bluuebunny/crossref_metadata_2025 huggingface dataset:
Motions of Molecules in Condensed Systems: I. Selection Rules, Relative Intensities, and Orientation Effects for Raman and Infra-Red Spectra
Authors: Halford Ralph S.
Published: January 2005

A method is described that permits deduction of selection rules for Raman and infra-red spectra of crystals from a knowledge of just the space group designation and the population of the unit cell. When implemented by an appendix to this article showing, in a convenient arrangement, the distribution of point symmetries within space groups the method can be applied by a user familiar with point group manipulation but having no special knowledge of space groups. There are no selection rules operating in the liquid state. Some tentative qualitative conclusions are drawn concerning the relative intensities of certain kinds of components appearing in spectra of condensed systems. Sharpened selection rules are expected, and have been observed by others, to operate for single crystals containing non-rotating molecules when these are given special orientations in the light path. These special orientations and the appropriate selection rules can be predicted from information mentioned above along with inspection of the external form of a crystal. All procedures have a desirable advantage of simplicity over prior ones directed toward the same ends.
The Experimental Determination of the Intensities of Infra-Red Absorption Bands I. Theory of the Method
Authors: Wilson E. Bright , Wells A. J.
Published: January 2005

With spectrographs of available resolving power, the apparent integrated absorption coefficients of infra-red bands usually differ greatly from the true coefficients because the spectrograph yields at each setting a weighted average of the fractional light transmitted in a band of frequencies, whereas what is desired is the unweighted average of the logarithm of the fractional light transmitted. It is shown that true absorption coefficients can be obtained by (a) eliminating the violent fluctuations in intensity with frequency by broadening the rotational lines with a non-absorbing foreign gas and (b) eliminating the error due to the intensity variation of the envelope by extrapolating the apparent integrated absorption coefficient divided by the partial pressure to zero partial pressure of the absorbing gas. These two steps permit vibrational intensities to be measured to a reasonable accuracy even with a spectrograph of low resolving power.
Ultrasonic Propagation in Liquids. II. Theoretical Study of the Free Volume Model of the Liquid State
Authors: Kittel Charles
Published: January 2005

The free volume theory of liquids is applied to problems connected with sound propagation. The equation of state derived by Tonks for assemblies of hard elastic spheres may be written pVa≅3NkT, where the available volume Va is the difference between the actual volume and the volume corresponding to the closest possible packing. It is shown that the molar available volume is approximately given by the increase in molar volume of a substance on heating from 0°K to the temperature considered, so that the pictorial concept of the available volume has a real physical basis. Using the Tonks equation, expressions are derived for the sound velocity, and the temperature and pressure coefficients of sound velocity. These expressions are compared with experimental results. Expressions are also given for the cubic expansion coefficient and the difference of the specific heats on the free volume model. The Herzfeld-Rice derivation of the sound absorption caused by phase shifts in the vibrational specific heat is carried through for the Tonks equation of state. The temperature and pressure coefficients of sound absorption are discussed. Remarks are made on the relation of molecular structure to sound absorption.
Rotational Structure of Some Electronic Bands of Chlorine Dioxide
Authors: Coon Jesse B.
Published: January 2005

The rotational structure of several violet absorption bands of ClO2 has been studied. The bands considered correspond to the vibration transitions (v1′ 0 0)←(0 0 0) where v1′ = 1, 2, 3, 4, and 5. The most prominent identifiable feature of the rotational structure is a doublet qQ branch resolved in the tail of each band. For v1′ = 1, 2, and 3, the rotational wave numbers of these branches are given by ν1,2r=−12(0.0615+0.0042v1′)L(L+1) −12(1.188+0.008v1′)K2+0.13L+0.174(v1′−0.4)K ∓12[0.109(2K−L)−0.045(v1′−1)].This fits the spectrum in the observed range of quantum numbers, K = 11 to 20 and L = K to K+8. The first two terms are associated with that prolate symmetrical top which comes closest to the actual ClO2 molecule. Hence, neglecting the effect of asymmetry on the spectrum, 2ΔB̄=−(0.0615+0.0042v1′) cm−1and 2(ΔA−ΔB̄)=−(1.188+0.008v1′) cm−1.The last term is associated with spin doubling. The coupling is shown to be close to Hund's ``case b.'' The linear terms in L and K may be a consequence of the doubling. The term in v1′K describes an outstanding feature of the spectrum but is not interpreted. The bands are shown to be of the parallel type. A few qP and qR branches have also been identified and Δ1F values have been found. Assuming the spectrum to be that of a prolate symmetrical top, the resulting values of 2B̄ for the vibrationless states are 2B̄″ = 0.612 cm−1 and 2B̄′ = 0.550 cm−1. The value of 2B̄″ with the Cl–O distance s″ = 1.53A from electron diffraction is sufficient to determine the lower electronic state. The upper state is then determined from the values of 2ΔB̄ and 2(ΔA — ΔB̄) for the vibrationless states. The resulting model of both states is designated as model (2). Error in this model, because of the fact that the effect of asymmetry was neglected, is found to be small. Asymmetry correction made on model (2) leads to the following model: 2θ′′=109∘±3∘,2θ′=92∘±6∘, s′′=1.53A±0.02A, s′=1.805A±0.05A.The rotational isotope effect is observed and explained but gives no additional information about the molecular model.
Mass Spectrometric Study of CH3D. Dissociation Probabilities of C–H and C–D Bonds by Electron Impact
Authors: Evans Marjorie W. , Bauer Norman , Beach J. Y.
Published: January 2005

A pure sample of CH3D has been prepared and studied in the mass spectrometer. The mass spectrum of CH3D cannot be predicted from that of CH4 by assuming complete similarity between H and D (except mass). More information has been obtained on the difference in behavior of H and D on electron impact previously noticed by Delfosse and Hipple for C2H2D2. There is a difference in dissociation probability of C–H and C–D bonds which is approximately constant for several ionization processes and over a considerable range in electron energy. In addition to this there is a difference in behavior of H atoms in CH4 and H atoms in CH3D. In order to account for the mass spectrum of CH3D in terms of the mass spectrum of CH4 both these effects must be taken into consideration. Re-examination of the data of Delfosse and Hipple points not only to the difference between H and D atoms in C2H2D2, but also to a difference in behavior of H atoms in C2H4 and H atoms in C2H2D2.
Investigation of Oxidation of Copper by Use of Radioactive Cu Tracer
Authors: Bardeen J. , Brattain W. H. , Shockley W.
Published: January 2005

A very thin layer of radioactive copper was electrolytically deposited on a copper blank. The surface was then oxidized in air at 1000°C for 18 minutes, giving an oxide layer with a thickness of 1.25×10−2 cm. After quenching, successive layers of the oxide were removed chemically, and the copper activity in each layer was measured. The observed self-diffusion of radioactive copper in the oxide agrees quantitatively with a theory based on the following assumptions: (a) The oxide grows by diffusion of vacant Cu+ sites from the outer surface of the oxide inward to the metal. (b) The concentration of vacant sites as the oxygen-oxide interface is independent of the oxide thickness, and drops linearly from this constant value to zero at the metal boundary. (c) Accompanying the inward flow of vacant sites, there is a flow of positive electron holes such as to maintain electrical neutrality. (d) Self-diffusion of copper ions takes place only by motion into vacant sites. The results give a fairly direct confirmation of the theory of oxidation first suggested by Wagner.
Chemical Forms Assumed by C14 Produced by Neutron Irradiation of Nitrogenous Substances
Authors: Yankwich P. E. , Rollefson G. K. , Norris T. H.
Published: January 2005

The nature of the radioactive compounds formed in the production of C14 by neutron irradiation of nitrogen-containing material of various types has been investigated. Of the radioactive compounds sought, carbon dioxide and carbon monoxide were found in all samples which contained oxygen, either in the parent compound or in the solvent water. Formic acid and methanol were found only in water solutions. Hydrogen cyanide, on the other hand, was found in no solution, but was found in all the pure substance samples except ammonium nitrate and aniline. Methane was found only in those hydrogenous substances containing no oxygen. Formaldehyde was never found. An attempt is made to interpret the results in terms of the processes in which the C14 atoms must be involved in the dissipation of the recoil energy associated with their formation. The fact that several compounds are produced in radioactive form in the original irradiation eliminates the necessity for the synthesis from radioactive carbon dioxide of these compounds for use in tracer work. A simplified method for the preparation of C14 is suggested.
Motions of Molecules in Condensed Systems. II. The Infra-Red Spectra for Benzene Solid, Liquid, and Vapor in the Range from 3 to 16.7 μ
Authors: Halford Ralph S. , Schaeffer Oliver A.
Published: January 2005

Infra-red spectra are compared throughout the interval from 600 to 3300 wave numbers for the same amount of benzene existing separately as solid at 3°, liquid at 8°, and vapor at 20°C. The comparison demonstrates convincingly that (1) predictable selection rules for the solid are obeyed strictly, (2) there are no selection rules operating in the liquid phase, (3) the shift of molecular frequencies induced by successive stages of condensation is small and can be ignored in the approximation that regards the vibrations of an isolated molecule as harmonic ones, (4) the intensification of components of the spectrum accompanying changes of state proceeds as predicted qualitatively in an earlier paper, (5) all proposed complete assignments of the fundamental frequencies of benzene require revision. The components appearing in the several spectra are classified in accordance with a scheme suggested by the selection rules for the vapor and solid.
Heat-Capacity Lag in Gas Dynamics
Authors: Kantrowitz Arthur
Published: January 2005

The existence of energy dissipations in gas dynamics, which must be attributed to a lag in the vibrational heat capacity of the gas, has been established both theoretically and experimentally. The flow about a very small impact tube is discussed. It is shown that total-head defects due to heat-capacity lag during and after the compression of the gas at the nose of an impact tube are to be anticipated. Experiments quantitatively verifying these anticipations in carbon dioxide are discussed. A general theory of the dissipations in a more general flow problem is developed and applied to some special cases. It is pointed out that energy dissipations due to this effect are to be anticipated in turbines. Dissipations of this kind might also introduce errors in cases in which the flow of one gas is used in an attempt to simulate the flow of another gas. Unfortunately, the relaxation times of most of the gases of engineering importance have not been studied. A new method of measuring the relaxation time of gases is introduced in which the total-head defects observed with a specially shaped impact tube are compared with theoretical considerations. A parameter is thus evaluated in which the only unknown quantity is the relaxation time of the gas. This method has been applied to carbon dioxide and has given consistent results for two impact tubes at a variety of gas velocities.
Application of the Methods of Molecular Distribution to Solutions of Large Molecules
Authors: Zimm Bruno H.
Published: January 2005

The equations for the thermodynamic potentials of the solvent in solutions of ordinary organic molecules are extended to solutions of large molecules by methods using continuous molecular distribution functions. Particular attention is given to the coefficient, A2, of the second term in the expansion of the osmotic pressure in terms of the concentration, since this coefficient has a simple molecular meaning and is sufficient to describe the deviation of the system from ideality at low concentrations. A2 is calculated by direct integration for two rigid shapes, the sphere and the long thin rod. A general expression is then developed for A2 for flexible chain molecules in terms of the interactions of the segments of the chains. In favorable cases it is found possible to relate A2 for a chain molecule to the solution properties of its small molecule homologues by an equation very similar to those developed by Flory, Huggins, and Miller. In general, however, interactions that depend both on the local structure and also on the over-all shape of the chain molecules seriously modify such a relationship. The nature of these interactions, including the effects of branching and of limited flexibility, is discussed. It is also shown that higher coefficients in the expansion of the osmotic pressure in terms of concentration can be treated in a similar way. Comparison with experimental data confirms the general predictions of the theory both for proteins and chain polymers.
The Non-Planar Vibrations of Benzene
Authors: Miller Foil A. , Crawford Bryce L.
Published: January 2005

A thorough normal coordinate treatment has been carried out for the non-planar vibrations of benzene and its deuterium derivatives. The analysis has led to a critical survey of the assignments, and has furnished strong independent support for those advanced by Pitzer and Scott. It has also furnished values for all eight force constants of the complete harmonic potential function. For four of the constants two sets of values were obtained, both of which are physically reasonable and which reproduce the frequencies equally well. It was not possible to decide between them. The force constants were then used to calculate frequencies for the various deuterobenzenes. Nearly sixty such frequencies were evaluated, of which about forty could be checked against experiment. With four exceptions the errors are less than 1.5 percent. Lastly, certain valence-force constants pertaining to hydrogen vibrations have been determined.
Force Constant of the Association Bond in the Formic and Acetic Acid Dimers
Authors: Halford J. O.
Published: January 2005

The entropies of the formic and acetic acid dimers are used to find a minimum value for the force constant of the association bond. With the aid of a normal coordinate treatment the probable value is found to be 4×104 dynes/cm. This agrees with a rough estimate from the heat of dimerization. A Raman band in the formic acid spectrum at 200 cm−1 may be due to the association vibrations.
The Behavior of Macromolecules in Inhomogeneous Flow
Authors: Kramers H. A.
Published: January 2005

Following up the ideas of W. Kuhn's skein theory, J. J. Hermans has recently elaborated a theory of the properties of dilute solutions of highly polymerized molecules. Hermans describes the behavior of linear molecules by the diffusion of their end points. He finds that the contribution of a single molecule to the viscosity coefficient as well as to the double refraction of the solution is proportional to the square of the degree of polymerization, in agreement with the experiments. In the present article we investigate the statistical behavior of the individual links of the solved molecules. In this way we arrive at results equivalent with those of Hermans. Our method, however, can also be applied to molecules which possess branching points and rings.
Statistical Mechanics of Multimolecular Adsorption II. Localized and Mobile Adsorption and Absorption
Authors: Hill Terrell L.
Published: January 2005

It is shown that the transition from localized to mobile adsorption (in the first adsorbed layer) takes place at rather low temperatures for potential barriers of around 1000 cal./mole or less, so that localized physical adsorption should be a very rare phenomenon at the temperatures usually employed in adsorption experiments. Since the BET (and Langmuir) isotherm actually assumes localized adsorption, a new approximate isotherm equation is derived here on the basis of a mobile first layer obeying a two-dimensional van der Waals' equation. This isotherm is in semiquantitative agreement with the two-dimensional phase changes observed by Jura and co-workers, and by earlier workers, at very low pressures. The localized-mobile transition in the absorption of hydrogen by metals is mentioned briefly. The relationships between (1) the two-dimensional equation of state of a monolayer, (2) the adsorption isotherm of the monolayer, and (3) the ordinary three-dimensional equation of state of the gas being adsorbed are discussed in some detail.
Contraction of Lie Groups
Authors: Saletan Eugene J.
Published: January 2005

Contraction is defined for a Lie group to coincide on its Lie algebra with a generalization of contraction as first introduced by Inönü and Wigner. This is accomplished with a sequence of nonsingular coordinate transformations on the group (or nonsingular linear coordinate transformations on its Lie algebra), whose limit is a singular one. Essentially all of the calculations are performed in the algebra. It is assumed that in the limit the association in the algebra (the multiplication law in the group) converges, and this gives a necessary and sufficient condition on the algebra. Once it is satisfied, the new (contracted) algebra is uniquely determined in terms of the original one. It is found that the contracted algebra can be further contracted in the same way, and likewise the algebra so obtained. In this way one obtains a terminating sequence of algebras. Inönü-Wigner contraction corresponds to a sequence terminating at the first contraction. Some properties of the original and contracted algebras are studied, and some specific examples are given. Contraction of a Lie algebra induces contraction of any of its representations. This is examined for the case of finite-dimensional representations. Ray representations are discussed in general. It is shown how the trivial exponent of the Lorentz group changes under contraction to the nontrivial one of the Galilei group.
Asymptotic Properties of a Boson Field with Given Source
Authors: Cook J. M.
Published: January 2005

Some mathematical questions are answered in the theory of quantized boson fields with given linear source distribution. It is proved that the total Hamiltonian in the Fock representation is self-adjoint. Unitary operators governing the evolution of the system in the interaction picture are defined, and conditions given for their weak and strong convergence at remote times.
Analyticity of Wightman Functions
Authors: Fronsdal Christian
Published: January 2005

The problem of mapping certain domains in the space of complex four-vectors onto the space of their inner products is solved by a novel method. The ``primitive'' domains of regularity of the three and four point Wightman functions are determined. The domains whose X-space characterization was obtained by Streater, for the holomorphy envelope of the union of several primitive domains (both for the three and four point function) are also determined. The corresponding problem in perturbation theory is examined, and the analytic form of the perturbation theory boundary of the three and four point functions is obtained by the same method. The problem of the four-point function is reduced to constructing the holomorphy envelope of the union of three domains. It is shown that the boundary of the domain of four-point singularitites in perturbation theory bears no resemblance to any part of the boundary of the primitive domains, or the domains found or conjectured by Streater.
Dynamical Mappings of Density Operators in Quantum Mechanics
Authors: Jordan Thomas F. , Sudarshan E. C. G.
Published: January 2005

The most general dynamical law for a quantum mechanical system is studied with particular reference to the necessary and sufficient conditions for such a law to represent Hamiltonian dynamics. The main results are stated in the form of three theorems.
Foundations for a Treatment of the Scattering of Light by the Hydrodynamical and Statistical Atom Model
Authors: Wakano Masami
Published: January 2005

A study has been made on the scattering of light by the hydrodynamical and statistical atom model. Bloch's treatment of the hydrodynamical equations of motion for this model is supplemented here by inclusion of the interaction with the electromagnetic field. We limited attention to oscillations of small amplitude. By correspondence principle arguments, general expressions were derived for the cross sections for absorption, coherent and incoherent scattering.

The energy can be expressed—following Bloch—as the energy of a Thomas-Fermi atom plus a Hamiltonian which is associated with departures from the Thomas-Fermi distribution. Using Bloch's quantization of this Hamiltonian and applying the method of quantum field theory, we rederived the correspondence principle results for elementary cross sections.

Then applying the correspondence theoretical argument to the matrix element, we rederived Heisenberg's result for the total intensity of the Compton scattering. We also apply the method of stationary phase to the hydrodynamical treatment and show that this method gives the same result as does the linear term in the momentum transfer in Heisenberg's expression, except for a numerical factor 3½/2—a point that was discussed by Bloch many years ago.

Application of the general formulas given here for angular distribution of Rayleigh and Compton scattering will require electronic machine calculations of higher modes of oscillation of the gas model of the atom analogous to those made by Wheeler and Fireman for l=1.
Note on the Albertoni-Bocchieri-Loinger Theorem of Classical Statistical Mechanics
Authors: Lomont J. S.
Published: January 2005

The A-B-L theorem states that for almost all density functions the time-averaged probability of a system being in a region R of an energy shell in phase space is mR/m, where m is the volume of the energy shell and mR is the volume of R. In the present note a stronger form of the A-B-L theorem is proved. It is proved that for almost all density functions, the probability of the system being in R is mR/m. In particular, it is proved that the time averaging of the original A-B-L theorem is unnecessary.
Method for Defining Principal Modes of Nonlinear Systems Utilizing Infinite Determinants
Authors: Magiros Demetrios G.
Published: January 2005

A method for calculation of ``principal modes'' of linear or nonlinear systems is discussed. The physical definition of ``principal modes'' is formulated mathematically in two ways. The trial solution of the differential equation of the motion of the system is taken in an appropriate structure. The calculation of principal modes leads to infinite determinants of Hill's and von Koch's type, which are analyzed. The above method yields the possibility of getting the ``principal modes'' in the form of a series, all the coefficients of which can be calculated.
General Perturbational Solution of the Harmonically Forced van der Pol Equation
Authors: Struble Raimond A. , Fletcher John E.
Published: January 2005

Some formal techniques for the study of nonlinear oscillations are illustrated through a development of a general perturbational solution of the harmonically forced van der Pol equation. These techniques provide for the study of perturbations of nearly linear oscillations in almost complete generality. The intricate resonance problems associated with small divisors, and in this case leading to the entrainment phenomena, are treated with unusual ease and serve to illustrate both the versatility and the generality of the techniques.
Green's Distributions and the Cauchy Problem for the Iterated Klein-Gordon Operator
Authors: Bowman John Judson , Harris Joseph David
Published: January 2005

An explicit form of the homogeneous Green's function for the multi-dimensional iterated Klein-Gordon operator is obtained. By a direct calculation from its Fourier representation, the Green's function is expressed as a one-dimensional, infinite integral of the Sonine type. Although this integral is classically divergent when the order of the operator is less than the number of space dimensions, it can be treated rigorously under these conditions using the concepts of distribution analysis. A generalized Sonine integral is developed and the result applied to obtaining an explicit expression for the Green's function, which is now to be regarded as a distribution in the sense of Schwartz. Using a distribution introduced for this purpose, the Green's function is written in a form which explicitly displays its singularities on the light cone. The well-known difference between even- and odd-dimensional spaces is reflected in the nature of these singularities. The singularities appearing for an odd number of space dimensions consist of a finite linear combination of derivatives of the Dirac delta function δ(s2) where s is the space-time distance. The highest derivative appearing is of order ½(n − 2l − 1) with n giving the number of space dimensions and 2l giving the order of the operator. The singular part for even-dimensional spaces consists of a polynomial in 1/s of degree n − 2l + 1. No singularities appear when the order of the operator is greater than the number of dimensions. The general solution of Cauchy's problem for the iterated Klein-Gordon operator is obtained in convolution form. An explicit solution for the ordinary Klein-Gordon equation is presented in a form which exhibits separately the contributions due to the singular part and the regular part of the Green's function.
Theory of High-Energy Potential Scattering
Authors: Lynch P. J. , Carlson B. C.
Published: January 2005

The exact amplitude for scattering of a Schrödinger or Dirac particle by a static potential is rewritten in a two-potential form by splitting the potential into two parts, one of which contributes only to exactly forward scattering. Replacement of the exact wave function by a modified plane wave gives a high-energy approximation that is shown to be equivalent to the Saxon-Schiff approximation in the Schrödinger case. Corrections to the approximation are obtained in principle from a simplified series expansion of the exact wave function having the modified plane wave as leading term. The approximate amplitude reduces at small scattering angles to a well-known result; at large angles, it reduces to Schiff's stationary-phase approximation in the Dirac case but not, as shown by the example of a Gaussian potential, in the Schrödinger case.
Double-Tensor Operators for Configurations of Equivalent Electrons
Authors: Judd B. R.
Published: January 2005

To every irreducible representation W of the rotation group in 2l + 1 dimensions that is used to classify states of the electronic configurations ln, there correspond two couples (v, S), where v and S stand for the seniority number and total spin, respectively. Determinantal product states are introduced to examine this correspondence in detail. It is shown that for two double tensors W(κk) and W(κ′k), the set of reduced matrix elements (lnv1WξS1L ‖W(κk)‖ lnv1′W′ξ′S1′L′),for fixed n, v1, v1′, W, and W′, is proportional to the set (lmv2WξS2L ‖W(κ′k)‖ lmv2′W′ξ′S2′L′),where ξ and ξ′ are additional labels that may be required to define the states uniquely, provided (a) the two couples (v1, S1) and (v2, S2) are distinct, (b) the two couples (v1′, S1′) and (v2′, S2′) are distinct, and (c) the sum κ + κ′ + k is odd. The amplitudes of the double tensors are chosen so that the constant of proportionality is equal to the ratio of two 3-j symbols, multiplied by a phase factor. An explicit expression for this factor is given for f electrons, and a number of applications are discussed.
An Approach to Gravitational Radiation by a Method of Spin Coefficients
Authors: Newman Ezra , Penrose Roger
Published: January 2005

A new approach to general relativity by means of a tetrad or spinor formalism is presented. The essential feature of this approach is the consistent use of certain complex linear combinations of Ricci rotation coefficients which give, in effect, the spinor affine connection. It is applied to two problems in radiation theory; a concise proof of a theorem of Goldberg and Sachs and a description of the asymptotic behavior of the Riemann tensor and metric tensor, for outgoing gravitational radiation.
Reduction of Operator Rings and the Irreducibility Axiom in Quantum Field Theory
Authors: Jordan Thomas F. , Sudarshan E. C. G.
Published: January 2005

The mathematical theory of the reduction of operator rings is used to investigate some structures which can occur in quantum field theory when the postulate that the field operators generate an irreducible ring is relaxed. In particular, it is shown that if a quantum field theory has a commutator which commutes with all field operators it is a direct integral of theories in each of which the commutator is a scalar. If in addition it satisfies the postulates of Lorentz covariance, existence of an invariant vacuum, and mass and energy spectra, then it is a direct integral of generalized free field theories whenever the unitary representation of the Lorentz group can be constructed in terms of functions of the field operators and every state can be constructed by applying field operators on the vacuum. It is also shown that the latter two assumptions together with the requirement of a unique invariant vacuum state are sufficient to prove that the ring generated by the field operators is irreducible. In other words, under these conditions the irreducibility postulate is redundant.
Singularities of the Second Type
Authors: Fairlie D. B. , Landshoff P. V. , Nuttall J. , Polkinghorne J. C.
Published: January 2005

A new class of singularities associated with Landau-Cutkosky diagrams is investigated. They correspond to solutions of the Landau equations with infinite internal momenta. Some of their properties depend on the spins of the participating particles. It is shown that a subset of these singularities, called pure second-type singularities, does not appear on the physical sheet.
Renormalization of a Classical Gravitational Field Interacting with Quantized Matter Fields
Authors: Utiyama Ryoyu , DeWitt Bryce S.
Published: January 2005

The behavior of the mean value of the energy-momentum tensor of a set of quantized matter fields interacting with a classical gravitational field which is, in turn, produced by this mean value, is investigated. Singularities appear in the energy-momentum tensor corresponding to divergences of three different orders: ∞4, ∞2, and log ∞. These can be removed by the introduction of counter terms into Einstein's equation. The ∞4 singularity is removed by a ``cosmological term,'' the ∞2 singularity by a renormalization of the gravitation constant, and the log ∞ singularity by a counter term derivable from a Lagrangian which is quadratic in the Riemann tensor.

The gravitational Green's function corresponding to this semiclassical approximation to the fully quantized theory is found to have the asymptotic behavior 1/p4 instead of 1/p2, and thus to have a much weaker singularity in the coordinate representation than the Green's function of the ``bare'' linearized theory.
Definition of Commutators via the Uncertainty Principle
Authors: DeWitt Bryce S.
Published: January 2005

A general demonstration is given of the fact that the quantization of any system, as expressed by the uncertainty principle, implies the quantization of all other systems to which it can be coupled. It is also shown that the precise form which the uncertainty principle takes is uniquely specified. That is to say, the uncertainty principle effectively defines the commutator of any pair of observables. The argument serves to detach completely from its canonical origins a previously given definition of the commutator (which is a generalization of one given by Peierls) applicable to systems with infinite-dimensional invariance groups, for which the identification of canonical variables is an inconvenience. The measurements of a single observable and of a pair of observables are analyzed according to the Bohr-Rosenfeld scheme.
Mass Singularities of Feynman Amplitudes
Authors: Kinoshita Toichiro
Published: January 2005

Feynman amplitudes, regarded as functions of masses, exhibit various singularities when masses of internal and external lines are allowed to go to zero. In this paper, properties of these mass singularities, which may be defined as pathological solutions of the Landau condition, are studied in detail. A general method is developed that enables us to determine the degree of divergence of unrenormalized Feynman amplitudes at such singularities. It is also applied to the determination of mass dependence of a total transition probability. It is found that, although partial transition probabilities may have divergences associated with the vanishing of masses of particles in the final state, they always cancel each other in the calculation of total probability. However, this cancellation is partially destroyed if the charge renormalization is performed in a conventional manner. This is related to the fact that interacting particles lose their identity when their masses vanish. A new description of state and a new approach to the problem of renormalization seem to be required for a consistent treatment of this limit.
Schrödinger Scattering Amplitude. III
Authors: Grossmann Alex , Wu Tai Tsun
Published: January 2005

The methods of an earlier paper are used to obtain a domain of analyticity for the Schrödinger scattering amplitude minus the first Born term. The connection between the scattering integral equation and the Schrödinger equation is also studied.
On a General Class of Scattering Problems
Authors: Twersky Victor
Published: January 2005

This paper considers a class of scattering problems corresponding to a wave of propagation number K1 exciting an object characterized by K2, and giving rise to a scattered field of propagation number K3. In addition, the boundary conditions on the wave functions and their normal derivatives at the object involve three sets of two parameters associated with the three propagation numbers. This generalized class of scattering problems can be treated by conventional analytical procedures (as for the usual problems K1 = K3, etc.) to obtain series solutions for symmetrical shapes and Green's function representations for general shapes. The numerators of the new series coefficients differ from the usual ones, and the Green's function representation has an additional inhomogeneous term, i.e., a volume integral containing the source term in its kernel. These results are applied for special cases to obtain explicit approximations for the ``two-external-space'' scattering amplitude g(K3, K1) (e.g., for small spheres with three sets of ε's and μ's, and for large tenuous scatterers).
A Symmetric Representation for Three-Body Problems. I. Motion in a Plane
Authors: Smith Felix T.
Published: January 2005

A symmetric representation is sought for the motion of three particles in the limit of weak interaction. Operators with the desired symmetry can be obtained from the 6-dimensional generalized angular momentum tensor [F. T. Smith, Phys. Rev. 120, 1058 (1960)]. Here, the 4-dimensional problem of motion in a plane is worked out. Symmetric angular coordinates are found, operators and eigenfunctions are constructed, and the coupling coefficients connecting this with more familiar representations are discovered. Formally, the eigenfunctions are similar to the symmetric rotor functions, but with different arguments.
Ordering Energy Levels of Interacting Spin Systems
Authors: Lieb Elliott , Mattis Daniel
Published: January 2005

The total spin S is a good quantum number in problems of interacting spins. We have shown that for rather general antiferromagnetic or ferrimagnetic Hamiltonians, which need not exhibit translational invariance, the lowest energy eigenvalue for each value of S [denoted E(S) ] is ordered in a natural way. In antiferromagnetism, E(S + 1) > E(S) for S ≥ 0. In ferrimagnetism, E(S + 1) > E(S) for S≥S, and in addition the ground state belongs to S≤S. S is defined as follows: Let the maximum spin of the A sublattice be SA and of the B sublattice SB; then S≡|SA−SB|. Antiferromagnetism is treated as the special case of S=0. We also briefly discuss the structure of the lowest eigenfunctions in an external magnetic field.
Dynamic Behavior of a Set of Weakly Coupled Ising Spins
Authors: Meijer Paul H. E. , Tanaka Tomoyasu , Barry Jeremiah
Published: January 2005

The master equation of a set of independent equivalent spins contains only one undetermined constant, the rate constant. If one assumes the local field to be altered by the field produced by the two neighboring spins, one can formulate a set of equations for the average of one, two, three, etc., spins. On assuming an Ising interaction between the spins, weak compared to the coupling with the heat bath, we can terminate the hierarchy and solve the problem of a linear chain with periodic boundary conditions by Fourier-transformation. The resulting secular equation determines two sets of relaxation times and two sets of eigensolutions. An explicit solution for the spin averages is given for the initial condition describing a localized excitation. Similarities with, and differences between, this and the random walk problem is pointed out.
Normal Forms of Complex Matrices
Authors: Zumino Bruno
Published: January 2005

Simple normal forms are given for symmetric, skew symmetric, and general complex matrices. These three cases can be combined into a single one, if one considers matrices with elements in a suitable ground field.
On Covariant Formulations of the Maxwell-Lorentz Theory of Electromagnetism
Authors: Sachs Mendel , Schwebel Solomon L.
Published: January 2005

Maxwell's equations are formulated in a number of different representations: (a) As a single four-component spinor equation whose transformation properties are almost identical with those of the Dirac equation. (b) As a pair of uncoupled two-component spinor equations, in two different representations. One of these is similar to the Weyl equation for the neutrino field and the other to the two-component spinor form of the Dirac equation. (c) As a single equation in which the field variables are 2 × 2 matrices. In terms of these new field variables corresponding conservation laws are derived. Identification of these conservation laws with the standard ones is made. The transformations that are appropriate to the maintainance of covariance of the field equations are discussed.
On a Paper of Green and Lanford
Authors: Kuroda S. T.
Published: January 2005

The phase shift formulas for the wave and scattering operators in the potential scattering of a single nonrelativistic particle are proved under weaker assumptions on the potential than the one used previously by Green and Lanford.
Introduction to Momentum Space Integration Techniques in Perturbation Theory
Authors: Fowler M.
Published: January 2005

Generalized ``pinch'' techniques are developed for analyzing singularity configurations of any Feynman integral in the product space of the loop momenta. The Landau equations follow immediately, and the dual diagrams arise naturally as geometric singularity criteria. Cutkosky's formula for the discontinuity is derived by an elementary method, and its structure is clearly exhibited by this approach. The basic differences between Landau and non-Landau singularities for single loop diagrams are discussed, and it is shown why the presence of non-Landau singularities, in contrast to those in the Landau scheme, depends on the dimensionality of space.
Distortion in the Metric of a Small Center of Gravitational Attraction due to its Proximity to a Very Large Mass
Authors: Manasse F. K.
Published: January 2005

The problem of two bodies in general relativity in an especially restricted mode is analyzed. One body is of small mass, and, under the influence of gravitational attraction, moves toward a much larger mass whose field produces ``tidal'' deformations in the geometry of the smaller one. To evaluate these deformations, we treat the Schwarzschild metric of the particle by a perturbation analysis similar to that performed by Regge and Wheeler. The boundary conditions for this analysis are obtained from the metric of the background field expressed in a novel set of comoving coordinates, here called Fermi normal coordinates. These coordinates have been described in the previous paper. In this paper we use the Schwarzchild metric given there in comoving coordinates as an asymptotic approximation for the full solution which is evaluated here.

A perturbation analysis using the ``tidal deformation'' (background curvature) as an expansion parameter is performed on the metric of a small Schwarzschild particle. The solution so obtained satisfies Einstein's equations for empty space for small deviations from a nonflat metric. This solution also reduces to the Fermi metric in the appropriate limit, namely, when the ``distance'' from the geodesic is large compared to the radius of the small mass but small compared to the separation of the two masses. Thus only the quadratic terms in the distance are important. This solution is then used to find the deformations in shape of the throat of the wormhole and to locate this region of symmetry (the throat) by finding a coordinate transformation which leaves the metric invariant.
Principle of General Q Covariance
Authors: Finkelstein D. , Jauch J. M. , Schiminovich S. , Speiser D.
Published: January 2005

In this paper the physical implications of quaternion quantum mechanics are further explored. In a quanternionic Hilbert space HQ, the lattice of subspaces has a symmetry group which is isomorphic to the group of all co-unitary transformations in HQ. In contrast to the complex space HC (ordinary Hilbert space), this group is connected, while for HC it consists of two disconnected pieces.

The subgroup of transformations in HQ which associates with every quaternion q of magnitude 1, the correspondence ψ → qψq−1 for all ψ∈HQ (called Q conjugations), is isomorphic to the three-dimensional rotation group. We postulate the principle of Q covariance: The physical laws are invariant under Q conjugations. The full significance of this postulate is brought to light in localizable systems where it can be generalized to the principle of general Q covariance: Physical laws are invariant under general Q conjugations. Under the latter we understand conjugation transformations which vary continuously from point to point.

The implementation of this principle forces us to construct a theory of parallel transport of quaternions. The notions of Q-covariant derivative and Q curvature are natural consequences thereof.

There is a further new structure built into the quaternionic frame through the equations of motion. These equations single out a purely imaginary quaternion η(x) which may be a continuous function of the space—time coordinates. It corresponds to the i in the Schrödinger equation of ordinary quantum mechanics. We consider η(x) as a fundamental field, much like the tensor gμν in the general theory of relativity. We give here a classical theory of this field by assuming the simplest invariant Lagrangian which can be constructed out of η and the covariant Q connection. It is shown that this theory describes three vector fields, two of them with mass and charge, and one massless and neutral. The latter is identifiable with the classical electromagnetic field.
Theorem on Energy Shifts Due to a Potential in a Large Box
Authors: Iwamoto Fumiaki
Published: January 2005

A theorem conjectured by Cohen, which gives a concise form to the Rayleigh—Schrödinger perturbation series for energy shifts due to a potential in a large box, is proved.
Lorentz—Lorenz and Sellmeier Formulas in Irregular Gases
Authors: Taylor Leonard S.
Published: January 2005

The integral equation expressing the condition of dynamical equilibrium of dipoles in an applied electromagnetic field is solved to determine the index of refraction for the ensemble-average wave in a media in which the polarizability varies irregularly. The solution closely follows that employed in the usual derivation of the Ewald—Oseen theorem for isotropic media. A Lorentz—Lorenz formula, modified by the appearance of an additional term, is determined. The additional term is real when the scale length of the irregularities is small compared to the wavelength. For larger irregularities, however, the correction leads to a complex index of refraction, expressing the attenuation due to scattering. Applying the concept of the depolarizing effect of electron—ion collisions, these results are extended to the case of an ionized gas. In this latter instance, a similar modification of the Sellmeier formula for the index of refraction is determined. Explicit formulas are given for the case of irregularities with scale large compared to a wavelength.
Quadratic Invariants of Surface Deformations and the Strain Energy of Thin Elastic Shells
Authors: Serbin Hyman
Published: January 2005

A conceptual theory of thin, elastic shells is developed. A general form for the strain energy is postulated in terms of parameters which characterize, in the sense of Euclid, the geometry of the shell. The strain energy permits a formal definition of generalized stresses and then a statement of equilibrium. The physical interpretation of the stresses is obtained from the boundary conditions. The energy coefficients, which are not derivable within the scope of this theory, are related to material constants by using classical results derived from the three-dimensional theory of elasticity. The result is a complete, self-contained theory of shells which can be used for prediction. The methods used depend on a combination of Cartesian tensors and exterior differential forms.
Antireciprocity and Memory in the Statistical Approach to Irreversible Thermodynamics
Authors: Nettleton R. E.
Published: January 2005

The macroscopic state of a large, closed system is specified by the ensemble averages αj of a set of even dynamical variables Aj, together with the averages vj of the quantities iLAj, obtained by operating on the Aj with the self-adjoint Liouville operator L. Phenomenological equations for the time rates of change, α̇j and v̇j, are derived by a technique due to Zwanzig, in which one operates on Liouville's equation with a projection operator which projects out the part which is ``relevant'' to the phenomenological description employed. These phenomenological equations are shown to exhibit the Onsager—Casimir reciprocity relations, including antisymmetry relations whose derivation is found to require a slight modification of Zwanzig's mathematical assumptions. Since vj=α̇j, these equations also show that irreversible thermodynamics can be extended to the case where second-order time derivatives appear representing memory effects, as well as nonlinear terms in the αj, provided the equations are still required to be linear in the vj. Furthermore, Onsager's equations are obtained while allowing the phenomenological matrices to be functions of the variables αj, and not merely of constants of the motion. This serves to generalize and extend Zwanzig's earlier treatment.