Time-dependent Gauge Transformation Research Articles

Schwinger-Dyson equations in Coulomb gauge consistent with numerical simulation

In the present work we undertake a study of the Schwinger-Dyson equation (SDE) in the Euclidean formulation of local quantum gauge field theory, with Coulomb gauge condition $\partial_i A_i = 0$. We continue a previous study which kept only instantaneous terms in the SDE that are proportional to $\delta(t)$ in order to calculate the instantaneous part of the time component of the gluon propagator $D_{A_0 A_0}(t, R)$. We compare the results of that study with a numerical simulation of lattice gauge theory and find that the infrared critical exponents and related quantities agree to within 1\% to 3\%. This raises the question, "Why is the agreement so good, despite the systematic neglect of non-instantaneous terms?" We discovered the happy circumstance that all the non-instantaneous terms are in fact zero. They are forbidden by the symmetry of the local action in Coulomb gauge under time-dependent gauge transformations $g(t)$. This remnant gauge symmetry is not fixed by the Coulomb gauge condition. The numerical result of the present calculation is the same as in the previous study; the novelty is that we now demonstrate that all the non-instantaneous terms in the SDE vanish. We derive some elementary properties of propagators which are a consequence of the remnant gauge symmetry. In particular the time component of the gluon propagator is found to be purely instantaneous $D_{A_0 A_0}(t, R) = \delta(t) V(R)$, where $V(R)$ is the color-Coulomb potential. Our results support the simple physical scenario in which confinement is the result of a linearly rising color-Coulomb potential, $V(R) \sim \sigma R$ at large $R$.

Constructing a boosted, spinning black hole in the damped harmonic gauge

The damped harmonic gauge is important for numerical relativity computations based on the generalized harmonic formulation of Einstein's equations, and is used to reduce coordinate distortions near binary black hole mergers. However, currently there is no prescription to construct quasiequilibrium binary black hole initial data in this gauge. Instead, initial data are typically constructed using a superposition of two boosted analytic single black hole solutions as free data in the solution of the constraint equations. Then, a smooth time-dependent gauge transformation is done early in the evolution to move into the damped harmonic gauge. Using this strategy to produce initial data in damped harmonic gauge would require the solution of a single black hole in this gauge, which is not known analytically. In this work we construct a single boosted, spinning, equilibrium BH in damped harmonic coordinates as a regular time-independent coordinate transformation from Kerr-Schild coordinates. To do this, we derive and solve a set of 4 coupled, nonlinear, elliptic equations for this transformation, with appropriate boundary conditions. This solution can now be used in the construction of damped harmonic initial data for binary black holes.

Equations of motion as constraints: superselection rules, Ward identities

The meaning of local observables is poorly understood in gauge theories, not to speak of quantum gravity. As a step towards a better understanding we study asymptotic (infrared) transformations in local quantum physics. Our observables are smeared by test functions, at first vanishing at infinity. In this context we show that the equations of motion can be seen as constraints, which generate a group, the group of space and time dependent gauge transformations. This is one of the main points of the paper. Infrared nontrivial effects are captured allowing test functions which do not vanish at infinity. These extended operators generate a larger group. The quotient of the two groups generate superselection sectors, which differentiate different infrared sectors. The BMS group changes the super-selection sector, a result long known for its Lorentz subgroup. It is hence spontaneously broken. Ward identities implied by the gauge invariance of the S-matrix generalize the standard results and lead to charge conservation and low energy theorems. Their validity does not require Lorentz invariance.

Correspondences between quantum and classical orbits Berry phases and Hannay angles for harmonic oscillator system

On the basis of quantum-classical correspondence for two-dimensional anisotropic oscillator, we study quantum-classical correspondence for two-dimensional rotation and translation harmonic oscillator system from both quantum-classical orbits and geometric phases. Here, the two one-dimensional oscillators refer to a common harmonic oscillator and a rotation and translation harmonic oscillator. In terms of the generalized gauge transformation, we obtain the stationary Lissajous orbits and Hannay's angle. On the other hand, the eigenfunctions and Berry phases are derived analytically with the help of time-dependent gauge transformation. We may draw the conclusion that the nonadiabatic Berry phase in the original gauge is-n times the classical Hannay's angle, here n is the eigenfunction index. As a matter of fact, the quantum geometric phase and the classical Hannay's angle have the same nature according to Berry. Finally, by using the SU(2) coherent superposition of degenerate two-dimensional eigenfunctions for a fixed energy value, we construct the stationary wave functions and show that the spatial distribution of wave-function probability clouds is in excellent accordance with the classical orbits, indicating the exact quantum-classical correspondence. We also demonstrate the quantum-classical correspondences for the geometric phase-angle and the quantum-classical orbits in a unified form.

Some Remarks on Quantum Mechanics in a Curved Spacetime, Especially for a Dirac Particle

Some precisions are given about the definition of the Hamiltonian operator H and its transformation properties, for a linear wave equation in a general spacetime. In the presence of time-dependent unitary gauge transformations, H as an operator depends on the gauge choice. The other observables of QM and their rates also become gauge-dependent unless a proper account for the gauge choice is done in their definition. We show the explicit effect of these non-uniqueness issues in the case of the Dirac equation in a general spacetime with the Schwinger gauge. We show also in detail why, the meaning of the energy in QM being inherited from classical Hamiltonian mechanics, the energy operator and its mean values ought to be well defined in a general spacetime.

Equations of motion for natural orbitals of strongly driven two-electron systems

Natural orbital theory is a computationally useful approach to the few and many-body quantum problem. While natural orbitals are known and applied since many years in electronic structure applications, their potential for time-dependent problems is being investigated only since recently. Correlated two-particle systems are of particular importance because the structure of the two-body reduced density matrix expanded in natural orbitals is known exactly in this case. However, in the time-dependent case the natural orbitals carry time-dependent phases that allow for certain time-dependent gauge transformations of the first kind. Different phase conventions will, in general, lead to different equations of motion for the natural orbitals. A particular phase choice allows us to derive the exact equations of motion for the natural orbitals of any (laser-) driven two-electron system explicitly, i.e., without any dependence on quantities that, in practice, require further approximations. For illustration, we solve the equations of motion for a model helium system. Besides calculating the spin-singlet and spin-triplet ground states, we show that the linear response spectra and the results for resonant Rabi flopping are in excellent agreement with the benchmark results obtained from the exact solution of the time-dependent Schr\"odinger equation.

Nonadiabatic Geometric Phase in Composite Systems and Its Subsystem

We point out that the time-dependent gauge transformation technique may be effective in investigating the nonadiabatic geometric phase of a subsystem in a composite system. As an example, we consider two uniaxially coupled spin —1/2 particles with one of particles driven by rotating magnetic Geld. The influences of coupling and precession frequency of the magnetic Geld on geometric phase are also discussed in detail.

THE CONSISTENCY OF THE ADIABATIC AND EXACT GEOMETRIC PHASES

With the help of the time-dependent gauge transformation, by which a set of time-invariant eigenstates of the Hamiltonian can be obtained, the geometric phase of the time-dependent system is obtained analytically. In the limit of the adiabatic condition, this phase exactly reduces to the adiabatic geometric phase. To illustrate the present theory, two solvable time-dependent systems are considered.

The geometric phase of the quantum systems with slow but finite rate of the external time-dependent field

With the help of the time-dependent gauge transformation technique, we have studied the geometric phase of a spin-half particle in a rotating magnetic field. We have found that the slow but finite frequency of the rotating magnetic field will make the difference between the adiabatic geometric phase and the exact geometric phase. When the frequency is much smaller than the energy space and the adiabatic condition is perfectly guaranteed, the adiabatic approximation geometric phase is exactly consistent with the adiabatic geometric phase. A simple relation for the accuracy of the adiabatic approximation is given in terms of the changing rate of the frequency of the rotating magnetic field and the energy level space.

Spin polarized localization between two quantum dots

We study the dynamics of two electrons confined in double quantum dots driven by a rotating magnetic field. An exact solution of the Schrödinger equation with explicitly time-dependent Hamiltonian is obtained in terms of the time-dependent gauge transformation. The spin separation is investigated for various initial states.

Spin‐charge rotating local reference frames: a unified U(2) = U(1) ⊗ SU(2) approach to the interacting electrons

AbstractA spin‐charge unifying description for the Hubbard model based on the time dependent local gauge transformations is developed. The collective variables for charge and spin are isolated in the form of the space‐time fluctuating U(1) phase field and rotating spin quantization axis governed by the SU(2) symmetry, respectively. As a result interacting electrons appear as a composite objects consisting of bare fermions with attached U(1) and SU(2) gauge fields. We elaborate on the microscopic origins of the effective action with the Coulomb interaction that contain topological theta terms. Furthermore, we unravel the link between non‐trivial multiply‐connected topological structure of the U(2) = U(1) ⊗ SU(2) configurational space for gauge fields and the instanton contribution to the statistical sum. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Effective theory of Wilson lines and deconfinement

To study the deconfining phase transition at nonzero temperature, I outline the perturbative construction of an effective theory for straight, thermal Wilson lines. Certain large, time dependent gauge transformations play a central role. They imply the existence of interfaces, which can be used to determine the form of the effective theory as a gauged, nonlinear sigma model of adjoint matrices. Especially near the transition, the Wilson line may undergo a Higgs effect. As an adjoint field, this can generate eigenvalue repulsion in the effective theory.

Dynamics and geometric phase of two spins with exchange coupling in a rotating magnetic field

An exact solution of time-dependent Schrödinger equation for two coupled spins of arbitrary values in a rotating magnetic field is obtained by means of a time-dependent gauge transformation technique which results in both exact and adiabatic geometric phases without the questionable request of remaining in the initial instantaneous eigenstate during the time-evolution. Thus our derivation of the Berry phase is free of inconsistency caused by the standard adiabatic theorem.

BERRY PHASE IN AN EFFECTIVE SU(1,1) SYSTEM

We study the dynamics of a quantum system with a time-dependent Hamiltonian which is given by a linear combination of SU(1,1) generators and coupled with a heat bath. An effective Hamiltonian of this dissipative system is derived. With the help of a time-dependent gauge transformation, we obtain the exact solutions of the time-dependent Schrödinger equation, from which the time-evolution operator and the non-adiabatic and adiabatic Berry phases, which depend on time, are calculated explicitly. In the weak dissipation limit, an additional term besides the original Berry phase is found. The additional phase does not have a geometrical meaning due to the dissipation.

Gauge-covariant properties of a linear nonautonomous quantum system: time-dependent even and odd coherent states

A new method for finding the dynamical invariant for a general time-dependent harmonic oscillator is proposed by making use of two linearly independent solutions to the classical equation of motion. It is shown that the dynamical invariant for different gauged Hamiltonians are connected by time-dependent gauge transformations. Therefore, the representation whose bases are the instantaneous eigenstates of the invariant operator is a good representation for the quantum nonautonomous system. In this representation, the wavefunction of the system is gauge covariant and thus any observable physical effect is naturally independent of the choice of gauged Hamiltonians. The exact even and odd coherent states for a time-dependent harmonic oscillator are constructed in terms of these gauge covariant bases. The harmonic oscillator with periodically varying frequency is treated as a demonstration of our general approach.

Time Evolution and Berry Phases of a Time－Dependent Oscillator in Fin ite－Dimensional Hilbert Space

In this paper we investigate the time evolution of a time－dependent oscillator in finite－dimensional Hilbert space. By performing appropriate time－dependent gauge transformation closed formulas for the time evolution of quantum states an d Berry phases are obtained. Finally we discuss the influence of the driving te rm in the Hamiltonian on the squeezed feature of coherent states.

Introducing nonlinear gauge transformations in a family of nonlinear Schrödinger equations.

In earlier work we proposed a family of nonlinear time-evolution equations for quantum mechanics associated with certain unitary group representations [Doebner and Goldin, Phys. Lett. A 162, 397 (1992); J. Phys. A 27, 1771 (1994)]. Such nonlinear Schr\"odinger equations are expected to describe irreversible and dissipative quantum systems. Here we introduce and justify physically the group of nonlinear gauge transformations necessary to interpret our equations. We determine the parameters that are actually gauge invariant and describe some of their properties. Our conclusions contradict, at least in part, the view that any nonlinearity in quantum mechanics leads to unphysical predictions. We also show how time-dependent nonlinear gauge transformations connect our equations to those proposed by Kostin [J. Chem. Phys. 57, 3589 (1972)] and by Bialynicki-Birula and Mycielski [Ann. Phys. 100, 62 (1976)]. We believe our approach to be a fundamental generalization of the usual notions about gauge transformations in quantum mechanics. \textcopyright{} 1996 The American Physical Society.

On topological soliton dynamics in multidimensional ferromagnetic continuum

A multidimensional model for a ferromagnetic continuum with hydrodynamical properties, which can be regarded as a modified Landau-Lifshitz equation, is presented. The treatment of some physical examples suggests that the fluid vorticity has to be proportional to the magnetic topological current. The model can be written in the Hirota bilinear form. In two spatial dimensions, the existence of a positive definite ‘energy’ functional is shown. The Bogomol'nyi inequality leads to the self-dual equations of the model, which can be expressed by the Liouville equation. By using time-dependent gauge transformations, a wide class of solutions can be generated. These are, in general, associated with the linear problem of the modified Kadomtsev-Petviashvili equation. In some particular cases, the isolated vortices can move along arbitrary trajectories on the plane. The quantization problem of the time-dependent vortex configurations is briefly discussed, in relation to the possible evaluation of their energy spectrum.

Time-dependent gauge transformations and Berry's phase

The time-dependent gauge transformation is discussed in the context of modified Lagrangian and Hamiltonian formalisms in which time is treated as a dynamical coordinate. It is shown that Berry's phase is invariant under both time-dependent unitary and generalized gauge transformations. The Berry phase is therefore unique. It is then demonstrated that the Berry phase can be shifted into the dynamical part of the total phase by choosing a new set of instantaneous eigenstates which are associated with a different gauge. This procedure breaks the gauge equivalence and also changes the dynamics of the system considered.

Time-dependent gauge transformations and Berry's phase: Jiu-Qing Liang. Department of Physics, University of Kaiserslautern, 6750 Kaiserslautern, Germany; China Center of Advanced Science and Technology (World Laboratory), P.O. Box 8730, Beijing, People's Republic of China; and Institute of Theoretical Physics, Shanxi University, Taiyuan, Shanxi 030006, People's Republic of China; and H. J. W. Müller-Kirsten. Department of Physics, University of Kaiserslautern, 6750

Time-dependent gauge transformations and Berry's phase: Jiu-Qing Liang. Department of Physics, University of Kaiserslautern, 6750 Kaiserslautern, Germany; China Center of Advanced Science and Technology (World Laboratory), P.O. Box 8730, Beijing, People's Republic of China; and Institute of Theoretical Physics, Shanxi University, Taiyuan, Shanxi 030006, People's Republic of China; and H. J. W. Müller-Kirsten. Department of Physics, University of Kaiserslautern, 6750