Non-relativistic expansion of single-nucleon Dirac equation:Comparison between Foldy-Wouthuysen transformation and similarity renormalization group

By following the Foldy-Wouthuysen(FW) transformation of the Dirac equation,we derive the exact analytic expression up to the 1/M~4 order for general cases in the covariant density functional theory.The results are compared with the corresponding ones derived from another novel non-relativistic expansion method,the similarity renormalization group(SRG).Based on this comparison,the origin of the difference between the results obtained with the FW transformation and the SRG method is explored.     

Fluid-dynamical representations of the Dirac equation

A relative kinetic mass operator is defined bym =c ⁻²·(E −eΦ), and it is shown that bt using it in a symmetric form one can correlate the (charge) velocity operatorα in the Dirac theory exactly with the general quantum mechanical momentum —ih∇. Then the net force, defined as the rate of change of the relative momentum with time, is exactly equal to the Lorentz force. The contribution due to the time variation of mass equals the negative of space variation of the scalar potential, the Newtonian force, whereas the time variation of the charge current absorbs the entire vector potential dependence. The analogous Euler equations can be written either in terms of the charge current or in terms of the mass current. For a many particle system one needs the usual net single particle parameters and the consideration of both the direct and exchange contributions of the two particle interaction. These Euler equations yield two different conditions of the stationary state. It is shown that the charge-current condition is necessary but not sufficient, whereas the mass-current condition retains the appropriate scalar potential dependence. These two conditions are compared for the spherically symmetric case. The charge density, charge current and relative mass current are tabulated for atomic spinors. Differences between the quantum and classical forces for the H ₂ ⁺ molecular ion exhibit the inadequacy of ordinary atomic spinor basis in forming molecular spinors.     

On the excited state wave functions of Dirac fermions in the random gauge potential

In the last decade, it was shown that the Liouville field theory is an effective theory of Dirac fermions in the random gauge potential (FRGP). We show that the Dirac wave functions in FRGP can be written in terms of descendents of the Liouville vertex operator. In the quasiclassical approximation of the Liouville theory, our model predicts that the localization length ξ scales with the energy E as $ \xi \sim E^{{{ - b^2 } \mathord{\left/ {\vphantom {{ - b^2 } {(1 + b^2 )^2 }}} \right. \kern-\nulldelimiterspace} {(1 + b^2 )^2 }}} $ \xi \sim E^{{{ - b^2 } \mathord{\left/ {\vphantom {{ - b^2 } {(1 + b^2 )^2 }}} \right. \kern-\nulldelimiterspace} {(1 + b^2 )^2 }}} , where b is the strength of the disorder. The self-duality of the theory under the transformation b → 1/b is discussed. We also calculate the distribution functions of t ₀ =|ψ ₀(x)|², (i.e. p(t ₀); ψ ₀(x) is the ground state wave function), which behaves as the log-normal distribution function. It is also shown that in small t ₀, p(t ₀) behaves as a chi-square distribution.     

Eigen spectra and wave functions of the massless Dirac fermions under the nonuniform magnetic fields in graphene

In this research, firstly, by using the new form of Dirac-Weyl equation and the series method with submitting more suitable details, the energy spectrum and wave functions of the massless Dirac fermions are calculated under the inhomogeneous and q-deformed spatially magnetic fields. Although, we discussed about the results of the energy levels, further, we obtained the wave function as the Hessenberg determinant with calculating the elements of it as exact. On the other hand, by using the Mellin-Barnes integral representation and Hurwitz zeta function, we have achieved the thermodynamic physical quantities of the Dirac-Weyl fermions in the absence of a magnetic field for inside of the graphene quantum dot. Finally, our numerical results for the wave functions and probability densities are presented too.     

On the geometry of some unitary Riemann surface braid group representations and Laughlin-type wave functions

In this note we construct the simplest unitary Riemann surface braid group representations geometrically by means of stable holomorphic vector bundles over complex tori and the prime form on Riemann surfaces. Generalised Laughlin wave functions are then introduced. The genus one case is discussed in some detail also with the help of noncommutative geometric tools, and an application of Fourier–Mukai–Nahm techniques is also given, explaining the emergence of an intriguing Riemann surface braid group duality.     

Dirac wave equation in the de Sitter universe

We present and discuss the Dirac wave equation in the de Sitter universe. This equation is obtained by factoring the second-order Casimir invariant operator associated to the Fantappié-de Sitter group.     

Duality between coordinates and wave functions on noncommutative space

The relation between coordinates and the solutions of the stationary Schrödinger equation in the noncommutative algebra of functions on R^2N is discussed. We derive this relation for a certain class of wave functions for which the quantum prepotentials depend linearly on the coordinates similarly to the commutative case. Also, the differential equation satisfied by the prepotentials is given.     

Solution of the Dirac equation in plane wave metric backgrounds

We present a new solution of the Dirac equation in the background of a plane wave metric. We examine the relation between sections of the exterior and Clifford bundles of a (pseudo-)Riemannian manifold. A spinor calculus is established and used to investigate a new solution of the Dirac equation lying in a minimal left ideal characterized by a certain idempotent projector.     

Connection formulas between the different QCD orders in application to the valence parton distribution functions

Formulas directly connecting parton distribution functions (PDFs) at the leading (LO) and next to leading (NLO) QCD orders are applied with respect to both unpolarized and polarized valence PDFs. It is shown that the connection formulas allow without any restriction on the allowed Q ² range for the analyzed data to produce improved LO results on valence PDFs, which strongly differ from the standard parametrizations on these quantities, and which could be obtained within the standard approach only by using the data produced at very high Q ² values (that is hardly possible in reality).     

非对易相空间中的狄拉克振子的能级和Wigner函数

非对易几何、弦论和圈量子引力理论的发展,使非对易空间受到越来越多的关注.非对易量子理论不同于平常的量子理论,它是弦尺度下的特殊的物理效应,处理非对易量子力学问题需要特殊方法.本文首先介绍了Moyal方程与Wigner函数,利用Moyal-Weyl乘法与Bopp变换将H(x,p)变换成^H(^x,^p),考虑坐标—坐标、动量—动量的非对易性,实现对非对易相空间中星乘本征方程的求解.并利用非对易相空间量子力学的代数关系,讨论了非对易相空间中狄拉克振子的Wigner函数和能级,研究结果发现非对易相空间中狄拉克振子的能级明显依赖于非对易参数.     

Connection between light cone singularities and the asymptotic behaviour of the moments of the structure functions

On the basis of the Dyson-Jost-Lehmann representation a light-cone type expansion of the matrix elements of the current commutator can be justified. Suitable moments of the structure functions being defined, the connection between the asymptotic behaviour of the moments as Q² → ∞ and the behaviour of the expansion coefficients as x² → 0 is investigated. To solve this problem a restriction of the allowed class of generalized functions is needed. Then a one-to-one correspondence between both limits can be established.     

Energy-level and wave functions of two moving charged particles with elastic coupling derived by virtue of the entangled state representations

We derive the energy-level and wave functions for two moving charged particles with elastic coupling, using two mutually conjugate entangled state representations.     

A relationship between the Dirac and Maxwell equations

It is shown that an analogue of Dirac's equation may be constructed from Milner's extension of Maxwell's equations and a comparison of the algebraic and physical features of the equations made.     

On the theory of interacting fields in the Foldy-Wouthuysen representation

The overview is devoted to quantum electrodynamics (QED) and the Standard Model in the Foldy-Wouthuysen representation. The Hamiltonian H _FW in the form of a power series in charge e is obtained as applied to the electromagnetic interaction in the FW representation. Quantum electrodynamics in lowest-order perturbation theory is examined. Calculations of specific QED processes are presented. For external fermion lines (p _f ² = m ²), a possibility to expand the scattering matrix, in powers of the coupling constant with matrix elements, not including fermion propagators, is shown. To take into account particle-antiparticle interaction, a modification of the Foldy-Wouthuysen representation is proposed. Fermions in the modified FW representation can be in two states that are characterized by the sign of a third component of the isotopic spin T _f ³ . The sign of T _f ³ is connected with the sign at mass terms in the modified Hamiltonian H _FW. Real fermions (p _f ² = m _f ² ), as well as antifermions, can interact with each other, while real fermions with a given sign of T _f ³ can only interact with real antifermions with the opposite sign of T _f ³ , and vice versa. The formulation of the Standard Model in the FW representation does not necessarily require an interaction of Higgs bosons with fermions. In this approach, the role of Higgs bosons narrows considerably as they are responsible only for gauge invariance of the theory and interact only with gauge bosons. Quantum electrodynamics in the FW representation is invariant under C, P, and T transformations. Weak interaction does not conserve C and P parity, but conserves combined CP parity. The theory allows a connection of CP violation and total or partial violation of isotopic symmetry in the modified Foldy-Wouthuysen representation.     

Phase representations of the density matrix and quantum distribution functions

A broad class of representations for the density matrix of phase-space functions is proposed. It is shown that all the known quantum distribution functions are particular cases of this representation. The corresponding equations are found. An examination is made of the conditions under which the phase representation may be a quantum distribution function.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii Fizika, Vol. 12, No. 4, pp. 111–115, April, 1969.The author thanks Professor Ya. P. Terletskii for valuable advice.     

Tau functions for the Dirac operator on the Poincaré disk

In this paper we define tau functions for holonomic fields associated with the Dirac operator on the Poincaré disk. The deformation analysis of the tau functions is worked out and in the case of the two point function, the tau function is expressed in terms of a Painlevé function of type VI.     

Dirac particle in a plane wave field and the semi‐classical approximation

In this paper we investigate the influence of photon represented by plane wave field on Dirac particle in the context of path integral approach given by Fradkin and Gitman formalism. In our case, although the action relative to Dirac particle in plane wave field seems to be non quadratic, the result obtained by semi‐classical approach is the same as that found by an exact calculation. Hence; when we add the plane wave field to any quadratic actions related to Fradkin and Gitman approach, the total action behaves like quadratic.