renormalisation, regularisation and number theory
"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 pathbreaking 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 WeinbergSalam 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. Nonrenormalizable 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) 800805
"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
SpaceTime and Microphysics  Towards a Theory of Scale Relativity (World
Scientific, 1993).
K. Shirish, Sduality 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 strongweak 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 BerryKeating 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 BerryKeating 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, "Logperiodic 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 Smatrix theories recently conjectured to describe a
cyclic regime of the KosterlitzThouless 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 ultraviolet, we argue that the massless version is
nonsingular 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,
logperiodicity, quantum groups (with real q), zeta function regularizations, number theory, etc.,
whose full significance needs to be clarified."
A. Petermann,
"The socalled renormalization group method applied to the specific
prime number logarithmic decrease"
"A socalled 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 nontrivial 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
secondorder 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
wavefield 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
Lfunctions 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
Lfunctions 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
(nontrivial) zeros of the Lfunctions. 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) 357367
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 2428, 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 noncoprime 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 RotaBaxter
operators. We stress the need for a characteristicfree 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 numbertheoretic level as outlined here.
Hence the occurrence of number theoretic functions in QFT becomes natural."
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 zScaling functions are constructed."
A. Abdesselam, A. Chandra and G. Guadagni, "Rigorous quantum field theory functional integrals over the $p$adics I: Anomalous dimensions" (preprint 01/2013)
[abstract:] "In this article we provide the complete proof of the result announced in arXiv:1210.7717 about the construction of scale invariant nonGaussian 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 infinitedimensional 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 selfcontained 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."
C. Castro, "On the Riemann hypothesis, area quantization, Dirac operators, modularity, and renomalization group", International Journal of Geometric Methods in Modern Physics 7 (2010) 1 31
[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 onedimensional Diraclike 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 fullfledged 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 selfadjoint Schroedinger operators. Crucial
is the introduction of an energydependent cutoff function $\Lambda(E)$. Finally, the natural
quantization of the phase space areas (associated to nonperiodic crystallike 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 onetoone 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."
V. Gayral, B. Iochum and D.V. Vassilevich, "Heat kernel and number theory
on NCtorus" (preprint 07/2006)
[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 numbertheoretical 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 nonlocal 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 RiemannHilbert correspondence,
and motivic Galois theory", from Frontiers in Number Theory,
Physics, and Geometry: On Random Matrices, Zeta Functions, and Dynamical Systems (Springer, 2006)
zeta function regularisation techniques
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 energymomentum tensor in de Sitter space", Phys. Rev. D 13 (1976) 32243232.
[abstract:] "The effective Lagrangian and vacuum energymomentum tensor $<T^{\mu\nu}>$ due to a
scalar field in a de Sitterspace background are calculated using the dimensionalregularization method...More
formally, a general zetafunction 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) 133148
[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 (SpringerVerlag, 1995)
"Zetafunction 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, hightemperature
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 zetafunction 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 zetafunction resummations" (report UBECMPF 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) 17571767
"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 RiemannHurwitz zeta functions,
which provide very good numerical approximations with just a few first
terms."
E. Elizalde, "Spectral
zeta functions in noncommutative spacetime", Nucl. Phys. Proc. Suppl. 104 (2002)
157160
"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 zetafunction 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 Ddimensional 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, "Zetafunction
regularization is welldefined and well", Journal of Physics A 27 (1994) L299304
[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 zetaregularization
in quantum gravity and cosmology", Grav. Cosmol. 8 (2002) 4348
"This is a short guide to some uses of the zetafunction regularization procedure as a a basic mathematical tool for quantum field theory
in curved spacetime (as is the case of NambuJonaLasinio 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, "Zetafunction regularization and oneloop
renormalization of field fluctuations in curved spacetimes", Phys. Lett. B425 (1998) 3340
"A method to regularize and renormalize the fluctuations of a quantum field in a curved background in the zetafunction approach is
presented. The method produces finite quantities directly and finite scaleparametrized counterterms at most. These finite couterterms
are related to the presence of a particular pole of the effectiveaction 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 BekensteinHawking 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 GinzburgLandau model to vortex lattice problems" (preprint 11/2010)
[abstract:] "We study minimizers of the twodimensional GinzburgLandau 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 energyminimizing configurations, once blownup at a suitable scale. This is a next order effect compared to the meanfield 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 ``2scale 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 Riemannzeta function regularization procedure has been studied intensively as a good method in the computation of the determinant for pseudodifferential 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 NJLtype 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 meanfield 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 EulerMacLaurin
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 braidpositive
knots. This establishes a knottonumber dictionary: if and only if a certain braidpositive
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 numberclass up to the sevenloop 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
BogoliubovParasuikHeppZimmermann (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, 2227 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 noninteracting harmonic oscillators with energy. The Riemann zeta function is
used to describe the CrisantiRitort 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) 157162
[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 schemescale ambiguity
and we fix the scale for which all of the criteria tested are satisfied. We find the fourloop
R(s) within the new scheme with flavor independent perturbative coefficients. . .
We find a remarkable cancellation of the Riemann zetafunctions at the 3loop 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 oneloop calculation of the interquark potential generated by the
PolyakovKleinert string. In this way, a justification of the formal treatment
of divergences by analytic continuation of the Riemann and EpsteinHurwitz
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
zetafunction 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,
"Nonzeta knots in the renormalization of the WessZumino model?"
"We solve the Schwinger Dyson equations of the O(N) symmetric
WessZumino model at O(1/N^{3}) at the nontrivial fixed
point of the ddimensional betafunction and deduce a critical
exponent for the wave function renormalization at this order. By
developing the epsilonexpansion 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 counterterms. 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 BolliniGiambiagi and
SalamStrathdee in order to gain finite part of functions with a simple pole."
R. Cianci and A. Khrennikov, "Can padic numbers be useful to
regularize divergent expectation values of quantum observables?",
International Journal of Theoretical Physics, 33, no. 6
(1994) 12171228.
S. Albeverio and A. Khrennikov, "A regularization of quantum field
Hamiltonians with the aid of padic numbers", Acta Appl. Math.
50 (1998) 225251.
R. Cianci, A. Khrennikov, "padic numbers and the renormalization
of eigenfunctions in quantum mechanics", Physics Letters B, no. 1/2,
(1994) 109112.
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 numbertheoretic
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 numbertheoretically 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 numbertheoretic 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 zeropoint energy" (preprint 11/2013)
[abstract:] "We postulate the existence of a selfadjoint 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 spacetime. The scalar field is confined to the interval $[0,a]$ in one dimension and is not restricted in the other dimensions. The renormalized zeropoint energy of this system is presented using techniques of dimensional and analytic regularization. In even dimensional spacetime, the series that defines the regularized vacuum energy is finite. For the odddimensional 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 zetafunction regularization for the mass of the SUSY kink"
[abstract:] "We apply zetafunction 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) 71100
[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 KaluzaKlein 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 1loop effective
potential using a zetafunction 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 oneloop 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 extradimensional models. The contributions from bulk fermions and bulk bosons are not offset because the general boundary conditions break the supersymmetry. The nonzero 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 selfinteracting scalar field in a spherical shell"
[abstract:] "In this paper we calculate the Casimir energy for
spherical shell with massless selfinteracting 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 selfinteracting 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 Spin1/2 Heisenberg Antiferromagnet"
[abstract:] "We exactly evaluate the third neighbor correlator < S_{j}^{z} S_{j}+3^{z} >
and all the possible nonzero correlators <S^{alpha}_{j} S^{beta}_{j}+1S^{gamma}
_{j}+2S^{delta}_{j}+3 > of the spin1/2 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 densitymatrix renormalization group method
and the numerical diagonalization."
J. Fiala and P. Kleban, "Thermodynamics of the Farey Fraction
Spin Chain", J. Stat. Physics 116 (2004) 14711490
[abstract:] "We consider the Farey fraction spin chain, a onedimensional 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 upperhalf 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 p_{1}q_{2} 
p_{2}q_{1}=1 for quantum Hall transitions between
filling factors nu_{1} = p_{1}/q_{1} and
nu_{2} = p_{2}/q_{2}; critical values for the conductivity
tensor; and Farey sequences of transitions. Extra assumptions about the form of the renormalisation group
flow lead to the semicircle 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 zScaling 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 nonpositive integers. When restricted to Chen cones, this yields yet another way to renormalize multiple zeta values at nonpositive 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 selftaught 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 Dfunction 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 Dfunction of electronpositron annihilation to hadrons and Bjorken polarized deepinelastic 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 Cscheme, 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.StuermerDaitx, "Similarity
in the Statistics of Prime Numbers", Physica A 296 (2001) 523528
[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:
1+2+3+4+...
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
1+2^{n} +3^{n} +...
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.
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