Zeta Function Regularization Research Articles

Deriving measurement collapse using zeta function regularisation and speculative measurement theory

This paper shows how an application of zeta function regularisation to a physical model of quantum measurement yields a solution to the problem of wavefunction collapse. Realistic measurement dynamics based on a particle becoming non-isolated are introduced and, based on this, an outcome function is derived using the method of maximum entropy. It is shown how regularisation of an information theoretic quantity related to this outcome function leads to apparent collapse of the wavefunction. The physical principles and key assumptions that underlie this theory are discussed. Some possible experimental approaches are described.

Thermal Casimir effect for a Dirac field on flat space with a nontrivial circular boundary condition

This work investigates the thermal Casimir effect associated with a massive spinor field defined on a four-dimensional flat space with a circularly compactified spatial dimension whose periodicity is oriented along a vector in the xy plane. We employ the generalized zeta function method to establish a finite definition for the vacuum free energy density. This definition conveniently separates into the zero-temperature Casimir energy density and additional terms accounting for temperature corrections. The structure of existing divergences is analyzed from the asymptotic behavior of the spinor heat kernel function and removed in the renormalization by subtracting scheme. The only non-null heat coefficient is the one associated with the Euclidean divergence. We also address the need for a finite renormalization to treat the ambiguity in the zeta function regularization prescription associated with this Euclidean heat kernel coefficient and ensure that the renormalization procedure is unique. The high- and low-temperature asymptotic limits are also explored. In particular, we explicitly show that free energy density lacks a classical limit at high temperatures, and the entropy density agrees with the Nernst heat theorem at low temperatures. Published by the American Physical Society 2024

Zeta Regularization in the Chiral Separation Effect

It has been shown that an expression for the chiral separation effect can be obtained by means of the zeta regularization, which is dictated by the thermodynamic description of the system.

Sampling the lattice Nambu-Goto string using Continuous Normalizing Flows

Effective String Theory (EST) represents a powerful non-perturbative approach to describe confinement in Yang-Mills theory that models the confining flux tube as a thin vibrating string. EST calculations are usually performed using the zeta-function regularization: however there are situations (for instance the study of the shape of the flux tube or of the higher order corrections beyond the Nambu-Goto EST) which involve observables that are too complex to be addressed in this way. In this paper we propose a numerical approach based on recent advances in machine learning methods to circumvent this problem. Using as a laboratory the Nambu-Goto string, we show that by using a new class of deep generative models called Continuous Normalizing Flows it is possible to obtain reliable numerical estimates of EST predictions.

Casimir energy for spinor fields with δ-shell potentials

This work analyzes the Casimir energy of a massive spinor field propagating in flat space endowed with a spherically symmetric δ-function potential. By utilizing the spectral zeta function regularization method, the Casimir energy is evaluated after performing a suitable analytic continuation. Explicit numerical results are provided for specific cases in which the Casimir energy is unambiguously defined. The results described in this work represent a generalization of the MIT bag model for spinor fields.

Quantum vacuum, rotation, and nonlinear fields

In this paper, we extend previous results on the quantum vacuum or Casimir energy, for a noninteracting rotating system and for an interacting nonrotating system, to the case where both rotation and interactions are present. Concretely, we first reconsider the noninteracting rotating case of a scalar field theory and propose an alternative and simpler method to compute the Casimir energy based on a replica trick and the Coleman-Weinberg effective potential. We then consider the simultaneous effect of rotation and interactions, including an explicit breaking of rotational symmetry. To study this problem, we develop a numerical implementation of zeta-function regularization. Our work recovers previous results as limiting cases and shows that the simultaneous inclusion of rotation and interactions produces nontrivial changes in the quantum vacuum energy. Besides expected changes (where, as the size of the ring increases for fixed interaction strength, the angular momentum grows with the angular velocity), we notice that the way rotation combines with the coupling constant amplifies the intensity of the interaction strength. Interestingly, we also observe a departure from the typical massless behavior where the Casimir energy is proportional to the inverse size of the ring.

Casimir self-energy of a δ−δ′ sphere

We extend previous work on the vacuum energy of a massless scalar field in the presence of singular potentials. We consider a single sphere defined by the so-called $\ensuremath{\delta}\text{\ensuremath{-}}{\ensuremath{\delta}}^{\ensuremath{'}}$ interaction. Contrary to the Dirac $\ensuremath{\delta}$-potential, we find a nontrivial one-parameter family of potentials such that the regularization procedure gives an unambiguous result for the Casimir self-energy. The procedure employed is based on the zeta function regularization and the cancellation of the heat kernel coefficient ${a}_{2}$. The results obtained are in agreement with particular cases, such as the Dirac $\ensuremath{\delta}$ or Robin and Dirichlet boundary conditions.

Kazimirov efekat

Introduction/purpose: The quantization of the electromagnetic field gives rise to quantum fluctuations which in turn produce a force on macroscopic boundaries. This phenomenon is called the Casimir effect. Method: The second quantization of the electromagnetic field is employed. The Zeta function regularization technique has been applied. Results: Because of the electromagnetic field quantization, a force on macroscopic boundaries is observed. Conclusions: Vacuum fluctuations due to quantum effects give macroscopic results.

On virtual scalar fields in a conformally flat FLRW spacetime

The case of a virtual bosonic scalar field ϕ(x) gravitationally coupled in a conformally flat spacetime is investigated. The action S is known to be Weyl invariant only for specific expressions of the potential V(ϕ). In the conformally flat FLRW case the harmonic angular frequency ωk (η) becomes uniquely temporally independent. Therefore, any inertial observers embedded in a conformally flat FLRW spacetime all agree on the choice of virtual vacuum states. We postulate that the wavenumber k must be quantized in order to be able to regulate the vev by zeta regularization. Some fundamental implications of this result are derived, and likely conclusions drawn.

Resolution of challenging problems in quantum cosmology with electromagnetic radiation

We investigate the quantum cosmology of a closed spatially homogeneous and isotropic Friedmann–Lemaître–Robertson–Walker (FLRW) minisuperspace model with electromagnetic radiation as matter content. We solve the corresponding Wheeler–DeWitt equation by utilizing Riemann's zeta function regularization method. We demonstrate that the regularized vacuum energy of the electromagnetic field can overcome factor ordering, boundary conditions, and singularity problems.

Free fermions, KdV charges, generalised Gibbs ensembles and modular transforms

In this paper we consider the modular properties of generalised Gibbs ensembles in the Ising model, realised as a theory of one free massless fermion. The Gibbs ensembles are given by adding chemical potentials to chiral charges corresponding to the KdV conserved quantities. (They can also be thought of as simple models for extended characters for the W-algebras). The eigenvalues and Gibbs ensembles for the charges can be easily calculated exactly using their expression as bilinears in the fermion fields. We re-derive the constant term in the charges, previously found by zeta-function regularisation, from modular properties. We expand the Gibbs ensembles as a power series in the chemical potentials and find the modular properties of the corresponding expectation values of polynomials of KdV charges. This leads us to an asymptotic expansion of the Gibbs ensemble calculated in the opposite channel. We obtain the same asymptotic expansion using Dijkgraaf’s results for chiral partition functions. By considering the corresponding TBA calculation, we are led to a conjecture for the exact closed-form expression of the GGE in the opposite channel. This has the form of a trace over multiple copies of the fermion Fock space. We give analytic and numerical evidence supporting our conjecture.

Zeta function regularization technique in the electrostatics context for discrete charge distributions

Spectral functions, such as the zeta functions, are widely used in quantum field theory to calculate physical quantities. In this work, we compute the electrostatic potential and field due to an infinite discrete distribution of point charges, using the zeta function regularization technique. This method allows us to remove the infinities that appear in the resulting expression. We found that the asymptotic behavior dependence of the potential and field is similar to the cases of continuous charge distribution. Finally, this exercise can be useful for graduate students to explore spectral and special functions.

Ultraviolet sensitivity of Casimir energy

Abstract We quantitatively estimate the effect of ultraviolet (UV) physics on the Casimir energy in a five-dimensional (5D) model on S1/Z2. If the cutoff scale of the 5D theory is not far from the compactification scale, the UV physics may affect the low-energy result. We work in the cutoff regularization scheme by introducing two independent cutoff scales for the spatial momentum in the non-compact space and for the Kaluza–Klein masses. The effects of the UV physics are incorporated as a damping effect of the contributions to the vacuum energy around the cutoff scales. We numerically calculate the Casimir energy and evaluate the deviation from the result obtained in the zeta-function regularization, which does not include information on the UV physics. We find that the result agrees well with the latter for Gaussian-type damping, while it can deviate for kink-type damping.

Magnetizations and de Haas-van Alphen oscillations in massive Dirac fermions

We theoretically study magnetic field, temperature, and energy band-gap dependences of magnetizations in the Dirac fermions. We use the zeta function regularization to obtain analytical expressions of thermodynamic potential, from which the magnetization of graphene for strong field/low temperature and weak field/high temperature limits are calculated. Further, we generalize the result by considering the effects of impurity on orbital susceptibility of graphene. In particular, we show that in the presence of impurity, the susceptibility follows a scaling law which can be approximated by the Faddeeva function. In the case of the massive Dirac fermions, we show that a large band-gap gives a robust magnetization with respect to temperature and impurity. In the doped Dirac fermion, we discuss the dependence of the band-gap on the period and amplitude of the de Haas-van Alphen effect.

Vacuum Polarization with Zero-Range Potentials on a Hyperplane

The quantum vacuum fluctuations of a neutral scalar field induced by background zero-range potentials concentrated on a flat hyperplane of co-dimension 1 in (d+1)-dimensional Minkowski spacetime are investigated. Perfectly reflecting and semitransparent surfaces are both taken into account, making reference to the most general local, homogeneous and isotropic boundary conditions compatible with the unitarity of the quantum field theory. The renormalized vacuum polarization is computed for both zero and non-zero mass of the field, implementing a local version of the zeta regularization technique. The asymptotic behaviors of the vacuum polarization for small and large distances from the hyperplane are determined to leading order. It is shown that boundary divergences are softened in the specific case of a pure Dirac delta potential.

Zeta Functions and the Cosmos—A Basic Brief Review

This is a very basic and pedagogical review of the concepts of zeta function and of the associated zeta regularization method, starting from the notions of harmonic series and of divergent sums in general. By way of very simple examples, it is shown how these powerful methods are used for the regularization of physical quantities, such as quantum vacuum fluctuations in various contexts. In special, in Casimir effect setups, with a note on the dynamical Casimir effect, and mainly concerning its application in quantum theories in curved spaces, subsequently used in gravity theories and cosmology. The second part of this work starts with an essential introduction to large scale cosmology, in search of the observational foundations of the Friedmann-Lemaître-Robertson-Walker (FLRW) model, and the cosmological constant issue, with the very hard problems associated with it. In short, a concise summary of all these interrelated subjects and applications, involving zeta functions and the cosmos, and an updated list of the pioneering and more influential works (according to Google Scholar citation counts) published on all these matters to date, are provided.

Intrinsic time gravity, heat kernel regularization, and emergence of Einstein’s theory

The Hamiltonian of intrinsic time gravity is elucidated. The theory describes Schrödinger evolution of our universe with respect to the fractional change of the total spatial volume. Gravitational interactions are introduced by extending Klauder’s momentric variable with similarity transformations, and explicit spatial diffeomorphism invariance is enforced via similarity transformation with exponentials of spatial integrals. In analogy with Yang–Mills theory, a Cotton–York term is obtained from the Chern–Simons functional of the affine connection. The essential difference is the fundamental variable for geometrodynamics is the metric rather than a gauge connection; in the case of Yang–Mills, there is also no analog of the integral of the spatial Ricci scalar curvature. Heat kernel regularization is employed to isolate the divergences of coincidence limits; apart from an additional Cotton–York term, a prescription in which Einstein’s Ricci scalar potential emerges naturally from the positive-definite self-adjoint Hamiltonian of the theory is demonstrated.

Semiclassical p-branes in hyperbolic space

The one-loop effects to the Dirac action of p-branes in a hyperbolic background from the path integral and the solution of the Wheeler–DeWitt equation are analysed. The objective of comparing the equivalent quantization procedures is to study in detail the validity of the semiclassical approximation and divergences associated to one-loop corrections. This is in line with a bottom-up approach to holographic Wilson loops. We employ the heat kernel regularization method for both quantization procedures and we study in great detail one-loop corrections to geodesics in a two-dimensional hyperbolic space and semi-spheres in a three-dimensional hyperbolic space. We show that the divergences, given by the high energy expansion of the heat kernel, can be classified by their compatibility with the semiclassical approximation and geometric nature.

Renormalization of Feynman amplitudes on manifolds by spectral zeta regularization and blow-ups

Our goal in this paper is to present a generalization of the spectral zeta regularization for general Feynman amplitudes on Riemannian manifolds. Our method uses complex powers of elliptic operators but involves several complex parameters in the spirit of analytic renormalization by Speer, to build mathematical foundations for the renormalization of perturbative interacting quantum field theories. Our main result shows that spectrally regularized Feynman amplitudes admit analytic continuation as meromorphic germs with linear poles in the sense of the works of Guo–Paycha and the second author. We also give an explicit determination of the affine hyperplanes supporting the poles. Our proof relies on suitable resolution of singularities of products of heat kernels to make them smooth. As an application of the analytic continuation result, we use a universal projection from meromorphic germs with linear poles on holomorphic germs to construct renormalization maps which subtract singularities of Feynman amplitudes of Euclidean fields. Our renormalization maps are shown to satisfy consistency conditions previously introduced in the work of Nikolov–Todorov–Stora in the case of flat space-times.

6d mathcal{N} = (1, 0) anomalies on S1 and F-theory implications

We show that the pure gauge anomalies of 6d mathcal{N} = (1, 0) theories compactified on a circle are captured by field-dependent Chern-Simons terms appearing at one-loop in the 5d effective theories. These terms vanish if and only if anomalies are canceled. In order to obtain this result, it is crucial to integrate out the massive Kaluza-Klein modes in a way that preserves 6d Lorentz invariance; the often-used zeta-function regularization is not sufficient. Since such field-dependent Chern-Simons terms do not arise in the reduction of M-theory on a threefold, six-dimensional F-theory compactifications are automatically anomaly free, whenever the M/F-duality can be used. A perfect match is then found between the 5d mathcal{N} = 1 prepotentials of the classical M-theory reduction and one-loop circle compactification of an anomaly free theory. Finally, from this potential, we read off the quantum corrections to the gauge coupling functions.