Operator Gauge Transformation Research Articles

Twisted fracton models in three dimensions

We study novel three-dimensional gapped quantum phases of matter which support quasiparticles with restricted mobility, including immobile "fracton" excitations. So far, most existing fracton models may be instructively viewed as generalized Abelian lattice gauge theories. Here, by analogy with Dijkgraaf-Witten topological gauge theories, we discover a natural generalization of fracton models, obtained by twisting the gauge symmetries. Introducing generalized gauge transformation operators carrying an extra phase factor depending on local configurations, we construct a plethora of exactly solvable three-dimensional models, which we dub "twisted fracton models." A key result of our approach is to demonstrate the existence of rich non-Abelian fracton phases of distinct varieties in a three-dimensional system with finite-range interactions. For an accurate characterization of these novel phases, the notion of being inextricably non-Abelian is introduced for fractons and quasiparticles with one-dimensional mobility, referring to their new behavior of displaying braiding statistics that is, and remains, non-Abelian regardless of which quasiparticles with higher mobility are added to or removed from them. We also analyze these models by embedding them on a three-torus and computing their ground state degeneracies, which exhibit a surprising and novel dependence on the system size in the non-Abelian fracton phases. Moreover, as an important advance in the study of fracton order, we develop a general mathematical framework which systematically captures the fusion and braiding properties of fractons and other quasiparticles with restricted mobility.

Solving the constrained modified KP hierarchy by gauge transformations

In this paper, we mainly investigate two kinds of gauge transformations for the constrained modified KP hierarchy in Kupershmidt-Kiso version. The corresponding gauge transformations are required to keep not only the Lax equation but also the Lax operator. For this, by selecting the special generating eigenfunction and adjoint eigenfunction, the elementary gauge transformation operators of modified KP hierarchy \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${T_D}(\\Phi ) = ({\\Phi ^{ - 1}})_x^{ - 1}\\partial {\\Phi ^{ - 1}}$$\\end{document} and TI(Ψ) = Ψ−1∂−1Ψx, become the ones in the constrained case. Finally, the corresponding successive applications of TD and TI on the eigenfunction Φ and the adjoint eigenfunction Ψ are discussed.

The gauge transformations of the constrainedq-deformed KP hierarchy

In this paper, we mainly study the gauge transformations of the constrained q-deformed Kadomtsev–Petviashvili (q-KP) hierarchy. Different from the usual case, we have to consider the additional constraints on the Lax operator of the constrained q-deformed KP hierarchy, since the form of the Lax operator must be kept when constructing the gauge transformations. For this reason, the selections of generating functions in elementary gauge transformation operators [Formula: see text] and [Formula: see text] must be very special, which are from the constraints in the Lax operator. At last, we consider the successive applications of n-step of [Formula: see text] and k-step of [Formula: see text] gauge transformations.

The gauge transformation of the modified KP hierarchy

In this paper, we firstly investigate the successive applications of three elementary gauge transformation operators Ti with i = 1, 2, 3 for the mKP hierarchy in Kupershmidt-Kiso version, and find that the gauge transformation operators Ti can not commute with each other. Then two types of gauge transformation operators TD and TI constructed from Ti are proved that they can commute with each other. In particular, TI is introduced for the first time in the literature. And the successive applications of TD and TI in the form of T(n,k), which is the product of n terms of TD and k terms of TI, are derived in three cases for different n and k. At last, the corresponding successive applications of TD and TI on the eigenfunction Φ, the adjoint eigenfunction Ψ and the tau functions τ0 and τ1 are considered.

Gauge transformations of constrained discrete modified KP systems with self-consistent sources

In this paper, we firstly recall some basic facts about the discrete KP(d-KP) and discrete modified KP(d-mKP) hierarchies, and then we find that d-KP hierarchy and d-mKP hierarchy are linked by a gauge transformation. What’s more, we give three gauge transformation operators of the d-mKP hierarchy and give their successive applications. We further construct the ghost symmetry and use this symmetry to give the definition the d-mKP hierarchy with self-consistent sources. We also give gauge transformations of a newly defined constrained d-mKP(cd-mKP) hierarchy and the constrained d-mKP hierarchy with self-consistent sources(cd-mKPHSCSs).

The Successive Application of the Gauge Transformation for the Modified Semidiscrete KP Hierarchy

Abstract In this article, the successive application of three gauge transformation operators for the modified semidiscrete Kadomtsev–Petviashvili(mdKP) hierarchy has been provided. The commutativity of the Bianchi diagram of these gauge transformation operators is investigated.

Multifold Darboux Transformations of the Extended Bigraded Toda Hierarchy

Abstract With the extended logarithmic flow equations and some extended Vertex operators in generalized Hirota bilinear equations, extended bigraded Toda hierarchy (EBTH) was proved to govern the Gromov-Witten theory of orbiford c NM in literature. The generating function of these Gromov-Witten invariants is one special solution of the EBTH. In this article, the multifold Darboux transformations and their determinant representations of the EBTH are given with two different gauge transformation operators. The two Darboux transformations in different directions are used to generate new solutions from known solutions which include soliton solutions of (N, N)-EBTH, i.e. the EBTH when N=M. From the generation of new solutions, we can find the big difference between the EBTH and the extended Toda hierarchy (ETH). Also, we plotted the soliton graphs of the (N, N)-EBTH from which some approximation analysis is given. From the analysis on velocities of soliton solutions, the difference between the extended flows and other flows are shown. The two different Darboux transformations constructed by us might be useful in Gromov-Witten theory of orbiford c NM .

Coulomb Solutions from Improper Pseudo-Unitary Free Gauge Field Operator Translations

Fundamental problems of quantum field theory related to the representation problem of canonical commutation relations are discussed within a gauge field version of a van Hove-type model. The Coulomb field generated by a static charge distribution is described as a formal superposition of time-like pseudo-photons in Fock space with a Krein structure. In this context, a generalization of operator gauge transformations is introduced to generate coherent states of abelian gauge fields interacting with a charged background.

The gauge transformation of the q-deformed modified KP hierarchy

In this paper, we mainly study three types of gauge transformation operators for the q-mKP hierarchy. The successive applications of these gauge transformation operators are derived. And the corresponding communities between them are also investigated.

Aspects of the derivative coupling model in four dimensions

A concise discussion of a \(3+1\)-dimensional derivative coupling model, in which a massive Dirac field couples to the four-gradient of a massless scalar field, is given in order to elucidate the role of different concepts in quantum field theory like the regularization of quantum fields as operator-valued distributions, correlation distributions, locality, causality, and field operator gauge transformations.

Gauge transformations for the constrained CKP and BKP hierarchies

A special chain of the gauge transformations of the two-component constrained KP is introduced, and then the transformed components and the τ-function of this hierarchy are presented. Based on this, by the different reductions, the gauge transformations of the constrained BKP and the constrained CKP hierarchies are obtained. Furthermore, the transformed component functions and τ-functions of them are expressed by the determinants with the help of the determinant representation of the gauge transformation operators.

Operator Gauge Symmetry in QED

In this paper, operator gauge transformation, first introduced by Kobe, is applied to Maxwell's equations and continuity equation in QED. The gauge invariance is satisfied after quantization of electromagnetic fields. Inherent nonlinearity in Maxwell's equations is obtained as a direct result due to the nonlinearity of the operator gauge trans- formations. The operator gauge invariant Maxwell's equations and corresponding charge conservation are obtained by defining the generalized derivatives of the f irst and second kinds. Conservation laws for the real and virtual charges are obtained too. The additional terms in the field strength tensor are interpreted as electric and magnetic polarization of the vacuum.

THE DETERMINANT REPRESENTATION OF THE GAUGE TRANSFORMATION OPERATORS

The determinant representation of the gauge transformation operators is establised. In this process, the generalized Wronskian determinant is introduced. As a simple application, the authors present a construction of the special τ-function obtained firstly by Chau et al. (Commun. Math. Phys., 149(1992), 263), which involves the generalized Wronskian determinant. Also, some properties of this determinant are given.

Neutrino polarization as a consequence of gauge invariance

There are gauge-transformation operators applicable to massless spin-1/2 particles within the little-group framework of internal space-time symmetries of massive and massless particles. It is shown that two of the SL(2, c) spinors are invariant under gauge transformations while the remaining two are not. The Dirac equation contains only the gauge-invariant spinors leading to polarized neutrinos. It is shown that the gauge-dependent SL(2, c) spinor is the origin of the gauge dependence of electromagnetic four-potentials. It is noted also that, for spin-1/2 particles, the symmetry group for massless particles is an infinite-momentum/zero-mass limit of the symmetry group for massive particles.

Generalized quantum dynamics

We propose a generalization of Heisenberg picture quantum mechanics in which a lagrangian and hamiltonian dynamics is formulated directly for dynamical systems on a manifold with noncommuting coordinates, which act as operators on an underlying Hilbert space. This is accomplished by defining the lagrangian and hamiltonian as the real part of a graded total trace over the underlying Hilbert space, permitting a consistent definition of the first variational derivative with respect to a general operator-valued coordinate. The hamiltonian form of the equations is expressed in terms of a generalized bracket operation, which obeys a Jacobi identity. The formalism permits the natural implementation of gauge invariance under operator-valued gauge transformations. When an operator hamiltonian exists as well as a total trace hamiltonian, as is generally the case in complex quantum mechanics, one can make an operator gauge transformation from the Heisenberg to the Schrödinger picture. When applied to complex quantum mechanical systems with one bosonic or fermionic degree of freedom, the formalism gives the usual operator equations of motion, with the canonical commutation relations emerging as constraints associated with the operator gauge invariance. More generally, our methods permit the formulation of quaternionic quantum field theories with operator-valued gauge invariance, in which we conjecture that the operator constraints act as a generalization of the usual canonical commutators.

Causal construction of Yang-Mills theories

Pure quantized Yang-Mills theories are constructed by causalperturbation theory. We study operator gauge transformations which lead in a natural way to the introduction of ghost fields. Considering fermionic as well as bosonic ghosts, we find that only the former save gauge invariance in second order. We work with free quantum fields throughout, so that all expressions are mathematically well defined.

Solving the KP hierarchy by gauge transformations

We show that it is convenient to use “gauge” transformations (sometimes called explicit Bäcklund transformations) to generate new solutions for the KP hierarchy. Two particular kinds of gauge transformation operators, constructed out of the initial wave functions, are of fundamental importance in this approach. Through such gauge transformations, a very simple formula for the tau-function is obtained, encompassing and unifying all kinds of existing solutions. The corresponding free fermion representation and Baker functions for the new τ function can also be constructed.

Операторные калибровочные преобразования в нерелятивистской квантовой электродинамике Применения к мультипольному Гамильтониану

Operator gauge transformations on the electromagnetic potentials in quantum electrodynamics are obtained by requiring form invariance of the Schrodinger equation of a single particle under arbitrary space- and time-dependent unitary transformations. These operator gauge transformations are found to have the same form as in non-Abelian gauge field theories. By considering the Lorentz force on the particle, we show that the definition of the electromagnetic-field strength operator in terms of the potentials must also have the same form as in non-Abelian gauge field theory. Since the Hamiltonian is gauge dependent, a distinction is made between the Hamiltonian and the gauge-invariant energy operator of the particle. Local conservation of energy for the particle is derived and it is shown that the particle gains (loses) energy at the same rate as expected classically. An operator gauge transformation is made to obtain potentials in the multipolar gauge and the multipolar Hamiltonian is shown to be the minimally coupled Hamiltonian in the multipolar gauge.

Gauge equivalence of the electrodynamics of charged bosons

The quantum electrodynamics of charged scalar and vector bosons is formulated in the Lorentz gauge, and the effect of the charged particle–photon interaction on the subsidiary condition is explicitly taken into account. The results are extensions of earlier work on spinor quantum electrodynamics, but the presence of seagull vertices and anomalous current commutators in the case of the charged bosons make the extensions nontrivial. An operator gauge transformation that encompasses equations of motion as well as the commutator algebra of the field operators is developed; it is used to transform the theory from the Lorentz gauge to the Coulomb gauge.

Gauge invariance, time-dependent Foldy-Wouthuysen transformations, and the Pauli Hamiltonian

The Foldy-Wouthuysen derivation of the generalized Pauli Hamiltonian is reexamined, and described as an operator gauge transformation. The unitary inequivalence of the Pauli and Dirac Hamiltonians is shown to occur, because in general it is only the entire coupled field Hamiltonian which is gauge invariant, not the particle Hamiltonian alone.