Table of contents

John Stuart Dowker was born in Sheffield, Yorkshire, on 18 March 1937. His life, therefore, was very much influenced by the Second World War. This is evident as his father died on active service in 1945, after being called up in 1941. His grandfather also died shortly afterwards, so he did not know either of them very well. Nevertheless, it seems that he picked up a positive attitude towards natural sciences as both were technically minded. His mother later provided, often from borrowed money, all the necessary intellectual food in forms of chemistry sets, slide rules and other things that a boy needed to develop his interests. Stuart scored excellently in the 11-plus exam, which was used to decide the type of school a pupil should attend after primary school. Although Stuart was generally allowed to do what he wanted, his mother insisted that he chose King Edward VII Grammar School (KES), the top school in Sheffield at the time. KES allowed Stuart to fully develop his intellectual abilities, and after the S-level exam he received a prestigious state scholarship which allowed him to study at any university in the country.

He picked Nottingham over other possibilities, mainly because of his interest in electronics and because of the relative proximity to his family. In Nottingham, where he stayed from 1955 to 1958, his research concentration turned out to be mostly solid state physics. But with time on his hands, Stuart raided the library and taught himself things like complex analysis and quantum mechanics, with de Broglie's La mécanique ondulatoire [1] as one of his favorites. Remarkably, this book already contains a discussion of quantization on curved configuration spaces, a setting so relevant in Stuart's later career.

Stuart wanted to investigate quantum field theory for his doctoral thesis. So he wrote, among others, to Rudolph Peierls in Birmingham, and, after being interviewed by Peierls himself and J G Valatin, he received an offer of a PhD position. He went to Birmingham in 1958, where his supervisor was Leonardo Castillejo, best known from the Castillejo–Dalitz–Dyson-ambiguity; see [2]. His thesis involved stripping theory, and on this topic he wrote his first ever publication [3]. Takeaways from that time were that sometimes one simply has to do what one is told, and even more importantly, sometimes one has to get on with a calculation and 'just roll through it'. It was also around that period that Stuart started to learn quantum field theory properly following the developments of Feynman.

After his PhD, it was Peierls who helped him to find a position at the University of Pennsylvania in Philadelphia in 1961, which he accepted before he received an offer from CERN. In retrospect, this was probably fortunate as it is where he met his wife, Pwu Yih, to whom he has been married for nearly 50 years. Scientifically he remained somewhat isolated there and spent much of his time learning more about quantum field theory, for example by reading Bogoliubov and Shirkov [4]. His entry point for general relativity was Eddington's The Mathematical Theory of Relativity [5], which together with the works of Julian Schwinger and Bryce de Witt were the most influential ones at that time. A shift of focus to questions in quantum field theory in curved space time took place.

After his stay in Philadelphia, in 1963, helped by connections between Abe Klein and Brian Flowers, he went to the University of Manchester, where he remains to this day. This period was only briefly interrupted by a sabbatical in 1978 to 1979 at Austin following an invitation by Bryce de Witt.

Looking back, it seems there were two major driving forces or principles that determined much of Stuart's selection of research projects. One of them is the method of images, about which he says: '...it intrigued me that one problem (charge + plane) could be got from another (just charge) by geometrical reasoning plus uniqueness. Thomson's book [6] took this further. Chapter 5 is devoted to the image and inversion methods and I must have read this closely as there are lots of marginal notes...' The second principle can be read off from the following quote: 'After reading Eddington circa 1960 it was clear to me (and others of course) there is a strong analogy (at least) between gravitation and electromagnetism ... (His work has very strongly influenced me.) So I played a game of asking for the gravitational analogues of existing electromagnetic concepts. The basic analogue is between field strength/charge and curvature/spin... [in that] ...spin, in general relativity, plays the passive role that charge plays in electromagnetism in the sense that it is the spin–curvature coupling that knocks a particle off a geodesic.' It is quite amazing how much of Stuart's work can be traced back to these principles. This is briefly explained in the following by describing some of his most important works.

His most cited work [7] fits this bill and can be seen as doing what Julian Schwinger did in [8], for a constant electromagnetic field in the gravitational setting. In more detail he noticed Schulman's work on propagators on the three sphere [9], which he extended to Lie groups; see [10, 11]. He then noticed that having exact propagators, work like Schwinger's could be done and de Sitter space was a natural 'curved' candidate. The paper is best known for the mathematical technique introduced, namely for the zeta function method much used since for the computation of singular quantum field theoretic quantities like effective actions and the Casimir energy. The motivation for introducing this scheme goes back to reading an article of I M Gel'fand about some number theory problem involving zeta functions, image sums, propagators etc. How could an object occurring with other objects physicists were using all the time not be useful? It turned out to be very useful, although in this paper the method was only introduced but actually not used! This is probably the reason that his paper did not receive the same recognition as the one by Hawking [12], where zeta function regularization was mentioned in the title and where it was actually applied to examples in the article. Of course, Stuart's and Raymond's paper hit the pulse of the time in that after Stephen Hawking's announcement about Hawking radiation [13], the quantum field theory in curved space-time frenzy began. He had been interested in that subject for a while, the most important influence being de Witt's Les Houches lectures and also Chris Isham, who introduced him to Dennis Sciama's Oxford group, including Philip Candelas and Derek Raine, in 1973. The project about de Sitter space got somewhat delayed by work on path integrals and the ordering problem until Ray Critchley came by looking for a PhD topic.

Several more of his best known papers are in the context of quantum field theory in curved spacetime. In the article [14], conformal transformations play a fundamental role. They are used to transform static manifolds to ultra-static ones where a high-temperature expansion of the effective action can be done. The result has since been rediscovered many times for special cases. In the process they also showed results about the conformal transformation of heat kernel coefficients claimed later on by mathematicians.

The paper by Kennedy, Critchley and Dowker [15] belongs in the same context. Deutsch and Candelas had shown the occurrence of non-integrable singularities near boundaries of the Casimir energy densities, which made it impossible to obtain global energies by naively integrating local quantities. To resolve that problem, the needed surface counterterms for an arbitrarily shaped smooth boundary in curved space were computed.

How does the paper [16] fit into the general scheme? The seed to considering this topic was probably planted when Yakir Aharonov was visiting Birmingham in 1958. In the Aharonov–Bohm effect we have an electromagnetic field with finite extension that impacts particles never entering that region. What is the gravitational analogue for that situation? The analogue concerns the impact a localized curvature has, and the cone is an excellent example to shed light on that question.

Related to the method of images, Stuart has done an enormous amount of work on the influence of topology and curvature on quantum field theory. An example is [17], where the vacuum stress–energy tensor for Clifford–Klein forms of the flat or spherical type were computed.

Another strand we would like to mention is Stuart's interest in higher spin equations. In [18], Steven Weinberg wrote down a set of higher spin equations that took his fancy. They involved angular momentum theory, which has always pleased Stuart, and the description was an alternative to Roger Penrose's use of two-spinors. Investigating the inconsistencies that arose on coupling to gauge theories, Stuart extended the classic results in [19], from electromagnetism to gravity in accordance with his general philosophy; see, e.g., [20, 21, 22].

Lately, Stuart is best known for his many applications in the context of zeta function regularization and its applications to quantum field theory under external conditions and spectral theory. He can be considered the world expert on particular case calculations with a knowledge of the literature, old and recent, that is not seen very often and which originated in the many hours spent at different (mostly British) libraries. His attitude towards explicit computations is nicely summarized by himself: 'I have always been interested in exact solutions, even if unphysical, so long as they are pretty. They seem to be working mechanisms that fit together, complete in themselves, like a watch.'

The following issue in honour of Stuart's 75th birthday contains contributions that touch upon the various topics he has worked on.

References

[1] de Broglie L 1928 La mécanique ondulatoire (Paris: Gauthier-Villars)

[2] Castillejo L, Dalitz R H and Dyson F J 1956 Low's scattering equation for the charged and neutral scalar theories Phys. Rev.101 453

[3] Dowker J S 1961 Application of the Chew and Low extrapolation procedure to K^- + d → Y + N + π absorption reactions Il Nuovo Cimento10 182

[4] Bogoliubov N N and Shirkov D V 1959 Introduction to the Theory of Quantized Fields (New York: Interscience)

[5] Eddington A S 1923 The Mathematical Theory of Relativity (Cambridge: Cambridge University Press)

[6] Thomson J J 1909 Elements of Electricity and Magnetism 4th edn (Cambridge: Cambridge University Press)

[7] Dowker J S and Critchley R 1976 Effective Lagrangian and energy momentum tensor in de Sitter space Phys. Rev. D 13 3224

[8] Schwinger J 1951 On gauge invariance and vacuum polarization Phys. Rev.82 664

[9] Schulman L S 1968 A path integral for spin Phys. Rev.176 1558

[10] Dowker J S 1970 When is the sum over classical paths exact? J. Phys. A: Math. Gen.3 451

[11] Dowker J S 1971 Quantum mechanics on group space and Huygens' principle Ann. Phys.62 361

[12] Hawking S W 1977 Zeta function regularization of path integrals in curved space-time Comm. Math. Phys.55133

[13] Hawking S W 1974 Black hole explosions Nature248 30

[14] Dowker J S and Kennedy G 1978 Finite temperature and boundary effects in static space-times J. Phys. A: Math. Gen.11 895

[15] Kennedy G, Critchley R and Dowker J S 1980 Finite temperature field theory with boundaries: stress tensor and surface action renormalization Ann. Phys.125 346

[16] Dowker J S 1977 Quantum field theory on a cone J. Phys. A: Math. Gen.10 115

[17] Dowker J S and Banach R 1978 Quantum field theory on Clifford–Klein space-times. The effective Lagrangian and vacuum stress-energy tensor J. Phys. A: Math. Gen.11 2255

[18] Weinberg S 1964 Feynman rules for any spin Phys. Rev.133 B1318

[19] Fierz M and Pauli W 1939 On relativistic wave equations for particles of arbitrary spin in an electromagnetic field Proc. Roy. Soc. A 173 221

[20] Dowker J S and Dowker Y P 1966 Particles of arbitrary spin in curved spaces Proc. Phys. Soc. (Lond.)87 65

[21] Dowker J S and Dowker Y P 1966 Interactions of massless particles of arbitrary spin Proc. Roy. Soc. A 294 175

[22] Dowker J S 1972 Propagators for arbitrary spin in an Einstein universe Ann. Phys.71 577

We discuss various issues associated with the calculation of the reduced functional determinant of a special second-order differential operator $\boldsymbol{F}=-{\rm d}^2/{\rm d}\tau ^2+\ddot{g}/g$, $\ddot{g}\equiv {\rm d}^2g/{\rm d}\tau ^2$ with a generic function g(τ), subject to periodic and Dirichlet boundary conditions. These issues include the gauge-fixed path integral representation of this determinant, the monodromy method of its calculation and the combination of the heat kernel and zeta-function technique for the derivation of its period dependence. Motivations for this particular problem, coming from applications in quantum cosmology, are also briefly discussed. They include the problem of microcanonical initial conditions in cosmology driven by a conformal field theory, cosmological constant and cosmic microwave background problems.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'.

This review paper contains a concise introduction to highest weight representations of infinite-dimensional Lie algebras, vertex operator algebras and Hilbert schemes of points, together with their physical applications to elliptic genera of superconformal quantum mechanics and superstring models. The common link of all these concepts and of the many examples considered in this paper is to be found in a very important feature of the theory of infinite-dimensional Lie algebras: the modular properties of the characters (generating functions) of certain representations. The characters of the highest weight modules represent the holomorphic parts of the partition functions on the torus for the corresponding conformal field theories. We discuss the role of the unimodular (and modular) groups and the (Selberg-type) Ruelle spectral functions of hyperbolic geometry in the calculation of elliptic genera and associated q-series. For mathematicians, elliptic genera are commonly associated with new mathematical invariants for spaces, while for physicists elliptic genera are one-loop string partition function. (Therefore, they are applicable, for instance, to topological Casimir effect calculations.) We show that elliptic genera can be conveniently transformed into product expressions, which can then inherit the homology properties of appropriate polygraded Lie algebras.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'.

The main part of this paper is to present an updated review of the Casimir energy at zero and finite temperature for the transverse oscillations of a piecewise uniform closed string. We make use of three different regularizations: the cutoff method, the complex contour integration method and the zeta-function method. The string model is relativistic, in the sense that the velocity of sound is for each string piece set equal to the velocity of light. In this sense the theory is analogous to the electromagnetic theory in a dielectric medium in which the product of permittivity and permeability is equal to unity (an isorefractive medium). We demonstrate how the formalism works for a two-piece string, and for a 2N-piece string, and show how in the latter case a compact recursion relation serves to facilitate the formalism considerably. The Casimir energy turns out to be negative, and the more so the larger the number of pieces in the string. The two-piece string is quantized in D-dimensional spacetime, in the limit when the ratio between the two tensions is very small. We calculate the free energy and other thermodynamic quantities, demonstrate scaling properties, and comment finally on the meaning of the Hagedorn critical temperature for the two-piece string. Thereafter, as a novel development we present a scalar field theory for a real field in three-dimensional space in a potential rising linearly with a longitudinal coordinate z in the interval 0 < z < 1, and which is thereafter held constant on a horizontal plateau. The potential is taken as a rough model of the two-piece string potential under simplifying conditions, when the length ratio between the pieces is replaced formally with the mentioned length parameter z.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'.

We review the application of the spectral zeta function to the one-loop properties of quantum field theories on manifolds with boundary, with emphasis on Euclidean quantum gravity and quantum cosmology. As was shown in the literature some time ago, the only boundary conditions that are completely invariant under infinitesimal diffeomorphisms on metric perturbations suffer from a drawback, i.e. lack of strong ellipticity of the resulting boundary-value problem. Nevertheless, at least on the Euclidean 4-ball background, it remains possible to evaluate the ζ(0) value, which describes in this case a universe which, in the limit of small 3-geometry, has vanishing probability of approaching the cosmological singularity. An assessment of this result is performed here, discussing its physical and mathematical implications.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'.

The spectral operator was introduced by Lapidus and van Frankenhuijsen (2006 Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings) in their reinterpretation of the earlier work of Lapidus and Maier (1995 J. Lond. Math. Soc.52 15–34) on inverse spectral problems and the Riemann hypothesis. In essence, it is a map that sends the geometry of a fractal string onto its spectrum. In this review, we present the rigorous functional analytic framework given by Herichi and Lapidus (2012) and within which to study the spectral operator. Furthermore, we give a necessary and sufficient condition for the invertibility of the spectral operator (in the critical strip) and therefore obtain a new spectral and operator-theoretic reformulation of the Riemann hypothesis. More specifically, we show that the spectral operator is quasi-invertible (or equivalently, that its truncations are invertible) if and only if the Riemann zeta function ζ(s) does not have any zeros on the vertical line Re(s) = c. Hence, it is not invertible in the mid-fractal case when $c=\frac{1}{2}$, and it is quasi-invertible everywhere else (i.e. for all c ∈ (0, 1) with $c\ne \frac{1}{2}$) if and only if the Riemann hypothesis is true. We also show the existence of four types of (mathematical) phase transitions occurring for the spectral operator at the critical fractal dimension $c=\frac{1}{2}$ and c = 1 concerning the shape of the spectrum, its boundedness, its invertibility as well as its quasi-invertibility.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'.

Casimir and Casimir–Polder repulsions have been known for more than 50 years. The general 'Lifshitz' configuration of parallel semi-infinite dielectric slabs permits repulsion if they are separated by a dielectric fluid that has a value of permittivity that is intermediate between those of the dielectric slabs. This was indirectly confirmed in the 1970s, and more directly by Capasso's group recently. It has also been known for many years that electrically and magnetically polarizable bodies can experience a repulsive quantum vacuum force. More amenable to practical application are situations where repulsion could be achieved between ordinary conducting and dielectric bodies in vacuum. The status of the field of Casimir repulsion with emphasis on some recent developments will be surveyed. Here, stress will be placed on analytic developments, especially on Casimir–Polder (CP) interactions between anisotropically polarizable atoms, and CP interactions between anisotropic atoms and bodies that also exhibit anisotropy, either because of anisotropic constituents, or because of geometry. Repulsion occurs for wedge-shaped and cylindrical conductors, provided the geometry is sufficiently asymmetric, that is, either the wedge is sufficiently sharp or the atom is sufficiently far from the cylinder.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'.

We review an exact analytical resolution method for general one-dimensional quantal anharmonic oscillators: stationary Schrödinger equations with polynomial potentials. It is an exact form of WKB treatment involving 'spectral' (usual) versus 'classical' (newer) zeta-regularizations in parallel. The central results are a set of Bohr–Sommerfeld-like but exact quantization conditions, directly drawn from Wronskian identities, and appearing to extend the Bethe-ansatz formulae of integrable systems. Such exact quantization conditions do not just select the eigenvalues; some evaluate the spectral determinants, and others the wavefunctions, for the spectral parameter in general position.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'.

We consider some applications (old and new) of the Patterson–Selberg zeta function to 3D gravity with a negative cosmological constant. We also consider 2D black hole vacua with a parabolic generator of their holonomy.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'.

We study the stability of a non-Abelian chromomagnetic vacuum in Yang–Mills theory in Euclidean Einstein universe S¹ × S³. We assume that the gauge group is a simple compact group G containing the group SU(2) as a subgroup and consider static covariantly constant gauge fields on S³ taking values in the adjoint representation of the group G and forming a representation of the group SU(2). We compute the heat kernel for the Laplacian acting on fields on S³ in an arbitrary representation of SU(2) and use this result to compute the heat kernels for the gluon and the ghost operators and the one-loop effective action. We show that the only configuration of the covariantly constant Yang–Mills background that is stable is the one that contains only spinor (fundamental) representations of the group SU(2); all other configurations contain negative modes and are unstable. For the stable configuration we compute the asymptotics of the effective action, the energy density, the entropy and the heat capacity in the limits of low/high temperature and small/large volume and show that the energy density has a non-trivial minimum at a finite value of the radius of the sphere S³.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'.

We examine the local super trace asymptotics for the de Rham complex defined by an arbitrary super connection on the exterior algebra. We show, in contrast to the situation in which the connection in question is the Levi-Civita connection, that these invariants are generically non-zero in positive degree and that the critical term is not the Pfaffian.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'.

We evaluate the renormalized vacuum expectation values (VEVs) of electric and magnetic field squared and the energy–momentum tensor for the electromagnetic field in the geometry of two parallel conducting plates on the background of cosmic string spacetime. On the basis of these results, the Casimir–Polder force acting on a polarizable particle and the Casimir forces acting on the plates are investigated. The VEVs are decomposed into the pure string and plate-induced parts. The VEV of the electric field squared is negative for points with the radial distance to the string smaller than the distance to the plates, and positive for the opposite situation. On the other hand, the VEV for the magnetic field squared is negative everywhere. The boundary-induced part in the VEV of the energy–momentum tensor is different from zero in the region between the plates only. Moreover, this part only depends on the distance from the string. The boundary-induced part in the vacuum energy density is positive for points with a distance to the string smaller than the distance to the plates and negative in the opposite situation. The Casimir stresses on the plates depend non-monotonically on the distance from the string. We show that the Casimir forces acting on the plates are always attractive.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'.

We calculate the quantum vacuum interaction energy between two kinks of the sine-Gordon equation. Using the TGTG formula, the problem is reduced to the known formulas for quantum fluctuations in the background of a single kink. This interaction induces an attractive force between the kinks in parallel to the Casimir force between conducting mirrors.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'.

We calculate the expectation values of the stress–energy bitensor defined at two different spacetime points x, x' of a massless, minimally coupled scalar field with respect to a quantum state at finite temperature T in a flat N-dimensional spacetime by means of the generalized zeta-function method. These correlators, also known as the noise kernels, give the fluctuations of energy and momentum density of a quantum field which are essential for the investigation of the physical effects of negative energy density in certain spacetimes or quantum states. They also act as the sources of the Einstein–Langevin equations in stochastic gravity which one can solve for the dynamics of metric fluctuations as in spacetime foams. In terms of constitutions these correlators are one rung above (in the sense of the correlation—BBGKY or Schwinger-Dyson—hierarchies) the mean (vacuum and thermal expectation) values of the stress–energy tensor which drive the semiclassical Einstein equation in semiclassical gravity. The low- and the high-temperature expansions of these correlators are also given here: at low temperatures, the leading order temperature dependence goes like T^N while at high temperatures they have a T² dependence with the subleading terms exponentially suppressed by e^−T. We also discuss the singular behavior of the correlators in the x' → x coincident limit as was done before for massless conformal quantum fields.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'.

Motivated by the dark energy issue, the one-loop quantization approach for a class of relativistic higher order theories is discussed in some detail. A specific F(R, P, Q) gravity model at the one-loop level in a de Sitter universe is investigated, extending the similar program developed for the case of F(R) gravity. The stability conditions under arbitrary perturbations are derived.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'.

Through a unified and relatively simple approach which uses complex contour integrals, particularly convenient integration contours and calculus of residues, closed-form summation formulas for 12 very general families of trigonometric sums are deduced. One of them is a family of cosecant sums which was first summed in closed form in a series of papers by Dowker (1987 Phys. Rev. D 36 3095–101; 1989 J. Math. Phys.30 770–3; 1992 J. Phys. A: Math. Gen.25 2641–8), whose method has inspired our work in this area. All of the formulas derived here involve the higher-order Bernoulli polynomials.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'.

On fractals, spectral functions such as heat kernels and zeta functions exhibit novel features, very different from their behaviour on regular smooth manifolds, and these can have important physical consequences for both classical and quantum physics in systems having fractal properties.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'.

One of the many problems to which Dowker devoted his attention is the effect of a conical singularity in the base manifold on the behavior of the quantum fields. In particular, he studied the small-t asymptotic expansion of the heat-kernel trace on a cone and its effects on physical quantities as the Casimir energy. In this paper, we review some peculiar results found in the last decade, regarding the appearance of non-standard powers of t, and even negative integer powers of log t, in this asymptotic expansion for the self-adjoint extensions of some symmetric operators with singular coefficients. Similarly, we show that the ζ-function associated with these self-adjoint extensions presents an unusual analytic structure.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'.

One of J Stuart Dowker's most significant achievements has been to observe that the theory of diffraction by wedges developed a century ago by Sommerfeld and others provided the key to solving two problems of great interest in general-relativistic quantum field theory during the last quarter of the 20th century: the vacuum energy associated with an infinitely thin, straight cosmic string, and (after an interchange of time with a space coordinate) the apparent vacuum energy of empty space as viewed by an accelerating observer. In a sense the string problem is more elementary than the wedge, since Sommerfeld's technique was to relate the wedge problem to that of a conical manifold by the method of images. Indeed, Minkowski space, as well as all cone and wedge problems, are related by images to an infinitely sheeted master manifold, which we call Dowker space. We review the research in this area and exhibit in detail the vacuum expectation values of the energy density and pressure of a scalar field in Dowker space and the cone and wedge spaces that result from it. We point out that the (vanishing) vacuum energy of Minkowski space results, from the point of view of Dowker space, from the quantization of angular modes, in precisely the way that the Casimir energy of a toroidal closed universe results from the quantization of Fourier modes; we hope that this understanding dispels any lingering doubts about the reality of cosmological vacuum energy.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'.

We review a few rigorous and partly unpublished results on the regularization of the stress–energy in quantum field theory on curved spacetimes: (1) the symmetry of the Hadamard/Seeley–DeWitt coefficients in smooth Riemannian and Lorentzian spacetimes, (2) the equivalence of the local ζ-function and the Hadamard-point-splitting procedure in smooth static spacetimes and (3) the equivalence of the DeWitt–Schwinger- and the Hadamard-point-splitting procedure in smooth Riemannian and Lorentzian spacetimes.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'.

The principal object in noncommutative geometry is the spectral triple consisting of an algebra $\mathcal {A}$, a Hilbert space $\mathcal {H}$ and a Dirac operator $\mathcal {D}$. Field theories are incorporated in this approach by the spectral action principle, which sets the field theory action to ${\rm Tr}\,f(\mathcal {D}^2/\Lambda ^2)$, where f is a real function such that the trace exists and Λ is a cutoff scale. In the low-energy (weak-field) limit, the spectral action reproduces reasonably well the known physics including the standard model. However, not much is known about the spectral action beyond the low-energy approximation. In this paper, after an extensive introduction to spectral triples and spectral actions, we study various expansions of the spectral actions (exemplified by the heat kernel). We derive the convergence criteria. For a commutative spectral triple, we compute the heat kernel on the torus up to the second order in gauge connection and consider limiting cases.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'.

Theory of the Casimir effect is presented in several examples. Casimir–Polder-type formulas, Lifshitz theory and theory of the Casimir effect for two gratings separated by a vacuum slit are derived. Equations for the electromagnetic field in the presence of a medium and dispersion are discussed. The Casimir effect for systems with a layer of 2 + 1 fermions is studied.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'.

In the context of quantum field theory, an anomaly exists when a theory has a classical symmetry which is not a symmetry of the quantum theory. This short exposition aims at introducing a new point of view, which is that the proper setting for anomaly calculations is the 'in–in', or closed-time path formulation of quantum field theory. There are also some new results for anomalies in the context of boundary value problems, and a new correction to the a₅ heat-kernel coefficient.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'.

I describe the logical basis of the method that I developed in 1962 and 1963 to define a quantum operator corresponding to the observable particle number of a quantized free scalar field in a spatially-flat isotropically expanding (and/or contracting) universe. This work also showed for the first time that particles were created from the vacuum by the curved spacetime of an expanding spatially-flat Friedmann–Lemaître–Robertson–Walker (FLRW) universe. The same process is responsible for creating the nearly scale-invariant spectrum of quantized perturbations of the inflaton scalar field during the inflationary stage of the expansion of the universe. I explain how the method that I used to obtain the observable particle number operator involved adiabatic invariance of the particle number (hence, the name adiabatic regularization) and the quantum theory of measurement of particle number in an expanding universe. I also show how I was led in a surprising way, to the discovery in 1964 that there would be no particle creation by these spatially-flat FLRW universes for free fields of any integer or half-integer spin satisfying field equations that are invariant under conformal transformations of the metric. The methods I used to define adiabatic regularization for particle number were based on generally-covariant concepts like adiabatic invariance and measurement that were fundamental and determined results that were unique to each given adiabatic order.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'.

We consider the situation when a globally defined four-dimensional field system is separated on two entangled sub-systems by a dynamical (random) two-dimensional surface. The reduced density matrix averaged over ensemble of random surfaces of fixed area and the corresponding average entropy are introduced. The average entanglement entropy is analyzed for a generic conformal field theory in four dimensions. Two important particular cases are considered. In the first, both the intrinsic metric on the entangling surface and the spacetime metric are fluctuating. An important example of this type is when the entangling surface is a black hole horizon, the fluctuations of which cause necessarily the fluctuations in the spacetime geometry. In the second case, the spacetime is considered to be fixed. The detailed analysis is carried out for the random entangling surfaces embedded in flat Minkowski spacetime. In all cases, the problem reduces to an effectively two-dimensional problem of random surfaces which can be treated by means of the well-known conformal methods. Focusing on the logarithmic terms in the entropy, we predict the appearance of a new ln ln(A) term.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'.

We attempt the construction of perturbative rotating hairy black holes and boson stars, invariant under a single helical Killing field, in (2 + 1) dimensions to complete the perturbative analysis in arbitrary odd dimension recently put forth in Stotyn et al (2012 Phys. Rev. D 85 044036). Unlike the higher dimensional cases, we find evidence for the non-existence of hairy black holes in (2 + 1) dimensions in the perturbative regime, which is interpreted as another mass gap, within which the black holes cannot have hair. The boson star solutions face a similar impediment in the background of a conical singularity with a sufficiently high angular deficit, most notably in the zero-mass BTZ background where boson stars cannot exist at all. We construct such boson stars in the AdS₃ background as well as in the background of conical singularities of periodicities $\pi ,\frac{2\pi }{3},\frac{\pi }{2}$.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'.

We study a self-interacting scalar field theory in the presence of a δ-function background potential. The role of surface interactions in obtaining a renormalizable theory is stressed and demonstrated by a two-loop calculation. The necessary counterterms are evaluated by adopting dimensional regularization and the background field method. We also calculate the effective potential for a complex scalar field in a non-simply connected spacetime in the presence of a δ-function potential. The effective potential is evaluated as a function of an arbitrary phase factor associated with the choice of boundary conditions in the non-simply connected spacetime. We obtain asymptotic expansions of the results for both large and small δ-function strengths, and stress how the non-analytic nature of the small strength result vitiates any analysis based on standard weak field perturbation theory.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'.

We study the asymptotic behaviour of the heat content on a compact Riemannian manifold with boundary and with singular specific heat and singular initial temperature distributions. Assuming the existence of a complete asymptotic series, we determine the first three terms in that series. In addition to the general setting, the interval is studied in detail.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'.