arXiv subjects

André LeClair

Publications and source records attributed to André LeClair.

At least 19 recordsLinked to original sources

arXiv2026

Quantum Vacuum energy as the origin of Gravity

We explore the idea that quantum vacuum energy $ρ_{\rm vac} $ is at the origin of Gravity. We formulate a gravitational version of the electromagnetic Casimir effect, and provide an argument for how gravity can arise from $ρ_{\rm vac} $ by showing how Einstein's field equations emerge in the form of Friedmann's equations. This leads to the idea that Newton's constant $G_N$ is environmental, namely it depends on the total mass-energy of the Universe $M_\infty $ and its size $R_\infty $, with $G_N = c^2 R_\infty /2 M_\infty$. This leads to a new interpretation of the Gibbons-Hawking entropy of de Sitter space, and also the Bekenstein-Hawking entropy for black holes, wherein the quantum information bits are quantized massless particles at the horizon with wavelength $λ= 2 πR_\infty$. We assume a recently proposed formula for $ρ_{\rm vac} \sim m_z^4/\mathfrak{g}$, where $m_z$ is the mass of the lightest particle, and $\mathfrak{g}$ is a marginally irrelevant coupling. This leads to an effective, induced RG flow for Newton's constant $G_N$ as a function of an energy scale, which indicates that $G_N$ decreases at higher energies until it reaches a Landau pole at a minimal value of the cosmological scale factor $a(t) > a_{\rm min}$, thus avoiding the usual geometric singularity at $a=0$. The solution to the scale factor satisfies an interesting symmetry between the far past and far future due to $a(t) = a(-t + 2 t_{\rm min})$, where $a(t_{\rm min}) = a_{\rm min}$. We propose that this energy scale dependent $G_N$ can explain the Hubble tension and we thereby constrain the coupling constant $\mathfrak{g}$ and its renormalization group parameters. For the $Λ{\rm CDM}$ model we estimate $a_{\rm min} \approx e^{-1/\hat{b} }$ where $\hat{b} \approx 0.02$ based on the Hubble tension data.

hep-th↗

arXiv2026

A rich structure of renormalization group flows for Higgs-like models in 4 dimensions

We consider $2$ coupled Higgs doublets which transform in the usual way under SU(2). By constructing marginal operators which satisfy an operator product expansion based on the SU(2) Lie algebra, we can obtain a rich pattern of renormalization group (RG) flows which includes lines of fixed points and more interestingly, cyclic RG flows which are unavoidable in this model. The hamiltonian is pseudo-hermitian, $H^\dagger = {\cal K} H {\cal K}^\dagger $ with ${\cal K}$ unitary satisfying ${\cal K}^2 =1$, thus the model is non-unitary. The hamiltonian still has real eigenvalues, but the non-unitarity is manifested in negative norm states. Based on a generalized optical theorem for pseudo-hermitian hamiltonians, we show that our model is in fact unitary below the threshold for particle/anti-particle pair production. It is thus unitary in the non-relativistic limit, which opens up some potential applications to condensed matter physics. We argue that our model breaks ${\cal C}{\cal P}$ symmetry. Upon spontaneous symmetry breaking, the Higgs-like fields have an infinite number of vacuum expectation values $v_n$ which satisfy ``Russian Doll" scaling $v_n \sim e^{2 n λ}$ where $n=1,2,3,\ldots$ and $λ$ is the period of one RG cycle which is an RG invariant. We speculate that this Russian Doll RG flow can perhaps resolve the so-called hierarchy problem and may shed light on the origin of ``families" in the Standard Model of particle physics. If after spontaneous symmetry breaking of the SU(2) to U(1) a cyclic RG with period $λ$ is operative up to the electro-weak scale, then this admits 3 RG cycles, i.e. 3 families of quarks and leptons. The strongest constraints on the RG period $λ$ comes from the phenomenological Koide formula, wherein $λ\approx π/2$.

hep-th↗

arXiv2026

Non-perturbative renormalization group for pseudo-hermitian scalar fields in 4D

We define a model of 2 coupled SU(2) doublets of scalar fields in $4$ spacetime dimensions which have a rich structure of renormalization group (RG) flows to 1-loop when the SU(2) is broken to U(1). The model is pseudo-hermitian, $H^\dagger = {\cal K} H {\cal K}^\dagger$ with ${\cal K} ^\dagger {\cal K} = {\cal K}^2 =1$, which makes it non-unitary, however in a very specific manner with some desirable properties. We compute the beta functions to 3 loops from the operator product expansion and show that the 1-loop structure of flows persists to higher orders. For $SU(2)$ broken to $U(1)$, we conjecture a beta function to all orders. The flows can be extended to large coupling using a strong-weak coupling symmetry $g \to 1/g$ of the beta functions. One finds a line of fixed points which are non-unitary conformal field theories in 4 spacetime dimensions that were previously unknown. We also find massless flows between 2 non-trivial fixed points, and a regime with a cyclic RG flow, which is allowed since the model is non-unitary. For the flows between fixed points on the critical line, we compute the anomalous dimensions of the perturbations in the UV and IR, and identify some special points where anomalous dimensions are rational numbers.

hep-th↗

arXiv2025

Comment on the cosmological constant for $λϕ^4$ theory in $d$ spacetime dimensions

In a recent article we showed that the analog of the cosmological constant in two spacetime dimensions for a wide variety of integrable quantum field theories has the form $ρ_{\rm vac} = - m^2 /2 \mathfrak{g} $ where $m$ is a physical mass and $\mathfrak{g} $ is a generalized coupling, where in the free field limit $\mathfrak{g} \to 0$, $ρ_{\rm vac}$ diverges. We speculated that in four spacetime dimensions $ρ_{\rm vac} $ takes a similar form $ρ_{\rm vac} = - m^4/2 \mathfrak{g}$, but did not support this idea in any specific model. In this article we study this problem for $λϕ^4$ theory in $d$ spacetime dimensions. We show how to obtain the exact $ρ_{\rm vac}$ for the sinh-Gordon theory in the weak coupling limit by using a saddle point approximation. This calculation indicates that the cosmological constant can be well-defined, positive or negative, without spontaneous symmetry breaking. We also show that $ρ_{\rm vac}$ satisfies a Callan-Symanzik type of renormalization group equation. For the most interesting case physically, $ρ_{\rm vac}$ is positive and can arise from a marginally relevant negative coupling $\mathfrak{g}$ and the cosmological constant flows to zero at low energies.

hep-th↗

arXiv2025

Spectral Flow for the Riemann zeros

Recently, with Mussardo we defined a quantum mechanical problem of a single particle scattering with impurities wherein the quantized energy levels $E_n (σ)$ are exactly equal to the zeros of the Riemann $ζ(s)$ where $σ= \Re (s)$ in the limit $σ\to 1/2$. The S-matrix is based on the Euler product and is unitary by construction, thus the underlying hamiltonian is hermitian and all eigenvalues must be real. Motivated by the Hilbert-Pólya idea we study the spectral flows for $\{ E_n (σ) \}$. This leads to a simple criterion for the validity of the Riemann Hypothesis. The spectral flow arguments are simple enough that we present analogous results for the Generalized and Grand Riemann Hypotheses. We also illustrate our results for a counter example where the Riemann Hypothesis is violated since there is no underlying unitary S-matrix due to the lack of an Euler product.

math-ph↗

arXiv2024

A theory for the zeros of Riemann $ζ$ and other $L$-functions (updated)

In these lectures we first review the important properties of the Riemann $ζ$-function that are necessary to understand the nature and importance of the Riemann hypothesis (RH). In particular this first part describes the analytic continuation, the functional equation, trivial zeros, the Euler product formula, Riemann's main result relating the zeros on the critical strip to the distribution of primes, the exact counting formula for the number of zeros on the strip $N(T)$, and the GUE statistics of the zeros on the critical line. We then turn to presenting some new results obtained in the past year and describe several strategies towards proving the RH. First we describe an electrostatic analogy and argue that if the electric potential along the line $\Re (z) =1$ is a regular alternating function, the RH would follow. The main new result is that the zeros on the critical line are in one-to-one correspondence with the zeros of the cosine function, and this leads to a transcendental equation for the $n$-th zero on the critical line that depends only on $n$. If there is a unique solution to this equation for every $n$, denoting $N_0 (T)$ the number of zeros on the critical line, then $N_0(T) = N(T)$, i.e. all zeros are on the critical line. These results are generalized to two infinite classes of functions, Dirichlet $L$-functions and $L$-functions based on modular forms. We present extensive numerical analysis of the solutions of these equations. We apply these methods to the Davenport-Heilbronn $L$-function, which is known to have zeros off of the line, and explain why the RH fails in this case. We also present a new approximation to the $ζ$-function that is analogous to the Stirling approximation to the $Γ$-function. In this updated version we added references to subsequent publications that made progress on open questions posed in the previous version from 10 years ago.

math.NT↗

arXiv2024

Vacuum energy density from the form factor bootstrap

The form-factor bootstrap is incomplete until one normalizes the zero-particle form factor. For the stress energy tensor we describe how to obtain the vacuum energy density $ρ_{\rm vac}$, defined as $\langle 0| T_{μν} | 0 \rangle = ρ_{\rm vac} \, g_{μν}$, from the form-factor bootstrap. Even for integrable QFT's in D=2 spacetime dimensions, this prescription is new, although it reproduces previously known results obtained in a different and more difficult thermodynamic Bethe ansatz computation. We propose a version of this prescription in D=4 dimensions. For these even dimensions, the vacuum energy density has the universal form $ρ_{\rm vac} \propto m^D/\mathfrak{g}$ where $\mathfrak{g}$ is a dimensionless interaction coupling constant which can be determined from the high energy behavior of the S-matrix. In the limit $\mathfrak{g} \to 0$, $ρ_{\rm vac} $ diverges due to well understood UV divergences in free quantum field theories. If we assume the the observed Cosmological Constant originates from the vacuum energy density $ρ_{\rm vac}$ computed as proposed here, then this suggests there must exist a particle which does not obtain its mass from spontaneous symmetry breaking in the electro-weak sector, which we designate as the "zeron". A strong candidate for the zeron is a massive Majorana neutrino.

hep-th↗

arXiv2024

Thermodynamic formulation of vacuum energy density in flat spacetime and potential implications for the cosmological constant

We propose a thermodynamical definition of the vacuum energy density $ρ_{\rm vac}$, defined as $\langle 0| T_{μν} |0\rangle = - ρ_{\rm vac} \, g_{μν}$, in quantum field theory in flat Minkowski space in $D$ spacetime dimensions, which can be computed in the limit of high temperature, namely in the limit $β= 1/T \to 0$. It takes the form $ρ_{\rm vac} = {\rm const} \cdot m^D$ where $m$ is a fundamental mass scale and ${\rm "const"}$ is a computable constant which can be positive or negative. Due to modular invariance $ρ_{\rm vac}$ can also be computed in a different non-thermodynamic channel where one spatial dimension is compactifed on a circle of circumference $β$ and we confirm this modularity for free massive theories for both bosons and fermions for $D=2,3,4$. We list various properties of $ρ_{\rm vac}$ that are generally required, for instance $ρ_{\rm vac}=0$ for conformal field theories, and others, such as the constraint that $ρ_{\rm vac}$ has opposite signs for free bosons verses fermions of the same mass, which is related to constraints from supersymmetry. Using the Thermodynamic Bethe Ansatz we compute $ρ_{\rm vac}$ exactly for 2 classes of integrable QFT's in $2D$ and interpreting some previously known results. We apply our definition of $ρ_{\rm vac}$ to Lattice QCD data with two light quarks (up and down) and one additional massive flavor (the strange quark), and find it is negative, $ρ_{\rm vac} \approx - ( 200 \, {\rm MeV} )^4$. Finally we make some remarks on the Cosmological Constant Problem since $ρ_{\rm vac}$ is central to any discussion of it.

hep-th↗

arXiv2023

Mingling of the infrared and ultraviolet and the "cosmological constant" for interacting QFT in 2d

We propose a proper definition of the vacuum expectation value of the stress energy tensor $\langle 0 | T_{μν} |0 \rangle$ for integrable quantum field theories in two spacetime dimensions, which is the analog of the cosmological constant in 4d. For a wide variety of models, massive or massless, we show $ρ_{\rm vac} = - m^2/2 \mathfrak{g} $ exactly, where $\mathfrak{g} $ is a generalized coupling which we compute and $m$ is a basic mass scale. The kinds of models we consider are the massive sinh-Gordon and sine-Gordon theories and perturbations of the Yang-Lee and 3-state Potts models, pure $T\bar{T}$ perturbations of infra-red QFT's, and UV completions of the latter which are massless flows between UV and IR fixed points. In the massive case $m$ is the mass of the lightest particle and $\mathfrak{g} $ is related to parameters in the 2-body S-matrix. In some examples $ρ_{\rm vac} =0$ due to an unbroken fractional supersymmetry. For massless cases, $m$ can be a scale of spontaneous symmetry breaking. The "cosmological constant problem" generically arises in the free field limit $\mathfrak{g} \to 0$, thus interactions can potentially resolve the problem at least for most cases considered in this paper. We speculate on extensions of these results to 4 spacetime dimensions and propose $ρ_{\rm vac} =- m^4/2 \mathfrak{g}$,however without integrability we cannot yet propose a precise manner in which to calculate $\mathfrak{g}$. Nevertheless, based on cosmological data on $ρ_{\rm vac} $, if $\mathfrak{g} \approx 1$ then the lightest mass particle is on the order of experimental values of proposed neutrino masses.

hep-th↗

arXiv2022

$T \bar{T}$ deformation of the Ising model and its ultraviolet completion

Pure $T\bar{T}$ deformations of conformal field theories are generally asymptotically incomplete in the ultra-violet (UV) due to square-root singularities in the ground state energy on a cylinder of circumference $R$, such that the theory is ill-defined for distances shorter than some critical $R_*$. In this article we show how a theory can be completed if one includes an infinite number of additional irrelevant perturbations. This is fully demonstrated in the case of the Ising model at $c_{IR}= 1/2$ in the infra-red (IR), where we find two completions with central charges $c_{UV} = 3/2$ and $c_{UV} = 7/10$, the latter being the tri-critical Ising model. Both of these UV completions have ${\cal N}=1$ supersymmetry which is broken in the renormalization group flow to low energies. We also consider multiple $T\bar{T}$ deformations of a free massless boson, where we cannot find a UV completion that is consistent with the c-theorem. For negative coupling $g$, which violates the c-theorem, in both cases we find $c_{UV} = -c_{IR}$ as $g \to -\infty$. Finally we also study pure $T\bar{T}$ deformations of the off-critical Ising model.

hep-th↗

arXiv2022

Phenomenological formula for Quantum Hall resistivity based on the Riemann zeta function

We propose a formula constructed out of elementary functions that captures many of the detailed features of the transverse resistivity $ρ_{xy}$ for the integer quantum Hall effect. It is merely a phenomenological formula in the sense that it is not based on any transport calculation for a specific class of physical models involving electrons in a disordered landscape, thus, whether a physical model exists which realizes this resistivity remains an open question. Nevertheless, since the formula involves the Riemann zeta function and its non-trivial zeros play a central role, it is amusing to consider the implications of the Riemann Hypothesis in light of it.

cond-mat.mes-hall↗

arXiv2022

On the classification of UV completions of integrable $T \bar{T}$ deformations of CFT

It is well understood that 2d conformal field theory (CFT) deformed by an irrelevant $T\bar{T}$ perturbation of dimension $4$ has universal properties. In particular, for the most interesting cases, the theory develops a singularity in the ultra-violet (UV), signifying a shortest possible distance, with a Hagedorn transition in applications to string theory. We show that by adding an infinite number of higher $[T\bar{T}]_{s>1}$ irrelevant operators with tuned couplings, this singularity can be resolved and the theory becomes UV complete with UV central charge $c_{\rm UV}$. We propose an approach to classifying the possible UV completions that are integrable. For the Ising model with $c_{\rm IR} = \tfrac{1}{2}$ we find 3 UV completions based on a single massless Majorana fermion description with $c_{\rm UV} = \tfrac{7}{10}$ and $ \tfrac{3}{2}$, which both have ${\cal N}=1$ supersymmetry and were previously known, and we argue that these are the only solutions to our classification problem based on this spectrum of particles. We find 3 additional ones with a spectrum of 8 massless particles related to the Lie group $E_8$ appropriate to a magnetic perturbation with $c_{\rm UV} = \tfrac{21}{22} , \tfrac{15}{2}$ and $\tfrac{31}{2}$. We argue that it is likely there are more cases for this $E_8$ spectrum. We also study simpler cases based on ${\rm su(3)}$ and ${\rm su(4)}$ where we can propose complete classifications. For ${\rm su(3)}$ the infra-red (IR) theory is the 3-state Potts model with $c_{\rm IR} = \tfrac{4}{5}$ and we find 3 completions with $\tfrac{4}{5} < c_{\rm UV} \leq \tfrac{16}{5}$. For the ${\rm su(4)}$ case, which has $3$ particles and $c_{\rm IR} =1$, and we find $11$ UV completions with $1 < c_{\rm UV} \leq 5$, most of which were previously unknown.

hep-th↗

arXiv2022

The sinh-Gordon model beyond the self dual point and the freezing transition in disordered systems

The S-matrix of the well-studied sinh-Gordon model possesses a remarkable strong/weak coupling duality $b \to 1/b$. Since there is no understanding nor evidence for such a duality based on the quantum action of the model, it should be questioned whether the properties of the model for $b>1$ are simply obtained by analytic continuation of the weak coupling regime $0 1$. Namely, we propose that in this region one needs to introduce a background charge $Q_\infty = b + 1/b -2$ which differs from the Liouville background charge by the shift of $-2$. We propose that in this regime the model has non-trivial massless renormalization group flows between two different conformal field theories. This is in contrast to the weak coupling regime which is a theory of a single massive particle. Evidence for our proposal comes from higher order beta functions. We show how our proposal correctly reproduces the freezing transitions in the multi-fractal exponents of a Dirac fermion in $2+1$ dimensions in a random magnetic field, which provides a strong check since such transitions have several detailed features. We also point out a connection between a semi-classical version of this transition and the so-called Manning condensation phenomena in polyelectrolyte physics.

hep-th↗

arXiv2021

Thermodynamics of $T \bar{T}$ perturbations of some single particle field theories

We study the Thermodynamic Bethe Ansatz (TBA) equations for pure $T\bar{T}$ perturbations of some simple integrable quantum field theories with a single bosonic or fermionic particle, in particular the massive sinh-Gordon model and its ultraviolet (UV) limit which is a deformation of the conformally invariant free massless boson. Whereas the TBA equations for $T\bar{T}$ deformations of massive theories are in principle known, the TBA equations we propose for the deformations of conformal field theories (CFT's) are relatively new and require a special factorization in rapidity variables of the CDD factor for the scattering of the massless particles. The latter TBA equations can be solved exactly and reproduce the known results for the ground state energy on a cylinder of circumference $R$ which were previously obtained using different methods based for instance on the Burgers differential equation. Special attention is paid to the c-theorem in this context which is discussed in some detail. For positive infra-red (IR) central charge $c_{IR}$, for flows consistent with the c-theorem the ground state energy develops a (previously known) square-root singularity towards the UV, which strongly suggests the theories are UV incomplete in these physically important cases. We suggest that the singularity indicates a tachyonic vacuum instability. Other cases with $c_{IR} < 0$ do not have this singularity and are interpreted as being UV complete with $c_{UV} = 0$. We extend our results to a continuously variable $c_{IR}$ by introducing a chemical potential and suggest this as a possible toy model for the $T\bar{T}$ perturbed Liouville theory.

hep-th↗

arXiv2021

Riemann Hypothesis and Random Walks: the Zeta case

In previous work it was shown that if certain series based on sums over primes of non-principal Dirichlet characters have a conjectured random walk behavior, then the Euler product formula for its $L$-function is valid to the right of the critical line $\Re (s) > \tfrac{1}{2}$, and the Riemann Hypothesis for this class of $L$-functions follows. Building on this work, here we propose how to extend this line of reasoning to the Riemann zeta function and other principal Dirichlet $L$-functions. We apply these results to the study of the argument of the zeta function. In another application, we define and study a 1-point correlation function of the Riemann zeros, which leads to the construction of a probabilistic model for them. Based on these results we describe a new algorithm for computing very high Riemann zeros, and we calculate the googol-th zero, namely $10^{100}$-th zero to over 100 digits, far beyond what is currently known.

math.NT↗

arXiv2019

Extensions of the Schwarzschild solution into regions of non-zero energy density and pressure

We present solutions of the Einstein equations that extend the static Schwarzschild solution in empty space into regions of non-zero energy density $ρ$ and radial pressure $p= w ρ$, where $w$ is a constant equation of state parameter. For simplicity we focus mainly on solutions with constant $ρ$. For $w=0$ we find solutions both with and without a singularity at the origin. Possible applications to galaxies are considered, where we find enhanced velocity rotation curves towards the edge of a galaxy. We propose that our explicit non-singular solution with $w=-1$ describes the interior of a black hole, which is a form of vacuum energy, and verify that its entropy is consistent with the Bekenstein-Hawking entropy. We propose that this idea can perhaps be applied to dark energy, if one views the latter as arising from black holes as pockets of vacuum energy. We estimate the density of the resulting dark energy to be $ρ_Λ\approx 10^{-30} g/{\rm cm}^3$, which is close to the measured value for the observable universe. We also present solutions with non-constant $ρ\propto 1/r^2$.

physics.gen-ph↗

arXiv2019

Early inflation based on an aboriginal black hole

It has been recently proposed that the interior of a black hole is vacuum energy, and this offers a possible explanation for the dark energy of the current observable universe. In this article we explore the idea that black holes are a source of vacuum energy in the early universe as well, and could possibly explain early inflation. Our scenario pre-supposes a single aboriginal black hole as the source of vacuum energy that drives inflation and eventually evaporates due to Hawking radiation. In such a model we can estimate $t_i $ and $t_f $ which are the times when inflation began and ended respectively. Our simplified model leads to $t_i \approx 10^{-45}$ sec, $t_f \approx 10^{-43} $ sec and the number of e-foldings during inflation to be about $70$.

gr-qc↗

arXiv2018

Generalized Riemann Hypothesis, Time Series and Normal Distributions

$L$ functions based on Dirichlet characters are natural generalizations of the Riemann $ζ(s)$ function: they both have series representations and satisfy an Euler product representation, i.e. an infinite product taken over prime numbers. In this paper we address the Generalized Riemann Hypothesis relative to the non-trivial complex zeros of the Dirichlet $L$ functions by studying the possibility to enlarge the original domain of convergence of their Euler product. The feasibility of this analytic continuation is ruled by the asymptotic behavior in $N$ of the series $B_N = \sum_{n=1}^N \cos \bigl( t \log p_n - \arg χ(p_n) \bigr)$ involving Dirichlet characters $χ$ modulo $q$ on primes $p_n$. Although deterministic, these series have pronounced stochastic features which make them analogous to random time series. We show that the $B_N$'s satisfy various normal law probability distributions. The study of their large asymptotic behavior poses an interesting problem of statistical physics equivalent to the Single Brownian Trajectory Problem, here addressed by defining an appropriate ensemble $\mathcal{E}$ involving intervals of primes. For non-principal characters, we show that the series $B_N$ present a universal diffusive random walk behavior $B_N = O(\sqrt{N})$ in view of the Dirichlet theorem on the equidistribution of reduced residue classes modulo $q$ and the Lemke Oliver-Soundararajan conjecture on the distribution of pairs of residues on consecutive primes. This purely diffusive behavior of $B_N$ implies that the domain of convergence of the infinite product representation of the Dirichlet $L$-functions for non-principal characters can be extended from $\Re(s) > 1$ down to $\Re (s) = \frac{1}{2}$, without encountering any zeros before reaching this critical line.

math.NT↗