Volume 7, Issue 8, August 1966
Index of content:

Laws like Newton's
View Description Hide DescriptionInstantaneous interaction for n particles is defined in terms of kinematic concepts only. A set of ``laws,'' reminiscent of, but much weaker than, Newton's three laws, is formulated in solely kinematic terms. Invariance under the Euclidean group, the Galilean group, and the special Galilean group is defined, and the most general interactions satisfying the laws, and invariant under these groups, are found. It is shown that they all satisfy Newton's laws. It is shown that the interaction between moving charges cannot be instantaneous and Galilean‐invariant.

Doppler Measurement of Space—Time Curvature
View Description Hide DescriptionIn special relativity, the Doppler shift between two freely moving identical oscillators is constant if their world lines are coplanar. We show that in general relativity, instead, the rate of change of their Doppler shift is proportional to a component of the space—time curvature, averaged along the light ray. A possible application to the detection of gravitational waves is discussed.

A Two‐Meson Solution of the Charged Scalar Static Model
View Description Hide DescriptionPresented in this paper is a solution of the charged scalar static model, in which the scattering amplitude is crossing symmetric and has two‐ and three‐particle intermediate states (two‐meson approximation). The production and six‐point amplitudes are also obtained and are shown to lead to a two‐ and three‐particle unitary scattering matrix.

A Class of Representations of the Generalized Bondi—Metzner Group
View Description Hide DescriptionA procedure is given for the construction of a faithful linear representation of the generalized Bondi—Metzner group from each faithful linear representation of the inhomogeneous orthochronous Lorentz group. Unitary representations can be obtained in this way.

Role of Causality in Quantum Field Theory and the Dynamical Postulate
View Description Hide DescriptionIt is shown that Bogoliubov causality together with the usual assumptions of quantum field theory suffice to determine the off‐mass‐shell behavior of the S‐operator as that resulting from the φ‐product. The same assumptions lead to an equation for the current which replaces the field equation for the interpolating field. The specification of the interaction term in this equation corresponds to the specification of an interaction Hamiltonian in the usual formalism. This interaction term is an operator distribution whose support is a single point and which is otherwise severly restricted.

Wave Mechanics in Classical Phase Space, Brownian Motion, and Quantum Theory
View Description Hide DescriptionA wave dynamics of fields φ(p, q; t) ∈ L _{2}(Γ) over the phase space Γ(p, q) of a classical system S is derived from the Liouville theorem. We define the energy contained in a given field φ(p, q; t). We show that for a special class of fields, selected on physical grounds, the energy spectrum is given by a time‐independent Schrödinger equation. This allows us to associate with S an ordinary quantum system Q such that the values of the quantized energy coincide for the fields in the phase space of S and for Q. Then we make use of Wiener's stochastic integral based on the theory of Brownian motion to derive probabilities which are the same as those one would obtain through Born's statistical postulate of quantum theory. From this it follows that we can regard normalized fields φ(p, q; t) as ``probability amplitudes'' leading to a probability density function ρ(p, q; t) = φφ* in the sense of Gibbs' statistical mechanics. Our work therefore appears as a bridge between a statistical theory (in the sense of Gibbs) of a mechanical system S and the usual quantum theory of the related quantum system Q.

Anisotropic Linear Magnetic Chain
View Description Hide DescriptionThe ground‐state and the spin‐wave states of the Hamiltonian,,are studied for all values of ρ, and analytical expressions are given for their energies. On the other hand, by using a canonical transformation which changes H(ρ) into ‐H(‐ ρ), the states of highest energy can also be obtained. The ground state is ferromagnetic for ρ ≤ − 1 and antiferromagnetic for ρ ≥ −1. For ρ = ±1, the energy has singularities, but it remains continuous. For ρ = 1, all its derivatives are also continuous. In the range − 1 ≤ ρ ≤ 1, the spin‐wave states of given momentum are degenerate but for ρ ≥ 1; this degeneracy is removed, and an energy gapG(ρ) appears.

Degeneracy of Cyclotron Motion
View Description Hide DescriptionThe classical problem of planar cyclotron motion of a charged particle in a uniform magnetic field possesses symmetries which account for the ``accidental'' degeneracies of the analogous nonrelativistic Schrödinger equation, as found by Johnson and Lippman. The essentially quadratic nature of the Hamiltonian is not changed by considering the particle moving in a harmonicoscillator potential, a ``Zeeman effect'' for the harmonicoscillator. The transitions to the limiting cases of a weak magnetic field (pure harmonicoscillator) or a strong field (pure cyclotron motion) involve the contraction of the corresponding symmetry groups, yielding Larmor precession of the oscillator orbits in the first case, and the drift of the cyclotron orbit in the second. The constants of the motion generate the unitary unimodular group SU _{2} in all cases except for pure cyclotron motion, in which case one obtains the commutation rules of creation and annihilation operators. Only for certain ratios of magnetic field strength to the oscillator frequency does one obtain bounded closed orbits, and presumably only in these cases do degeneracies exist quantum‐mechanically. A transition to a rotating coordinate system reduces the problem to that of a plane harmonicoscillator; however, the time dependencies of the transformation must be allowed for interpreting the constants thereby arising. Moreover, the velocity‐dependent forces introduce gauge transformations which also affect the interpretation of the symmetries. There are two kinds of symmetries—inner symmetries involving the canonical coordinates and governing the shape of the orbits, and outer symmetries involving the mechanical coordinates and governing the location of the orbits.

The Definition of States in Quantum Statistical Mechanics
View Description Hide DescriptionThe controversy relative to the use of normal states in quantum physics is discussed in the light of ergodic theory. The nature of the spectrum of the Hamiltonian is shown to play a central role in the decision to enlarge the ordinary frame provided by the traditional density‐matrix formalism. The connection of these considerations with the infinite‐time, infinite‐volume limits in nonequilibrium statistical mechanics is pointed out.

Wave‐Packet Derivation of Field‐Theoretic Scattering Amplitude
View Description Hide DescriptionA formal expression is derived for the field‐theoretic scattering amplitude in a Brillouin—Wigner perturbation expansion. Wave packets are used to introduce the initial conditions, thereby avoiding the necessity of adiabatically switching the coupling constant. The field of incident particles is second‐quantized, and the target is first‐quantized. The principal improvement on previous derivations is that the number of incident quanta, although finite, is otherwise unrestricted. The result is thus applicable, for example, to the nonrelativistic description of the scattering of a photonbeam of arbitrary intensity by an atom or a charged particle.

Concise Derivation of the Formulas for Lattice Thermal Conductivity
View Description Hide DescriptionA perturbationexpansion for the correlation‐function formula (or Kubo formula) for thermal conductivity was presented in a previous paper. The formulas obtained there for the lowest‐order contribution are used here to derive the transport equations for the thermal conductivity for a lattice with imperfections and/or anharmonic forces. The result has the same form as the familiar Boltzmann equation for phonons.

Higher‐Order Corrections to the Lattice Thermal Conductivity
View Description Hide DescriptionIn a previous paper the correlation function formula, or Kubo formula, was used to derive formulas for the lowest‐order (λ^{−2}) and first‐order (λ^{−1}) corrections to the lattice thermal conductivity (the Hamiltonian is written H = H° + λH′). The formulas obtained there are used here to derive the transportequations for the calculation of the first‐order correction to the conductivity. These transportequations have the same homogeneous part as the familiar Boltzmann equation for phonons; however, their inhomogeneous parts are different and depend on the nature of the perturbation. Formulas for these inhomogeneous parts are given for both the scattering due to randomly distributed point imperfections and that due to anharmonic forces. At high temperature, the first‐order correction for anharmonic scattering is independent of the temperature.

General Spherical Harmonic Tensors in the Boltzmann Equation
View Description Hide DescriptionThe irreducible velocity space direction cosine tensors associated with velocity magnitude spherical harmonic expansion of the distribution function are manipulated in the Boltzmann‐Vlasov flow terms to yield a linked chain of equations whose general (lth) equation is given explicitly. This generalizes earlier results for l = 0, 1, 2, 3.

Exact Solutions for a Semi‐Infinite Square Lattice Gas
View Description Hide DescriptionThe problem of a two‐dimensional lattice gas with nearest neighbor infinite repulsion is considered by obtaining exact solutions for a sequence of semi‐infinite spaces. The exact solutions are obtained for M × ∞ spaces with 2 ≤ M ≤ 14 in even steps, and although there are no phase transitions in these spaces, a criterion for the point of ``closest approach'' to a phase transition is established. The values of the thermodynamic variables evaluated at this point for each M are extrapolated to obtain the properties of the two‐dimensionally infinite space. The data indicate that a phase transition occurs with possibly infinite compressibility at an activity z = 3.799, a density ρ/ρ_{max} = 0.7356, and a pressure given by P/k_{B}T = 0.7914. The density is obtained by a rigorous differentiation of the secular determinant that determines the value of the pressure for a given z, thus securing the accuracy of the calculations and enabling the extrapolated values of the thermodynamic variables to be estimated with good precision.

Extraction of Singularities from the S Matrix
View Description Hide DescriptionTwo methods are described for extracting triangle singularities from matrix elements. The first is comparatively simple but involves the use of off‐mass‐shell amplitudes; the second is rather involved. A proof of the Cutkosky discontinuity formula is given independent of perturbation theory, and it is shown that the Riemann‐sheet properties of the singularity in the physical region agree with perturbation theory. The connection between this and a causality requirement is discussed. The relevance of the work to practical computations is explained.

Generalized Ward Identity and Unified Treatment of Conservation Laws
View Description Hide DescriptionA technique for deriving conservation laws directly from field equations without recourse to the Lagrangians or Noether's theorem is reviewed and extended. The method allows a simple treatment of the so‐called ``generalized'' conservation laws including Lipkin's ``zilch.'' An interesting feature which results from our approach is the existence of conserved currents for discrete as well as continuous symmetries. It is also pointed out that conservation laws do not always follow from the invariance of equation of motion if it is not derivable from a Lagrangian. Finally, we show how our method can be applied to the normalization of wavefunctions of composite particles such as Bethe‐Salpeter wavefunctions.

Lattice Statistics of Hydrogen Bonded Crystals. I. The Residual Entropy of Ice
View Description Hide DescriptionThe lattice statistical problem of calculating the residual entropy of ice has been considered in some detail for the hexagonal and cubic ice lattices as well as for a two‐dimensional icelike lattice. Even for the two‐dimensional lattice, this problem appears to be intractable using exact methods, so an approximation method is in order. The series method of DiMarzio and Stillinger has been developed so that the series is completely characterized by the numbers of various kinds of cycles on the lattice. The first five terms of the series have been evaluated and used to extrapolate values of the residual entropyS(0) within rather narrow limits for all practical purposes. The result for hexagonal ice and cubic ice is S(0) = .8145 ± .0002 cal/deg/mole which agrees with experiment even better than Pauling's original approximation. Some other methods are also discussed, and their results tend to confirm the series results.

Lattice Statistics of Hydrogen Bonded Crystals. II. The Slater KDP Model and the Rys F‐Model
View Description Hide DescriptionThe Slater KDP model, a simple hydrogen bonded ferroelectric model, and the Rys F‐model, a simple hydrogen bonded antiferroelectric model, have been treated using both high‐ and low‐temperature series for the partition function. The high‐temperature series is a modification of the residual entropy of ice series discussed in I. For each model a temperature is found at which the high‐temperature series and the low‐temperature series are identically equal. For the KDP model this equality gives a transition temperature and a latent heat is easily calculated, both of which are exact. It so happens that these exact results agree with previous analyses which used only mean field approximations. For the F‐model the formal equality of the series gives the first evidence for a phase transition. Although the latent heat calculation throws some doubt on the existence of a transition, after further discussion of the series it is concluded that there most probably is a phase transition.

Nonrelativistic Quantum Theory of an Electron in an Arbitrarily Intense Laser Field
View Description Hide DescriptionAn essentially exact treatment of the time‐dependent Schrödinger equation for a Bloch electron (or a free electron) in the presence of an arbitrarily intense laser field is described. Expressions for the wavefunction,current density, and energy of the electron state are presented in closed form for the case when the effective mass approximation is valid. The limitations of an ``almost exact'' solution of very simple form are investigated, the corrections to the almost exact solution being determined by the WKB approximation method. The exact solution for the wavefunction turns out to be quite different from that given by perturbation theory. However, the changes in the values of the current density and energy due to the presence of the laser field turn out to be, within the limitations imposed by the nonrelativistic nature of the Schrödinger theory, linear and quadratic in the field amplitude, and therefore agree with the results of perturbation theory.

A New Cluster Scheme in Statistical Physics
View Description Hide DescriptionA new cluster scheme is proposed, and some of its advantages over the Mayer expansion method are demonstrated. An explicit equation is derived for the s‐particle correlation in a large equilibrium system of particles interacting through a two‐body potential. This equation is solved to the leading order in the plasma limit. It is found that, in several systems, the equilibrium s‐particle correlation is, to the leading order, a functional of the two‐particle correlation, independent of the detailed form of the potential function.