Scale Invariance Symmetry Research Articles

Symmetry, Music and Water

There is a deep link existing between symmetry and nature that may be best apprehended by using group theory. In physics, giving a symmetry group is entirely equivalent to fix natural laws (Noether’s theorem). Within such a framework, it follows that all fundamental mechanical laws may be given by defining a group of symmetry operations belonging to the Gal (3,1) symmetry group having ten generators (3 + 3 + 3 + 1 = 10). The move to special relativity amounts simply to substitute another group ISO (3,1) to Gal (3,1) and having the same number of generators. In both cases, one immediately perceives that we live in universe with 3 dimensions of space and 1 dimension of time. A further progress could be made by looking for the symmetry group that leaves Maxwell’s equations invariant. The answer to this fundamental question is the group ISO (4,2) ⊗U (2) ⊗U (2) characterized by 23 generators: 15 = 10 + 5 for the conformal group ISO(4,2), 4 for U(2) and hence 8 = 4 + 4 for the compact part U(2) ⊗U(2) that could be related to the existence of quantum physics besides relativity. As ISO (3,1) is a sub-group of ISO (4,2), the two numbers 4 and 2 suggests that, taking into account the existence of light besides matter, we live in an universe having 4 spacelike dimensions and 2 time-like dimensions. Here, it is proposed that, as music is characterized by a scale-invariant symmetry operation (transposition by octaves) and that as time may be perceived differently in shamanic trance for instance, the two “new” dimensions of our universe may be related to music and consciousness. Related to the fifth dimension of music, the notion of diapason linking physical frequencies to a musical note is revisited. It is thus proposed to adopt a new A4 = 429.62 Hz value based on universal physical constants of our universe and on the mass of the water molecule H₂O, "The most abundant heteronuclear molecule in the entire universe". Demonstration is done that this new diapason is superior to other diapasons based on frequencies of 440 Hz or even 432 Hz.

Electroweak symmetry breaking by gravity

We consider a simple scale-invariant action coupling the Higgs field to the metric scalar curvature R and containing an R2 term that exhibits spontaneous breaking of scale invariance and electroweak symmetry. The coefficient of the R2 term in this case determines the self-coupling of the Higgs boson in the Einstein frame, and the scalaron becomes a dilaton weakly coupled to the Higgs boson. Majorana mass terms for right-handed neutrinos can be generated in a scale-invariant manner by using the Higgs-field invariant; in this case, the existing experimental limits on the Higgs-boson total width rule out Majorana mass values in a certain range. The model inherits the naturalness issues of general relativity connected with the smallness of the gravitational and cosmological constants.

Soft pion scattering in infrared-conformal gauge-fermion theories

We consider the problem of soft scattering for the analogue of pion states in gauge-fermion theories which approach a conformal fixed point in the infrared limit. Introducing a fermion mass into such a theory will explicitly break both scale invariance and chiral symmetry, leading to confinement and a spectrum of bound states. We argue that in such a theory, the pion scattering length diverges in the limit of zero fermion mass, in sharp contrast to QCD-like theories where the chiral Lagrangian predicts a vanishing scattering length. We demonstrate this effect both with a simple dimensional argument, and in a generalized linear sigma model which we argue can be used to describe the interactions of light scalar and pseudoscalar bound states in the soft limit of a mass-deformed infrared-conformal theory. As a result, lattice calculations of pion scattering lengths could be a sensitive probe for infrared scale invariance in gauge-fermion theories.

Hidden scale invariance in the Gay-Berne model.

This paper presents a numerical study of the Gay-Berne liquid crystal model with parameters corresponding to calamitic (rod-shaped) molecules. The focus is on the isotropic and nematic phases at temperatures above unity, where we find strong correlations between the virial and potential-energy thermal fluctuations, reflecting the hidden scale invariance symmetry. This implies the existence of isomorphs, which are curves in the thermodynamic phase diagram of approximately invariant physics. We study numerically one isomorph in the isotropic phase and one in the nematic phase. In both cases, good invariance of the dynamics is demonstrated via data for the mean-square displacement and the reduced-unit time-autocorrelation functions of the velocity, angular velocity, force, torque, and first- and second-order Legendre polynomial orientational order parameters. Deviations from isomorph invariance are observed at short times for the orientational time-autocorrelation functions, which reflects the fact that the moment of inertia is assumed to be constant and thus not isomorph-invariant in reduced units. Structural isomorph invariance is demonstrated from data for the radial distribution functions of the molecules and their orientations. For comparison, all quantities were also simulated along an isochore of similar temperature variation, in which case invariance is not observed. We conclude that the thermodynamic phase diagram of the calamitic Gay-Berne model is essentially one-dimensional in the studied regions as predicted by isomorph theory, a fact that potentially allows for simplifications of future theories and numerical studies.

In search of large signals at the cosmological collider

We look for oscillating signals in the primordial bispectrum from new physics heavy particles which are visibly large for next generation large scale structures (LSS) survey. We show that in ordinary inflation scenarios where a slow-rolling inflaton generates density fluctuations and with no breaking of scale invariance or spacetime symmetry, there exist no naturally large signals unless the rolling inflaton generates a parity-odd chemical potential for the heavy particles. We estimate the accessibility of this signal through observations. While current CMB data are already sensitive in the most optimistic scenario, future probes, including LSS survey and 21 cm observation, can cover interesting regions of the model space.

Scale-invariant unconstrained online learning

We consider an online supervised learning problem, in which both the instances (input vectors) and the comparator (weight vector) are unconstrained. We exploit a natural scale invariance symmetry in our unconstrained setting: the predictions of the optimal comparator are invariant under any linear transformation of the instances. Our goal is to design online algorithms which also enjoy this property, i.e. are scale-invariant. We start with the case of coordinate-wise invariance, in which the individual coordinates (features) can be arbitrarily rescaled. We give an algorithm, which achieves essentially optimal regret bound in this setup, expressed by means of a coordinate-wise scale-invariant norm of the comparator. We then study general invariance with respect to arbitrary linear transformations. We first give a negative result, showing that no algorithm can achieve a meaningful bound in terms of scale-invariant norm of the comparator in the worst case. Next, we compliment this result with a positive one, providing an algorithm which “almost” achieves the desired bound, incurring only a logarithmic overhead in terms of the relative size of the instances.

Entropic Dynamics of Stocks and European Options.

We develop an entropic framework to model the dynamics of stocks and European Options. Entropic inference is an inductive inference framework equipped with proper tools to handle situations where incomplete information is available. The objective of the paper is to lay down an alternative framework for modeling dynamics. An important information about the dynamics of a stock’s price is scale invariance. By imposing the scale invariant symmetry, we arrive at choosing the logarithm of the stock’s price as the proper variable to model. The dynamics of stock log price is derived using two pieces of information, the continuity of motion and the directionality constraint. The resulting model is the same as the Geometric Brownian Motion, GBM, of the stock price which is manifestly scale invariant. Furthermore, we come up with the dynamics of probability density function, which is a Fokker–Planck equation. Next, we extend the model to value the European Options on a stock. Derivative securities ought to be prices such that there is no arbitrage. To ensure the no-arbitrage pricing, we derive the risk-neutral measure by incorporating the risk-neutral information. Consequently, the Black–Scholes model and the Black–Scholes-Merton differential equation are derived.

Two scalar fields inflation from scale-invariant gravity with modified measure

A new class of gravity-matter model defined with an independent non-Riemannian volume form is studied, in the second order formalism. The action has a global scale invariance symmetry, which can be broken by the equation of motion of the measure fields. By a conformal transformation the theory transforms into a theory which governed by two independent scalar fields and a specific potential. When the scale invariance symmetry is not broken also in the equations of motion, only one field appear in the effective potential. This effective potential has a flat region which is responsible for the inflation, and it ends with a minimum, which is responsible for the present vacuum energy. In the general case, with a spontaneous symmetry breaking of the scale symmetry, one scalar field is responsible for the inflation and the other could be responsible for the particle creation. The first field (inflaton) can slowly roll, while the second field (curvaton) is more strongly coupled to the potential. The number of e-folds for both models is also dissuaded and could be constraint in the future.

CMF-iteMS: An automatic threshold selection for detection of copy-move forgery

Taking into consideration that the prior CMF detection methods rely on several fixed threshold values in the filtering process, we propose an efficient CMF detection method with an automatic threshold selection, named as CMF-iteMS. The CMF-iteMS recommends a PatchMatch-based CMF detection method that adapts Fourier-Mellin Transform (FMT) as the feature extraction technique while a new automatic threshold selection based on iterative means of regions size (iteMS) procedure is introduced to have flexibility in changing the threshold value for various characteristics (quality, sizes, and attacks) in each input image. To ensure the reliability of the proposed CMF-iteMS, the method is compared with four state-of-the-art CMF detection methods based on Scale Invariant Feature Transform (SIFT), patch matching, multi-scale analysis and symmetry techniques using three available datasets that cover the variety of characteristics in CMF images. The results show that the F-score of the CMF-iteMS outperformed existing CMF detection methods by exceeding an average of 90% F-score values for image-level evaluation and 82% of F-score value for pixel-level evaluation for all datasets in original size. As special attention is given to the image resizing attack, the method is able to maintain the highest performance even if the images in the datasets are resized to 0.25 parameter.

Polarity formation in crystals with long range molecular interactions: A Monte Carlo study

Stochastic formation of a bi-polar state in three dimensional arrays of polar molecules with full Coulomb interactions is reproduced by Monte Carlo simulation. The size of the system is comparable to that of a real crystal seed. The spatial decay of the average order parameter is significantly slowed down by the long range interactions and the exact representation of correlation effects in terms of a single characteristic length becomes impossible. Finite size effects and possible scale invariance symmetry of the order parameter are extensively discussed.

Inflation and reheating in theories with spontaneous scale invariance symmetry breaking

We study a scale-invariant model of quadratic gravity with a non-minimally coupled scalar field. We focus on cosmological solutions and find that scale invariance is spontaneously broken and a mass scale naturally emerges. Before the symmetry breaking, the Universe undergoes an inflationary expansion with nearly the same observational predictions of Starobinsky's model. At the end of inflation, the Hubble parameter and the scalar field converge to a stable fixed point through damped oscillations and the usual Einstein-Hilbert action is recovered. The oscillations around the fixed point can reheat the Universe in various ways and we study in detail some of these possibilities.

Scalar dark matter in scale invariant standard model

We investigate single and two-component scalar dark matter scenarios in classically scale invariant standard model which is free of the hierarchy problem in the Higgs sector. We show that despite the very restricted space of parameters imposed by the scale invariance symmetry, both single and two-component scalar dark matter models overcome the direct and indirect constraints provided by the Planck/WMAP observational data and the LUX/Xenon100 experiment. We comment also on the radiative mass corrections of the classically massless scalon that plays a crucial role in our study.

Self-acceleration with quasidilaton

Quasidilaton massive gravity is an extension of massive General Relativity to a theory with additional scale invariance and approximate internal Galilean symmetry. The theory has a novel self-accelerated solution with the metric indistinguishable (in the decoupling limit) from the de Sitter space, its curvature set by the graviton mass. The spectra of tensor, vector, and scalar perturbations on this solution contain neither ghosts, nor gradient instabilities or superluminal modes, for a range of the parameter space. This represents an example of a self-accelerated solution with viable perturbations, attainable within a low energy effective field theory.

Universality and scaling in theN-body sector of Efimov physics

Universal behaviour has been found inside the window of Efimov physics for systems with $N=4,5,6$ particles. Efimov physics refers to the emergence of a number of three-body states in systems of identical bosons interacting {\it via} a short-range interaction becoming infinite at the verge of binding two particles. These Efimov states display a discrete scale invariance symmetry, with the scaling factor independent of the microscopic interaction. Their energies in the limit of zero-range interaction can be parametrized, as a function of the scattering length, by a universal function. We have found, using a particular form of finite-range scaling, that the same universal function can be used to parametrize the energies of $N\le6$ systems inside the Efimov-physics window. Moreover, we show that the same finite-scale analysis reconciles experimental measurements of three-body binding energies with the universal theory.

DIMENSIONAL CROSSOVER AND DIMENSIONAL EFFECTS IN QUASI-TWO-DIMENSIONAL BOSE GASES

This paper gives a systematic review on studies of dimensional effects in pure- and quasi-two-dimensional (2D) Bose gases, focusing on the role of dimensionality in the fundamental relation among the universal behavior of breathing mode, scale invariance and dynamic symmetry. First, we illustrate the emergence of universal breathing mode in the case of pure 2D Bose gases, and elaborate on its connection with the scale invariance of the Hamiltonian and the hidden SO(2, 1) symmetry. Next, we proceed to quasi-2D Bose gases, where excitations are frozen in one direction and the scattering behavior exhibits a 3D to 2D crossover. We show that the original SO(2, 1) symmetry is broken by arbitrarily small 2D effects in scattering, which consequently shifts the breathing mode from the universal frequency. The predicted shift rises significantly from the order of 0.5% to more than 5% in transiting from the 3D-scattering to the 2D-scattering regime. Observing this dimensional effect directly would present an important step in revealing the interplay between dimensionality and quantum fluctuations in quasi-2D.

Scaling symmetry and conserved charge for shape-invariant optical fields

In this work we present an extensive study of the scaling symmetry typical of a paraxial wave theory. In particular, by means of a Lagrangian approach we derive the conservation law and the corresponding generalized charge associated with the scale invariance symmetry. In general, such a conserved charge, qs say, can take any value that remains constant during propagation. However, it is explicitly proven that for the whole class of physically realizable shape-invariant fields, that is, fields whose intensity distribution maintains its shape on propagation, qs must necessarily vanish. Finally, an interesting relation between such charge qs and the effective radius of a beam, as introduced by Siegman some years ago, is derived.

Quasicrystals and the Quest for Next Generation Thermoelectric Materials

Since the lattice thermal conductivity of most thermoelectric materials of interest has already been greatly reduced, next generation bulk thermoelectric materials will rely on the ability to properly tailor the electronic band structure in the vicinity of the Fermi level in order to optimize their thermoelectric performance. To this end, a main theoretical goal focuses on obtaining a fundamental guiding principle, able to properly replace previous methods in terms of a band structure engineering approach. Following this approach in this review we will consider the case of quasicrystals: a class of materials based on new lattice geometries, characterized by scale-invariance symmetry and long-range aperiodic order. As a consequence, quasicrystalline alloys exhibit semiconductor-like, rather than metallic electronic transport properties, along with extremely low thermal conductivity values. Thus, quasicrystals occupy a very promising position in the quest for novel thermoelectric materials, naturally bridging the gap between semiconducting materials and metallic ones. Accordingly, by keeping into account both experimental data and solid state physics theoretical aspects, we will provide a prospective study on the potential use of quasicrystals and their related phases in actual thermoelectric devices in the years to come.

N = 4 SYM low-energy effective action in N = 4 harmonic superspace

We apply the N = 4 harmonic superspace with USp(4) harmonic variables for describing the N = 4 SYM low-energy effective action. Scale invariance and gauge symmetry fix the leading term in the low-energy effective action uniquely, up to a constant. The value of the remaining constant can be fixed by the topological quantization condition for the Wess-Zumino term which is present in the component structure of this action.

Radial Symmetry Detection and Shape Characterization with the Multiscale Area Projection Transform

AbstractWe present a novel method to characterize 3D surfaces through the computation of a function called (multiscale) area projection transform, measuring the likelihood of points in the 3D space to be center of radial symmetry at selected scales (radii). The function is derived through a simple geometric framework based on parallel surfaces and can be easily computed on triangulated meshes. It measures locally the area of the surface well approximated by a sphere of radius R centered in the point and can be normalized in order to obtain a scale invariant radial symmetry enhancement transform. This transform can therefore be used to detect and characterize salient regions like approximately spherical and approximately cylindrical surface parts and, being robust against holes and missing parts, it is suitable for real world applications e.g. anatomical features detection. Furthermore, its histograms can be effectively used to build a global shape descriptor that provides very good results in shape retrieval experiments.

Model building in AdS/CMT: DC conductivity and Hall angle

Using the bottom-up approach in a holographic setting, we attempt to study both the transport and thermodynamic properties of a generic system in $3+1$-dimensional bulk spacetime. We show the exact $1/T$ and ${T}^{2}$ dependence of the longitudinal conductivity and Hall angle, as seen experimentally in most copper-oxide systems, which are believed to be close to quantum critical point. This particular temperature dependence of the conductivities are possible in two different cases: (1) background solutions with scale invariant and broken rotational symmetry and (2) solutions with pseudoscaling and unbroken rotational symmetry, but only at low density limit. Generically, the study of the transport properties in a scale-invariant background solution, using the probe brane approach, at high density and at low temperature limit suggests we consider only metrics with two exponents. More precisely, the spatial part of the metric components should not be same, i.e. ${g}_{xx}\ensuremath{\ne}{g}_{yy}$. In doing so, we have generalized the above-mentioned behavior of conductivity with a very special behavior of specific heat which at low temperature goes as: ${C}_{V}\ensuremath{\sim}{T}^{3}$. However, if we break the scaling symmetry of the background solution by including a nontrivial dilaton, axion, or both and keep the rotational symmetry, then also we can generate such a behavior of conductivity, but only in the low density regime. As far as we are aware, this particular temperature dependence of both the conductivity and Hall angle is being shown for the first time using holography.