Energy-Levels Crossing and Radial Dirac Equation: Supersymmetry and Quasi-Parity Spectral Signatures

The (3+1)-dimensional Dirac equation with position dependent mass in 4-vector electromagnetic fields is considered. Using two over-simplified examples (the Dirac-Coulomb and Dirac-oscillator fields), we report energy-levels crossing as a spectral property or as an effect of the hidden supersymmetric quantum mechanical language and/or quasi-parity signatures. Under different settings of the related interactions’ way-of-coupling into Dirac equation, it is observed that the two ultimate/effective descendents, Dirac-Coulomb and Dirac-oscillator, exhibit different conditions on the energy-levels crossings.     

A Remark on the Kramers Problem

We present new point of view on the old problem, the Kramers problem. The passages from the Fokker–Planck equation to the Smoluchowski equation, including corrections to the Smoluchowski current, is treated through an asymptotic expansion of the solution of the stochastic dynamical equations. The case of an extremely weak force of friction is also discussed.     

Kramers time in weakly nonpotential systems

We consider a bistable Fokker-Planck system with a known stationary distribution and a small nonpotential part in the drift force. We perform a perturbation calculation of its Kramers time, _K, and compare it with the corresponding time, _K ⁽⁰⁾ , for the potential system which has the same stationary distribution. We show that _K/ _K ⁽⁰⁾ depends only on the properties of the drift force close to the saddle-point.The authors would like to dedicate this work to their colleagues Y. Orlov, R. Nazarian, and V. Brailovski.     

Supersymmetry, PT-symmetry and spectral bifurcation

We demonstrate that large class of PT-symmetric complex potentials, which can have isospectral real partner potentials, possess two different superpotentials. In the parameter domain, where the superpotential is unique, the spectrum is real and shape-invariant, leading to translational shift in a suitable parameter by real units. The case of two different superpotentials, leading to same potential, yields broken PT-symmetry, the energy spectra in the two phases being separated by a bifurcation. Interestingly, these two superpotentials generate the two disjoint sectors of the Hilbert space. In the broken case, shape invariance produces complex parametric shifts.     

Simple generalization of Kramers theory to finite barrier height in spatial diffusion regime

Motivated by the escape process at low reduced barrier heights (measured in units of $k_{B}T$) is still a stationary one, the Kramers theoretical method in spatial diffusion regime should be applicable to this process. The Kramers theory is generalized to finite barrier height in a simple manner. The integration constant is redetermined by introducing metastable equilibrium state concept and continuous condition of the probability at the joint point of the potential barrier and potential well. The parabolic barrier with local frequency is replaced by a parabolic barrier with nonlocal frequency. The modified Kramers theory is confirmed by a cubic potential case. The maximal relative error in the spatial diffusion regime is less than 3% for the applied parameters.     

Charged Brownian particles: Kramers and Smoluchowski equations and the hydrothermodynamical picture

We consider a charged Brownian gas under the influence of external and non-uniform electric, magnetic and mechanical fields, immersed in a non-uniform bath temperature. With the collision time as an expansion parameter, we study the solution to the associated Kramers equation, including a linear reactive term. To the first order we obtain the asymptotic (overdamped) regime, governed by transport equations, namely: for the particle density, a Smoluchowski-reactive like equation; for the particle’s momentum density, a generalized Ohm’s-like equation; and for the particle’s energy density, a Maxwell-Cattaneo-like equation. Defining a nonequilibrium temperature as the mean kinetic energy density, and introducing Boltzmann’s entropy density via the one particle distribution function, we present a complete thermohydrodynamical picture for a charged Brownian gas. We probe the validity of the local equilibrium approximation, Onsager relations, variational principles associated to the entropy production, and apply our results to: carrier transport in semiconductors, hot carriers and Brownian motors. Finally, we outline a method to incorporate non-linear reactive kinetics and a mean field approach to interacting Brownian particles.     

Supersymmetry and Integrability in Planar Mechanical Systems

We present an N=2-supersymmetric mechanical system whose bosonic sector, with two degrees of freedom, exhibits the most general possible supersymmetric fourth order potential, including the interesting case of SU(2) Yang–Mills theory. The Painlevé test is adopted to discuss integrability and we focus on the rôle of supersymmetry and parity invariance in two space dimensions for the attainment of integrable or non-integrable models, with some remarks on the chaotic behavior. Our result shows that, for the model studied here, the relationships among the parameters, as imposed by supersymmetry, restrict the parameter space in such a way that the reduction on its non-integrable sector is much more severe than on its integrable sector (especially on the non-separable subset of the latter), thus suggesting that supersymmetry may favor (mainly non-separable) integrability.     

High pressure studies of elementary chemical dynamics

Abstract

Studies of chemical reactions over large density ranges provide a simple access to a variety of static and dynamic interactions between reactants and solvents. The interpretation of the observed “activation volumes” by transition state theory is shown to be inadequate. Kramers-Smoluchowski theory provides a more suitable starting point of the analysis. However, multidimensional barrier and cluster effects also have to be taken into account. Viscoelastic effects describe the transition into the solid phase. Photoisomerization studies of diphenylpolyenes illustrate barrier crossing processes. The effects of cage dynamics and diffusion control are demonstrated with bromine and iodine photolysis experiments.     

Supersymmetry and Solution of Dirac Equation with Vector and Scalar Potentials

The Dirac equations with vector and scalar potentials of the Coulomb types in two and three dimensions are solved using the supersymmetric quantum mechanics method. For the system of such potentials, the analytical expressions of the matrix dements for both position and momentum operators are obtained.     

Supersymmetry for nuclear cluster systems

A supersymmetry scheme is proposed for nuclear cluster systems. The bosonic sector of the superalgebra describes the relative motion of the clusters, while its fermionic sector is associated with their internal structure. An example of core + α configurations is discussed in which the core is a p-shell nucleus and the underlying superalgebra is U(4|12). The α-cluster states of the nuclei ²⁰Ne and ¹⁹F are analyzed and correlations between their spectra, electric quadrupole transitions, and one-nucleon transfer reactions are interpreted in terms of U(4|12) supersymmetry. Received: 14 September 2001 / Accepted: 26 October 2001     

Quaternion Dirac Equation and Supersymmetry

Quaternion Dirac equation has been analyzed and its supersymmetrization has been discussed consistently. It has been shown that the quaternion Dirac equation automatically describes the spin structure with its spin up and spin down components of two component quaternion Dirac spinors associated with positive and negative energies. It has also been shown that the supersymmetrization of quaternion Dirac equation works well for different cases associated with zero mass, nonzero mass, scalar potential and generalized electromagnetic potentials. Accordingly we have discussed the splitting of supersymmetrized Dirac equation in terms of electric and magnetic fields.     

Supersymmetry and SWKB Approach to Dirac Equation with Hyperbolic Scarf Potential

By using the supersymmetric quantum mechanics and shape invariance concept, we study the Dirac equation with the hyperbolic Scarf potential and the exact energy spectrum is obtained. Also, we calculate the bound state energy eigenvalues by using the supersymmetric WKB approximation approach so that we get the same results.     

Microcanonical variational transition-state theory for reaction rates in dissipative systems

Upper bounds for the classical escape rate of a particle trapped in a metastable well and interacting with a dissipative medium are derived based on the periodic orbits of a reduced two-degree-of-freedom Hamiltonian involving the unstable normal mode and a collective bath mode. It is shown that even in what is usually thought of as the spatial diffusion limit the reactive flux can involve an energy diffusion term due to energy transfer from the dissipative media, in addition to the standard spatial diffusion term.     

Generalized spin and pseudo-spin symmetry: Relativistic extension of supersymmetric quantum mechanics

We consider the Dirac equation in 1+1$1+1$ space–time dimension with vector, scalar and pseudo-scalar coupling. In the traditional spin (or pseudo-spin) symmetry, the difference between (or sum of) the scalar and vector potentials is a constant. Here, however, we introduce an alternative symmetry where the scalar or pseudo-scalar potential is proportional to the vector potential. This leads to a model with significant extensions to supersymmetric quantum mechanics. We present a formal solution of the problem but give explicit analytic results for specific examples.     

两能级原子主方程和激光通道主方程的解之间的超对称性

提出了研究原子演化的Ket-Bra纠缠态方法,并用此方法给出了原子主方程的Kraus算符形式的解.在得到此新解后,发现它和激光通道主方程的解形式相似,表现了光场算符a,a~(+)与原子算符σ_-,σ_+之间具有某种超对称性.通过进一步的探讨,寻找到了Pauli算符的多种Bose表示.     

On “Comment on Supersymmetry, PT-symmetry and spectral bifurcation”

In “Comment on Supersymmetry, PT-symmetry and spectral bifurcation” [1], Bagchi and Quesne correctly show the presence of a class of states for the complex Scarf-II potential in the unbroken PT-symmetry regime, which were absent in [2]. However, in the spontaneously broken PT-symmetry case, their argument is incorrect since it fails to implement the condition for the potential to be PT-symmetric: C^PT[2(A − B) + α] = 0. It needs to be emphasized that in the models considered in [2], PT is spontaneously broken, implying that the potential is PT-symmetric, whereas the ground state is not. Furthermore, our supersymmetry (SUSY)-based ‘spectral bifurcation’ holds independent of the sl(2) symmetry consideration for a large class of PT-symmetric potentials.     

Dynamical Field Inference and Supersymmetry

Knowledge on evolving physical fields is of paramount importance in science, technology, and economics. Dynamical field inference (DFI) addresses the problem of reconstructing a stochastically-driven, dynamically-evolving field from finite data. It relies on information field theory (IFT), the information theory for fields. Here, the relations of DFI, IFT, and the recently developed supersymmetric theory of stochastics (STS) are established in a pedagogical discussion. In IFT, field expectation values can be calculated from the partition function of the full space-time inference problem. The partition function of the inference problem invokes a functional Dirac function to guarantee the dynamics, as well as a field-dependent functional determinant, to establish proper normalization, both impeding the necessary evaluation of the path integral over all field configurations. STS replaces these problematic expressions via the introduction of fermionic ghost and bosonic Lagrange fields, respectively. The action of these fields has a supersymmetry, which means there exists an exchange operation between bosons and fermions that leaves the system invariant. In contrast to this, measurements of the dynamical fields do not adhere to this supersymmetry. The supersymmetry can also be broken spontaneously, in which case the system evolves chaotically. This affects the predictability of the system and thereby makes DFI more challenging. We investigate the interplay of measurement constraints with the non-linear chaotic dynamics of a simplified, illustrative system with the help of Feynman diagrams and show that the Fermionic corrections are essential to obtain the correct posterior statistics over system trajectories.     

Diffusion equations and the time evolution of foreign exchange rates

We investigate which type of diffusion equation is most appropriate to describe the time evolution of foreign exchange rates. We modify the geometric diffusion model assuming a non-exponential time evolution and the stochastic term is the sum of a Wiener noise and a jump process. We find the resulting diffusion equation to obey the Kramers–Moyal equation. Analytical solutions are obtained using the characteristic function formalism and compared with empirical data. The analysis focus on the first four central moments considering the returns of foreign exchange rate. It is shown that the proposed model offers a good improvement over the classical geometric diffusion model.     

The Complex Bateman Equation

The general solution to the complex Bateman equation is constructed. It is given in implicit form in terms of a functional relationship for the unknown function. The known solution of the usual Bateman equation is recovered as a special case.