Relativistic n-body wave equations in scalar quantum field theory

The variational method in a reformulated Hamiltonian formalism of Quantum Field Theory (QFT) is used to derive relativistic n-body wave equations for scalar particles (bosons) interacting via a massive or massless mediating scalar field (the scalar Yukawa model). Simple Fock-space variational trial states are used to derive relativistic n-body wave equations. The equations are shown to have the Schrödinger non-relativistic limits, with Coulombic interparticle potentials in the case of a massless mediating field and Yukawa interparticle potentials in the case of a massive mediating field. Some examples of approximate ground state solutions of the n-body relativistic equations are obtained for various strengths of coupling, for both massive and massless mediating fields.     

Dynamical symmetries of two-dimensional systems in relativistic quantum mechanics

The two-dimensional Dirac Hamiltonian with equal scalar and vector potentials has been proved commuting with the deformed orbital angular momentum L. When the potential takes the Coulomb form, the system has an SO(3) symmetry, and similarly the harmonic oscillator potential possesses an SU(2) symmetry. The generators of the symmetric groups are derived for these two systems separately. The corresponding energy spectra are yielded naturally from the Casimir operators. Their non-relativistic limits are also discussed.     

Hamiltonian Map to Conformal Modification of Spacetime Metric: Kaluza-Klein and TeVeS

It has been shown that the orbits of motion for a wide class of non-relativistic Hamiltonian systems can be described as geodesic flows on a manifold and an associated dual by means of a conformal map. This method can be applied to a four dimensional manifold of orbits in spacetime associated with a relativistic system. We show that a relativistic Hamiltonian which generates Einstein geodesics, with the addition of a world scalar field, can be put into correspondence in this way with another Hamiltonian with conformally modified metric. Such a construction could account for part of the requirements of Bekenstein for achieving the MOND theory of Milgrom in the post-Newtonian limit. The constraints on the MOND theory imposed by the galactic rotation curves, through this correspondence, would then imply constraints on the structure of the world scalar field. We then use the fact that a Hamiltonian with vector gauge fields results, through such a conformal map, in a Kaluza-Klein type theory, and indicate how the TeVeS structure of Bekenstein and Saunders can be put into this framework. We exhibit a class of infinitesimal gauge transformations on the gauge fields U_m(x){\mathcal{U}}_{\mu}(x) which preserve the Bekenstein-Sanders condition U_mU^m=-1{\mathcal{U}}_{\mu}{\mathcal{U}}^{\mu}=-1. The underlying quantum structure giving rise to these gauge fields is a Hilbert bundle, and the gauge transformations induce a non-commutative behavior to the fields, i.e. they become of Yang-Mills type. Working in the infinitesimal gauge neighborhood of the initial Abelian theory we show that in the Abelian limit the Yang-Mills field equations provide residual nonlinear terms which may avoid the caustic singularity found by Contaldi et al.     

Generalized coordinates molecular Hamiltonian

The transformation of the complete (non-relativistic) molecular Hamiltonian from Cartesian to generalized coordinates by the quantum-mechanical path outlined in our previous works is reviewed. Explicit expressions for vibrational kinetic energy matrix and vibration-rotation coupling matrices have been derived and some remarkable simplifications have been made to bring the Hamiltonian into a more practical form.     

BOUND STATE WAVEFUNCTIONS AND ANOMALOUS MAGNETIC MOMENTS OF LEPTONS

Assuming that leptons are composed of a heavy fermion and a heavy scalar boson and using Bethe-Salpeter equation, we conclude that in the non-relativistic limit the radius and , in particular, the anomalous magnetic moment of leptons can be sufficiently small provided that the interaction of the constituents is of vector type and that the fermion is much beavier than the scalar boson. Whereas the scalar type interaction can only give wavefunctions with large radius and anomalous magnetic moment.     

Adiabatic switching approach to multidimensional supersymmetric quantum mechanics for several excited states

We present a time-dependent method for determining several approximate excited-state energies and wave functions using a vectorial approach to multidimensional supersymmetric quantum mechanics. First, a vectorial approach is used to generate the tensor sector two Hamiltonian, which is isospectral with the original scalar sector one Hamiltonian above the ground state of the sector one Hamiltonian. We construct a time-dependent Hamiltonian interpolating between the scalar sector one Hamiltonian and the tensor sector two Hamiltonian. Then, we can adiabatically switch from the ground state of the sector one Hamiltonian to the ground state of the sector two Hamiltonian by solving the time-dependent Schrödinger equation. In addition, by employing an initial wave packet orthogonal to that leading to the ground state of sector two, we also obtain the first-excited state of sector two. Construction of the orthogonal sector one states is trivial due to the tensor nature of sector two. The ground and first-excited states of the sector two Hamiltonian can be used with the charge operator to obtain the first two excited state wave functions of the sector one Hamiltonian. Excellent computational results are obtained for two-dimensional nonseparable degenerate and nondegenerate systems.     

Semi-Relativistic Two-Body States of Spinless Particles with a Scalar-Type Interaction Potential

A semi-relativistic quantum approximation for mutual scalar interaction potentials is outlined and discussed.Equations are consistent with two-body Dirac equations for bound states of zero total angular momentum. Two-body effects near the non-relativistic limit for a linear scalar potential is studied in some detail.     

U(1) Invariant $F(\tilde{R})$ Hořava–Lifshitz gravity

This paper is devoted to the study of various aspects of projectable F(R) Hořava–Lifshitz (HL) gravity. We show that some versions of F(R) HL gravity may have stable de Sitter solution and unstable flat-space solution. In this case, the problem of scalar graviton does not appear because flat space is not vacuum state. Generalizing the U(1) HL theory proposed in , we formulate U(1) extension of scalar theory and of F(R) Hořava–Lifshitz gravity. The Hamiltonian approach for such the theory is developed in full detail. It is demonstrated that its Hamiltonian structure is the same as for the non-relativistic covariant HL gravity. The spectrum analysis performed around the flat background indicates the consistency of the theory because it contains a graviton with only transverse polarization. Finally, we analyze the spatially flat FRW equations for U(1) invariant F(R) Hořava–Lifshitz gravity.     

Relativistic breakdown of a non-linear classical field theory

It is shown that the relativistic generalization of a certain classical, non-linear, scalar, field theory displays unphysical behavior not found for the non-relativistic case. An explicit, unphysical solution of the relativistic equation is given. The different properties of relativistic and non-relativistic particle densities affect the behavior of the solutions in the respective cases.     

The molecular Hamiltonian with two-rotational-angle embedding

Our approach for the derivation of the exact (non-relativistic) translational–rovibronic Hamiltonian, based on the Hamiltonian operator in tensor form, is now extended to cases in which a body-fixed frame is defined via the introduction of two Euler angles (two-rotational-angle embedding). Diatomic molecules and diatom–diatom systems are considered as examples for this general formulation. For comparison, the three-rotational-angle embedding version of the diatom–diatom Hamiltonian is also derived.     

A non-relativistic wave equation of second order in time for scalar fields

In this paper a generalization of the traditional non-relativistic Schrödinger equation is considered. It is a wave equation of second order in time and fourth order in the space coordinates for scalar fields. The equation has certain features, which make it a closer analogue of the Klein-Gordon equation than the traditional Schrödinger equation. However, the equation maintains the non-relativistic relation between energy and momentum.The implications of this generalized wave equation and the quantized field theory based on it are studied. The theory can be shown to be charge symmetric and allows to introduce anti-particles and pair creation. We compare the Green functions for this theory with those of conventional non-relativistic quantum theory.The theory allows to formulate a transformation for charge conjugation. The PCT-theorem is valid for it. The usual spin-statistics connection holds.     

Relativistic and separable classical hamiltonian particle dynamics

We show within the Hamiltonian formalism the existence of classical relativistic mechanics of N scalar particles interacting at a distance which satisfies the requirements of Poincaré invariance, separability, world-line invariance and Einstein causality. The line of approach which is adopted here uses the methods of the theory of systems with constraints applied to manifestly covariant systems of particles. The study is limited to the case of scalar interactions remaining weak in the whole phase space and vanishing at large space-like separation distances of the particles. Poincaré invariance requires the inclusion of many-body, up to N-body, potentials. Separability requires the use of individual or two-body variables and the construction of the total interaction from basic two-body interactions. Position variables of the particles are constructed in terms of the canonical variables of the theory according to the world-line invariance condition and the subsidiary conditions of the non-relativistic limit and separability. Positivity constraints on the interaction masses squared of the particles ensure that the velocities of the latter remain always smaller than the velocity of light.     

The molecular Hamiltonian in internal coordinates

A systematic approach for the derivation of the exact translational–rovibronic (non-relativistic) Hamiltonian for a polyatomic molecule consisting of N nuclei and n electrons is presented. All coupling terms which contribute to the total energy are identified. The Hamiltonian is greatly simplified by taking the internal coordinates (bond lengths and bond angles) as the vibrational variables. The translational–rovibronic Hamiltonian of triatomic molecules are considered as an application for this general formulation.     

Perturbations of unitary representations of the Galilei group: Mass operators with continuous spectra

We present a systematic procedure for constructing mass operators with continuous spectra for a system of particles in a manner consistent with Galilean relativity. These mass operators can be used to construct what may be called point-form Galilean dynamics. As in the relativistic case introduced by Dirac, the point-form dynamics for the Galilean case is characterized by both the Hamiltonian and momenta being altered by interactions. An interesting property of such perturbative terms to the Hamiltonian and momentum operators is that, while having well-defined transformation properties under the Galilei group, they also satisfy Maxwell’s equations. This result is an alternative to the well-known Feynman-Dyson derivation of Maxwell’s equations from non-relativistic quantum physics.     

三价原子的非相对论能级结构理论

本文以原子结构的拉卡方法为基础,从三价原子的拉卡基出发,反复利用三个角动量耦合的基本关系以及3j、6j、9j符号的性质,具体推导了(n1 l1)(n2 l2)(n3l3)和(n1 l1)2(n3l3)组态下三价原子非相对论哈密顿矩阵元的计算式,除径向部分用Slater-Condon径向积分表示以外,完成了所有的角向积分与自旋求和;简要举例说明了如何利用该矩阵元计算式推导三价原子非相对论谱项能量表达式以及利用变分原理确定其中的Slater-Condon径向积分,从而求出具体的谱项能量数值.可以说,本文已建立起三价原子非相对论性能级结构的一般理论.     

Perturbations of unitary representations of finite dimensional Lie groups

In quantum physical theories, interactions in a system of particles are commonly understood as perturbations to certain observables, including the Hamiltonian, of the corresponding interaction-free system. The manner in which observables undergo perturbations is subject to constraints imposed by the overall symmetries that the interacting system is expected to obey. Primary among these are the spacetime symmetries encoded by the unitary representations of the Galilei group and Poincaré group for the non-relativistic and relativistic systems, respectively. In this light, interactions can be more generally viewed as perturbations to unitary representations of connected Lie groups, including the non-compact groups of spacetime symmetry transformations. In this paper, we present a simple systematic procedure for introducing perturbations to (infinite dimensional) unitary representations of finite dimensional connected Lie groups. We discuss applications to relativistic and non-relativistic particle systems.