quantum field theory and number theory

"I was immediately led to the idea that somehow passing from the integers to the primes is very similar to passing from quantum field theory, as we observe it, to the elementary particles, whatever they are."

A. Connes quoted in K. Sabbagh, Dr. Riemann's Zeros (Atlantic, 2002), p.204

any thoughts on what Connes is implying here?

P. Woit, "Towards a grand unified theory of mathematics and physics" (preprint 06/2015)

[abstract:] "Wigner's "unreasonable effectiveness of mathematics" in physics can be understood as a reflection of a deep and unexpected unity between the fundamental structures of mathematics and of physics. Some of the history of evidence for this is reviewed, emphasizing developments since Wigner's time and still poorly understood analogies between number theory and quantum field theory."

D.J. Broadhurst, "Euler sums in quantum field theory"

The above is an appendix to the following:

J.M. Borwein and R. Girgensohn, "Evaluation of triple Euler sums", Electronic J. Combinatorics 3 (1996), R23

D.J. Broadhurst, "On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory", Journal of Mathematical Physics (to appear?)

[abstract:] "A generating function is given for the number, E(l,k), of irreducible k-fold Euler sums, with all possible alternations of sign, and exponents summing to l. Its form is remarkably simple: $\sum_n E(k+2n,k) x^n = \sum_{d|k}\mu(d) (1-x^d)^{-k/d}/k$, where $\mu$ is the Möbius function. Equivalently, the size of the search space in which k-fold Euler sums of level l are reducible to rational linear combinations of irreducible basis terms is $S(l,k) = \sum_{nk-fold sums with l < 7; combined methods yield bases for all remaining search spaces with $S(l,k)\leq34$. These findings confirm expectations based on Dirk Kreimer's connection of knot theory with quantum field theory. The occurrence in perturbative quantum electrodynamics of all 12 irreducible Euler sums with l < 7 is demonstrated. It is suggested that no further transcendental occurs in the four-loop contributions to the electron's magnetic moment. Irreducible Euler sums are found to occur in explicit analytical results, for counterterms with up to 13 loops, yielding transcendental knot-numbers, up to 23 crossings."

D.J. Broadhurst, "Conjectured enumeration of irreducible multiple zeta values, from knots and Feynman diagrams" (preprint, 12/96)

[abstract:] "Multiple zeta values (MZVs) are under intense investigation in three arenas - knot theory, number theory, and quantum field theory - which unite in Kreimer's proposal that field theory assigns MZVs to positive knots, via Feynman diagrams whose momentum flow is encoded by link diagrams. Two challenging problems are posed by this nexus of knot/number/field theory: enumeration of positive knots, and enumeration of irreducible MZVs. Both were recently tackled by Broadhurst and Kreimer (BK). Here we report large-scale analytical and numerical computations that test, with considerable severity, the BK conjecture that the number, D{n,k}, of irreducible MZVs of weight n and depth k, is generated by $\prod_{n\ge3}\prod_{k\ge1}(1-x^n y^k) ^{D_{n,k}}=1-\frac{x^3y}{1-x^2}+ \frac{x^{12}y^2(1-y^2)}{(1-x^4)(1-x^6)}$, which is here shown to be consistent with all shuffle identities for the corresponding iterated integrals, up to weights n = 44, 37, 42, 27, at depths k = 2, 3, 4, 5, respectively, entailing computation at the petashuffle level. We recount the field-theoretic discoveries of MZVs, in counterterms, and of Euler sums, from more general Feynman diagrams, that led to this success."

D.J. Broadhurst, "Where do the tedious products of zetas come from?", Nucl. Phys. Proc. Suppl. 116 (2003) 432-436

[abstract:] "Lamentably, the full analytical content of the epsilon-expansion of the master two-loop two-point function, with arbitrary self-energy insertions in 4-2epsilon dimensions, is still unknown. Here we show that multiple zeta values (MZVs) of weights up to 12 suffice through O(epsilon^9). Products of primitive MZVs are generated by a processes of "pseudo-exponentiation"" whose combinatorics faithfully accord with expectations based on Kreimer's modified shuffle product and on the Drinfeld-Deligne conjecture. The existence of such a mechanism, relating thousands of complicated rational numbers, enables us to identify precise and simple combinations of MZVs specific to quantum field theories in even numbers of spacetime dimensions."

D.J. Broadhurst and D. Kreiemer, "Knots and numbers in $\phi^4$ theory to 7 loops and beyond", Int. J. Mod. Phys. C6 (1995) 519-524

[abstract:] "We evaluate all the primitive divergences contributing to the 7-loop $\beta$-function of $\phi^4$ theory, i.e. all 59 diagrams that are free of subdivergences and hence give scheme--independent contributions. Guided by the association of diagrams with knots, we obtain analytical results for 56 diagrams. The remaining three diagrams, associated with the knots 10124, 10139, and 10152, are evaluated numerically, to 10 sf. Only one satellite knot with 11 crossings is encountered and the transcendental number associated with it is found. Thus we achieve an analytical result for the 6-loop contributions, and a numerical result at 7 loops that is accurate to one part in 1011. The series of 'zig-zag' counterterms, $\{6\zeta_3,\,20\zeta_5,\, \frac{441}{8}\zeta_7,\,168\zeta_9,\,\ldots\}$, previously known for n = 3,4,5,6 loops, is evaluated to 10 loops, corresponding to 17 crossings, revealing that the n-loop zig-zag term is $4C_{n-1} \sum_{p>0}\frac{(-1)^{p n - n}}{p^{2n-3}}$, where $C_n=\frac{1}{n+1}{2n \choose n}$ are the Catalan numbers, familiar in knot theory. The investigations reported here entailed intensive use of REDUCE, to generate O(104) lines of code for multiple precision FORTRAN computations, enabled by Bailey's MPFUN routines, running for ${\rm O}(10^3)$ CPUhours on DecAlpha machines."

J.M. Borwein and D.J. Broadhurst, "Determinations of rational Dedekind-zeta invariants of hyperbolic manifolds and Feynman knots and links" (preprint 11/98)

[abstract:] "We identify 998 closed hyperbolic 3-manifolds whose volumes are rationally related to Dedekind zeta values, with coprime integers a and b giving $a/b vol(M)=(-D)^{3/2}/(2\pi)^{2n-4} (\zeta_K(2))/(2\zeta(2))$ for a manifold M whose invariant trace field K has a single complex place, discriminant D, degree n, and Dedekind zeta value $\zeta_K(2)$. The largest numerator of the 998 invariants of Hodgson-Weeks manifolds is, astoundingly, a = 24 x 23 x 37 x 691 = 9,408,656; the largest denominator is merely b = 9. We also study the rational invariant a/b for single-complex-place cusped manifolds, complementary to knots and links, both within and beyond the Hildebrand-Weeks census. Within the censi, we identify 152 distinct Dedekind zetas rationally related to volumes. Moreover, 91 census manifolds have volumes reducible to pairs of these zeta values. Motivated by studies of Feynman diagrams, we find a 10-component 24-crossing link in the case n = 2 and D = -20. It is one of 5 alternating platonic links, the other 4 being quartic. For 8 of 10 quadratic fields distinguished by rational relations between Dedekind zeta values and volumes of Feynman orthoschemes, we find corresponding links. Feynman links with D = -39 and D = -84 are missing; we expect them to be as beautiful as the 8 drawn here. Dedekind-zeta invariants are obtained for knots from Feynman diagrams with up to 11 loops. We identify a sextic 18-crossing positive Feynman knot whose rational invariant, a/b=26, is 390 times that of the cubic 16-crossing non-alternating knot with maximal D9 symmetry. Our results are secure, numerically, yet appear very hard to prove by analysis."

D.J. Broadhurst and D. Kreimer, "Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops", Phys. Lett. B 393 (1997) 403

D.J. Broadhurst, "Massive 3-loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity", Eur. Phys. J. C8 (1999) 311-333

D. Broadhurst, "Feynman integrals, $L$-series and Kloosterman moments" (preprint 02/2016)

[abstract:] "This work lies at an intersection of three subjects: quantum field theory, algebraic geometry and number theory, in a situation where dialogue between practitioners has revealed rich structure. It contains a theorem and 7 conjectures, tested deeply by 3 optimized algorithms, on relations between Feynman integrals and $L$-series defined by products, over the primes, of data determined by moments of Kloosterman sums in finite fields. There is an extended introduction, for readers who may not be familiar with all three of these subjects. Notable new results include conjectural evaluations of non-critical $L$-series of modular forms of weights 3, 4 and 6, by determinants of Feynman integrals, an evaluation for the weight 5 problem, at a critical integer, and formulas for determinants of arbitrary size, tested up to 30 loops. It is shown that the functional equation for Kloosterman moments determines much but not all of the structure of the $L$-series. In particular, for problems with odd numbers of Bessel functions, it misses a crucial feature captured in this work by novel and intensively tested conjectures. For the 9-Bessel problem, these lead to an astounding compression of data at the primes."

J.M. Borwein, D.M. Bradley, D.J. Broadhurst, and P. Lisonek, "Combinatorial aspects of multiple zeta values", Electronic Journal of Combinatorics 5 (1998), R38

D.J. Broadhurst, J.M. Borwein, and D.M. Bradley, "Evaluation of k-fold Euler/Zagier sums: a compendium of results for arbitrary k", Electronic Journal of Combinatorics 4 (2) (1997), R5

J. Borwein, D. Broadhurst, J. Kamnitzer, "Central binomial sums, multiple Clausen values and zeta values" (preprint 04/00)

D.J. Broadhurst and Dirk Kreimer's number/field/knot theory bibliography

"Construing renormalization as a skein operation on link diagrams that encode momentum flow, new connections between field theory, knot theory, and number theory, have been forged, and intensively investigated to 7-loop order. This has also given a better understanding of which Euler sums are irreducible."

D. Kreimer, Knots and Feynman Diagrams (Cambridge U.P., 2000)

[excerpt from publisher's description:] "Beginning with a summary of key ideas from perturbative quantum field theory and an introduction to the Hopf algebra structure of renormalization, early chapters discuss the rationality of ladder diagrams and simple link diagrams. The necessary basics of knot theory are then presented and the number-theoretic relationship between the topology of Feynman diagrams and knot theory is explored. Later chapters discuss four-term relations motivated by the discovery of Vassiliev invariants in knot theory and draw a link to algebraic structures recently observed in noncommutative geometry."

C. Bergbauer and D. Kreimer, "New algebraic aspects of perturbative and non-perturbative Quantum Field Theory" (preprint 04/2007)

[abstract:] "In this expository article we review recent advances in our understanding of the combinatorial and algebraic structure of perturbation theory in terms of Feynman graphs, and Dyson-Schwinger equations. Starting from Lie and Hopf algebras of Feynman graphs, perturbative renormalization is rephrased algebraically. The Hochschild cohomology of these Hopf algebras leads the way to Slavnov-Taylor identities and Dyson-Schwinger equations. We discuss recent progress in solving simple Dyson-Schwinger equations in the high energy sector using the algebraic machinery. Finally there is a short account on a relation to algebraic geometry and number theory: understanding Feynman integrals as periods of mixed (Tate) motives."

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

"Feynan 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.

Combinatorics of perturbative Quantum Field Theory

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."

D.J. Broadhurst's homepage

popular article by Ivars Peterson on PSLQ algorithm as a generalisation of Euclid's algorithm, which refers to Broadhurt's application to QFT

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)

Yu. Manin and M. Marcolli, "Holography principle and arithmetic of algebraic curves", Adv. Theor. Math. Phys. 5 (2001), no. 3, 617–650.

[abstract:] "According to the holography principle (due to G. 't Hooft, L. Susskind, J. Maldacena, et al.), quantum gravity and string theory on certain manifolds with boundary can be studied in terms of a conformal field theory on the boundary. Only a few mathematically exact results corroborating this exciting program are known. In this paper we interpret from this perspective several constructions which arose initially in the arithmetic geometry of algebraic curves. We show that the relation between hyperbolic geometry and Arakelov geometry at arithmetic infinity involves exactly the same geometric data as the Euclidean AdS3 holography of black holes. Moreover, in the case of Euclidean AdS2 holography, we present some results on bulk/boundary correspondence where the boundary is a non-commutative space."

M. Marcolli's survey article "Number Theory in Physics" contains material on the role of multiple zeta values in QFT, etc.

J. G. Dueñas, N. F. Svaiter and G. Menezes, "One-loop effective action and the Riemann zeros" (preprint 05/2014)

[abstract:] "We present a remarkable connection between the asymptotic behavior of the Riemann zeros and one-loop effective action in Euclidean scalar field theory. We show that in a two-dimensional space, the asymptotic behavior of the Fourier transform of two-point correlation functions fits the asymptotic distribution of the non-trivial zeros of the Riemann zeta function. We work out an explicit example, namely the non-linear sigma model in the leading order in $1/N$ expansion."

G. Menezes and N.F. Svaiter, "Quantum field theories and prime numbers spectrum" (preprint 11/2012)

[abstract:] "The Riemann hypothesis states that all nontrivial zeros of the zeta function lie on the critical line $\Re(s)=1/2$. Hilbert and Pólya suggested a possible approach to prove it, based on spectral theory. Within this context, some authors formulated the question: is there a quantum mechanical system related to the sequence of prime numbers? In this Letter we assume that there is a class of hypothetical physical systems described by self-adjoint operators with countable infinite number of degrees of freedom with spectra given by the sequence of primes numbers. We prove a no-go theorem. We show that the generating functional of connected Schwinger functions of such theories cannot be constructed."

T. Ohsaku, "The anomalous Nambu–Goldstone Theorem in relativistic/nonrelativistic quantum field theory" (preprint 12/2013)

[abstract:] "The anomalous Nambu–Goldstone (NG) theorem which is found as a violation of counting law of the number of NG bosons of the normal NG theorem in nonrelativistic and Lorentz-symmetry-violated relativistic theories is studied in detail, with emphasis on its mathematical aspect from Lie algebras, geometry to number theory. The basis of counting law of NG bosons in the anomalous NG theorem is examined by Lie algebras (local) and Lie groups (global). A quasi-Heisenberg algebra is found generically in various symmetry breaking schema of the anomalous NG theorem, and it indicates that it causes a violation/modification of the Heisenberg uncertainty relation in an NG sector which can be experimentally confirmed. The formalism of effective potential is presented for understanding the mechanism of anomalous NG theorem with the aid of our result of Lie algebras. After an investigation on a bosonic kaon condensation model with a finite chemical potential as an explicit Lorentz-symmetry-breaking parameter, a model Lagrangian approach on the anomalous NG theorem is given for our general discussion. Not only the condition of the counting law of true NG bosons, but also the mechanism to generate a mass of massive NG boson is also found by our examination on the kaon condensation model. Furthermore, the generation of a massive mode in the NG sector is understood by the quantum uncertainty relation of the Heisenberg algebra, obtained from a symmetry breaking of a Lie algebra, which realizes in the effective potential of the kaon condensation model. Hence the relation between a symmetry breaking scheme, a Heisenberg algebra, a mode-mode coupling, and the mechanism of mass generation in an NG sector is established. Finally, some relations between the Riemann hypothesis and the anomalous NG theorem are presented."

M.W. Coffey, "On a three-dimensional symmetric Ising tetrahedron, and contributions to the theory of the dilogarithm and Clausen functions" (preprint 01/2008)

[abstract:] "Perturbative quantum field theory for the Ising model at the three-loop level yields a tetrahedral Feynman diagram $C(a,b)$ with masses $a$ and $b$ and four other lines with unit mass. The completely symmetric tetrahedron $C^Tet \equiv C(1,1)$ has been of interest from many points of view, with several representations and conjectures having been given in the literature. We prove a conjectured exponentially fast convergent sum for $C(1,1)$, as well as provide further analytic support for an empirical relation for $C(1,1)$ as a remarkable difference of Clausen function values. Our presentation includes Propositions extending the theory of the dilogarithm $Li_2$ and Clausen $Cl_2$ functions, as well as their relation to other special functions of mathematical physics. The results strengthen connections between Feynman diagram integrals, volumes in hyperbolic space, number theory, and special functions and numbers, specifically including dilogarithms, Clausen function values, and harmonic numbers."

M.W. Coffey, "Alternative evaluation of a ln tan integral arising in quantum field theory" (preprint, 10/2008)

[Abstract:] "A certain dilogarithmic integral $I_7$ turns up in a number of contexts including Feynman diagram calculations, volumes of tetrahedra in hyperbolic geometry, knot theory, and conjectured relations in analytic number theory. We provide an alternative explicit evaluation of a parameterized family of integrals containing this particular case. By invoking the Bloch--Wigner form of the dilogarithm function, we produce an equivalent result, giving a third evaluation of $I_7$. We also alternatively formulate some conjectures which we pose in terms of values of the specific Clausen function $Cl_2$."

H. Furusho, "p-adic multiple zeta values I - p-adic multiple polylogarithms and the p-adic KZ equation"

[abstract:] "Our main aim in this paper is to give a foundation of the theory of p-adic multiple zeta values. We introduce (one variable) p-adic multiple polylogarithms by Coleman's p-adic iterated integration theory. We define p-adic multiple zeta values to be special values of p-adic multiple polylogarithms. We consider the p-adic KZ equation and introduce the p-adic Drinfel'd associator by using certain two fundamental solutions of the p-adic KZ equation. We show that our p-adic multiple polylogarithms appear on coefficients of a certain fundamental solution of the p-adic KZ equation and our p-adic multiple zeta values appear on coefficients of the p-adic Drinfel'd associator. We show various properties of p-adic multiple zeta values, which are sometimes analogous to the complex case and are sometimes peculiar to the p-adic case, via the p-adic KZ equation."

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.H.E. Duchamp, V.H.N. Minh, A.I. Solomon and S. Goodenough, "An interface between physics and number theory" (preprint 11/2010)

[abstract:] "We extend the Hopf algebra description of a simple quantum system given previously, to a more elaborate Hopf algebra, which is rich enough to encompass that related to a description of perturbative quantum field theory (pQFT). This provides a {\em mathematical} route from an algebraic description of non-relativistic, non-field theoretic quantum statistical mechanics to one of relativistic quantum field theory. Such a description necessarily involves treating the algebra of polyzeta functions, extensions of the Riemann Zeta function, since these occur naturally in pQFT. This provides a link between physics, algebra and number theory. As a by-product of this approach, we are led to indicate {\it inter alia} a basis for concluding that the Euler gamma constant $\gamma$ may be rational."

M.-A. Sanchis-Lozano, J. Fernando Barbero, J. Navarro-Salas, "Prime numbers, quantum field theory and the Goldbach conjecture" (preprint 01/2012)

[abstract:] "Motivated by the Goldbach and Polignac conjectures in Number Theory, we propose the factorization of a classical non-interacting real scalar field (on a two-cylindrical spacetime) as a product of either two or three (so-called primer) fields whose Fourier expansion exclusively contains prime modes. We undertake the canonical quantization of such primer fields and construct the corresponding Fock space by introducing creation operators $a_p^{\dag}$ (labeled by prime numbers $p$) acting on the vacuum. The analysis of our model, based on the standard rules of quantum field theory, suggests intriguing connections between different topics in Number Theory, notably the Riemann hypothesis and the Goldbach and Polignac conjectures. Our analysis also suggests that the (non) renormalizability properties of the proposed model could be linked to the possible validity or breakdown of the Goldbach conjecture for large integer numbers."

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 algebra. 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."

U. Müller and C. Schubert, "A quantum field theoretical representation of Euler-Zagier sums" (preprint 08/99)

[abstract:] "We establish a novel representation of arbitrary Euler-Zagier sums in terms of weighted vacuum graphs. This representation uses a toy quantum field theory with infinitely many propagators and interaction vertices. The propagators involve Bernoulli polynomials and Clausen functions to arbitrary orders. The Feynman integrals of this model can be decomposed in terms of an algebra of elementary vertex integrals whose structure we investigate. We derive a large class of relations between multiple zeta values, of arbitrary lengths and weights, using only a certain set of graphical manipulations on Feynman diagrams. Further uses and possible generalizations of the model are pointed out."

D. Wohl, "Selberg integrals, multiple zeta values and Feynman diagrams" (preprint 06/02)

[Abstract:] "We prove that there is an isomorphism between the Hopf Algebra of Feynman diagrams and the Hopf algebra corresponding to the Homogenous Multiple Zeta Value ring H in C<<X,Y>> . In other words, Feynman diagrams evaluate to Multiple Zeta Values in all cases. This proves a recent conjecture of Connes-Kreimer, and others including Broadhurst and Kontsevich.

The key step of our theorem is to present the Selberg integral as discussed in Terasoma [22] as a Functional from the Rooted Trees Operad to the Hopf algebra of Multiple Zeta Values. This is a new construction which provides illumination to the relations between zeta values, associators, Feynman diagrams and moduli spaces. An immediate implication of our Main Theorem is that by applying Terasoma's result and using the construction of our Selberg integral-rooted trees functional, we prove that the Hermitian matrix integral as discussed in Mulase [18] evaluates to a Multiple Zeta Value in all 3 cases: Asymptotically, the limit as N goes to infinity, and in general.

Furthermore, this construction provides for a positive resolution to Goncharov's conjecture (see [7] pg. 30). The Selberg integral functional can be extended to map the special values to depth m multiple polylogarithms on X."

D. Wohl, "Analysis of zeta functions, Multiple zeta values, and related integrals" (preprint 06/02)

[Abstract:] "In this work, we begin to uncover the architecture of the general family of zeta functions and multiple zeta values as they appear in the theory of integrable systems and conformal field theory. One of the key steps in this process is to recognize the roles that zeta functions play in various arenas using transform methods. Other logical connections are provided by the the appearance of the Drinfeld associator, Hopf algebras, and techniques of conformal field theory and braid groups. These recurring themes are subtly linked in a vast scheme of a logically woven tapestry.

An immediate application of this framework is to provide an answer to a question of Kontsevich regarding the appearance of Drinfeld type integrals and in particular, multiple zeta values in: a) Drinfeld's work on the KZ equation and the associator; b) Etingof-Kazhdan's quantization of Poisson-Lie algebras; c) Tamarkin's proof of formality theorems; d) Kontsevich's quantization of Poisson manifolds. Combinatorial arguments relating Feynman diagrams to Selberg integrals, multiple zeta values, and finally Poisson manifolds provide an additional step in this framework. Along the way, we provide additional insight into the various papers and theorems mentioned above. This paper represents an overall introduction to work currently in progress. More details to follow. See our paper math.QA/0206030 for a proof of the Connes-Kreimer Conjecture."

A.I. Solomon, G.E.H. Duchamp, P. Blasiak, A. Horzela, K.A. Penson, "Hopf algebra structure of a model quantum field theory" (Talk presented by first-named author at 26th International Colloquium on Group Theoretical Methods in Physics, New York, June 2006. See cs.OH/0609107 for follow-up talk delivered by second-named author.)

[abstract:] "Recent elegant work on the structure of Perturbative Quantum Field Theory (PQFT) has revealed an astonishing interplay between analysis (Riemann Zeta functions), topology (Knot theory), combinatorial graph theory (Feynman Diagrams) and algebra (Hopf structure). The difficulty inherent in the complexities of a fully-fledged field theory such as PQFT means that the essential beauty of the relationships between these areas can be somewhat obscured. Our intention is to display some, although not all, of these structures in the context of a simple zero-dimensional field theory; i.e. a quantum theory of non-commuting operators which do not depend on spacetime. The combinatorial properties of these boson creation and annihilation operators, which is our chosen example, may be described by graphs, analogous to the Feynman diagrams of PQFT, which we show possess a Hopf algebra structure. Our approach is based on the partition function for a boson gas. In a subsequent note in these Proceedings we sketch the relationship between the Hopf algebra of our simple model and that of the PQFT algebra."

I. Gálvez-Carrillo, R.M. Kaufmann and A. Tonks, "Three Hopf algebras and their common simplicial and categorical background" (preprint 07/2016)

[abstract:] "We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. These are the Hopf algebras of Goncharov for multiple zeta values, that of Connes–Kreimer for renormalization, and a Hopf algebra constructed by Baues to study double loop spaces. We show that these examples can be successively unified by considering simplicial objects, cooperads with multiplication and Feynman categories at the ultimate level. These considerations open the door to new constructions and reinterpretation of known constructions in a large common framework."

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 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."

L. N. Lipatov, A. Sabio Vera, V. N. Velizhanin and G.G. Volkov, "Reflexive numbers and Berger graphs from Calabi-Yau spaces" (preprint 01/05)

[abstract:] "A novel relation between number theory and recently found Berger graphs is studied. The Berger graphs under investigation are constructed mainly for the CY_3 space. The method of analysis is based on the slice classification of CY_l polyhedra in the so-called Universal Calabi-Yau algebra. The concept of reflexivity in these polyhedra is reviewed and translated into the theory of reflexive numbers. A new approach based on recurrence relations and Quantum Field Theory methods is applied to the simply-laced and quasi-simply-laced subsets of the reflexive numbers. In the correspondence between the reflexive vectors and Berger graphs the role played by the generalized Coxeter labels is shown to be important. We investigate the positive roots of some of the Berger graphs to guess the algebraic structure hidden behind them."

V.N. Velizhanin, "Twist-2 at five loops: Wrapping corrections without wrapping computations" (preprint 11/2013)

[abstract:] "Using known all-loop results from the BFKL and generalized double-logarithmic equations and large spin limit we have computed the five-loop anomalous dimension of twist-2 operators without consideration of any wrapping effects. One part of the anomalous dimension was calculated in a usual way with the help of Asymptotic Bethe Ansatz. The rest part, related with the wrapping effects, was reconstructed from known constraints with the help of methods from number theory."

V. Gayral, B. Iochum and D.V. Vassilevich, "Heat kernel and number theory on NC-torus" (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 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."

S.K. Ashok, F. Cachazo, E. Dell'Aquila, "Children's drawings from Seiberg-Witten curves", Communications in Number Theory and Physics 1 no. 2 (2007) 237-305

[abstract:] "We consider $N=2$ supersymmetric gauge theories perturbed by tree level superpotential terms near isolated singular points in the Coulomb moduli space. We identify the Seiberg-Witten curve at these points with polynomial equations used to construct what Grothendieck called "dessins d'enfants" or "children's drawings" on the Riemann sphere. From a mathematical point of view, the dessins are important because the absolute Galois group $Gal(\bar{Q}/Q)$ acts faithfully on them. We argue that the relation between the dessins and Seiberg-Witten theory is useful because gauge theory criteria used to distinguish branches of $N=1$ vacua can lead to mathematical invariants that help to distinguish dessins belonging to different Galois orbits. For instance, we show that the confinement index defined in hep-th/0301006 is a Galois invariant. We further make some conjectures on the relation between Grothendieck's programme of classifying dessins into Galois orbits and the physics problem of classifying phases of $N=1$ gauge theories. "

T. Huber, "The Sudakov form factor to three loops in $N=4$ super Yang–Mills" (preprint 10/2012, from proceedings of the 11th DESY workshop "Loops and Legs in Quantum Field Theory", April 2012, Wernigerode, Germany)

[abstract:] "We review the results for the Sudakov form factor in $N=4$ super Yang–Mills theory up to the three-loop level. At each loop order, the form factor is expressed as a linear combination of only a handful scalar integrals, with small integer coefficients. Working in dimensional regularisation, the expansion coefficients of each integral exhibit homogeneous transcendentality in the Riemann zeta-function. We find that the logarithm of the form factor reproduces the correct values of the cusp and collinear anomalous dimensions. Moreover, the form factor in $N=4$ super Yang–Mills can be related to the leading transcendentality pieces of the QCD quark and gluon form factor. Finally, we comment briefly on the ultraviolet properties of the $N=4$ form factor in $D>4$ dimensions."

M. Banagl, "Positive topological quantum field theories" (preprint 03/2013)

[abstract:] "We propose a new notion of positivity for topological field theories (TFTs), based on S. Eilenberg's concept of completeness for semirings. We show that a complete ground semiring, a system of fields on manifolds and a system of action functionals on these fields determine a positive TFT. The main feature of such a theory is a semiring-valued topologically invariant state sum that satisfies a gluing formula. The abstract framework has been carefully designed to cover a wide range of phenomena. For instance, we derive Polya's counting theory in combinatorics from state sum identities in a suitable positive TFT. Several other concrete examples are discussed, among them Novikov signatures of fiber bundles over spacetimes and arithmetic functions in number theory. In the future, we will employ the framework presented here in constructing a new differential topological invariant that detects exotic smooth structures on spheres."

J. Andrade, "Hilbert–Pólya conjecture, zeta-functions and bosonic quantum field theories" (preprint 05/2013)

[abstract:] "The original Hilbert and Pólya conjecture is the assertion that the non-trivial zeros of the Riemann zeta function can be the spectrum of a self-adjoint operator. So far no such operator was found. However the suggestion of Hilbert and Pólya, in the context of spectral theory, can be extended to approach other problems and so it is natural to ask if there is a quantum mechanical system related to other sequences of numbers which are originated and motivated by number theory.

In this paper we show that the functional integrals associated with a hypothetical class of physical systems described by self-adjoint operators associated with bosonic fields whose spectra is given by three different sequence of numbers cannot be constructed. The common feature of the sequence of numbers considered here, which causes the impossibility of zeta regularization, is that the various Dirichlet series attached to such sequences – such as those which are sums over "primes" of $(\mathrm{norm}\ P)^{-s}$ have a natural boundary, i.e., they cannot be continued beyond the line $\Re(s)=0$. The main argument is that once the regularized determinant of a Laplacian is meromorphic in $s$, it follows that the series considered above cannot be a regularized determinant. In other words we show that the generating functional of connected Schwinger functions of the associated quantum field theories cannot be constructed."

Mike Hoffman's notes on multiple zeta values, referring to QFT

M. Hoffman, "Algebras of multiple zeta values, quasi-symmetric functions, and Euler sums" (a talk given at UQAM, 05/98)

J.-F. Burnol, "The Explicit Formula and a propagator" (preprint, 09/98)

[abstract:] "I give a new derivation of the Explicit Formula for the general number field K, which treats all primes in exactly the same way, whether they are discrete or archimedean, and also ramified or not. In another token, I advance a probabilistic interpretation of Weil's positivity criterion, as opposed to the usual geometrical analogies or goals. But in the end, I argue that the new formulation of the Explicit Formula signals a specific link with Quantum Fields, as opposed to the Hilbert-Pólya operator idea (which leads rather to Quantum Mechanics)."

[excerpt:] "There is an essential concept from Quantum Physics which I believe will play an important role in the problem. This is the notion of correlation functions of a quantum field. Well-known advances in the field of knot invariants and many other geometric arenas have shown that these correlation functions can be used to represent intersection products or linking numbers and many other geometrical things. On the other hand they have of course a probabilistic interpretation (being complex amplitudes) and rigorous mathematical developments of constructive quantum field theory have as their goal the construction of probability measures on spaces of distributions, as considered above.

For these reasons, and other more precise reflections on the subject, moving towards quantum fields is an urgent goal. In this context it is reassuring that the "-log(|x|)" formulation of the Explicit Formula enables a few additional comments.

Indeed as is well-known "-log(|x|)" is the propagator of the free Boson field in 2 dimensions (here we look at the complex place and |x| so that we have both holomorphic and anti-holomorphic sectors). But let us rather consider the non-archimedean completion Kv, of the number field K. here too "-log(|x|)" can be seen as a propagator associated to an action..."

[conclusion:] "This paper finds its roots in the conviction that the Riemann Hypothesis has a lot to do with (suitably envisioned) Quantum Fields. The belief in a possible link between the Riemann Hypothesis and Quantum Mechanics seems to be widespread and is a modern formulation of the Hilbert-Pólya operator approach. I believe that techniques and philosophy more organic to Quantum Fields will be most relevant."

These notes provide a good introduction to Burnol's programme of research.

D. Spector, "Supersymmetry and the Möbius inversion function", Communications in Mathematical Physics 127 (1990) 239.

"We show that the Möbius inversion function of number theory can be interpreted as the operator (-1)F in quantum field theory...We will see in this paper that the function...has a very natural interpretation. In the proper context, it is equivalent to (-1)F, the operator that distinguishes fermionic from bosonic states and operators, with the fact that $mu(n) = 0$ when n is not squarefree being equivalent to the Pauli exclusion principle..."

Although nothing particularly new is added to number theory or supersymmetric QFT, and the claim about the PNT is spurious, this paper introduces a 'number theoretical QFT' (analogous to B. Julia's 'number theoretical gas') which provides a surprising natural 'physical' framework for certain manipulations on arithmetic functions. A standard calculation of a Witten index produces the Möbius function, and Euler's totient function and Möbius inversion emerge easily.

I. Ya. Aref'eva, I.V. Volovich, "Quantization of the Riemann zeta-function and cosmology" (preprint 12/2006)

[abstract:] "Quantization of the Riemann zeta-function is proposed. We treat the Riemann zeta-function as a symbol of a pseudodifferential operator and study the corresponding classical and quantum field theories. This approach is motivated by the theory of p-adic strings and by recent works on stringy cosmological models. We show that the Lagrangian for the zeta-function field is equivalent to the sum of the Klein-Gordon Lagrangians with masses defined by the zeros of the Riemann zeta-function. Quantization of the mathematics of Fermat-Wiles and the Langlands program is indicated. The Beilinson conjectures on the values of L-functions of motives are interpreted as dealing with the cosmological constant problem. Possible cosmological applications of the zeta-function field theory are discussed."

B. Dragovich, "On p-adic and adelic generalization of quantum field theory", Nuclear Physics B - Proceedings Supplements 102-103 (2001) 150-155

"A brief review of p-adic and adelic quantum mechanics is presented, and a new approach to p-adic and adelic quantum field theory is proposed. Path integral method turns out to be generic for quantum dynamics on both archimedean and nonarchimedean spaces."

B. Dragovich, "Some Lagrangians with zeta function nonlocality" (preprint, 05/2008)

[abstract:] "Some nonlocal and nonpolynomial scalar field models originated from $p$-adic string theory are considered. Infinite number of spacetime derivatives is governed by the Riemann zeta function through d'Alembertian $\Box$ in its argument. Construction of the corresponding Lagrangians begins with the exact Lagrangian for effective field of $p$-adic tachyon string, which is generalized replacing $p$ by arbitrary natural number $n$ and then taken a sum of over all $n$. Some basic classical field properties of these scalar fields are obtained. In particular, some trivial solutions of the equations of motion and their tachyon spectra are presented. Field theory with Riemann zeta function nonlocality is also interesting in its own right."

B. Dragovich, "Zeta nonlocal scalar fields" (preprint, 04/2008)

[abstract:] "We consider some nonlocal and nonpolynomial scalar field models originated from p-adic string theory. Infinite number of spacetime derivatives is determined by the operator valued Riemann zeta function through d'Alembertian $\Box$ in its argument. Construction of the corresponding Lagrangians L starts with the exact Lagrangian $\mathcal{L}_p$ for effective field of p-adic tachyon string, which is generalized replacing p by arbitrary natural number n and then taken a sum of $\mathcal{L}_n$ over all n. The corresponding new objects we call zeta scalar strings. Some basic classical field properties of these fields are obtained and presented in this paper. In particular, some solutions of the equations of motion and their tachyon spectra are studied. Field theory with Riemann zeta function dynamics is interesting in its own right as well."

B.D.B. Roth, "A general approach to quantum fields and strings on adeles", Physics Letters B 213 No. 3 (1988) 263-268

[abstract:] "We develop a general framework for constructing quantum theories on the space of adeles. We consider complementary approaches utilizing both the functional integral and generalized Poincaré (translation) invariant Hilbert space methods. As a starting point, we seek to construct the most general quantum theory on the rational numbers. Very general arguments land us in the adeles. Quantum amplitudes generically will not factorize into separate p-adic sectors. The vacuum amplitude, in particular, factorizes only in a linear regime. We conclude with implications for the cosmological constant and adelic string sigma models."

The following three items contain reviews of work done on p-adic and adelic approaches to QFT:

L. Brekke and P. Freund, "p-adic numbers in physics", Physics Reports 233, (1993) 1-66.

V.S. Vladimirov, I.V. Volovich, E.I. Zelenov, p-Adic Analysis and Mathematical Physics (World Scientific Publishing, 1994)

A. Khrennikov, p-Adic Valued Distributions in Mathematical Physics, (Kluwer, 1994).

The literature covered includes:

S. Albeverio, A. Khrennikov, and R. Cianci, "A representation of quantum field Hamiltonian in a p-adic Hilbert space", Theor. Mathem. Physics 112 no. 3 (1997) 355-374.

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.

A. Khrennikov, "Statistical interpretation of p-adic valued quantum field theory", Dokl. Akad. Nauk 328, no. 1 (1993) 46-50.

I. Ya. Aref'eva, "Physics at the Planck length and p-adic field theory", from Differential Geometry Methods in Theoretical Physics, eds. LL. Chau and W. Nahm (Plenum, 1990)

V.A. Smirnov, "Renormalization in p-adic quantum field theory", Mod. Phys. Lett. A6 (1991) 1421-1427

V.A. Smirnov, "p-Adic Feynman amplitudes", Preprint MPI-Ph/91-48 (1991) 1-13

B. Grossman, "Adelic conformal field theory", Physics Letters B215 (1988) 14-19

B. Grossman, "Arithmetic directions in topological quantum field theory and strings", pr3print, Rockerfeller University DOE/ER40 325-52-Task.B. (1988)

E. Yu. Lerner and M.D. Missarov, "Scalar models in p-adic quantum field theory and hierarchical models", Theor. math. Phys. 78 (1989) 248-257.

H. Nishino, Y. Okada and M.R. Ubriaco, "Effective field theory and a p-adic string", Phys. Rev. D40 (1989) 1153-1157.

M.R. Ubriaco, "Fermions on the field of p-adic numbers", Phys. Rev. D41 (1990) 2631-2636

M.R. Ubriaco, "Field quantization in nonarchimedean field theory", preprint LTP-012-UPR

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, "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."

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."

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 counterterms 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."

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."

J.S. Dowker, "Sphere Renyi entropies" (preprint 12/2012)

[abstract:] "I give some scalar field theory calculations on a $d$-dimensional lune of arbitrary angle, evaluating, numerically, the effective action which is expressed as a simple quadrature, for conformal coupling. Using this, the entanglement and Renyi entropies are computed. Massive fields are also considered and a renormalisation to make the (one-loop) effective action vanish for infinite mass is suggested and used, not entirely successfully. However a universal coefficient is derived from the large mass expansion. For the round sphere, I show how to convert the quadrature form of the conformal Laplacian determinant into the more usual sum of Riemann zeta functions (and $\log 2$)."

K.V. Shajesh, I. Brevik, I. Cavero-Peláez and P. Parashar, "Self-similar plates: Casimir energies" (07/2016)

[abstract:] "We construct various self-similar configurations using parallel $\delta$-function plates and show that it is possible to evaluate the Casimir interaction energy of these configurations using the idea of self-similarity alone. We restrict our analysis to interactions mediated by a scalar field, but the extension to electromagnetic field is immediate. Our work unveils an easy and powerful method that can be easily employed to calculate the Casimir energies of a class of self-similar configurations. As a highlight, in an example, we determine the Casimir interaction energy of a stack of parallel plates constructed by positioning $\delta$-function plates at the points constituting the Cantor set, a prototype of a fractal. This, to our knowledge, is the first time that the Casimir energy of a fractal configuration has been reported. Remarkably, the Casimir energy of some of the configurations we consider turn out to be positive, and a few even have zero Casimir energy. For the case of positive Casimir energy that is monotonically decreasing as the stacking parameter increases the interpretation is that the pressure of vacuum tends to inflate the infinite stack of plates. We further support our results, derived using the idea of self-similarity alone, by rederiving them using the Green's function formalism. These expositions gives us insight into the connections between the regularization methods used in quantum field theories and regularized sums of divergent series in number theory."

G. Moore, "Arithmetic and attractors" (preprint 07/03)

[abstract:] "We study relations between some topics in number theory and supersymmetric black holes. These relations are based on the "attractor mechanism" of N=2 supergravity. In IIB string compactification this mechanism singles out certain "attractor varieties". ' We show that these attractor varieties are constructed from products of elliptic curves with complex multiplication for N=4 and N=8 compactifications. The heterotic dual theories are related to rational conformal field theories. In the case of N=4 theories U-duality inequivalent backgrounds with the same horizon area are counted by the class number of a quadratic imaginary field. The attractor varieties are defined over fields closely related to class fields of the quadratic imaginary field. We discuss some extensions to more general Calabi-Yau compactifications and explore further connections to arithmetic including connections to Kronecker's Jugendtraum and the theory of modular heights. The paper also includes a short review of the attractor mechanism. A much shorter version of the paper summarizing the main points is the companion note entitled "Attractors and Arithmetic""

R. Padma and H. Gopalkrishna Gadiyar, "Renormalisation and the density of prime pairs" (preprint 06/98)

"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."

J. Fine, "Some notes on the inverse problem for braids" (preprint 08/2006)

[abstract:] "The Kontsevich integral $Z$ associates to each braid $b$ (or more generally knot $k$) invariants $Z_i(b)$ lying in finite dimensional vector spaces, for $i = 0,1,2,...$. These values are not yet known, except in special cases. The inverse problem is that of determining $b$ from its invariants $Z_i(b)$. In this paper we study the case of braids on two strands, which is already sufficient to produce interesting and unexpected mathematics. In particular, we find connections with number theory, numerical analysis and field theory in physics. However, we will carry this study out with an eye to the more general case of braids on $n$ strands. We expect that solving the inverse problem even for $n=3$ will present real difficulties. Most of the concepts in this paper also apply to knots, but to simplify the exposition we will rarely mention this. The organisation and bulk of the writing of this paper predates its most significant results. We hope later to present better and develop further these results."

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

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."

The elements of the Monster involve Ogg's supersingular primes. Here are some other instances of The Monster in quantum field theory, as pointed out by M. Thomas:

A. N. Schellekens, "Seventy relatives of the monster module" (preprint, 04/93)

M. Jankiewicz and T.W. Kephart, "Transformations among large c conformal field theories" (preprint, 02/05)

Yang-Hui He and V. Jejjala, "Modular matrix models" (preprint, 07/03)

R.I. Ivanov and M.P. Tuite, "Some irrational generalised Moonshine from orbifolds", Nucl. Phys. B 635 (2002) 473-491

T. Gannon, "Postcards from the edge, or snapshots of the theory of generalised Moonshine" (talk at Banff conference in honour of R.V. Moody)

T. Gannon, "Monstrous moonshine and the classification of CFT" (based on lectures given in Istanbul)

M. Weiner, "Bosonic construction of vertex operator para-algebras from symplectic affine Kac-Moody Algebras" (preprint 06/94)

P. Montague, "Ternary codes and $Z_3$-orbifold constructions of conformal field theories" (talk presented at the "Monster Bash", Ohio State University, May 1993)

P. Montague, "Third and higher order NFPA twisted constructions of conformal field theories from lattices", Nucl. Phys. B 441 (1995) 337-382

P. Montague, "Discussion of self-dual c = 24 conformal field theories" (preprint 05/92)

M.P. Tuite, "Monstrous Moonshine and the uniqueness of the Moonshine module" (talk presented at workshop on 'Low dimensional topology and quantum field theory', Cambridge, 6-13 Sept. 1992)

M.P. Tuite, "On the relationship between the uniqueness of the Moonshine module and Monstrous Moonshine", Commun. Math. Phys. 166 (1995) 495-532

M.P. Tuite, "Generalised Moonshine and abelian orbifold constructions" (Talk presented at the AMS meeting on Moonshine, the Monster and related topics, Mt. Holyoke, June 1994)

L. Dolan, P. Goddard and P. Montague, "Conformal field theories, representations and lattice constructions", Comm. Math. Phys. 179 (1996) 61-120

F. Toppan, "Exceptional structures in mathematics and physics and the role of the octonions" (talk given at workshop "Supersymmetry and Quantum Symmetries", July 2003, Dubna - to appear in the proceedings)

E. Jurisich, J. Lepowsky and R. L. Wilson, "Realizations of the Monster Lie algebra" (preprint 08/94)

Yi-Zhi Huang and J. Lepowsky, "Tensor products of modules for a vertex operator algebra and vertex tensor categories" (preprint 01/94)

R.W. Gebert, "Introduction to vertex algebras, Borcherds algebras, and the Monster Lie algebra", Int. J. Mod. Phys. A 8 (1993) 5441-5504

number theory, renormalisation and zeta-function regularisation techniques

Multiple Zeta Values, Multiple Polylogarithms and Quantum Field Theory, October 7–11, 2013, ICMAT, Campus de Cantoblanco, Madrid

The Erwin Schrödinger Institute's 1998 seminar on Quantum Field Theory and the Statistical Distribution of Prime Numbers

M. Pitkänen's speculative notes on 'arithmetic QFT'


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