A Higher-Order Calculation of The SiH3 ESR Spectrum

The e.s.r. spectrum of the simple SiH₃ radical in a low temperature matrix was first reported by Gordy and coworkers¹. They concluded that the large ²⁹Si splitting of 266 gauss is due to the strong deviation from planarity of the radical. This deviation from the simple methyl radical, which is known to be essentially planar, can be qualitatively explained by the electronegativity difference between silicon and carbon as suggested by Pauling². Several other silyl radicals have since been reported³, all have ³ splittings relatively larger than the corresponding ¹³C analogs. Recently several workers questioned the interpretation of the ²⁹Si splittings on the geometry of simple silyl radicals^4,5. Recent evidence appears to show that the silyl radical is not as strongly pyramidal as previously supposed. Biddles and Hudson⁴ pointed out that the ²⁹Si splittings are negative due to a negative magnetic moment, but the proton splittings can be either negative for a quasi-planar structure or positive for a pyramidal geometry. Satisfactory interpretation of the ²⁹Si splitting can be obtained from the INDO calculation, provided the proton splittings are negative. The large ²⁹Si splittingin SiH₃ calls for treatment of second and higher-order effects of the e.s.r. parameters⁶. A full calculation of SiH₃ is reported below.     

Foldy-Wouthuysen transformations for any spin

We give the generalized Foldy-Wouthuysen transformation for any 2(2J + 1)-component Poincaré-invariant Hamiltonian theory that describes free massive spin-J particles and that is subject to the conditions: (a) every observable $\text{O}$ is either Hermitian ($\text{O}$ = $\text{O}$⁺) or pseudo-Hermitian ($\text{O}$ = ?₃$\text{O}$⁺?₃) and (b) the theory is invariant under the discrete symmetries. The requirement that the Hamiltonian be defined in the rest frame specifies one and only one boost generator that is also defined at p = 0.     

Higher-Order Lagrangian Equations of Higher-Order Motive Mechanical System

In this paper, if the condition of variation δt=0 is satisfied, the higher-order Lagrangian equations and higher-order Hamilton's equations, which show the consistency with the results of traditional analytical mechanics, are obtained from the higher-order Lagrangian equations and higher-order Hamilton's equations. The results can enrich the theory of analytical mechanics.     

I. The isotopic Foldy-Wouthuysen representation and chiral symmetry

The paper introduces the isotopic Foldy-Wouthuysen representation. This representation was used to derive equations for massive interacting fermion fields. When the interaction Hamiltonian commutes with the matrix γ⁵, these equations possess chiral invariance irrespective of whether fermions have mass or are massless. The isotopic Foldy-Wouthuysen representation preserves the vector and axial currents irrespective of the fermion mass value. In the Dirac representation, the axial current is preserved only for massless fermions. In the isotopic Foldy-Wouthuysen representation, the ground state of fermions (vacuum) turns out to be degenerate, and therefore there is the possibility of spontaneously breaking parity (P — symmetry). This study considers the example of constructing a chirally symmetric quantum electrodynamics framework in the isotopic Foldy-Wouthuysen representation. A number of physical processes are calculated in the lowest orders of the perturbation theory. Final results of the calculations agree with the results of the standard quantum electrodynamics.     

Validation of the Eriksen method for the exact Foldy-Wouthuysen representation

The Eriksen method is proven to yield a correct and exact result when a sufficient condition of exact transformation to the Foldy-Wouthuysen (FW) representation is satisfied. Therefore, the Eriksen method is confirmed as valid. This makes it possible to establish the limits within which the approximate “step-by-step” methods are applicable. The latter is done by comparing the relativistic formulas for a Hamiltonian operator in FW representation (obtained using those methods) and the known expression for the first terms of a series, which defines the expansion of this operator in powers of v/c as found by applying the Eriksen method.     

用改进的哈密顿量计算Wilson费米子真空凝聚

用改进的哈密顿量和变分法计算格点(1+1)维QED(Schwinger模型)中Wilson费米子真空凝聚<ψψ>，结果(同使用未改进的哈密顿量的计算结果相比较)与Wilson参数r的依赖关系明显地减小了.     

Transformation of the effective rotational Hamiltonian of nonrigid X 2 Y molecules

The transformation of the effective rotational Hamiltonian H of nonrigid X ₂ Y molecules to the form having a minimum number of diagonals in the basis of rotational functions of a symmetric top is discussed. Such a transformation is a generalization of the reduction transformation performed for the polynomial effective Hamiltonian H. It is shown that in the general case the transformation substantially changes the form of the initial Hamiltonian, which restricts the region of applicability (J<J*) of the reduced Hamiltonian represented in a class of elementary functions in terms of angular momentum operators. The values of the rotational quantum number J* are estimated for the (000) ground and (010) vibrational states of the H₂O molecule.     

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.     

Unified Field Theory from Enlarged Transformation Group. The Consistent Hamiltonian

A theory has been presented previously in which the geometrical structure of a real four-dimensional space time manifold is expressed by a real orthonormal tetrad, and the group of diffeomorphisms is replaced by a larger group. The group enlargement was accomplished by including those transformations to anholonomic coordinates under which conservation laws are covariant statements. Field equations have been obtained from a variational principle which is invariant under the larger group. These field equations imply the validity of the Einstein equations of general relativity with a stress-energy tensor that is just what one expects for the electroweak field and associated currents. In this paper, as a first step toward quantization, a consistent Hamiltonian for the theory is obtained. Some concluding remarks are given concerning the need for further development of the theory. These remarks include discussion of a possible method for extending the theory to include the strong interaction.     

Feynman Transformation Matrix Element for a type of Hamiltonian of Angular Momentum system

Using the technique of integration within ordered product (IWOP) of operators and the Schwinger boson representation for the angular momentum theory as well as the coherent state's method, we evaluate the Feynman transformation matrix element for the angular momentum system whose Hamiltonian is H = AJ_x + BJ_y + CJ_x (A, B, C real). The eigenstate of H is derived and the comparison between it and the SU(2) coherent state is made.     

Connection between wave functions in the Dirac and Foldy-Wouthuysen representations

When the Foldy-Wouthuysen (FW) transformation is exact and the particle energy is positive, upper spinors in the Dirac and FW representations differ only by a constant factor, and lower spinors in the FW representation are zero. Deducing FW wave eigenfunctions directly from Dirac wave eigenfunctions allows one to use the FW representation to calculate expectation values of needed operators and to derive quantum and semiclassical equations of motion. The text was submitted by the author in English.