Symmetric Theory Research Articles

Hamiltonian formulation of higher rank symmetric gauge theories

Recent discussions of fractons have evolved around higher rank symmetric gauge theories with emphasis on the role of Gauss constraints. This has prompted the present study where a detailed hamiltonian analysis of such theories is presented. Besides a general treatment, the traceless scalar charge theory is considered in details. A new form for the action is given which, in 2+1 dimensions, yields area preserving diffeomorphisms. Investigation of global symmetries reveals that this diffeomorphism invariance induces a noncommuting charge algebra that gets exactly mapped to the algebra of coordinates in the lowest Landau level problem. Connections of this charge algebra to noncommutative fluid dynamics and magnetohydrodynamics are shown.

A grid-connected DVSI with features to enhance power quality

A dual-source voltage inverter system is proposed to improve the power and consistency aspects of using a microgrid. Evaluation of the Proportional Integral (PI), the intelligent Fuzzy Logic Controller (FLC), and the developed control method is known as the Instantaneous Symmetrical Component Theory (ISCT). A Dual Voltage Source Inverter (DVSI) uses Distributed Energy Resources (DER) to exchange power and compensate for unbalanced and non-linear loads in the system. The direct-quadrature-zero transformation (Dq0) conversion is used to obtain the positive sequence voltage. To evaluate the system control method, an inverter connected to a three-phase, four-wire distribution combination is used. The proposed system is verified by MATLAB simulation methods using a PI controller and an intelligent FLC system.

Isotropization of locally rotationally symmetric Bianchi-I universe in f(Q)-gravity

Despite having the somewhat successful description of accelerated cosmology, the early evolution of the universe always challenges mankind. Our promising approach lies in a new class of symmetric teleparallel theory of gravity named f(Q), where the non-metricity scalar Q is responsible for the gravitational interaction, which may resolve some of the issues. To study the early evolution of the universe, we presume an anisotropic locally rotationally symmetric (LRS) Bianchi-I spacetime and derive the motion equations. We discuss the profiles of energy density, equation of state and skewness parameter and observe that our models archive anisotropic spatial geometry in the early phase of the universe with a possible presence of anisotropic fluid and as time goes on, even in the presence of an anisotropic fluid, the universe could approach isotropy due to inflation and the anisotropy of the fluid fades away at the same time.

General covariant symmetric teleparallel cosmology

Symmetric teleparallel gravity theories, in which the gravitational interaction is attributed to the nonmetricity of a flat, symmetric, but not metric-compatible affine connection, have been a topic of growing interest in recent studies. Numerous works study the cosmology of symmetric teleparallel gravity assuming a flat Friedmann-Lema\^itre-Robertson-Walker metric, while working in the so-called "coincident gauge", further assuming that the connection coefficients vanish. However, little attention has been paid to the fact that both of these assumptions rely on the freedom to choose a particular coordinate system in order to simplify the metric or the connection, and that they may, in general, not be achieved simultaneously. Here we construct the most general symmetric teleparallel geometry obeying the conditions of homogeneity and isotropy, without making any assumptions on the properties of the coordinates, and present our results in both the usual cosmological coordinates and in the coincident gauge. We find that in general these coordinates do not agree, and that assuming both to hold simultaneously allows only for a very restricted class of geometries. For the general case, we derive the energy-momentum-hypermomentum conservation relations and the cosmological dynamics of selected symmetric teleparallel gravity theories. Our results show that the symmetric teleparallel connection in general contributes another scalar quantity into the cosmological dynamics, which decouples only for a specific class of theories from the dynamics of the metric, and only if the most simple geometry is chosen, in which both assumed coordinate systems agree. Most notably, the $f(Q)$ class of theories falls into this class.

Smooth entrywise positivity preservers, a Horn–Loewner master theorem, and symmetric function identities

A special case of a fundamental result of Loewner and Horn [Trans. Amer. Math. Soc. 136 (1969), pp. 269–286] says that given an integer n ⩾ 1 n \geqslant 1 , if the entrywise application of a smooth function f : ( 0 , ∞ ) → R f : (0,\infty ) \to \mathbb {R} preserves the set of n × n n \times n positive semidefinite matrices with positive entries, then f f and its first n − 1 n-1 derivatives are non-negative on ( 0 , ∞ ) (0,\infty ) . In a recent joint work with Belton–Guillot–Putinar [J. Eur. Math. Soc., in press], we proved a stronger version, and used it to strengthen the Schoenberg–Rudin characterization of dimension-free positivity preservers [Duke Math. J. 26 (1959), pp. 617–622; Duke Math. J. 9 (1942), pp. 96–108]. In recent works with Belton–Guillot–Putinar [Adv. Math. 298 (2016), pp. 325–368] and with Tao [Amer. J. Math. 143 (2021), pp. 1863-1929] we used local, real-analytic versions at the origin of the Horn–Loewner condition, and discovered unexpected connections between entrywise polynomials preserving positivity and Schur polynomials. In this paper, we unify these two stories via a Master Theorem (Theorem A) which (i) simultaneously unifies and extends all of the aforementioned variants; and (ii) proves the positivity of the first n n nonzero Taylor coefficients at individual points rather than on all of ( 0 , ∞ ) (0,\infty ) . A key step in the proof is a new determinantal / symmetric function calculation (Theorem B), which shows that Schur polynomials arise naturally from considering arbitrary entrywise maps that are sufficiently differentiable. Of independent interest may be the following application to symmetric function theory: we extend the Schur function expansion of Cauchy’s (1841) determinant (whose matrix entries are geometric series 1 / ( 1 − u j v k ) 1 / (1 - u_j v_k) ), as well as of a determinant of Frobenius [J. Reine Angew. Math. 93 (1882), pp. 53–68] (whose matrix entries are a sum of two geometric series), to arbitrary power series, and over all commutative rings.

Self-similar sequence transformation for critical exponents

Self-similar sequence transformation is an original type of nonlinear sequence transformations allowing for defining effective limits of asymptotic sequences. The method of self-similar factor transformations is shown to be regular. This method is applied for calculating the critical exponents of the O(N)-symmetric φ4 theory in three dimensions by summing asymptotic ε expansions. It is shown that this method is straightforward and essentially simpler than other summation techniques involving complicated numerical calculations, while enjoying comparable accuracy.

A symmetric DBI action theory and its applications to inflationary cosmology

The Dirac-Born-Infeld (DBI) field theory in string theory is important and can provide the field of the universe’s inflation. At the same time, it provides a causal mechanism for generating the original density perturbation, thereby providing the necessary density perturbation for existing the dense and sparse matter distributions of the universe. We deduce a symmetric DBI action, introduce it into inflationary cosmology to calculate various inflation parameters, further calculate the scalar perturbation spectrum and the tensor-scalar ratio, which are compared with Planck + WMAP9 + BAO data, the power spectrum predicted by the new general DBI inflation theory satisfies the CMB Experiment constraints, i.e., is consistent with the current theories and experimental observations. Consequently, the theory of this paper conforms to current experiments and is supplying the current theories, and also a new way of explaining the inflation of the universe.

An efficient compensation of modified DSTATCOM for improving microgrid operation

This paper proposes a design for a modified distribution static compensator (DSTATCOM) with a proposed current controller for microgrid (MG) applications. The modified DSTATCOM is incorporated in a three-phase four-wire network system to deal accurately at grid connected mode with balanced, unbalanced and excursion load for both nonlinear and inductive loads. Moreover, it handles the stability issues of the MG at fault caused islanding incidents. The design criteria of the modified DSTATCOM parameters is presented and discussed. The hysteresis current controller is introduced to drive the voltage source converter (VSC) and its reference voltage signal is generated based on the instantaneous symmetrical components theory (ISCT). Matlab/Simulink software is used to verify the performance of the modified DSTATCOM with the proposed controller under different load types and loading conditions for grid connected mode and islanded mode of MG operation. Based on the simulation results, the power factor is improved and the non-linear harmonics currents are compensated therefore the harmonic distortion of the supply currents and voltage are reduced, moreover the stability operation at fault cause islanding is maintained. The results show the superiority of the proposed modification of DSTATCOM in dealing with unbalanced, distorted and bad power factor system in comparing with conventional DSTATCOM.

A New Evaluation Method of Total Organic Carbon for Shale Source Rock Based on the Effective Medium Conductivity Theory

In the evaluation of source rocks, the total organic carbon (TOC) is an important indicator to evaluate the hydrocarbon generation potential of source rocks. At present, the commonly used methods for assessing TOC include △ log R and neural network method. However, practice shows that these methods have limitations in the application of unconventional intervals of sand-shale interbeds, and they cannot sufficiently reflect the variation of TOC in the vertical direction. Therefore, a total organic carbon (TOC) evaluation model suitable for shale and tight sandstone was established based on the effective medium symmetrical conduction theory. The model consists of four components: nonconductive matrix particles, clay minerals, organic components (solid organic matter and hydrocarbons), and pore water. The conductive phase in the model includes clay minerals and pore water, and other components are treated as nonconductive phases. When describing the conductivity of rock, each component in the model is completely symmetrical, and anisotropic characteristics of each component are considered. The model parameters are determined through the optimization method, and the bisection iteration method is used to solve the model equation. Compared with the classic TOC calculation method, the new model can evaluate the abundance of organic matter in shale and tight sandstone, which provides a new option to assess the TOC of rocks based on logging methods.

Symmetric and asymmetric theories of time

Symmetric and asymmetric theories of time

Simple structure dual-core photonic crystal fiber polarization beam splitter covering the O + E + S + C + L + U band based on the surface plasmon resonance effect

In this paper, a simple structure dual-core photonic crystal fiber (SS-DC-PCF) polarization beam splitter (PBS) based on the surface plasmon resonance (SPR) effect and symmetric dual-core coupling mode theory is proposed. For the proposed SS-DC-PCF-1 PBS, the coupling lengths (CLs) and coupling length ratio (CLR) are analyzed, and the variations of the normalized output powers with the propagation length are investigated for the chosen wavelengths 1.434, 1.451, and 1.469 µm. The extinction ratio in cores A and B are compared at the three splitting lengths (SLs) 170, 173, and 176 µm. When the optimal SL is 176 µm, the splitting bandwidth (SB) is 255 nm (1.437–1.692 µm), and the insertion loss (IL) is less than 0.043 dB. The regulations of the CLs and CLR with the change of the structure parameters are analyzed. For the proposed SS-DC-PCF-2 PBS, the optimal SL is 232 µm, the SB is 201 nm (1.249–1.450 µm), and the IL is less than 0.042 dB. Finally, it is demonstrated that the total SB of the proposed SS-DC-PCF-1 PBS and SS-DC-PCF-2 PBS is 443 nm (1.249–1.692 µm), which can cover theO+E+S+C+L+Uband, even if the gold film thickness changes by±1nm. The proposed SS-DC-PCF PBS has excellent performances, such as ultrashort SL, ultrawide SB, ultralow IL, and good error-tolerance rate. It will have important applications in all-fiber optical systems.

Study of the non-Hermitian PT -symmetric gϕ2(iϕ)ε theory: Analysis of all orders in ε and resummations

In a recent work the Green's functions of the $\mathcal{PT}$-symmetric scalar theory $g \phi^{2}(i\phi)^\epsilon$ were calculated at the first order of the logarithmic expansion, i.e. at first order in $\epsilon$, and it was proposed to use this expansion in powers of $\epsilon$ to implement a systematic renormalization of the theory. Using techniques that we recently developed for the analysis of an ordinary (hermitian) scalar theory, in the present work we calculate the Green's functions at $O(\epsilon^2)$, pushing also the analysis to higher orders. We find that, at each finite order in $\epsilon$, the theory is non-interacting for any dimension $d \geq 2$. We then conclude that by no means this expansion can be used for a systematic renormalization of the theory. We are then lead to consider resummations, and we start with the leading contributions. Unfortunately, the results are quite poor. Specifying to the physically relevant $i g \phi^3$ model, we show that this resummation simply gives the trivial lowest order results of the weak-coupling expansion. We successively resum subleading diagrams, but again the results are rather poor. All this casts serious doubts on the possibility of studying the theory $g \phi^{2}(i\phi)^\epsilon$ with the help of such an expansion. We finally add that the findings presented in this work were obtained by us some time ago (December 2019), and we are delighted to see that these results, that we communicated to C.M. Bender in December 2019, are confirmed in a recent preprint (e-Print:2103.07577) of C.M. Bender and collaborators.

Towards perturbative renormalization ofϕ2(iϕ)ϵquantum field theory

In a previous paper it was shown how to calculate the ground-state energy density $E$ and the $p$-point Green's functions $G_p(x_1,x_2,...,x_p)$ for the $PT$-symmetric quantum field theory defined by the Hamiltonian density $H=\frac{1}{2}(\nabla\phi)^2+\frac{1}{2}\phi^2(i\phi)^\varepsilon$ in $D$-dimensional Euclidean spacetime, where $\phi$ is a pseudoscalar field. In this earlier paper $E$ and $G_p(x_1,x_2,...,x_p)$ were expressed as perturbation series in powers of $\varepsilon$ and were calculated to first order in $\varepsilon$. (The parameter $\varepsilon$ is a measure of the nonlinearity of the interaction rather than a coupling constant.) This paper extends these perturbative calculations to the Euclidean Lagrangian $L= \frac{1}{2}(\nabla\phi)^2+\frac{1}{2}\mu^2\phi^2+\frac{1}{2} g\mu_0^2\phi^2\big(i\mu_0^{1-D/2}\phi\big)^\varepsilon-iv\phi$, which now includes renormalization counterterms that are linear and quadratic in the field $\phi$. The parameter $g$ is a dimensionless coupling strength and $\mu_0$ is a scaling factor having dimensions of mass. Expressions are given for the one-, two, and three-point Green's functions, and the renormalized mass, to higher-order in powers of $\varepsilon$ in $D$ dimensions ($0\leq D\leq2$). Renormalization is performed perturbatively to second order in $\varepsilon$ and the structure of the Green's functions is analyzed in the limit $D\to 2$. A sum of the most divergent terms is performed to {\it all} orders in $\varepsilon$. Like the Cheng-Wu summation of leading logarithms in electrodynamics, it is found here that leading logarithmic divergences combine to become mildly algebraic in form. Future work that must be done to complete the perturbative renormalization procedure is discussed.

Gauge-Invariant Post-Newtonian Perturbations in Symmetric Teleparallel Gravity

We present an extension to the gauge-invariant formulation of the parameterized post-Newtonian (PPN) formalism, which allows its application to symmetric teleparallel gravity theories. In its original formulation, the gauge-invariant PPN formalism makes use of a gauge-invariant perturbative expansion of tensor fields; however, one of the fundamental gravitational field variables in symmetric teleparallel gravity theories is a flat, torsion-free connection, hence not a tensor field. Since connections transform differently from tensor fields under diffeomorphisms, we introduce here an adapted notion of gauge-invariant perturbation variables for the symmetric teleparallel connection, and show how the lowest order terms can be expressed in terms of the common PPN potentials.

Symmetry, pattern formation, and finite-density QCD

A longstanding issue in the study of quantum chromodynamics (QCD) is its behavior at nonzero baryon density, which has implications for many areas of physics. The path integral has a complex integrand when the quark chemical potential is nonzero and therefore has a sign problem, but it also has a generalized symmetry. We review some new approaches to -symmetric field theories, including both analytical techniques and methods for lattice simulation. We show that -symmetric field theories with more than one field generally have a much richer phase structure than their Hermitian counterparts, including stable phases with patterning behavior. The case of a -symmetric extension of a φ4 model is explained in detail. The relevance of these results to finite density QCD is explained, and we show that a simple model of finite density QCD exhibits a patterned phase in its critical region.

Symmetry and renormalisation in quantum field theory

Quantum systems governed by non-Hermitian Hamiltonians with symmetry are special in having real energy eigenvalues bounded below and unitary time evolution. We argue that symmetry may also be important and present at the level of Hermitian quantum field theories because of the process of renormalisation. In some quantum field theories renormalisation leads to -symmetric effective Lagrangians. We show how symmetry may allow interpretations that evade ghosts and instabilities present in an interpretation of the theory within a Hermitian framework. From the study of examples -symmetric interpretation is naturally built into a path integral formulation of quantum field theory; there is no requirement to calculate explicitly the norm that occurs in Hamiltonian quantum theory. We discuss examples where -symmetric field theories emerge from Hermitian field theories due to effects of renormalisation. We also consider the effects of renormalisation on field theories that are non-Hermitian but -symmetric from the start.

The physical states of the Hybrid Formalism

String theory on AdS3 × S3 × mathbbm{T} 4 with one unit (k = 1) of NS-NS flux is considered in the hybrid formalism of Berkovits, Vafa & Witten (BVW). Using the free field realisation of the world-sheet theory at k = 1, we identify explicitly the BRST cohomology classes corresponding to some of the low-lying states of the dual CFT. In particular, we do this for the mathcal{N} = 4 superconformal generators of the symmetric orbifold theory, and we confirm these identifications by showing that the worldsheet correlators reproduce the expected dual CFT answer. Along the way we note that the physical vertex operators on the worldsheet have a simpler form if one works with a different, but equivalent, choice for the BRST operators relative to BVW.

An introductory review on resource theories of generalized nonclassical light

Quantum resource theory is perhaps the most revolutionary framework that quantum physics has ever experienced. It plays vigorous roles in unifying the quantification methods of a requisite quantum effect as wells as in identifying protocols that optimize its usefulness in a given application in areas ranging from quantum information to computation. Moreover, the resource theories have transmuted radical quantum phenomena like coherence, nonclassicality and entanglement from being just intriguing to being helpful in executing realistic thoughts. A general quantum resource theoretical framework relies on the method of categorization of all possible quantum states into two sets, namely, the free set and the resource set. Associated with the set of free states there is a number of free quantum operations emerging from the natural constraints attributed to the corresponding physical system. Then, the task of quantum resource theory is to discover possible aspects arising from the restricted set of operations as resources. Along with the rapid growth of various resource theories corresponding to standard harmonic oscillator quantum optical states, significant advancement has been expedited along the same direction for generalized quantum optical states. Generalized quantum optical framework strives to bring in several prosperous contemporary ideas including nonlinearity, -symmetric non-Hermitian theories, q-deformed bosonic systems, etc., to accomplish similar but elevated objectives of the standard quantum optics and information theories. In this article, we review the developments of nonclassical resource theories of different generalized quantum optical states and their usefulness in the context of quantum information theories.

General features of the thermalization of particle detectors and the Unruh effect

We study the thermalization of smeared particle detectors that couple locally to any operator in a quantum field theory in curved spacetimes. We show that if the field state satisfies the Kubo-Martin-Schwinger condition with inverse temperature $\ensuremath{\beta}$ with respect to the detector's local notion of time evolution, reasonable assumptions ensure that the probe thermalizes to the temperature $1/\ensuremath{\beta}$ in the limit of long interaction times. Our method also imposes bounds on the size of the system with respect to its proper acceleration and spacetime curvature in order to accurately probe the Kubo-Martin-Schwinger temperature of the field. We then apply this formalism to a uniformly accelerated detector probing the Minkowski vacuum of any $CPT$ symmetric quantum field theory, and show that the detector thermalizes to the Unruh temperature, independently of the operator it couples to. This exemplifies yet again the robustness of the Unruh effect, even when arbitrary smeared detectors are used to probe general operators in a quantum field theory.

A new method to incorporate three-phase power transformer model into distribution system load flow analysis

This paper proposes a new and simple method to incorporate three-phase power transformer model into distribution system load flow (DSLF) analysis. The objective of the present work is to find a robust and efficient technique for modeling and integrating power transformer in the DSLF analysis. The proposed transformer model is derived based on nodal admittance matrix and formulated by using the symmetrical component theory. Load flow formulation in terms of branch currents and nodal voltages is also proposed in this paper to enable integrating the model into the DSLF analysis. Singularity that makes the calculations in forward/backward sweep (FBS) algorithm is difficult to be carried out. It can be avoided in the method. The proposed model is verified by using the standard IEEE test system.