Complex Phase Factor Research Articles

Study of the complex fermion determinant in a U (1) L ⊗ U (1) R symmetric Yukawa model

Lattice theories that contain chiral multiplets of fermions can have complex fermion determinants. This is for example the case for the U (1) L ⊗ U (1) R symmetric Yukawa model with mirror fermions, if the number of generations of fermions and mirror fermions is odd. Whether a numerical simulation of such a model is possible depends on the magnitude of fluctuations of the complex phase factor of the fermion determinant. We investigate the fermion determinant of the U (1) Yukawa model with mirror fermions for a physically relevant choice of parameters. The argument of the complex phase turns out to fluctuate only very little and is at most of the order of 2×10 −3.

X-ray diffraction reconstruction of the inverted hexagonal (HII) phase in lipid-water systems

The structure of the inverted hexagonal (HII) phase in biological lipid-water systems is studied to examine the physical interactions which drive the polymorphic phase behavior and which are also thought to play a relevant role in biological membrane function. A method is derived which yields the complex phase factors of the HII phase diffraction amplitudes from examination of a single sample. This method is applied to a low-resolution Fourier reconstruction of the HII phase in dioleoylphosphatidylethanolamine (DOPE) + water, specifically to examine deviations from the presumed circular model of the HII phase. It is found that the average radius of the water core, Rw, as determined from a Fourier reconstruction, is in good agreement with previously measured values of Rw obtained from more time-consuming traditional methods [Tate, M. W., & Gruner, S. M. (1989) Biochemistry 28, 4245]. In addition to the average value of Rw, the Fourier reconstruction also can be used to determine the true shape of the water core. It is found that the water core is circular to within 5% of Rw when the unit cell size is less than approximately 75 A. Above 75 A, however, a definite shape deformation becomes apparent, with radial noncircularities of 5-10%, probably in response to the increased entropic cost of packing the hydrocarbon chains into the anisotropic environment of the HII unit cell [Kirk, G. L., Gruner, S. M., & Stein D. E. (1984) Biochemistry 23, 1093]. As a more direct probe of the packing anisotropy, Fourier reconstructions of DOPE + dodecane and DOPE + squalene systems were compared with the reconstruction of DOPE. These oils are known to promote the low temperature occurrence of the HII phase, presumably by a reduction in the hydrocarbon packing stress. In support of this hypothesis, the alkanes were observed to relax the water core to a circular shape for even large lattices. In addition, anisotropy of the electron density near the end of the lipid chains is reduced when alkane is added, implying a more uniform hydrocarbon packing environment, consistent with the results of neutron diffraction upon the addition of deuterated decane [Turner, D. C., Gruner, S. M., & Huang, J. (1992) Biochemistry (following paper in this issue)].

The zitterbewegung interpretation of quantum mechanics

Thezitterbewegung is a local circulatory motion of the electron presumed to be the basis of the electron spin and magnetic moment. A reformulation of the Dirac theory shows that thezitterbewegung need not be attributed to interference between positive and negative energy states as originally proposed by Schroedinger. Rather, it provides a physical interpretation for the complex phase factor in the Dirac wave function generally. Moreover, it extends to a coherent physical interpretation of the entire Dirac theory, and it implies azitterbewegung interpretation for the Schroedinger theory as well.

A generalized internal axis method for high barrier tunneling problems, as applied to the water dimer

When more than one large-amplitude vibrational motion is present in a molecule, it is often not possible to define a global internal-axis-method (IAM) coordinate system and set of basis functions. In the present work, a method is presented for extending the IAM treatment to tunneling problems in such cases, using as an illustration a model for the water dimer with three large-amplitude vibrational coordinates. The method involves the construction of two different sets of local IAM-like coordinate systems. The first of these contains n coordinate systems, one for the small neighborhood surrounding each of the n equilibrium frameworks. The second contains on the order of n 2 2 coordinate systems, one for each feasible tunneling path between each pair of frameworks. Basis functions written in the second set of local IAM-like coordinates are used to determine the complex phase factors associated in this method with tunneling matrix elements of the phenomenological rotational Hamiltonian in the high barrier limit. These phase factors govern the way in which the various real tunneling frequencies in the molecule constructively and/or destructively interfere in the Hamiltonian matrix elements and final energy expressions. Various mathematical approximations are involved in using the local IAM-like basis sets to obtain matrix elements; the full extent of the adverse effects of these approximations will not be known until an attempt to fit experimental data is carried out.

On the Application of Faddeev Equations and the CoulombtMatrix to Asymptotic Electron Capture in Hydrogen

Faddeev equations together with the Coulomb $t$ matrix have been used to determine the asymptotic amplitude for electron capture from neutral hydrogen by fast protons. The results show that in the high-energy limit the capture cross section should go down as ${v}^{\ensuremath{-}11}$, where $v$ is the velocity of the incident proton. The capture amplitude is identical to Drisko's second-Born-approximation calculation except for a complex energy-dependent phase factor which ultimately approaches unity with sufficiently high incident energy. The major contribution to the three-body capture amplitude can be shown to come from the on-energy-shell two-body $t$ matrix, in agreement with general theorems concerning scattering from complex systems. At high incident energies, the on-energy-shell contribution to the capture amplitude (not the cross section) will decrease as ${v}^{\ensuremath{-}5}$, while the off-energy-shell continuum contribution will decrease as ${v}^{\ensuremath{-}6}$. The contributions from the sum of the infinite number of two-body bound-state poles can be shown to converge, and the sum can be explicitly performed at high enough incident energies in all except the forward direction. The bound-state contributions to the capture amplitude go down as ${v}^{\ensuremath{-}11}$, which is much less than the continuum contributions.

Method of Complex Coordinates for Three-Body Calculations above the Breakup Threshold

A method is proposed for calculating elastic and inelastic three-particle scattering amplitudes in the breakup region using a coordinate-space variational approach, with each coordinate multiplied by a complex phase factor. The trial wave functions have relatively simple asymptotic forms. The method applies to short-range potentials analytic in coordinate space. A numerical example indicates that the method works well for two-particle scattering.

Complex wavefunctions and the conservation of total momentum in the Hartree-Fock theory

The conservation of total momentum is treated as a symmetry property connected with the antilinear complex conjugation operator, which for a real Hamiltonian has the character of a constant of motion. It is shown that either the real or imaginary part of a complex trial wave function optimizes its expectation value. The assumption of real wave functions is therefore justified provided the complex components of a wave function belong to the original set. For a nonlinear variation procedure, such as for instance the unrestricted Hartree-Fock scheme, with the set consisting of single determinant functions, the real or imaginary projection will in general not be contained in the given set. A real determinant (apart from a trivial complex phase factor) is shown to be characterized by a real Fock-Dirac density matrix, whose eigenfunctions the so-called natural spin-orbitals can be chosen real. Finally some cases where one has obtained complex single determinant solutions are pointed out and discussed.

On the Static and Spherically Symmetric Solutions of the Yang-Mills Field

The b field equations proposed by Yang and Mills are investigated from a classical point of view, and the general solution is obtained under the condition of the static and spherically symmetric field. It is found that the solution is reducible to the by means of the isotopic gauge transformation. From this canonical form it is very likely 'that the mass of the b quantum is zero in spite of the non-linearity of the field equations. § l. Introduction It was pointed out by Yang and Mills 1 l some years ago that the conven­ tional formalism of charge independence hypothesis is not consistent with the concept of the localized field that underlies usual physical theories. They explored the possibility of admitting the arbitrary orientations of the isospin axes at all space-time points. Such an arbitrary way of choosing the orientation of the isospin axes is called isotopic gauge. The isotopic gauge bears a ·close analogy with the electromagnetic gauge which represents an arbitrary way of choosing the complex phase factor of a charged field at all space-time points. In electrodynamics the introduction of the electromagnetic field enables us to counteract the gauge transformation of charged fields. In quite a similar manner Yang and Mills introduced a physical field of a new kind to ensure the invariance of the theory under the isotopic gauge transformation. This new field is called a b (or B) field, which is represented by twelve components bw Here the index p. (p.=l, 2, 3, 4) refers to the space-time coordinates and the bold face letter b indicates that it is a vector-like quantity in an isospace. Since the proposal of Yang and Mills, various investigations have been made 2 J-IO) concerning the isotopic gauge invariance and the properties of the b field from the standpoint of the classical as well as that of the quantized theory of fields. In spite of these studies, however, it is still difficult to draw any definite conclusion about their quantum properties on account of the non-linear

Some Theoretical Investigations on Metal Disc Delay Dielectrics

AbstractThe paper presents a brief report of the theoretical investigations on the properties of an artificial dielectric composed of a three-dimensional array of thin metal discs. If the dielectric sample is illuminated by a plane electromagnetic wave, it may be assumed that the incident field induces on the surface of each disc sheet electric and sheet magnetic currents. Electric and magnetic vector potentials are found in terms of and . The electric field intensity at a distant point P due to the radiation from a single disc is calculated from the geheralized relation of radiation intensity involving and . The resultant field intensity at P due to all the discs of the array acting as radiators is calculated by introducing the complex array factor in the expression for . The complex phase factor of the transmitted wave is derived from the expression for . Separating the real and imaginary parts of the complex phase factor, refractive index of the dielectric sample is calculated from the expression for t...