Gauge Transformations Research Articles

"My life can be said to form a circle" — An interview with Nobel laureate Professor Chen-Ning Yang

<abstract> <p><bold>Editor's note:</bold> Chen-Ning Yang (杨振宁, 1 October 1922 –), or C.N. Yang or by the English name Frank Yang received the 1957 Nobel Prize in Physics with Tsung-Dao Lee for their work on parity non-conservation of weak interaction. Yang and Robert Mills also proposed the non-abelian gauge theory, widely known as the Yang–Mills theory. Yang and Baxter found the Yang–Baxter equation that plays an important role in integrable models and has influenced several branches of physics and mathematics. Professor C.N. Yang is not only a great physicist, but also an outstanding educator and a thoughtful philosopher. This edited interview was conducted in 2014 by the author. Parts of the Chinese version were published in a few Chinese newspapers and magazines at the time. This English version, translated by Tyler Ross from New York, USA, provides readers worldwide with an opportunity to share and understand Professor Yang's long-lasting successful journey and thoughts on scientific preparation and innovation, cross-disciplinary integration, and effective collaboration, in particular how Yang's in-depth understanding and appreciation of mathematics in his successes in physics. This interview is also to celebrate Professor Yang's 101st birthday and his public speech at Zhejiang University in China in 2014.</p> </abstract>

Fractional Topological Charge Construction in a U(1) Lattice Gauge Theory Framework

We have constructed a fractional topological charge in the lattice Abelian gauge theory formalism, which avoids the issue of infinite degrees of freedom in continuum spacetime quantum field theory.

Anomalije u kvantnim teorijama polja

Introduction:purpose: Noether's theorem connects symmetry of the Lagrangian to conserved quantities. Quantum effects cancel the conserved quantities. Methods: Triangle diagram, Path integral, Pauli-Villars regularisation. Results: Quantum effects that spoil conserved quantities of local gauge symmetries endager renormalisability. Conclusion: A careful treatment of anomalies is needed in order to obtain correct results. The p 0 → gg decay is perhaps the most notable "impossible" effect allowed by anomalies.

Effective Hamiltonian for the system of the electromagnetic field and charged particles

A classical system of point charged particles and electromagnetic field is investigated in the general formulation, in which there is no direct interaction of the particles with each other. Particles are assumed to be non-relativistic for research simplicity. The point of view is defended that gauge choice determines the physical picture of processes in the system. The idea of extended gauge, in which scalar φk and longitudinal part Alnk of the vector potentials transform separately for large and small wave numbers, is proposed. Moreover, in the new gauge φk ≠ 0 only when k ≥ k0 and Alnk ≠ 0 only when k ≤ k0 , where k0 is a certain wave number. This greatly simplifies the study of the system dynamics, in which the transversecomponents Atnk of the vector potential are taken into account and do not change at the gauge transformation. Basing on the extended gauge idea, it is established that the system has a massive oscillatory mode described by a transverse potential Atnk and an oscillatory mod described by a longitudinal potential Alnk with the laws of dispersion (ωk2 + ω02 )1/2 і ω0, respectively (ωk = ck is the dispersion law for electromagnetic waves in vacuum, and ω0 is the plasma frequency). These modes can be called electromagnetic and plasma ones. At the same time, effective interaction between charged particles is introduced, which describes the screening effect. A connection of our work with Bohm and Pines investigations related to the presence of plasma modes in a Coulomb system, in which they do not use idea of gauge transformations, is analyzed.

Quintic vertices of spin 3, vector and scalar fields

As a result of special deformations of free gauge models of massless spin 3, massive vector and real scalar fields, quintic vertices within new approach Buchbinder and Lavrov (2021) [32]; Buchbinder and Lavrov (2021) [33]; Lavrov (2022) [34] are constructed. They are described by local functionals which are invariant under original gauge transformations.

Physical content of the bosonized light-front Schwinger model

The light-front Schwinger model in the bosonized form is analyzed focusing on its global properties. We point out that the gauge invariance of the original fermionic form of the model is lost after bosonization. Consequently, the previously found theta-vacuum structure based on the residual large gauge transformation is actually not an intrinsic property of the bosonized form of the model. We analyze the corresponding field equations without any “gauge fixing” in both the light-front and space-like (“instant-form”) versions of the model and find the coinciding results: the axial-vector anomaly is automatically present and the spectrum of the model is given by a massive boson with the mass [Formula: see text]. Finally, we characterize and critically assess properties of the bosonized LF Schwinger model, found in the previous studies.

Non-Abelian Gauge Theories with Composite Fields in the Background Field Method

Non-Abelian gauge theories with composite fields are examined in the background field method. Generating functionals of Green’s functions for a Yang–Mills theory with composite and background fields are introduced, including the generating functional of vertex Green’s functions (effective action). The corresponding Ward identities are obtained, and the issue of gauge dependence is investigated. A gauge variation of the effective action is found in terms of a nilpotent operator depending on the composite and background fields. On-shell independence from the choice of gauge fixing for the effective action is established. In the study of the Ward identities and gauge dependence, finite field-dependent BRST transformations with a background field are introduced and employed on a systematic basis. On the one hand, this involves the consideration of (modified) Ward identities with a field-dependent anticommuting parameter, also depending on a non-trivial background. On the other hand, the issue of gauge dependence is studied with reference to a finite variation of the gauge Fermion. The concept of a joint introduction of composite and background fields to non-Abelian gauge theories is exemplified by the Gribov–Zwanziger theory, including the case of a local BRST-invariant horizon, and also by the Volovich–Katanaev model of two-dimensional gravity with dynamical torsion.

Scalable fast benchmarking for individual quantum gates with local twirling

With the development of controllable quantum systems, fast and practical characterization of multi-qubit gates has become essential for building high-fidelity quantum computing devices. The usual way to fulfill this requirement via randomized benchmarking demands complicated implementation of numerous multi-qubit twirling gates. How to efficiently and reliably estimate the fidelity of a quantum process remains an open problem. This work thus proposes a character-cycle benchmarking protocol and a character-average benchmarking protocol using only local twirling gates to estimate the process fidelity of an individual multi-qubit operation. Our protocols were able to characterize a large class of quantum gates including and beyond the Clifford group via the local gauge transformation, which forms a universal gate set for quantum computing. We demonstrated numerically our protocols for a non-Clifford gate—controlled- ( T X ) and a Clifford gate—five-qubit quantum error-correcting encoding circuit. The numerical results show that our protocols can efficiently and reliably characterize the gate process fidelities. Compared with the cross-entropy benchmarking, the simulation results show that the character-average benchmarking achieves three orders of magnitude improvements in terms of sampling complexity.

Parallel spinors, pp-waves, and gravitational perturbations

We prove that any real, vacuum gravitational perturbation of a four-dimensional vacuum pp-wave space-time can be locally expressed, modulo gauge transformations, as the real part of a Hertz/Debye potential, where the scalar potential satisfies the wave equation. We discuss relations with complex perturbations, complex space-times, non-linear structures, and real spaces with split (ultra-hyperbolic/Kleinian) signature. Motivated by generalized notions of parallel spinors, we also discuss generalizations of the result to other space-times.

Geometry of nonrelativistic string

The nonrelativistic bosonic string theory in a curved manifold is formulated here using gauging of symmetry approach (Galilean Gauge theory). The corresponding model in flat space has some global symmetries. By localizing these symmetries as per Galilean Gauge theory, the action for the nonrelativistic string interacting with gravity is obtained. A canonical analysis of the model has been performed which demonstrate that the transformations of the basic field variables under gauge transformations in phase space are equivalent to the diffeomorphism parameters by an exact mapping. Thus complete consistency of our results from both Lagrangian and Hamiltonian procedures are established.

Vanishing Noether current in Weyl invariant gravities

We revisit the issue that the Noether current associated with a local scale symmetry, or equivalently the Weyl symmetry, identically vanishes. Based on only the second Noether theorem for a local symmetry, we prove that the Noether current associated with the Weyl symmetry is in general vanishing in any Weyl invariant gravitational theories in four-dimensional Riemannian geometry. We also clarify the reason: The Weyl transformation is nondynamical in the sense that it does not contain the derivative term of the transformation parameter as opposed to the conventional gauge transformation.

Lorentz Gauge and Coulomb Gauge for Tetrad Field of Gravity

In general relativity, an inertial frame can only be established in a small region of spacetime, and local inertial frames are mathematically represented by a tetrad field in gravity. The tetrad field is not unique due to the freedom to perform Lorentz transformations in local inertial frames, and there exists freedom to choose the local inertial frame at each spacetime point. The local Lorentz transformations are known as non-Abelian gauge transformations for the tetrad field, and to fix the gauge freedom corresponding to the Lorentz gauge ∂μAμ=0 and the Coulomb gauge ∂iAi=0 in electrodynamics, the Lorentz gauge and Coulomb gauge for tetrad fields are proposed in the present work. Moreover, properties of the Lorentz gauge and the Coulomb gauge for tetrad fields are discussed to show their similarities to those in electromagnetic fields.

Eigenfunctions of the van Diejen model generated by gauge and integral transformations

We present how explicit eigenfunctions of the principal Hamiltonian for the BCm relativistic Calogero-Moser-Sutherland model, due to van Diejen, can be constructed using gauge and integral transformations. In particular, we find that certain BC-type elliptic hypergeometric integrals, including elliptic Selberg integrals, of both Type I and Type II arise as eigenfunctions of the van Diejen model under some parameter restrictions. Among these are also joint eigenfunctions of so-called modular pairs of van Diejen operators. Furthermore, these transformations are related to reflections of the E8 Weyl group acting on the space of model parameters.

Maxwell to Photonics

The main topic to be addressed is the search for a new source of energy: light. Electromagnetism has been the energy that has most changed civilisation in the last two centuries. The emergence of photonics instead of electronics is a new challenge. Photonics is the clean energy to look for. The 20th century was that of electrons. Several innovations took place through electronics. However, despite these numerous innovations due to the electromagnetic properties of the electron, the 21st century will be that of the photon. The advent of a new generation of innovations arising from the electromagnetic properties of the photon is expected. There is a primordial photon from the light invariance still to be revealed, and a growing photonic market awaiting new properties of the photon. The new perspective lies in discovering electromagnetism where the photon is the own source of electromagnetic fields and self-interacting photons at the tree level are generated. Our proposal is the four bosons electromagnetism[1] . A model based on charge transfer. An enlargement to Maxwell supported upon a general electric charge triad {+,0,-} and an extension to gauge symmetry for a nonlinear abelian gauge theory[2] . Elementary particle physics shows several reactions interchanging positive, negative and zero charges. It yields a physicality considering the charges set {+,0,-} mediated by four gauge bosons. A quadruplet physics manifested by photon, massive photon and charged photons. A new EM energy is to be explored. Introducing new electromagnetic sectors beyond Maxwell as nonlinear EM, neutral EM, spintronics, weak interaction, and photonics. The basis for photonic engineering.

Fujikawa's Jacobian and Berry's phase for fermion-hopping models defined on lattices

After briefly reviewing Fujikawa's derivation of a chiral anomaly, we study fermions hopping on a lattice. Under the specific conditions considered in this paper, we show that a chiral gauge transformation induces a Berry's phase which is equal to Fujikawa's Jacobian.

Imaginary Gauge Transformation in Momentum Space and Dirac Exceptional Point.

An imaginary gauge transformation is at the core of the non-Hermitian skin effect. Here, we show that such a transformation can be performed in momentum space as well, which reveals that certain gain- and loss-modulated systems in their parity-time (PT) symmetric phases are equivalent to Hermitian systems with real potentials. Our analysis in momentum space also distinguishes two types of exceptional points (EPs) in the same system. Besides the conventional type that leads to a PT transition upon the continuous increase of gain and loss, we find real-valued energy bands connected at a Dirac EP in hybrid dimensions, consisting of a spatial dimension and a synthetic dimension for the gain and loss strength.

Spectral Parameter as a Group Parameter

A large class of integrable non-linear partial differential equations is characterized by the existence of the associated linear problem (in the case of two independent variables, known as a Lax pair) containing the so-called spectral parameter. In this paper, we present and discuss the conjecture that the spectral parameter can be interpreted as the parameter of some one-parameter groups of transformation, provided that it cannot be removed by any gauge transformation. If a non-parametric linear problem for a non-linear system is known (e.g., the Gauss–Weingarten equations as a linear problem for the Gauss–Codazzi equations in the geometry of submanifolds), then, by comparing both symmetry groups, we can find or indicate the integrable cases. We consider both conventional Lie point symmetries and the so-called extended Lie point symmetries, which are necessary in some cases. This paper is intended to be a review, but some novel results are presented as well.

Chiral Approach to Massive Higher Spins.

We propose a new, chiral description for massive higher-spin particles in four spacetime dimensions, which facilitates the introduction of consistent interactions. As proof of concept, we formulate three theories, in which higher-spin matter is coupled to electrodynamics, non-Abelian gauge theory, or gravity. The theories are chiral and have simple Lagrangians, resulting in Feynman rules analogous to those of massive scalars. Starting from these Feynman rules, we derive tree-level scattering amplitudes with two higher-spin matter particles and any number of positive-helicity photons, gluons, or gravitons. The amplitudes reproduce the arbitrary-multiplicity results that were obtained via on-shell recursion in a parity-conserving setting, and which chiral and nonchiral theories thus have in common. The presented theories are currently the only examples of consistent interacting field theories with massive higher-spin fields.

Supersymmetric gradient flow in 4d $${{\mathcal {N}}}=1$$ SQCD

A supersymmetric gradient flow for four-dimensional \({{\mathcal {N}}}=1\) supersymmetric QCD (SQCD) is proposed. The flow equation is given in both the superfield and component fields of the Wess–Zumino gauge. The superfield flow equation is defined for each of the gauge and matter multiplets individually. Adding a gauge fixing, the component-field flow equation is defined in the Wess–Zumino gauge in a gauge covariant manner. We find that the latter equation is supersymmetric in a sense that the commutator of the flow time derivative and the supersymmetry transformation vanishes up to a gauge transformation. We also discuss a simplified flow using the gradient of supersymmetric Yang–Mills (SYM) action instead of using SQCD action to define a gauge multiplet flow.

A Puzzle Concerning Local Symmetries and Their Empirical Significance

In the last 5 years, the controversy about whether or not gauge transformations can be empirically significant has intensified. On the one hand, Greaves and Wallace developed a framework according to which, under some circumstances, gauge transformations can be empirically significant—and Teh further supported this result by using the constrained Hamiltonian formalism. On the other hand, Friederich claims to have proved that gauge transformation can never be empirically significant. In this article, I accomplish two tasks. First, I argue that there are strong reasons to resist Friederich’s proof because one of its assumptions is, at the very least, highly controversial. Second, I argue that despite criticism by Brading and Brown and Friederich, ‘t Hooft’s beam-splitter experiment is indeed a concrete example of a case where a local gauge symmetry has empirical significance. By shedding light on these two points, this article shows that recent arguments that claim gauge transformations cannot be empirically significant are not satisfactory.