Ramanujan, Landau and Casimir, divergent series: a physicist approach

Vacuum is what remains when there is nothing left in space. But this "nothing" is still able to act on objects placed in vacuum. The archetype of such an action of vacuum is the so-called Casimir force.In fact vacuum consists in fluctuating electromagnetic fields propagating through space with the speed of light, and having the minimal fluctuation energy allowed by quantum theory. Since vacuum energy is a minimum, it cannot be used to extract work. But the fluctuations have observable consequences in atomic and subatomic physics. An atom interacting only with vacuum fields suffers spontaneous emission processes which can be considered as induced by these fields. When fallen in its ground state, the atom no longer emits photons but its coupling to vacuum results in a measurable Lamb shift of the absorption frequencies.Two atoms located at different places experience an attractive van der Waals force which plays an important role in physico-chemical and biological processes. Casimir was studying this effect when he discovered in 1948 that two mirrors placed in vacuum are attracted towards each other.There are many reasons for the growing interest in this Casimir force. Progress in experimental techniques have recently allowed for accurate measurements and have then led to tests of this crucial prediction of quantum electrodynamics. These tests have required theoretical developments for dealing satisfactory with the description of the optical response of mirrors or of geometry of the experiments. They have also made it possible to extend the search for hypothetical new weak forces into the distance range where the Casimir force dominates over other forces.Nevertheless, in spite of all this progress and renewal interest on the subject there are still basic questions that have not been answered and extensions that have not been achieved, for example a consistent treatment of temperature effects or a full theory of geometrical effects between mirrors made of real materials.Recently it has been recognized that vacuum induced forces played a considerable role in micro- and nano-electromechanical systems (MEMS or NEMS) with typical sizes in the µm range or below. The emerging interface between quantum Casimir forces and nanophysics is already clearly visible in this Focus Issue, and it will certainly gain more and more importance in the forthcoming years.The topics covered by the articles in this Focus Issue of New Journal of Physics reflect the increasing activity and broadening spectrum of the studies connected to the Casimir forces. We hope that the reader will find in this collection of papers new research results or proposals as well as reviews by experts of a large set of open issues which are currently addressed by an expanding community. We also hope that the issue will interest the non-expert readers and draw their attention to this exciting and interdisciplinary research field.The articles below represent the first contributions and further additions will appear.Focus on Casimir Forces ContentsCasimir energy with a Robin boundary: the multiple-reflection cylinder-kernel expansion Z Liu and S A Fulling On the use of hydrogen switchable mirrors in Casimir force experiments Sven de Man and Davide Iannuzzi Thermal corrections to the Casimir effect Iver Brevik, Simen Ellingsen and K Milton Casimir forces and non-Newtonian gravitation R Onofrio Sample dependence of the Casimir force I Pirozhenko, A Lambrecht and V B Svetovoy Casimir effect: a novel experimental approach at large separation P Antonini, G Bressi, G Carugno, Giuseppe Galeazzi, Giuseppe Messineo and G Ruoso Exact zero-point interaction energy between cylinders F D Mazzitelli, D A R Dalvit and F C Lombardo Stability and the proximity theorem in Casimir actuated nano devices R Esquivel-Sirvent, Luis Reyes and Jeffrey Bárcenas Casimir effect for arbitrary materials: contributions within and beyond the light cone W L Mochán and C Villarreal The Casimir effect within scattering theory A Lambrecht, Neto P A Maia and S Reynaud Quantum electrodynamical torques in the presence of Brownian motion Jeremy N Munday, Davide Iannuzzi and F Capasso Rubén G Barrera, Universidad Nacional Autónoma de México, Mexico Serge Reynaud, Université Pierre et Marie Curie, Paris, France

This theoretical work corresponds to the hope of extracting, without contradicting EMMY NOETHER’s theorem, an energy present throughout the universe: that of the spatial quantum vacuum. This article shows that it should be theoretically possible to maintain a continuous periodic vibration of a piezoelectric structure, which generates electrical power peaks during a fraction of the vibration period. Electronics without any power supply then transform these alternating current signals into a usable direct voltage. To manufacture these different structures, we also present an original microtechnology for producing the regulation and transformation electronics, as well as that necessary for controlling the very weak interfaces between the Casimir electrodes and that of the return electrodes. These vibrations are obtained by controlling automatically and at appropriate instants the action of the attractive Casimir force by a repulsive Coulomb force applied to return electrodes. The Casimir force appearing between the two electrodes of a reflector deforms a piezoelectric bridge, inducing a displacement of the barycenter of the ionic electric charges of the bridge. This internal piezoelectric field attracts opposing moving charges, from the mass, on either side of the piezoelectric bridge .They are used – after their homogenization - by a Coulomb force opposed and greater than the Casimir force. During the homogenization of the electric charges between the face 1 and the return electrode, periodic current peaks appear during a fraction of the vibration time of the device. These current peaks crossing an inductance, spontaneously induce voltage peaks at the terminals of this device, which are transformed into a usable DC voltage thanks to proposed electronics without a power supply. Casimir and Coulomb forces, vibrations, current, or voltage peaks appear spontaneously and without external energy input. Everything is only a consequence of the existence of the Casimir force due to the quantum fluctuations of the vacuum. This set does not seem to contradict Thermodynamics’ law and Emily Noether’s theorem.Casimir and Coulomb forces, vibrations, current, or voltage peaks appear spontaneously and without external energy input. Everything is only a consequence of the existence of the Casimir force due to the quantum fluctuations of the vacuum. This set does not seem to contradict Thermodynamics’ law and Emily Noether’s theorem.

The Casimir force, an attractive force between two parallel, uncharged, perfectly conducting plates separated by a small gap has attracted much attention not only because it is fascinating phenomenon in quantum electrodynamics, but also because of its important applications in microelectromechanical systems (MEMS). Although the Casimir force is extremely small, it plays a dominant role in MEMS when the distance between device components is also very small. The failure of MEMS due to stiction caused by attractive force motivates the study of repulsive Casimir force. As a result, more and more derivative concepts related to the Casimir force have emerged. Aside from the attractive and repulsive Casimir forces, the lateral Casimir force can occur between sinusoidally corrugated surfaces. Casimir friction arises when two plates are in relative motion even if they are not in direct contact. Such phenomenon not only reveals the non-intuitive nature of the quantum vacuum, but also demonstrate potential applications in nanoscience and quantum technology. In addition, a rotating nanoparticle can also experience a lateral Casimir force when it rotates near a plate, the underlying mechanism is different from that occurring between corrugated surfaces. In particular, when the nanoparticle rotates in the proximity of an optically anisotropic plate, an axial Casimir force along the direction of the rotation axis is exerted on the nanoparticle. At present, these novel phenomena related to the Casimir force form the foundation of Casimir physics research. Here, we focus on the recent progress in the theoretical and experimental research on the Casimir effect including the attractive Casimir force, repulsive Casimir force, lateral Casimir force, and Casimir friction. We also review the applications of various Casimir effects in the field of nanotechnology. The Casimir effect provides new perspectives on MEMS and quantum information processing, and may inspire new experimental approaches and technological applications. This phenomenon holds substantial scientific significance and promising application potential across both fundamental research and engineering fields.

The goal in this effort is twofold: (1) to develop an understanding of Casimir forces in geometries more complicated than the usual parallel-plate geometry and (2) to provide extensive numerical computations to elucidate quantitative and qualitative aspects of the vacuum fluctuation energy and Casimir forces for the rectangular cavity. We review geometries for which Casimir forces and vacuum energy have been computed, and point out some of the difficulties with the ideal-conductor boundary conditions and ideal-shape boundary conditions, e.g., infinitely sharp edges. We investigate the vacuum electromagnetic stress-energy tensor at 0 K for a perfectly conducting three-dimensional rectangular cavity with sides ${a}_{1}\ifmmode\times\else\texttimes\fi{}{a}_{2}\ifmmode\times\else\texttimes\fi{}{a}_{3}.$ The elements of the tensor are averaged over the appropriate spatial coordinates of the cavity. We first consider the average energy density ${T}^{00}=e(\mathit{a})/V$ from the viewpoint of symmetry, where ${e(a}_{1}{,a}_{2}{,a}_{3})=e(\mathit{a})$ is the finite change in the zero-point energy from the free-field case. The vacuum energy $e(\mathit{a})$ and the total vacuum force on the wall normal to the i direction, ${F}_{i}=\ensuremath{-}\ensuremath{\partial}e/\ensuremath{\partial}{a}_{i},$ are both homogeneous functions of the cavity dimensions. Because of this symmetry, the energy and forces are related by the equation $e(\mathit{a})=\mathit{a}\ensuremath{\cdot}\mathit{F}(\mathit{a}).$ We compute the vacuum forces and energy numerically for cavities with a broad range of dimensions. The implications of the perfect-conductor boundary conditions and the effects of the edges of the cavity are both considered. The ${C}_{3v}$ symmetry of the constant-energy surfaces is apparent. The zero-energy surface, which is invariant under dilations and therefore extends to infinity, separates the nested, concave, positive-energy surfaces from the open, negative-energy surfaces. The positive- (negative-) energy surfaces are mapped into each other by scale changes. The force $\mathit{F}(\mathit{a})$ is normal to the constant-energy surface at $\mathit{a}.$ The surfaces corresponding to zero forces, ${\mathit{F}}_{i}(\mathit{a})=0,$ are invariant under dilations and are therefore infinite. The zero-energy surface and the zero-force surfaces delineate the different geometries for which there are zero, one, or two negative (inward or attractive) forces on the cavity walls, along with the sign of the corresponding energy. There is no rectangular cavity geometry for which all forces are negative or zero; conversely, only geometries that are not too different from a cube have all positive (outward or repulsive) forces. Only for the last case is the energy $e(\mathit{a})$ necessarily positive. To provide an intuitive feeling for these vacuum energies, comparisons are made to other forms of energy in small cavities. We consider the energy balance for changes in cavity dimensions.

This essay explores the interrelationship between Ramanujan summation, the Riemann zeta function, and divergent series. the connection between Ramanujan summation, the Riemann zeta function, and divergent series will be explored. My objective is to illustrate how Ramanujan summation can be interpreted in a way to assign values to divergent series, thus extending classical convergence and gaining better understanding of infinite sums. Originally going back to the development of convergent series the Riemann zeta function is a fundamental element in analytic number theory and produces an architecture for understanding how prime numbers are distributed. Finally, I test the accuracy of the Ramanujan series and the analytical equation by presenting the special divergent series from the sum of all natural numbers. It may not be an intuitive outcome from a more elementary viewpoint, but it has been obtained in the overseas of some complicated mathematics and appears in the application of theory physics, for example; cable theory and Casimir effect computations involving infinite series sums that naturally emerge in quantum field theory.

This chapter opens the part of the book devoted to quantum vacuum in non-trivial gravitational background and to vacuum energy. There are several macroscopic phenomena, which can be directly related to the properties of the physical quantum vacuum. The Casimir effect is probably the most accessible effect of the quantum vacuum. The chapter discusses different types of Casimir effect in condensed matter in restricted geometry, including the mesoscopic Casimir effect and the dynamic Casimir effect resulting in the force acting on a moving interface between 3He-A and 3He-B, which serves as a perfect mirror for the ‘relativistic’ quasiparticles living in 3He-A. It also discusses the vacuum energy and the problem of cosmological constant. Giving the example of quantum liquids it is demonstrated that the perfect vacuum in equilibrium has zero energy, while the nonzero vacuum energy arises due to perturbation of the vacuum state by matter, by texture, which plays the role of curvature, by boundaries due to the Casimir effect, and by other factors. The magnitude of the cosmological constant is small, because the present universe is old and the quantum vacuum is very close to equilibrium. The chapter discusses why our universe is flat, why the energies of the true vacuum and false vacuum are both zero, and why the perfect vacuum (true or false) is not gravitating.

This paper discusses recent developments on quantum electrodynamical (QED) phenomena, such as the Casimir effect, and their use in nanomechanics and nanotechnology in general. Casimir forces and torques arise from quantum fluctuations of vacuum or, more generally, from the zero-point energy of materials and their dependence on the boundary conditions of the electromagnetic fields. Because the latter can be tailored, this raises the interesting possibility of designing QED forces for specific applications. After a concise review of the field in an historical perspective, high precision measurements of the Casimir force using microelectromechanical systems (MEMS) technology and applications of the latter to nonlinear oscillators are presented, along with a discussion of its use in nanoscale position sensors. Then, experiments that have demonstrated the role of the skin-depth effect in reducing the Casimir force are presented. The dielectric response of materials enters in a nonintuitive way in the modification of the Casimir-Lifshitz force between dielectrics through the dielectric function at imaginary frequencies epsiv(ixi). The latter is illustrated in a dramatic way by experiments on materials that can be switched between a reflective and a transparent state (hydrogen switchable mirrors). Repulsive Casimir forces between solids separated by a fluid with epsiv(ixi) intermediate between those of the solids over a large frequency range is discussed, including ongoing experiments aimed at its observation. Such repulsive forces can be used to achieve quantum floatation in a virtually frictionless environment, a phenomenon that could be exploited in innovative applications to nanomechanics. The last part of the paper deals with the elusive QED torque between birefringent materials and efforts to observe it. We conclude by highlighting future important directions

New developments in the Casimir effect

Casimir force in quantum electrodynamics is the representation of zero point energy of vacuum. Depending on the type of fluctuation medium, generalized Casimir force covers a wide spectrum of topics in physics, such as, quantum, critical, Goldstone mode, and non-equilibrium Casimir force. In general, long range correlated fluctuations and constraints are two conditions for generating the Casimir force. In this paper, through a survey of the development of Casimir physics, we discuss several types of Casimir forces and several regularization methods. We end the paper with an outlook for the further development of Casimir physics in the future.

Srinivasa Ramanujan was a great self-taught Indian mathematician, who died a century ago, at the age of only 32, one year after returning from England. Among his numerous achievements is the assignment of sensible, finite values to divergent series, which correspond to Riemann's $\zeta$-function with negative integer arguments. He hardly left any explanation about it, but following the few hints that he gave, we construct a direct justification for the best known example, based on analytic continuation. As a physical application of Ramanujan summation we discuss the Casimir effect, where this way of removing a divergent term corresponds to the renormalization of the vacuum energy density, in particular of the photon field. This leads to the prediction of the Casimir force between conducting plates, which has now been accurately confirmed by experiments. Finally we review the discussion about the meaning and interpretation of the Casimir effect. This takes us to the mystery surrounding the magnitude of Dark Energy.

A transformation that relates the Minkowskian space of Quantum Electrodynamics (QED) vacuum between parallel conducting plates and QED at finite temperature is obtained. From this formal analogy, the eigenvalues and eigenvectors of the photon self-energy for the QED vacuum between parallel conducting plates (Casimir vacuum) are found in an approximation independent form. It leads to two different physical eigenvalues and three eigenmodes. We also apply the transformation to derive the low energy photons phase velocity in the Casimir vacuum from its expression in the QED vacuum at finite temperature.

The possibilities of micro- and nanoelectromechanical systems analogs of structures with PCMS have been researchers, theoretically. The offered structure is considered, in which the series switching of devices from one state in another is implemented and the self-distribution of the switched state along structure and its transition to a new state (principle of a domino) is carried out. On the base of the solution of differential equations linking cantilever bias (sag) to its design parameters and an affixed voltage it was established, that in meshes with d>100-150 nm to the defining speaker of meshes and processes of failure (the switching) are forces of Coulomb and elasticity, the Casimir force become essential at d<100 nm, and Van-der-Vaals force at d<10 nm. If the exterior voltage is absent it is possible to establish the oscillations in the mesh by Casimir and elasticity forces which energy is ensured with energy of vacuum, and velocity of change of amplitude is ensured with parameters of a mesh and frictional force. The potential difference at a level 0.1 mV can cause switching of system from high-resistance unlocked in the conductive locked state. The implemented design parameters of a mesh at d 10-1000 nm allow to create controllable mediums, which on fundamental parameters are comparable or considerably exceed mediums on a PCMS in VO/sub 2/ basis.

We present a simple general proof that Casimir force cannot originate from the vacuum energy of electromagnetic (EM) field. The full QED Hamiltonian consists of 3 terms: the pure electromagnetic term $H_{\rm em}$, the pure matter term $H_{\rm matt}$ and the interaction term $H_{\rm int}$. The $H_{\rm em}$-term commutes with all matter fields because it does not have any explicit dependence on matter fields. As a consequence, $H_{\rm em}$ cannot generate any forces on matter. Since it is precisely this term that generates the vacuum energy of EM field, it follows that the vacuum energy does not generate the forces. The misleading statements in the literature that vacuum energy generates Casimir force can be boiled down to the fact that $H_{\rm em}$ attains an implicit dependence on matter fields by the use of the equations of motion and the illegitimate treatment of the implicit dependence as if it was explicit. The true origin of the Casimir force is van der Waals force generated by $H_{\rm int}$.

We consider a universe with a compact extra dimension and a cosmological constant emerging from a suitable ultraviolet cutoff on the zero-point energy of the vacuum. We derive the Casimir force between parallel conducting plates as a function of the following scales: plate separation, radius of the extra dimension and cutoff energy scale. We find that there are critical values of these scales where the Casimir force between the plates changes sign. For the cutoff energy scale required to reproduce the observed value of the cosmological constant, we find that the Casimir force changes sign and becomes repulsive for plate separations less than a critical separation ${d}_{0}=0.6\text{ }\text{ }\mathrm{mm}$, assuming a zero radius of the extra dimension (no extra dimension). This prediction contradicts Casimir experiments which indicate an attractive force down to plate separations of 100 nm. For a nonzero extra dimension radius, the critical separation ${d}_{0}$ gets even larger than 0.6 mm and remains inconsistent with Casimir force experiments. We conclude that with or without the presence of a compact extra dimension, vacuum energy with any suitable cutoff cannot play the role of the cosmological constant.

We study d-dimensional Conformal Field Theories (CFTs) on the cylinder, $$ {S}^{d-1}\times \mathrm{\mathbb{R}} $$ , and its deformations. In d = 2 the Casimir energy (i.e. the vacuum energy) is universal and is related to the central charge c. In d = 4 the vacuum energy depends on the regularization scheme and has no intrinsic value. We show that this property extends to infinitesimally deformed cylinders and support this conclusion with a holographic check. However, for $$ \mathcal{N}=1 $$ supersymmetric CFTs, a natural analog of the Casimir energy turns out to be scheme independent and thus intrinsic. We give two proofs of this result. We compute the Casimir energy for such theories by reducing to a problem in supersymmetric quantum mechanics. For the round cylinder the vacuum energy is proportional to a + 3c. We also compute the dependence of the Casimir energy on the squashing parameter of the cylinder. Finally, we revisit the problem of supersymmetric regularization of the path integral on Hopf surfaces.