Quantum Propagator Derivation for the Ring of Four Harmonically Coupled Oscillators

The ring model of the coupled oscillator has enormously studied from the perspective of quantum mechanics. The research efforts on this system contribute to fully grasp the concepts of energy transport, dissipation, among others, in mesoscopic and condensed matter systems. In this research, the dynamics of the quantum propagator for the ring of oscillators was analyzed anew. White noise analysis was applied to derive the quantum mechanical propagator for a ring of four harmonically coupled oscillators. The process was done after performing four successive coordinate transformations obtaining four separated Lagrangian of a one-dimensional harmonic oscillator. Then, the individual propagator was evaluated via white noise path integration where the full propagator is expressed as the product of the individual propagators. In particular, the frequencies of the first two propagators correspond to degenerate normal mode frequencies, while the other two correspond to non-degenerate normal mode frequencies. The full propagator was expressed in its symmetric form to extract the energy spectrum and the wave function.

Calculation of critical index η of the φ3-theory in four-loop approximation by the conformal bootstrap technique

The method of calculation of [Formula: see text]-expansion in model of scalar field with [Formula: see text]-interaction based on conformal bootstrap equations is proposed. This technique is based on self-consistent skeleton equations involving full propagator and full triple vertex. Analytical computations of the Fisher’s index [Formula: see text] are performed in four-loop approximation. The three-loop result coincides with one obtained previously by the renormalization group equations technique based on calculation of a larger number of Feynman diagrams. The four-loop result agrees with its numerical value obtained by other authors.

Optimal Perturbations in Quasigeostrophic Turbulence

This paper tests the hypothesis that optimal perturbations in quasigeostrophic turbulence are excited sufficiently strongly and frequently to account for the energy-containing eddies. Optimal perturbations are defined here as singular vectors of the propagator, for the energy norm, corresponding to the equations of motion linearized about the time-mean flow. The initial conditions are drawn from a numerical solution of the nonlinear equations associated with the linear propagator. Experiments confirm that energy is concentrated in the leading evolved singular vectors, and that the average energy in the initial singular vectors is within an order of magnitude of that required to explain the average energy in the evolved singular vectors. Furthermore, only a small number of evolved singular vectors (4 out of 4000) are needed to explain the dominant eddy structure when total energy exceeds a predefined threshold. The initial singular vectors explain only 10% of such events, but this discrepancy was similar to that of the full propagator, suggesting that it arises primarily due to errors in the propagator. In the limit of short lead times, energy conservation can be expressed in terms of suitable singular vectors to constrain the energy distribution of the singular vectors in statistically steady equilibrium. This and other connections between linear optimals and nonlinear dynamics suggests that the positive results found here should carry over to other systems, provided the propagator and initial states are chosen consistently with respect to the nonlinear system.

THE SU(2) NONLINEAR σ MODEL IN 2+1 DIMENSIONS: PERTURBATION THEORY IN A POLYNOMIAL FORMULATION

We construct a perturbation theory for the SU(2) nonlinear σ model in 2+1 dimensions using a polynomial, first-order formulation, where the variables are a non-Abelian vector field Lμ [the left SU(2) current], and a non-Abelian pseudovector field θμ, which imposes the condition Fμv(L)=0. The coordinates on the group do not appear in the Feynman rules, but their scattering amplitudes are easily related to those of the currents. We show that all the infinities affecting physical amplitudes at one-loop order can be cured by normal-ordering, presenting the calculation of the full propagator as an example of an application.

GAUGE INVARIANCE AND WARD-TAKAHASHI IDENTITY IN QUANTUM MECHANICS

In a Lagrange formulation of nonrelativistic quantum mechanics gauge invariance corresponds to a local U(1) symmetry of the Lagrangian density. We review how this gauge symmetry yields the electromagnetic interaction of the free and the locally interacting N-body system and why nonlocal interactions require additional exchange contributions. A nonrelativistic Ward-Takahashi identity is then derived. This relates electromagnetic operators to the full propagator of the particles in question, which may be off their energy shell.