arXiv subjects

Ariel Edery

Publications and source records attributed to Ariel Edery.

At least 19 recordsLinked to original sources

arXiv2026

Summing the reciprocal of the polynomial appearing in Fermat's Last Theorem

Consider the polynomial $f=x^N+y^N-z^N$ where $x,\,y$ and $z$ are positive integers and $N \ge 3$ is an integer. By Fermat's Last Theorem, $f$ is never zero so that its reciprocal, $1/f$, has no singularities. We therefore study the finite sum of the reciprocal: $S(m,N)=\sum_{x=1}^m\sum_{y=1}^m\sum_{z=1}^{m}\frac{1}{f}$. The terms $1/f$ can be positive, negative and their magnitude is less than unity. A key observation is that $S(m,N)$ can be split into two convenient parts: a dominant contribution $D(m,N)$ that has a simple analytical expression and a remainder $R(m,N)$ which is more complicated but negligible compared to $D(m,N)$. Therefore, $S(m,N)$ is almost identical to $D(m,N)$. The analytical expression for $D(m,N)$ is $(2\,m-1)\,H_m^{(N)}$ where $H_m^{(N)}=\sum_{x=1}^m\frac{1}{x^N}$ approaches quickly the Riemann zeta function $\zeta(N)$ as $m$ increases. Therefore, the original sum $S(m,N)$ has a simple expression: it is basically linear in $m$ with slope equal to $2\,\zeta(N)$. Its linear behavior is not an asymptotic result; plots of $S(m,N)$ vs. $m$ for different $N$ show a straight line starting at $m=1$. $S(m,N)$ deviates slightly from a straight line over a small interval $8\le m\le 12$ for the case $N=3$. This slight deviation is due to Fermat near misses where $x^3+y^3-z^3=\pm 1$ (for $z\ne x$ and $z\ne y$); these create a jump in the remainder $R(m,3)$ at $m=9$. We make a numerical and analytical study of the remainder $R(m,N)$. From the numerical analysis, $R(m,N)$ converges for $N\ge 4$ but it was harder to tell whether $N=3$ converged. An analytical study based on a comparison of $R(m,N)$ to its Cauchy principal value integral, shows that $R(m,3)$ likely diverges logarithmically. It also shows that $R(m,N)$ converges for $N\ge 4$ in agreement with the numerical analysis. We discuss in the conclusion some interesting questions for future investigation.

math.NT↗

arXiv2026

Finite path integral limits work in cases where the perturbative series is not Borel summable

The perturbative expansion in powers of the coupling of observables in quantum field theory and quantum mechanics is known to yield an asymptotic series. If the original physical system is well-behaved and a finite observable is expected, this can often be calculated via a Borel resummation of the asymptotic series. However, there are cases where a system is well-behaved and the series is not Borel summable. This typically occurs when the physical system has a non-trivial vacuum structure. It has recently been shown that if the perturbative series is carried out under finite path integral limits, one can obtain a convergent series that yields observables even at strong coupling. This was recently used to obtain the energy at strong coupling for the anharmonic oscillator. This is a Borel summable case so the question is whether finite path integral limits work when the series is not Borel summable. To begin answering this question we consider a simple non-Borel summable case: the series stemming from a basic integral where the function has a double-well shape and hence two minima. The integral has an exact analytical expression that the series can be compared to. Under finite integral limits that run from $-L$ to $L$, where $L$ is finite, positive and real, we develop two perturbative series in powers of the coupling: one by expanding the integral about the local maximum at the origin and the other by expanding it about one of the minima. In both cases, we obtain an absolutely convergent series and the series sums to the exact analytical expression of the original integral in the infinite $L$ limit. It is significant that a perturbative expansion about one of the minima reproduces the exact analytical expression because this implies that it captures the full effect of both minima.

hep-th↗

arXiv2026

Transition of a superposition of states under a delta-function pulse in a two-level system

Under a time-dependent perturbation, it is common to calculate the probability of a transition from one eigenstate to another eigenstate of a quantum system. Here we study the transition from a \textit{linear superposition of eigenstates} to an eigenstate under a delta-function pulse. We consider a two-level system with energy levels $E_1$ and $E_2$ and obtain exact analytical expressions for the coefficients $c_1$ and $c_2$ of the final state. The expressions are general since the coefficients $\alpha_1$ and $\alpha_2$ of the initial superposition state are free parameters constrained only by $|\alpha_1|^2+ |\alpha_2|^2=1$. This opens up new possibilities and in particular allows for an abrupt transition to a definite eigenstate with unit probability. We obtain a general analytical expression for the probability $P_{\alpha_1,\alpha_2 \to 2}$ of an initial superposition state to transition to the second eigenstate. Armed with this general expression we study some interesting special cases. With a delta-function pulse, the transitions are abrupt/instantaneous and we show that they do not depend on the energy gap $E_2-E_1$ and hence on the relative phase between the two eigenstates. For specific values of the interaction strength $\beta$, the initial superposition transitions abruptly to a definite eigenstate with unit probability. We discuss the similarities and differences such a transition has with the collapse of the wavefunction familiar in the context of a measurement.

quant-ph↗

arXiv2025

Convergent perturbative series via finite path integral limits: application to energy at strong coupling of the anharmonic oscillator

Solving quantum field theories at strong coupling remains a challenging task. The main issue is that the usual perturbative series are asymptotic series which can be useful at weak coupling but break down completely at strong coupling. In this work, we show that if the limits of integration in the path integral are finite, the perturbative series is remarkably an absolutely convergent series which works well at strong coupling. For now, we apply this perturbative approach to $\lambda \phi^4$ theory in 0+0 dimensions (a basic integral) and 0+1 dimensions (quartic anharmonic oscillator). As a further application, we also consider the sextic anharmonic oscillator. For the basic integral, we show that finite integral limits yields a convergent series whose values are in agreement with exact analytical results at any coupling. This worked even when the asymptotic series was not Borel summable. It is well known that the perturbative series expansion in powers of the coupling for the energy of the anharmonic oscillator yields an asymptotic series and hence fails at strong coupling. In quantum mechanics, if one is interested in the energy, it is often easier to use Schr\"odinger's equation to develop a perturbative series than path integrals. Finite path integral limits are then equivalent to placing infinite walls at positions -L and L in the potential where L is positive, finite and can be arbitrarily large. With walls, the series expansion for the energy is now convergent and approaches the energy of the anharmonic oscillator as the walls are moved further apart. We use the convergent series to calculate the ground state energy at weak, intermediate and strong coupling. At strong coupling, the result from the series agrees with the exact energy to within $0.1\%$, a remarkable result in light of the fact that at strong coupling the usual perturbative series diverges badly immediately.

hep-th↗

arXiv2024

Two types of series expansions valid at strong coupling

It is known that perturbative expansions in powers of the coupling in quantum mechanics (QM) and quantum field theory (QFT) are asymptotic series. This can be useful at weak coupling but fails at strong coupling. In this work, we present two types of series expansions valid at strong coupling. We apply the series to a basic integral as well as a QM path integral containing a quadratic and quartic term with coupling constant $\lambda$. The first series is the usual asymptotic one, where the quartic interaction is expanded in powers of $\lambda$. The second series is an expansion of the quadratic part where the interaction is left alone. This yields an absolutely convergent series in inverse powers of $\lambda$ valid at strong coupling. For the basic integral, we revisit the first series and identify what makes it diverge even though the original integral is finite. We fix the problem and obtain, remarkably, a series in powers of the coupling which is absolutely convergent and valid at strong coupling. We explain how this series avoids Dyson's argument on convergence. We then consider the QM path integral (discretized with time interval divided into $N$ equal segments). As before, the second series is absolutely convergent and we obtain analytical expressions in inverse powers of $\lambda$ for the $n$th order terms by taking functional derivatives of generalized hypergeometric functions. The expressions are functions of $N$ and we work them out explicitly up to third order. The general procedure has been implemented in a Mathematica program that generates the expressions at any order $n$. We present numerical results at strong coupling for different values of $N$ starting at $N=2$. The series matches the exact numerical value for a given $N$ (up to a certain accuracy). The continuum is formally reached when $N\to \infty$ but in practice this can be reached at small $N$.

hep-th↗

arXiv2023

Enlarging the symmetry of pure $R^2$ gravity, BRST invariance and its spontaneous breaking

Pure $R^2$ gravity was considered originally to possess only global scale symmetry. It was later shown to have the larger restricted Weyl symmetry where it is invariant under the Weyl transformation $g_{μν} \to Ω^2(x)\, g_{μν}$ when the conformal factor $Ω(x)$ obeys the harmonic condition $\Box Ω(x)=0$. Restricted Weyl symmetry has an analog in gauge theory. Under a gauge transformation $A_μ\to A_μ + \frac{1}{e}\partial_μ f(x)$, the gauge-fixing term $(\partial_μA^μ)^2$ has a residual gauge symmetry when $\Box f=0$. In this paper, we consider scenarios where the symmetry of pure $R^2$ gravity can be enlarged even further. In one scenario, we add a massless scalar field to the pure $R^2$ gravity action and show that the action becomes on-shell Weyl invariant when the equations of motion are obeyed. We then enlarge the symmetry to a BRST symmetry where no on-shell or restricted Weyl condition is required. The BRST transformations here are not associated with gauge transformations (such as diffeomorphisms) but with Weyl (local scale) transformations where the conformal factor consists of a product of Grassmann variables. BRST invariance in this context is a generalization of Weyl invariance that is valid in the presence of the Weyl-breaking $R^2$ term. In contrast to the BRST invariance of gauge theories like QCD, it is not preserved after quantization since renormalization introduces a scale (leading to the well-known Weyl (conformal) anomaly). We show that the spontaneous breaking of the BRST symmetry yields an Einstein action; this still has a symmetry which is also anomalous. This is in accord with previous work that shows that there is conformal anomaly matching between the unbroken and broken phases when conformal symmetry is spontaneously broken.

hep-th↗

arXiv2022

The non-minimally coupled gravitating vortex: phase transition at critical coupling $ξ_c$ in AdS$_3$

We consider the Nielsen-Olesen vortex non-minimally coupled to Einstein gravity with cosmological constant $Λ$. A non-minimal coupling term $ξ\,R\,|ϕ|^2$ is natural to add to the vortex as it preserves gauge-invariance (here $R$ is the Ricci scalar and $ξ$ a dimensionless coupling constant). This term plays a dual role: it contributes to the potential of the scalar field and to the Einstein-Hilbert term for gravity. As a consequence, the vacuum expectation value (VEV) of the scalar field and the cosmological constant in the AdS$_3$ background depend on $ξ$. This leads to a novel feature: there is a critical coupling $ξ_c$ where the VEV is zero for $ξ\ge ξ_c$ but becomes non-zero when $ξ$ crosses below $ξ_c$ and the gauge symmetry is spontaneously broken. Moreover, we show that the VEV near the critical coupling has a power law behaviour proportional to $|ξ-ξ_c|^{1/2}$. Therefore $ξ_c$ can be viewed as the analog of the critical temperature $T_c$ in Ginzburg-Landau (GL) mean-field theory where a second-order phase transition occurs below $T_c$ and the order parameter has a similar power law behaviour $|T-T_c|^{1/2}$ near $T_c$. The critical coupling exists only in an AdS$_3$ background; it does not exist in asymptotically flat spacetime (topologically a cone) where the VEV remains at a fixed non-zero value independent of $ξ$. However, the deficit angle of the asymptotic conical spacetime depends on $ξ$ and is no longer determined solely by the mass; remarkably, a higher mass does not necessarily yield a higher deficit angle. The equations of motion are more complicated with the non-minimal coupling term present. However, via a convenient substitution one can reduce the number of equations and solve them numerically to obtain exact vortex solutions.

hep-th↗

arXiv2021

Wave packets in QFT: leading order width corrections to decay rates and clock behaviour under Lorentz boosts

Decay rates in quantum field theory (QFT) are typically calculated assuming the particles are represented by momentum eigenstates (i.e. plane waves). However, strictly speaking, localized free particles should be represented by wave packets. This yields width corrections to the decay rate and to the clock behaviour under Lorentz boosts. We calculate the decay rate of a particle of mass $M$ modeled as a Gaussian wavepacket of width $a$ and centered at zero momentum. We find the decay rate to be $\Gamma_0 \big[1- \frac{3 a^2}{4 M^2} +\mathcal{O}\big(\tfrac{a^4}{M^4}\big)\big]$ where $\Gamma_0$ is the decay rate of the particle at rest treated as a plane wave. The leading correction is then of order $\tfrac{a^2}{M^2}$. We then perform a Lorentz boost of velocity $v$ on the above Gaussian and find that its decay rate does not decrease \textit{exactly} by the Lorentz factor $\sqrt{1-v^2}$. There is a correction of order $\tfrac{a^2v^2}{M^2}$. Therefore, the decaying wave packet does not act exactly like a typical clock under Lorentz boosts and we refer to it is a "WP clock" (wave packet clock). A WP clock does not move with a single velocity relative to an observer but has a spread in velocities (more specifically, a spread in momenta). Nonetheless, it is best viewed as a single clock as the wave packet represents a one-particle state in QFT. WP clocks do not violate Lorentz symmetry and are not based on new physics: they are a consequence of the combined requirements of special relativity, quantum mechanics and \textit{localized} free particles.

hep-th↗

arXiv2021

Non-singular vortices with positive mass in 2+1 dimensional Einstein gravity with AdS$_3$ and Minkowski background

In previous work, black hole vortex solutions in Einstein gravity with AdS$_3$ background were found where the scalar matter profile had a singularity at the origin $r=0$. In this paper, we find numerically static vortex solutions where the scalar and gauge fields have a non-singular profile under Einstein gravity in an AdS$_3$ background. Vortices with different winding numbers $n$, VEV $v$ and cosmological constant $Λ$ are obtained. These vortices have positive mass and are not BTZ black holes as they have no event horizon. The mass is determined in two ways: by subtracting the numerical values of two separate asymptotic metrics and via an integral that is purely over the matter fields. The mass of the vortex increases as the cosmological constant becomes more negative and this coincides with the core of the vortex becoming smaller (compressed). We then consider the vortex with gravity in asymptotically flat spacetime for different values of the coupling $α=1/(16 πG)$. At the origin, the spacetime has its highest curvature and there is no singularity. It transitions to an asymptotic conical spacetime with angular deficit that increases significantly as $α$ decreases. For comparison, we also consider the vortex without gravity in flat spacetime. For this case, one cannot obtain the mass by the first method (subtracting two metrics) but remarkably, via a limiting procedure, one can obtain an integral mass formula. In the absence of gauge fields, there is a well-known logarithmic divergence in the energy of the vortex. With gravity, we present this divergence in a new light. We show that the metric acquires a logarithmic term which is the $2+1$ dimensional realization of the Newtonian gravitational potential when General Relativity is supplemented with a scalar field. This opens up novel possibilities which we discuss in the conclusion.

hep-th↗

arXiv2019

Critical gravity from four dimensional scale invariant gravity

We show that a critical condition exists in four dimensional scale invariant gravity given by the pure quadratic action $β\,C_{μνσρ} C^{μνσρ} + α\,R^2$ where $C^μ_{\,\,νσρ}$ is the Weyl tensor, $R$ is the Ricci scalar and $β$ and $α$ are dimensionless parameters. The critical condition in a dS or AdS background is $β=6 α$. This leads to critical gravity where the massive spin two physical ghost becomes a massless spin two graviton. In contrast to the original work on critical gravity, no Einstein gravity with a cosmological constant is added explicitly to the higher-derivative action. The critical condition is obtained in two independent ways. In the first case, we show the equivalence between the initial action and an action containing Einstein gravity, a cosmological constant, a massless scalar field plus Weyl squared gravity. The scale invariance is spontaneously broken. The linearized Einstein-Weyl equations about a dS or AdS background yield the critical condition $β=6α$. In the second case, we work directly with the original quadratic action. After a suitable field redefinition, where the metric perturbation is traceless and transverse, we obtain linearized equations about a dS or AdS background that yield the critical condition $β= 6α$. As in the first case, we also obtain a propagating massless scalar field. Substituting $β=6α$ into the energy and entropy formula for the Schwarzschild and Kerr AdS or dS black hole in higher-derivative gravity yields zero, the same value obtained in the original work on critical gravity. We discuss the role of boundary conditions in relaxing the $β=6α$ condition.

hep-th↗

arXiv2019

Relativistic corrections to Landau levels in the presence of a parallel linear electric field

We consider an electron moving under a constant magnetic field (in the z-direction) and a \textit{linear} electric field parallel to the magnetic field above the z=0 plane and anti-parallel below the plane. Two frequencies characterize the system: the cyclotron frequency $ω_c$ corresponding to motion along the x-y plane and associated with the usual Landau levels, and a second frequency $ω_z$ corresponding to motion along the z-direction. In previous work, the non-relativistic energies of this system were obtained, and it was shown that an extra degeneracy (beyond the Landau degeneracy) occurs when the ratio $\text{w}=ω_c/ω_z$ is rational. In this paper, we use Dirac's equation to obtain compact formulas for the first and second order relativistic corrections to this system via perturbation theory. The formulas are expressed in terms of the two frequencies $ω_c$ and $ω_z$, and two quantum numbers, $n$ and $n_z$, both of which are non-negative integers. The first order correction is negative and lowers the original energies. We plot the energy (zeroth plus first order) versus the ratio $\text{w}$ and there are degeneracies at all points where lines intersect. However, the degeneracy does not occur at the same $\text{w}$ as before. To illustrate this, we show how the first order correction splits the energy levels for the case $ω_c=ω_z$.

quant-ph↗

arXiv2019

Palatini formulation of pure $R^2$ gravity yields Einstein gravity with no massless scalar

Pure $R^2$ gravity has been shown to be equivalent to Einstein gravity with non-zero cosmological constant and a massless scalar field. We show that the Palatini formulation of pure $R^2$ gravity is equivalent to Einstein gravity with non-zero cosmological constant as before but with no massless scalar field. This is an important new development because the massless scalar field is not readily identifiable with any known particle in nature or unknown particles like cold dark matter which are expected to be massive. We then include a non-minimally coupled Higgs field as well as fermions to discuss how the rest of the standard model fields fit into this paradigm. With Higgs field, Weyl invariance is maintained by using a hybrid formalism that includes both the Palatini curvature scalar $\mathcal{R}$ and the usual Ricci scalar $R$

hep-th↗

arXiv2019

New degeneracies and modification of Landau levels in the presence of a parallel linear electric field

We consider a three-dimensional system where an electron moves under a constant magnetic field (in the z-direction) and a \textit{linear} electric field parallel to the magnetic field above the z=0 plane and anti-parallel below the plane. The linear electric field leads to harmonic oscillations along the z-direction. There are therefore two frequencies characterizing the system: the usual cyclotron frequency $ω_c$ corresponding to motion along the x-y plane and associated with Landau levels and a second frequency $ω_z$ for motion along the z-direction. Most importantly, when the ratio $W=ω_c/ω_z$ is a rational number, the degeneracy of the energy levels does not remain always constant as the energy increases. At some energies, the degeneracy jumps i.e. it increases. In particular, when the two frequencies are equal, the degeneracy increases with each energy level. This is in stark contrast to the usual Landau levels where the degeneracy is independent of the energy. We derive compact analytical formulas for the degeneracy. We also obtain an analytical formula for the energy levels and plot them as a function of $W$. The increase in degeneracy can readily be seen in the plot at points where lines intersect. For concreteness, we consider the electric field produced by a uniformly charged ring. Besides a linear electric field in the z direction the ring produces an extra electric field in the xy plane which we treat via perturbation theory. The Landau degeneracy is now lifted and replaced by tightly spaced levels that come in "bands". The plot of the energy levels shows that there is still a degeneracy where the bands intersect.

cond-mat.mes-hall↗

arXiv2018

Gravitating magnetic monopole via the spontaneous symmetry breaking of pure $R^2$ gravity

The pure $R^2$ gravity is equivalent to Einstein gravity with cosmological constant and a massless scalar field and it further possesses the so-called restricted Weyl symmetry which is a symmetry larger than scale symmetry. To incorporate matter, we consider a restricted Weyl invariant action composed of pure $R^2$ gravity, SU(2) Yang-Mills fields and a non-minimally coupled massless Higgs field (a triplet of scalars). When the restricted Weyl symmetry is spontaneously broken, it is equivalent to an Einstein-Yang-Mills-Higgs (EYMH) action with a cosmological constant and a massive Higgs non-minimally coupled to gravity i.e. via a term $\tildeξ R |Φ|^2$. When the restricted Weyl symmetry is not spontaneously broken, linearization about Minkowski space-time does not yield gravitons in the original $R^2$ gravity and hence it does not gravitate. However, we show that in the broken gauge sector of our theory, where the Higgs field acquires a non-zero vacuum expectation value, Minkowski space-time is a viable gravitating background solution. We then obtain numerically gravitating magnetic monopole solutions for non-zero coupling constant $\tildeξ=1/6$ in three different backgrounds: Minkowski, anti-de Sitter (AdS) and de Sitter (dS), all of which are realized in our restricted Weyl invariant theory.

hep-th↗

arXiv2018

First and second-order relativistic corrections to the two and higher-dimensional isotropic harmonic oscillator obeying the spinless Salpeter equation

We study the relativistic version of the $d$-dimensional isotropic quantum harmonic oscillator based on the spinless Salpeter equation. This has no exact analytical solutions. We use perturbation theory to obtain compact formulas for the first and second-order relativistic corrections; they are expressed in terms of two quantum numbers and the spatial dimension $d$. The formula for the first-order correction is obtained using two different methods and we illustrate how this correction splits the original energy into a number of distinct levels each with their own degeneracy. Previous authors obtained results in one and three dimensions and our general formulas reduce to them when $d=1$ and $d=3$ respectively. Our two-dimensional results are novel and we provide an example that illustrates why two dimensions is of physical interest. We also obtain results for the two-dimensional case using a completely independent method that employs ladder operators in polar coordinates. In total, three methods are used in this work and the results all agree.

quant-ph↗

arXiv2015

Formation of a condensate during charged collapse

We observe a condensate forming in the interior of a black hole (BH) during numerical simulations of gravitational collapse of a massless charged (complex) scalar field. The magnitude of the scalar field in the interior tends to a non-zero constant; spontaneous breaking of gauge symmetry occurs and a condensate forms. This phenomena occurs in the presence of a BH without the standard symmetry breaking quartic potential; the breaking occurs via the dynamics of the system itself. We also observe that the scalar field in the interior rotates in the complex plane and show that it matches numerically the electric potential to within $1\%$. That a charged scalar condensate can form near the horizon of a black hole in the Abelian Higgs model without the standard symmetry breaking potential had previously been shown analytically in an explicit model involving a massive scalar field in an $AdS_4$ background. Our numerical simulation lends strong support to this finding, although in our case the scalar field is massless and the spacetime is asymptotically flat.

gr-qc↗

arXiv2015

Generating Einstein gravity, cosmological constant and Higgs mass from restricted Weyl invariance

Recently, it has been pointed out that dimensionless actions in four dimensional curved spacetime possess a symmetry which goes beyond scale invariance but is smaller than full Weyl invariance. This symmetry was dubbed {\it restricted Weyl invariance}. We show that starting with a restricted Weyl invariant action that includes a Higgs sector with no explicit mass, one can generate the Einstein-Hilbert action with cosmological constant and a Higgs mass. The model also contains an extra massless scalar field which couples to the Higgs field (and gravity). If the coupling of this extra scalar field to the Higgs field is negligibly small, this fixes the coefficient of the nonminimal coupling $R Φ^2$ between the Higgs field and gravity. Besides the Higgs sector, all the other fields of the standard model can be incorporated into the original restricted Weyl invariant action.

hep-th↗

arXiv2014

Restricted Weyl invariance in four-dimensional curved spacetime

We discuss the physics of {\it restricted Weyl invariance}, a symmetry of dimensionless actions in four dimensional curved space time. When we study a scalar field nonminimally coupled to gravity with Weyl(conformal) weight of $-1$ (i.e. scalar field with the usual two-derivative kinetic term), we find that dimensionless terms are either fully Weyl invariant or are Weyl invariant if the conformal factor $Ω(x)$ obeys the condition $g^{μν}\nabla_μ\nabla_νΩ=0$. We refer to the latter as {\it restricted Weyl invariance}. We show that all the dimensionless geometric terms such as $R^2$, $R_{μν}R^{μν}$ and $R_{μνστ}R^{μνστ}$ are restricted Weyl invariant. Restricted Weyl transformations possesses nice mathematical properties such as the existence of a composition and an inverse in four dimensional space-time. We exemplify the distinction among rigid Weyl invariance, restricted Weyl invariance and the full Weyl invariance in dimensionless actions constructed out of scalar fields and vector fields with Weyl weight zero.

hep-th↗