentum tensor in de Sitter space", Phys. Rev. D 13 (1976) 3224-3232.

[abstract:] "The effective Lagrangian and vacuum energy-momentum tensor $<T^{\mu\nu}>$ due to a scalar field in a de Sitter-space background are calculated using the dimensional-regularization method...More formally, a general zeta-function method is developed. It yields the renormalized effective Lagrangian as the derivative of the zeta function on the curved space. This method is shown to be virtually identical to a method of dimensional regularization applicable to any Riemann space."

S.W. Hawking, "Zeta function regularization of path integrals in curved spacetime", Communications in Mathematical Physics 55, No. 2 (1977) 133-148

[summary:] "Describes a technique for regularizing quadratic path integrals on a curved background spacetime. One forms a generalized zeta function from the eigenvalues of the differential operator that appears in the action integral. The zeta function is a meromorphic function and its gradient at the origin is defined to be the determinant of the operator. This technique agrees with dimensional regularization. The generalized zeta function can be expressed as a Mellin transform of the kernel of the heat equation which describes diffusion over the four dimensional spacetime manifold in a fifth dimension of parameter time."
 

Chapter 1 of the following book is also an excellent introduction to the subject:

E. Elizalde, Ten Physical Applications of Spectral Zeta Functions, Lecture Notes in Physics. New Series M, Monographs, M35 (Springer-Verlag, 1995)

"Zeta-function regularization is a powerful method in perturbation theory. This book is meant as a guide for the student of this subject. Everything is explained in detail, in particular the mathematical difficulties and tricky points, and several applications are given to show how the procedure works in practice (e.g. Casimir effect, gravity and string theory, high-temperature phase transition, topological symmetry breaking). The formulas some of which are new can be used for accurate numerical calculations. The book is to be considered as a basic introduction and a collection of exercises for those who want to apply this regularization procedure in practice."

E. Elizalde, S.D. Odintsov, A. Romeo and S. Zerbini, Zeta Regularization Techniques With Applications (World Scientific, 1994)

"This book is the result of several years of work by the authors on different aspects of zeta functions and related topics. The aim is twofold. On one hand, a considerable number of useful formulas, essential for dealing with the different aspects of zeta-function regularization (analytic continuation, asymptotic expansions), many of which appear here, in book format, for the first time are presented. On the other hand, the authors show explicitly how to make use of such formulas and techniques in practical applications to physical problems of very different nature. Virtually all types of zeta functions are dealt with in the book."

E. Elizalde, S. Leseduarte and S. Zerbini, "Mellin transform techniques for zeta-function resummations" (report UB-ECM-PF 92/7, 1993)

"Making use of inverse Mellin transform techniques for analytical continuation, an elegant proof and an extension of the zeta function regularization theorem is obtained...As an application of the method, the summation of the series which appear in the analytic computation (for different ranges of temperature) of the partition function of the string - basic in order to ascertain if QCD is some limit of a string theory - is performed."

E. Elizalde, S. Leseduarte and S.D. Odintsov, "Partition functions for the rigid string and membrane at any temperature", Phys. Rev. D48 (1993) 1757-1767

"Exact expressions for the partition functions of the rigid string and membrane at any temperature are obtained in terms of hypergeometric functions. By using zeta function regularization methods, the results are analytically continued and written as asymptotic sums of Riemann-Hurwitz zeta functions, which provide very good numerical approximations with just a few first terms."

E. Elizalde, "Spectral zeta functions in non-commutative spacetime", Nucl. Phys. Proc. Suppl. 104 (2002) 157-160

"Formulas for the most general case of the zeta function associated to a quadratic+linear+constant form (in {\bf Z}) are given. As examples, the spectral zeta functions $\zeta_\alpha (s)$ corresponding to bosonic ($\alpha =2$) and to fermionic ($\alpha =3$) quantum fields living on a noncommutative, partially toroidal spacetime are investigated. Simple poles show up at s = 0, as well as in other places (simple or double, depending on the number of compactified, noncompactified, and noncommutative dimensions of the spacetime). This poses a challenge to the zeta-function regularization procedure."

G. Cognola, E. Elizalde and S. Zerbini, "Fluctuations of quantum fields via zeta function regularization", Phys. Rev. D65 (2002)

[abstract:] "Explicit expressions for the expectation values and the variances of some observables, which are bilinear quantities in the quantum fields on a D-dimensional manifold, are derived making use of zeta function regularization. It is found that the variance, related to the second functional variation of the effective action, requires a further regularization and that the relative regularized variance turns out to be 2/N, where N is the number of the fields, thus being independent on the dimension D. Some illustrating examples are worked through."

E. Elizalde, "Zeta-function regularization is well-defined and well", Journal of Physics A 27 (1994) L299-304

[abstract:] "Hawking's zeta function regularization procedure is shown to be rigorously and uniquely defined, thus putting and end to the spreading lore about different difficulties associated with it. Basic misconceptions, misunderstandings and errors which keep appearing in important scientific journals when dealing with this beautiful regularization method - and other analytical procedures- are clarified and corrected."

E. Elizalde, "Some uses of zeta-regularization in quantum gravity and cosmology", Grav. Cosmol. 8 (2002) 43-48

"This is a short guide to some uses of the zeta-function regularization procedure as a a basic mathematical tool for quantum field theory in curved space-time (as is the case of Nambu-Jona-Lasinio models), in quantum gravity models (in different dimensions), and also in cosmology, where it appears e.g. in the calculation of possible 'contributions' to the cosmological constant coming through manifestations of the vacuum energy density."

V. Moretti and D. Iellici, "Zeta-function regularization and one-loop renormalization of field fluctuations in curved space-times", Phys. Lett. B425 (1998) 33-40

"A method to regularize and renormalize the fluctuations of a quantum field in a curved background in the zeta-function approach is presented. The method produces finite quantities directly and finite scale-parametrized counterterms at most. These finite couterterms are related to the presence of a particular pole of the effective-action zeta function as well as to the heat kernel coefficients. The method is checked in several examples obtaining known or reasonable results. Finally, comments are given for as it concerns the recent proposal by Frolov et.al. to get the finite Bekenstein-Hawking entropy from Sakharov's induced gravity theory."

M. Fujimoto and K. Uehara, "Regularization for zeta functions with physical applications I" (preprint 10/2006)

[abstract:] "We propose a regularization technique and apply it to the Euler product of zeta functions, mainly of the Riemann zeta function, to make unknown some clear. In this paper that is the first part of the trilogy, we try to demonstrate the Riemann hypotheses by this regularization technique and show conditions to realize them. In part two, we will focus on zeros of the Riemann zeta function and the nature of prime numbers in order to prepare ourselves for physical applications in the third part."

E. Sandier and S. Serfaty, "From the Ginzburg-Landau model to vortex lattice problems" (preprint 11/2010)

[abstract:] "We study minimizers of the two-dimensional Ginzburg-Landau energy with applied magnetic field, between the first and second critical fields. In this regime, minimizing configurations exhibit densely packed hexagonal vortex lattices, called Abrikosov lattices. We derive, in some asymptotic regime, a limiting interaction energy between points in the plane, $W$, which we prove has to be minimized by limits of energy-minimizing configurations, once blown-up at a suitable scale. This is a next order effect compared to the mean-field type results we previously established. The limiting ``Coulombian renormalized energy" $W$ is a logarithmic type of interaction, computed by a ``renormalization," and we believe it should be rather ubiquitous. We study various of its properties, and show in particular, using results from number theory, that among lattice configurations the hexagonal lattice is the unique minimizer, thus providing a first rigorous hint at the Abrikosov lattice. Its minimization in general remains open. The derivation of $W$ uses energy methods: the framework of $\Gamma$-convergence, and an abstract scheme for obtaining lower bounds for ``2-scale energies" via the ergodic theorem."

C. Jimenez and N. Vanegas, "Calculation of the determinant in the Wheeler–De Witt equation" (preprint 02/2013)

[abstract:] "The Riemann-zeta function regularization procedure has been studied intensively as a good method in the computation of the determinant for pseudo-differential operator. In this paper we propose a different approach for the computation of the determinant base on the Wheeler–De Witt equation."

S.S. Avancini, R.L.S. Farias, W.R. Tavares, "Neutral meson properties in hot and magnetized quark matter: A new magnetic field independent regularization scheme applied to NJL-type model" (preprint 12/2018)

[abstract:] "A magnetic field independent regularization scheme (zMFIR) based on the Hurwitz–Riemann zeta function is introduced. The new technique is applied to the regularization of the mean-field thermodynamic potential and mass gap equation within the $SU(2)$ Nambu–Jona–Lasinio model in a hot and magnetized medium. The equivalence of the new and the standard MFIR scheme is demonstrated. The neutral meson pole mass is calculated in a hot and magnetized medium and the advantages of using the new regularization scheme are shown."

J.J.G. Moreta, "A new approach to the renormalization of UV divergences using zeta regularization techniques" (preprint 04/2008)

[abstract:] "In this paper we present a method to deal with divergences in perturbation theory using the method of the zeta regularization, first of all we use the Euler-MacLaurin Sum formula to associate the divergent integral to a divergent sum in the form 1 + 2^m + 3^m + 4^m + .... After that we find a recurrence formula for the integrals and apply zeta regularization techniques to obtain finite results for the divergent series. (Through all the paper we use the notation "m" for the power of the modulus of p, so we must not confuse it with the value of the mass of the quantum particle)."

This is from D. Kreimer's summary of his research interests found on his homepage:

"Feynman Diagrams, Knot Theory and Number Theory

To what extent is a coefficient of ultraviolent divergence in a Feynman integral uniquely determined by the topology of the underlying graph?

It turned out to be true that the topology of a Feynman graph can be related to braid-positive knots. This establishes a knot-to-number dictionary: if and only if a certain braid-positive knot is obtained from a graph, the evaluation of this graph will produce a corresponding transcendental number as its coefficient of ultraviolet divergence

My results...led to the conclusion that Feynman diagrams obtained from a field theory in even dimensions all evaluate to the same number-class up to the seven-loop level, the limit of computational ability at this time, although we believe this result to be true in general. Recently Kontsevich conjectured a related result. The precise determination of this generic number class at high loop orders is an important open problem for number theorists and computational physicists alike.

The elimination of ultraviolet divergences by local counterterms, commonly known as Bogoliubov-Parasuik-Hepp-Zimmermann (BPHZ) renormalization, is achieved by a recursion whose solution is Zimmermann's forest formula. In the summer of 1997 I discovered that this algebraic structure establishes a Hopf algebra structure on Feynman graphs...The primitive elements of this Hopf algebra are primitive graphs considered in the previous section, and the determination of all the algebraic relations between them leads back to the number theory discussed above."

The number theory (and the "transcendental numbers") in question involves multiple zeta values and Euler sums.

V.L.Cartas, "The Riemann zeta function applied to the glassy systems and neural networks" (presented at International Conference on Theoretical Physics - Paris, UNESCO, 22-27 July 2002) [MS Word document]

[Abstract:] "In the present paper it is described how the Riemann zeta function could be a very useful tool in the analyze of the glassy systems and the neural networks. According to A. Crisanti and F. Ritort, this kind of complex systems could be analyzed using a simple solvable model of glass: "The oscillator model" which is defined by a set of N non-interacting harmonic oscillators with energy. The Riemann zeta function is used to describe the Crisanti-Ritort System. It has been also made a topological study in order to have a more intuitive representation of the critical points, where the states of the system changes."

[This draws heavily on Elizalde's work.]

L.R. Surguladze and M.A. Samuel, "On the renormalization group ambiguity of perturbative QCD for R(s) in e⁺e^- annihilation and $R_{\tau}$ in $\tau$-decay", Physics Letters B 309 (1993) 157-162

[abstract:] "The $O(\alpha_{s}^{3})$ perturbative QCD result for R(S) in e⁺e^- annihilation is given with explicit dependence on the scale parameter. We apply the three known approaches for resolving the scheme-scale ambiguity and we fix the scale for which all of the criteria tested are satisfied. We find the four-loop R(s) within the new scheme with flavor independent perturbative coefficients. . .

We find a remarkable cancellation of the Riemann zeta-functions at the 3-loop level. The theoretical uncertainty of the QCD effect in R(s) is estimated at 4%. The results of the analysis of $R_{\tau}$ in $\tau$-decay are presented."

V. V. Nesterenko and I. G. Pirozhenko, "Justification of the zeta function renormalization in rigid string model"

"A consistent procedure for regularization of divergences and for the subsequent renormalization of the string tension is proposed in the framework of the one-loop calculation of the interquark potential generated by the Polyakov-Kleinert string. In this way, a justification of the formal treatment of divergences by analytic continuation of the Riemann and Epstein-Hurwitz zeta functions is given. A spectral representation for the renormalized string energy at zero temperature is derived, which enables one to find the Casimir energy in this string model at nonzero temperature very easy."

V.V. Nesterenko, G. Lambiase and G. Scarpetta, "Casimir effect for a perfectly conducting wedge in terms of local zeta function"

[abstract:] "The vacuum energy density of electromagnetic field inside a perfectly conducting wedge is calculated by making use of the local zeta function technique. This regularization completely eliminates divergent expressions in the course of calculations and gives rise to a finite expression for the energy density in question without any subtractions. Employment of the Hertz potentials for constructing the general solution to the Maxwell equations results in a considerable simplification of the calculations. Transition to the global zeta function is carried out by introducing a cutoff nearby the cusp at the origin. Proceeding from this the heat kernel coefficients are calculated and the high temperature asymptotics of the Helmholtz free energy and of the torque of the Casimir forces are found. The wedge singularity gives rise to a specific high temperature behaviour $\sim T^2$ of the quantities under consideration. The obtained results are directly applicable to the free energy of a scalar massless field and electromagnetic field on the background of a cosmic string."

M. Schaden, "Sign and other aspects of semiclassical Casimir energies", Phys. Rev. A 73 (2006) 042102

[abstract:] "The Casimir energy of a massless scalar field is semiclassically given by contributions due to classical periodic rays. The required subtractions in the spectral density are determined explicitly. The semiclassical Casimir energies so defined coincide with those of zeta function regularization in the cases studied. Poles in the analytic continuation of zeta function regularization are related to nonuniversal subtractions in the spectral density. The sign of the Casimir energy of a scalar field on a smooth manifold is estimated by the sign of the contribution due to the shortest periodic rays only. Demanding continuity of the Casimir energy under small deformations of the manifold, the method is extended to integrable systems. The Casimir energy of a massless scalar field on a manifold with boundaries includes contributions due to periodic rays that lie entirely within the boundaries. These contributions in general depend on the boundary conditions. Although the Casimir energy due to a massless scalar field may be sensitive to the physical dimensions of manifolds with boundary. In favorable cases its sign can, contrary to conventional wisdom, be inferred without calculation of the Casimir energy."

C. Lousto, "Towards the solution of the relativistic gravitational radiation reaction problem for binary black holes"

"Here we present the results of applying the generalized Riemann zeta-function regularization method to the gravitational radiation reaction problem. We analyze in detail the headon collision of two nonspinning black holes with extreme mass ratio. The resulting reaction force on the smaller hole is repulsive. We discuss the possible extensions of these method to generic orbits and spinning black holes. The determination of corrected trajectories allows to add second perturbative corrections with the consequent increase in the accuracy of computed waveforms."

P.M. Ferreira, J.A. Gracey, "Non-zeta knots in the renormalization of the Wess-Zumino model?"

"We solve the Schwinger Dyson equations of the O(N) symmetric Wess-Zumino model at O(1/N³) at the non-trivial fixed point of the d-dimensional beta-function and deduce a critical exponent for the wave function renormalization at this order. By developing the epsilon-expansion of the result, which agrees with known perturbation theory, we examine the distribution of transcendental coefficients and show that only the Riemann zeta series arises at this order in 1/N."

J.A. Nogueira, A. Maia, Jr., "Demonstration of how the zeta function method for effective potential removes the divergences"

[abstract:] "The calculation of the minimum of the effective potential using the zeta function method is extremely advantagous, because the zeta function is regular at s = 0 and we gain immediately a finite result for the effective potential without the necessity of subtratction of any pole or the addition of infinite counter-terms. The purpose of this paper is to explicitly point out how the cancellation of the divergences occurs and that the zeta function method implicitly uses the same procedure used by Bollini-Giambiagi and Salam-Strathdee in order to gain finite part of functions with a simple pole."

R. Cianci and A. Khrennikov, "Can p-adic numbers be useful to regularize divergent expectation values of quantum observables?", International Journal of Theoretical Physics, 33, no. 6 (1994) 1217-1228.

S. Albeverio and A. Khrennikov, "A regularization of quantum field Hamiltonians with the aid of p-adic numbers", Acta Appl. Math. 50 (1998) 225-251.

R. Cianci, A. Khrennikov, "p-adic numbers and the renormalization of eigenfunctions in quantum mechanics", Physics Letters B, no. 1/2, (1994) 109-112.

P. Cvitanovic, "Circle Maps: Irrationally Winding" from Number Theory and Physics, eds. C. Itzykson, et. al. (Springer, 1992)

"The renormalization theory of critical circle maps demans at present rather tedious numerical computations, and our intuition is much facilitated by approximating circle maps by number-theoretic models. The model that we shall use here to illustrate the basic concepts might at first glance appear trivial, but we find it very instructive, as much that is obscured by numerical work required by the critical maps is here readily number-theoretically accessible. Indicative of the depth of mathematics lurking behind physicists' conjectures is that fact that the properties that one would like to establish about the renormalization theory of critical circle maps might turn out to be related to number-theoretic abysses such as the Riemann conjecture, already in the context of the 'trivial' models."

J.G. Dueñas and N.F. Svaiter, "Riemann zeta zeros and zero-point energy" (preprint 11/2013)

[abstract:] "We postulate the existence of a self-adjoint operator associated to a system with countably infinite number of degrees of freedom whose spectrum is the sequence of the nontrivial zeros of the Riemann zeta function. We assume that it describes a massive scalar field coupled to a background field in a $(d+1)$-dimensional flat space-time. The scalar field is confined to the interval $[0,a]$ in one dimension and is not restricted in the other dimensions. The renormalized zero-point energy of this system is presented using techniques of dimensional and analytic regularization. In even dimensional space-time, the series that defines the regularized vacuum energy is finite. For the odd-dimensional case, to obtain a finite vacuum energy per unit area we are forced to introduce mass counterterms. A Riemann mass appears, which is the correction to the mass of the field generated by the nontrivial zeros of the Riemann zeta function."

M. Bordag, A. S. Goldhaber, P. van Nieuwenhuizen and D. Vassilevich, "Heat kernels and zeta-function regularization for the mass of the SUSY kink"

[abstract:] "We apply zeta-function regularization to the kink and susy kink and compute its quantum mass. We fix ambiguities by the renormalization condition that the quantum mass vanishes as one lets the mass gap tend to infinity while keeping scattering data fixed. As an alternative we write the regulated sum over zero point energies in t erms of the heat kernel and apply standard heat kernel subtractions. Finally we discuss to what extent these procedures are equivalent to the usual renormalization conditions that tadpoles vanish."

V. Di Clemente, S. F. King and D.A.J. Rayner, "Supersymmetry and electroweak breaking with large and small extra dimensions", Nucl. Phys. B 617 (2001) 71-100

[abstract:] "We consider the problem of supersymmetry and electroweak breaking in a 5d theory compactified on an $S^{1}/Z_{2}$ orbifold, where the extra dimension may be large or small. We consider the case of a supersymmetry breaking 4d brane located at one of the orbifold fixed points with the Standard Model gauge sector, third family and Higgs fields in the 5d bulk, and the first two families on a parallel 4d matter brane located at the other fixed point. We compute the Kaluza-Klein mass spectrum in this theory using a matrix technique which allows us to interpolate between large and small extra dimensions. We also consider the problem of electroweak symmetry breaking in this theory and localize the Yukawa couplings on the 4d matter brane spatially separated from the brane where supersymmetry is broken. We calculate the 1-loop effective potential using a zeta-function regularization technique, and find that the dominant top and stop contributions are separately finite. Using this result we find consistent electroweak symmetry breaking for a compactification scale {$ 1/R \approx 830$ GeV} and a lightest Higgs boson mass $m_{h} \approx 170$ GeV."

H. Matsui and Y. Matsumoto, "Revisiting regularization with Kaluza–Klein states and Casimir vacuum energy from extra dimensional spacetime" (preprint 04/2018)

[abstract:] "In the present paper, we investigate regularization of the one-loop quantum corrections with infinite Kaluza–Klein (KK) states and evaluate Casimir vacuum energy from extra dimensions. The extra dimensional models always involve the infinite massless or massive KK states, and therefore, the regularization of the infinite KK corrections is highly problematic. In order to avoid the ambiguity, we adopt the proper time integrals and the Riemann zeta function regularization in evaluating the summations of infinite KK states. In the calculation, we utilized the dimensional regularization method without exchanging the summations and the loop integrals. At the same time, we also evaluate the correction by the KK regularization method. Then, we clearly show that the regularized Casimir corrections from the KK states have the form of $1/R^2$ for the Higgs mass and $1/R^4$ for the cosmological constant, where $R$ is the compactification radius. We also evaluate the Casimir energy in supersymmetric extra-dimensional models. The contributions from bulk fermions and bulk bosons are not offset because the general boundary conditions break the supersymmetry. The non-zero supersymmetric Casimir corrections from extra dimensions undoubtedly contribute to the Higgs mass and the cosmological constant. We conclude such corrections are enhanced compared to the case without bulk supersymmetry."

M.R. Setare and R. Mansouri, "Casimir energy for self-interacting scalar field in a spherical shell"

[abstract:] "In this paper we calculate the Casimir energy for spherical shell with massless self-interacting scalar filed which satisfying Dirichlet boundary conditions on the shell. Using zeta function regularization and heat kernel coefficients we obtain the divergent contributions inside and outside of Casimir energy. The effect of self-interacting term is similar with existing of mass for filed. In this case some divergent part arises. Using the renormalization procedure of bag model we can cancel these divergent parts."

K. Sakai, M. Shiroishi, Y. Nishiyama and M. Takahashi, "Third Neighbor Correlators of Spin-1/2 Heisenberg Antiferromagnet"

[abstract:] "We exactly evaluate the third neighbor correlator < S_j^z S_j+3^z > and all the possible non-zero correlators <S^alpha_j S^beta_j+1S^gamma _j+2S^delta_j+3 > of the spin-1/2 Heisenberg XXX antiferromagnet in the ground state without magnetic field. All the correlators are expressed in terms of certain combinations of logarithm ln2, the Riemann zeta function zeta(3), zeta(5) with rational coefficients. The results accurately coincide with the numerical ones obtained by the density-matrix renormalization group method and the numerical diagonalization."

J. Fiala and P. Kleban, "Thermodynamics of the Farey Fraction Spin Chain", J. Stat. Physics 116 (2004) 1471-1490

[abstract:] "We consider the Farey fraction spin chain, a one-dimensional model defined on (the matrices generating) the Farey fractions. We extend previous work on the thermodynamics of this model by introducing an external field h. From rigorous and renormalization group arguments, we determine the phase diagram and phase transition behavior of the extended model. Our results are fully consistent with scaling theory (for the case when a "marginal" field is present) despite the unusual nature of the transition for h=0."

T. Prellberg, J. Fiala and P. Kleban, "Cluster approximation for the Farey fraction spin chain" (prepring 07/05)

[abstract:] "We consider the Farey fraction spin chain in an external field h. Utilising ideas from dynamical systems, the free energy of the model is derived by means of an effective cluster energy approximation. This approximation is valid for divergent cluster sizes, and hence appropriate for the discussion of the magnetizing transition. We calculate the phase boundaries and the scaling of the free energy. At h = 0 we reproduce the rigorously known asymptotic temperature dependence of the free energy. For h <> 0, our results are largely consistent with those found previously using mean field theory and renormalization group arguments."

B.P. Dolan, "Duality and the modular group in the quantum Hall effect", J. Phys. A 32 (1999) L243

[abstract:] "We explore the consequences of introducing a complex conductivity into the quantum Hall effect. This leads naturally to an action of the modular group on the upper-half complex conductivity plane. Assuming that the action of a certain subgroup, compatible with the law of corresponding states, commutes with the renormalisation group flow, we derive many properties of both the integer and fractional quantum Hall effects, including: universality; the selection rule |p₁q₂ - p₂q₁|=1 for quantum Hall transitions between filling factors nu₁ = p₁/q₁ and nu₂ = p₂/q₂; critical values for the conductivity tensor; and Farey sequences of transitions. Extra assumptions about the form of the renormalisation group flow lead to the semi-circle rule for transitions between Hall plateaus."

N. Makhaldiani, "Renormdynamics, multiparticle production, negative binomial distribution and Riemann zeta function" (preprint 12/2010)

[abstract:] "Renormdynamic equations of motion and their solutions are given. New equation for NBD distribution and Riemann zeta function invented. Explicit forms of the z-Scaling functions are constructed."

L. Guo, S. Paycha and B. Zhang, "Renormalization of conical zeta values and the Euler–Maclaurin formula" (preprint 06/2013)

[abstract:] "We equip the space of convex rational cones with a connected coalgebra structure, which we further generalize to decorated cones by means of a differentiation procedure. Using the convolution product $\ast$ associated with the coproduct on cones we define an interpolator $\mu:= I^{\ast(-1)}\ast S$ as the $\ast$ quotient of an exponential discrete sum $S$ and an exponential integral $I$ on cones. A generalization of the algebraic Birkhoff decomposition to linear maps from a connected coalgebra to a space with a linear decomposition then enables us to carry out a Birkhoff–Hopf factorization $S:= S_-^{\ast (-1)}\ast S_+ $ on the map $S$ whose "holomorphic part" corresponds to $S_+$. By the uniqueness of the Birkhoff–Hopf factorization we obtain $\mu=S_+$ and $I=S_-^{\ast (-1)}$ so that this renormalization procedure à la Connes and Kreimer yields a new interpretation of the local Euler–Maclaurin formula on cones of Berline and Vergne. The Taylor coefficients at zero of the interpolating holomorphic function $\mu=S_+$ correspond to renormalized conical zeta values at non-positive integers. When restricted to Chen cones, this yields yet another way to renormalize multiple zeta values at non-positive integers previously investigated by the authors using other approaches.

In the present approach renormalized conical multiple zeta values lie at the cross road of three a priori distinct fields, the geometry on cones with the Euler–Maclaurin formula, number theory with multiple zeta values and renormalization theory with methods borrowed from quantum field theory."

G. Dattoli, M. Del Franco, "The Euler legacy to modern physics" (preprint 09/2010)

[abstract:] "Particular families of special functions, conceived as purely mathematical devices between the end of XVIII and the beginning of XIX centuries, have played a crucial role in the development of many aspects of modern Physics. This is indeed the case of the Euler gamma function, which has been one of the key elements paving the way to string theories, furthermore the Euler–Riemann Zeta function has played a decisive role in the development of renormalization theories. The ideas of Euler and later those of Riemann, Ramanujan and of other, less popular, mathematicians have therefore provided the mathematical apparatus ideally suited to explore, and eventually solve, problems of fundamental importance in modern Physics. The mathematical foundations of the theory of renormalization trace back to the work on divergent series by Euler and by mathematicians of two centuries ago. Feynman, Dyson, Schwinger... rediscovered most of these mathematical "curiosities" and were able to develop a new and powerful way of looking at physical phenomena."

W. Bietenholz, "From Ramanujan to renormalization: The art of doing away with divergences and arriving at physical results" (preprint 02/2021)

[abstract:] "A century ago Srinivasa Ramanujan – the great self-taught Indian genius of mathematics – died, shortly after returning from Cambridge, UK, where he had collaborated with Godfrey Hardy. Ramanujan contributed numerous outstanding results to different branches of mathematics, like analysis and number theory, with a focus on special functions and series. Here we refer to apparently weird values which he assigned to two simple divergent series, $\sum_{n\geq 1} n$ and $\sum_{n\geq 1}n^3$. These values are sensible, however, as analytic continuations, which correspond to Riemann's $\zeta$-function. Moreover, they have applications in physics: we discuss the vacuum energy of the photon field, from which one can derive the Casimir force, which has been experimentally measured. We also discuss its interpretation, which remains controversial. This is a simple way to illustrate the concept of renormalization, which is vital in quantum field theory."

I.O. Goriachuk and A.L. Kataev, "Riemann $\zeta(4)$ function contributions to $O(\alpha_s^5)$ terms of Adler D-function and Bjorken polarized sum rule in $SU(N_c)$ QCD: Results and consequences" (preprint 11/2020)

[abstract:] "Two renormalization group invariant quantities in quantum chromodinamics (QCD), defined in Euclidean space,namely, Adler D-function of electron-positron annihilation to hadrons and Bjorken polarized deep-inelastic scattering sum rule, are considered. It is shown, that the 5th order corrections to them in $\overline{MS}$-like renormalization prescriptions, proportional to Riemann $\zeta$-function $\zeta(4)$, can be restored by the transition to the C-scheme, with the $\beta$-function, analogous to Novikov, Shifman, Vainshtein and Zakharov exact $\beta$-function in $\mathcal{N} = 1$ supersymmetric gauge theories. The general analytical expression for these corrections in $SU(N_c)$ QCD is deduced and their scale invariance is shown. The $\beta$-expansion procedure for these contributions is performed and mutual cancellation of them in the 5th order of the generalized Crewther identity are discussed."

S.R.Dahmen, S.D.Prado and T.Stuermer-Daitx, "Similarity in the Statistics of Prime Numbers", Physica A 296 (2001) 523-528

[abstract:] "We present numerical evidence for regularities in the distribution of gaps between primes when these are divided into congruence families (in Dirichlet's classification). The histograms for the distribution of gaps of families are scale invariant."

Here is an excerpt from a posting by on the sci.physics newsgroup (02/98) by Dan Piponi:

"In (bosonic) string theory via the operator formalism you find an infinite looking zero point energy just like in QED except that you get a sum that looks like:
 

Now the naive thing to do is the same: subtract off this zero point energy. However later on you get into complications. In fact (if I remember correctly) you must replace this infinity with -1/12 (of all things!) to keep things consistent.

Now it turns out there is a nice mathematical kludge that allows you to see 1+2+3+4+... as equalling -1/12. What you do is rewrite it as
 

This is the Riemann Zeta function. This converges for large n but can be analytically continued to n = -1, even though the series doesn't converge there. Zeta(-1) is -1/12. So in some bizarre sense 1+2+3+4+... really is -1/12.

But even more amazingly is that you can get the -1/12 by a completely different route - using the path integral formalism rather than the operator formalism. This -1/12 is tied up in a deep way with the geometry of string theory so it's a lot more than simply a trick to keep the numbers finite.

However I don't know if the equivalent operation in QED is tied up with the same kind of interesting geometry."

John Baez has written extensively about the '-1/12 phenomenon', particularly highlighting the fact that -1/12 is the Euler characteristic of the group SL(2,Z) - see Week 213 of his This Week's Finds in Mathematical Physics series.