Showing papers in "Journal of Mathematical Physics in 1961"
TL;DR: In this paper, an action principle technique for the direct computation of expectation values is described and illustrated in detail by a special physical example, the effect on an oscillator of another physical system, which has the advantage of combining immediate physical applicability (e.g., resistive damping or maser amplification of a single electromagnetic cavity mode) with a significant idealization of the complex problems encountered in many particle and relativistic fieldtheory.
Abstract: An action principle technique for the direct computation of expectation values is described and illustrated in detail by a special physical example, the effect on an oscillator of another physical system. This simple problem has the advantage of combining immediate physical applicability (e.g., resistive damping or maser amplification of a single electromagnetic cavity mode) with a significant idealization of the complex problems encountered in many‐particle and relativistic fieldtheory. Successive sections contain discussions of the oscillator subjected to external forces, the oscillator loosely coupled to the external system, an improved treatment of this problem and, finally, there is a brief account of a general formulation.
2,222 citations
TL;DR: In this article, Utiyama's discussion is extended by considering the 10-parameter group of inhomogeneous Lorentz transformations, involving variation of the coordinates as well as the field variables.
Abstract: An argument leading from the Lorentz invariance of the Lagrangian to the introduction of the gravitational field is presented. Utiyama's discussion is extended by considering the 10‐parameter group of inhomogeneous Lorentz transformations, involving variation of the coordinates as well as the field variables. It is then unnecessary to introduce a priori curvilinear coordinates or a Riemannian metric, and the new field variables introduced as a consequence of the argument include the vierbein components hk μ as well as the ``local affine connection'' Ai jμ . The extended transformations for which the 10 parameters become arbitrary functions of position may be interpreted as general coordinate transformations and rotations of the vierbein system. The free Lagrangian for the new fields is shown to be a function of two covariant quantities analogous to Fμν for the electromagnetic field, and the simplest possible form is just the usual curvature scalar density expressed in terms of hk μ and Ai jμ . This Lagrangian is of first order in the derivatives, and is the analog for the vierbein formalism of Palatini's Lagrangian. In the absence of matter, it yields the familiar equationsRμν =0 for empty space, but when matter is present there is a difference from the usual theory (first pointed out by Weyl) which arises from the fact that Ai jμ appears in the matter field Lagrangian, so that the equation of motion relating Ai jμ to hk μ is changed. In particular, this means that, although the covariant derivative of the metric vanishes, the affine connection Γλ μν is nonsymmetric. The theory may be reexpressed in terms of the Christoffel connection, and in that case additional terms quadratic in the ``spin density'' Sk ij appear in the Lagrangian. These terms are almost certainly too small to make any experimentally detectable difference to the predictions of the usual metric theory.
1,498 citations
TL;DR: In this article, a criterion for the validity of approximate integral equations describing the various field quantities of interest was obtained by employing a ''configurational averaging'' procedure, and the extinction theorem was obtained and given rise to the forward-amplitude theorem of multiple scattering.
Abstract: Multiple scattering effects due to a random array of obstacles are considered. Employing a ``configurational averaging'' procedure, a criterion is obtained for the validity of approximate integral equations describing the various field quantities of interest. The extinction theorem is obtained and shown to give rise to the forward‐amplitude theorem of multiple scattering. In the limit of vanishing correlations in position, the complex propagation constant κ of the scattering medium is obtained. Under appropriate restrictions, the expression for κ is shown to include both the square‐root law of isotropic scatterers and the additive rule for cross sections valid for sufficiently low densities of anisotropic obstacles. Some specific examples from acoustics and electromagnetic theory then indicate that at least in the simplest cases the results remain valid for physically allowable densities of obstacles.
929 citations
TL;DR: In this article, the problem of cluster size distribution and percolation on a regular lattice or graph of bonds and sites is reviewed and its applications to dilute ferromagnetism, polymer gelation, etc., briefly discussed.
Abstract: The problem of cluster size distribution and percolation on a regular lattice or graph of bonds and sites is reviewed and its applications to dilute ferromagnetism, polymer gelation, etc., briefly discussed. The cluster size and percolation problems are then solved exactly for Bethe lattices (infinite homogeneous Cayley trees) and for a wide class of pseudolattices derived by replacing the bonds and/or sites of a Bethe lattice by arbitrary finite subgraphs. Explicit expressions are given for the critical probability (density), for the mean cluster size, and for the density of infinite clusters. The nature of the critical anomalies is shown to be the same for all lattices discussed; in particular, the density of infinite clusters vanishes as R(p) ≈ C(p−pc) (p≥pc).
421 citations
TL;DR: Contraction is defined for a Lie group to coincide on its Lie algebra with a generalization of contraction as first introduced by Inonu and Wigner as mentioned in this paper, which is accomplished with a sequence of nonsingular coordinate transformations on the group, whose limit is a singular one.
Abstract: Contraction is defined for a Lie group to coincide on its Lie algebra with a generalization of contraction as first introduced by Inonu 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 Inonu‐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
318 citations
TL;DR: In this article, it was shown that a field of null rays is geodetic and shear-free if and only if the associated family of null bivectors includes a solution to Maxwell's equations for charge-free space.
Abstract: It is shown that a field of null rays is geodetic and shear‐free if and only if the associated family of null bivectors includes a solution to Maxwell's equations for charge‐free space.
193 citations
TL;DR: In this article, a system consisting of an equal number of positively and negatively charged ''sheets'' is considered in thermal equilibrium, with motion restricted to one dimension, and the configurational part of the partition function can be represented as a sum of terms, each a simple algebraic expression.
Abstract: A system consisting of an equal number of positively and negatively charged ``sheets'' is considered in thermal equilibrium, with motion restricted to one dimension. The configurational part of the partition function can be represented as a sum of terms, each a simple algebraic expression. The summation is performed with the technique of generating functions. The asymptotic form in the limit of an infinite system is obtained from the pole of the generating function closest to the origin. This pole is the solution of a certain transcendental equation for which an explicit analytic representation in terms of an infinite continued fraction is available. It is shown that this equation is identical with the characteristic equation associated with the even Mathieu functions of even order.In the limit, when the ratio of interparticle force to pressure is small, the system behaves as an ideal gas, the deviations from this state being expandable in powers of the square root of this ratio. In the opposite limit of ...
190 citations
TL;DR: In this paper, it was shown that even though these points do not in general possess a limit point within the region of analyticity, one can still uniquely determine the Fourier transform of the Green's function directly from its values at these points.
Abstract: In the study of thermodynamic correlation functions or Green's functions, one is naturally led to a calculation of values of the Fourier transform of the Green's function on a discrete set of points in the complex energy plane. It is shown that even though these points do not in general possess a limit point within the region of analyticity, one may still uniquely determine the Fourier transform of the Green's function directly from its values at these points.
150 citations
TL;DR: In this paper, the problem of making the most effective use of the coefficients of series expansions for the Ising model and excluded volume problem in estimating critical behavior was discussed, and it was shown that after initial irregularities the coefficients appear to settle down to a smooth asymptotic behavior.
Abstract: The present paper discusses the problem of making the most effective use of the coefficients of series expansions for the Ising model and excluded volume problem in estimating critical behavior. It is shown that after initial irregularities the coefficients appear to settle down to a smooth asymptotic behavior. Alternative methods of analysis are considered for the provision of a steady series of approximations to the critical point. Numerical conclusions are drawn for particular lattices for which additional terms have recently become available.
136 citations
TL;DR: In this article, the R4 spherical harmonics and their properties are obtained as specializations of the general formulas, and their physical application to the problem of geometrizing the Coulomb field is briefly discussed.
Abstract: The local isomorphism of the R4 group to the group R3×R3 is utilized to obtain R4 Wigner coefficients for those representations in which the subgroup R3 is diagonal. The R4 Wigner coefficients so defined are then used to obtain recursion relations and differential equations for the representation coefficients, when the group is parametrized appropriately. The R4 spherical harmonics, and their properties, are explicitly obtained as specializations of the general formulas. Physical application to the problem of geometrizing the Coulomb field is briefly discussed.
133 citations
TL;DR: In this article, it was shown that the critical probabilities for the bond problem on the plane square and triangular lattice cannot exceed those for the corresponding site problems on a plane lattice without crossing bonds.
Abstract: When particles occupy the sites or bonds of a lattice at random with probability p, there is a critical probability pc above which an infinite connected cluster of particles forms. Rigorous bounds and inequalities are obtained for pc on a variety of lattices and compared briefly with previous numerical estimates. In particular, by extending Harris' work, it is proved that pc(s,L2)⩾12 for the site problem on a plane lattice L2 (without crossing bonds), while for the bond problem pc(b,L2)+pc(b,L2D)⩾1 where L2D is the dual lattice to L2. Simple arguments demonstrate that the bond problem is a special case of the site problem and that the critical probabilities for the bond problem on the plane square and triangular lattice cannot exceed those for the corresponding site problems.
TL;DR: In this paper, it was shown that a system of harmonic oscillators weakly coupled by nonlinear forces will not achieve equipartition of energy as long as the uncoupled frequencies ωk are linearly independent on the integers.
Abstract: A system of harmonic oscillators weakly coupled by nonlinear forces will not achieve equipartition of energy as long as the uncoupled frequencies ωk are linearly independent on the integers, i.e., as long as there is no collection of integers {nk} for which Σnkωk=0 other than all nk=0. This result is shown to follow from the general form of the Kryloff and Bogoliuboff series solution to the equations of motion. Physically, the linear independence of the uncoupled frequencies means that none of the interacting oscillators drives another at its resonant frequency, and this lack of internal resonance precludes appreciable energy sharing in the limit as the coupling tends to zero. It is shown that the lack of equipartition of energy observed by Ulam, Fermi, and Pasta for certain nonlinear systems may be explained in terms of the preceding remarks. Moreover, a Kryloff and Bogoliuboff series solution to the appropriate equations of motion is shown to yield qualitative agreement with the Ulam, Fermi, and Pasta c...
TL;DR: In this paper, the properties of the dipole antenna are studied by an approximate procedure that makes use of the Wiener-Hopf integral equation, in particular, the input admittance and the radiation pattern are found.
Abstract: The properties of the dipole antenna are studied by an approximate procedure that makes use of the Wiener‐Hopf integral equation. In particular, the input admittance and the radiation pattern are found. The present results thus supplement the existing theories, which are concerned mostly with short dipoles. The same procedure is then applied to several related problems. First, the back‐scattering cross section of a dipole antenna is found approximately for normal incidence. Secondly, the two‐wire transmission line is studied in detail by considering it to be two coupled dipole antennas. The capacitive end correction for an open end is evaluated, and the radiated power and the radiation resistance are found for a resonant section of transmission line with both ends open. Finally, the dielectric‐coated antenna is considered briefly.
TL;DR: A classification of four-dimensional Riemann spaces with signature + 2 is given in this paper, where the classification depends upon the differential as well as the algebraic properties of the riemann tensor.
Abstract: A classification of four‐dimensional Riemann spaces with signature +2 is given The classification depends upon the differential as well as the algebraic properties of the Riemann tensor The tool employed is the infinitesimal‐holonomy group of the space An introduction to the concept of the holonomy group is given, and the technique of classification is outlined A comparison with the classification of empty spaces given by A Z Petrov and with the recent work of E Newman is also given
TL;DR: In this paper, the exact enumeration of self-avoiding random walks on a lattice is studied and a theorem is derived that enables the number of such walks to be calculated recursively from a restricted class of closed graphs more easily enumerated than the walks themselves.
Abstract: The problem of the exact enumeration of self‐avoiding random walks on a lattice is studied and a theorem derived that enables the number of such walks to be calculated recursively from the number of a restricted class of closed graphs more easily enumerated than the walks themselves. The method of Oguchi for deriving a high‐temperature expansion for the zero‐field susceptibility of the Ising model is developed and a corresponding theorem enabling the successive coefficients to be calculated recursively from a restricted class of closed graphs deduced. The theorem relates the susceptibility to the configurational energy and enables the behavior of the antiferromagnetic susceptibility at the transition point to be inferred.
TL;DR: In this article, it was shown that the critical probability of an atom percolation process is not less than the critical probabilities of the corresponding bond percolations, and it was also shown that this is the case for the case of a pair of atoms percolating each other.
Abstract: Various inequalities, some of them strict, are proved concerning probabilities associated with percolation processes In particular, it is shown that the critical probability of an atom percolation process is not less than the critical probability of the corresponding bond percolation process
TL;DR: In this paper, the authors considered an n-dimensional cubic crystal with nearest-neighbor central and noncentral harmonic forces in which the mass M of one of the lattice particles is relatively large.
Abstract: The system considered is an n‐dimensional cubic crystal with nearest‐neighbor central and noncentral harmonic forces in which the mass M of one of the lattice particles is relatively large. It is assumed that the velocities and positions of the light particles in the system (mass m) are normally distributed, at time t=0, as in thermal equilibrium. The conditional velocity distribution for the heavy particle at time t is then a normal distribution with a time‐dependent mean value. This mean value is the velocity autocorrelation function. The dispersion of the distribution is shown to be a simple function of the autocorrelation. In the limit M/m≫1 in the one‐ and two‐dimensional lattices, the autocorrelation function is, respectively, a damped exponential and a damped oscillating exponential. These different types of statistical behavior are related to the different dynamic properties of the medium with which the heavy particle interacts.
TL;DR: In this paper, it was proved that the H theorem for an ensemble of isolated quantal systems with a discrete energy spectrum is false provided the systems satisfy certain broad conditions: the theorem is false for bounded many-particle systems with potential interaction, provided that interaction contains no repulsive singularities stronger than r−2 and no attractive singularities weaker than r −1.
Abstract: It is proved that the H theorem for an ensemble of isolated quantal systems with a discrete energy spectrum is false provided the systems satisfy certain broad conditions: the theorem is false for bounded many‐particle systems with potential interaction, provided that interaction contains no repulsive singularities stronger than r−2 and no attractive singularities stronger than r−1. Ensembles of such systems behave almost periodically, in the sense of H. Bohr. The entropy and the probability of finding an observable in a given range are both almost periodic functions of time.
TL;DR: In this article, it was shown that all elements of the S matrix can be obtained from a single analytic function of all channel momenta, the Fredholm determinant of the scattering and reaction integral equations.
Abstract: We are considering the nonrelativistic elastic and inelastic scattering of two particles with internal degrees of freedom, or reactions giving rise to two particles. It is shown under very general conditions that all elements of the S matrix can be simply obtained from a single analytic function of all channel momenta, the Fredholm determinant of the scattering and reaction integral equations. Its properties are investigated and the restrictions are established which are necessary and sufficient in order to assure that the unitarity condition is fulfilled. The square well and a superposition of Yukawa potentials are considered as examples.
TL;DR: In this paper, 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.
Abstract: 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
TL;DR: In this article, the frequency spectrum of a disordered one-dimensional lattice is calculated via an investigation of the phonon propagator, and the spectrum is evaluated in detail for a low concentration of light impurities inserted at random along a linear chain.
Abstract: The frequency spectrum of a disordered one‐dimensional lattice is calculated via an investigation of the phonon propagator. The spectrum is evaluated in detail for a low concentration of light impurities inserted at random along a linear chain. It is found that an impurity band occurs near the frequency of the local mode. Higher‐order effects resulting from clusters of impurities are calculated and discussed.
TL;DR: In this paper, a definition of strict localization of states in quantum field theory is presented, based on considering products of field operators as the primary measurable quantities of the theory, and an example of such a state arises when a free field interacts with an external current that is limited to a bounded region of space.
Abstract: A definition of strict localization of states in quantum field theory is presented. This definition is based on considering products of field operators as the primary measurable quantities of the theory. An example of a localized state is given, showing that such a state arises when a free field interacts with an external current that is limited to a bounded region of space‐time. It is shown by means of a graphical technique that a state having a finite number of particles cannot satisfy the definition of localization. A simple representation of localized states is investigated, and arguments are given to support its generality and uniqueness.
TL;DR: In this paper, the analytic n-point function in momentum space in quantum field theory is studied, and its boundary values for real value of the argument are determined, and a necessary and sufficient condition for them to be obtainable from the Wightman functions is given.
Abstract: The analytic n‐point function in momentum space in quantum field theory is studied. Its different boundary values for real value of the argument are determined, and a necessary and sufficient condition for them to be obtainable from the Wightman functions is given. The conditions are relativistic covariance, support properties in coordinate space (retardedness), two‐term identities for momentum below threshold (corresponding to spectrum conditions) and four‐term identities (Steinmann relations). The first three conditions are translatable into a statement about the domain of analyticity of the n‐point function: it is analytic in a union of various extended tubes plus the points of contact of two neighboring tubes for real part of one momentum below threshold.
TL;DR: In this article, the classifications of Einstein spaces by Schell and Petrov are combined and certain nonlocal results are obtained and it is shown that an Einstein space cannot be type I with a rank four Riemann tensor in a four-dimensional region.
Abstract: The classifications of Einstein spaces by Schell and Petrov are combined and certain nonlocal results are obtained. In particular, we show that an Einstein space cannot be type I with a rank four Riemann tensor in a four‐dimensional region. On using the notion of a perfect or imperfect infinitesimal‐holonomy group, we establish the conditions under which an Einstein space possesses a two‐, four‐, or six‐parameter group. We find that two‐ and four‐parameter groups are associated with special cases of type II null and type III, respectively.
TL;DR: The potential which minimizes the lowest eigenvalue of the one-dimensional Schrodinger equation is determined among all potentials V for which the integral of Vn has the prescribed value k.
Abstract: The potential which minimizes the lowest eigenvalue of the one‐dimensional Schrodinger equation is determined among all potentials V for which the integral of Vn has the prescribed value k. For each value of n and k this potential is found to be a special case of the Epstein‐Eckart potentials which were originally introduced because the Schrodinger equation for them could be solved explicitly. The minimum eigenvalue is determined and it provides a lower bound on the lowest eigenvalue of any potential for which ∫Vndx=k. The expression of this fact as an inequality yields an isoperimetric inequality. For an arbitrary potential, each value of n provides one lower bound on the lowest eigenvalue, the largest of which is the best. This best bound is determined for the square well, the exponential, and the inverse power potentials. In the case of the square well, it is compared with the exact value. In the limiting case n = 1 our result reduces to that previously obtained by Larry Spruch, who showed that the del...
TL;DR: In this paper, the pseudopotential method is used to study a special type of flow for a Bose system of hard spheres, where the wave function of the entire system is assumed to be the product of identical single-particle wave functions, which in general are time-dependent.
Abstract: The pseudopotential method is used to study a special type of flow for a Bose system of hard spheres. In the first‐order approximation, the wave function of the entire system is assumed to be the product of identical single‐particle wave functions, which in general are time‐dependent. Such a flow is necessarily irrotational, and the single‐particle wave function satisfies a Schrodinger equation with a nonlinear self‐coupling term. On the basis of this equation of motion, the following properties of the Bose system are discussed: the effect of the rigid wall, the moment of inertia, the compressional wave, and a type of ``vortex filament.'' In the second‐order approximation, the wave function of the system is expressed in terms of two functions such that one of them describes the single‐particle state suitable for most of the particles while the other one describes the pair excitations. The much more complicated equations of motion are found, but in this approximation the flow is no longer strictly irrotational. The compressional waves are also studied in the second‐order approximation.
TL;DR: In this article, it was shown that a class of Schwartz distributions on the real axis can be continued to holomorphic functions in the upper and lower complex half-planes such that the ''jump'' on real axis represents the distribution.
Abstract: It is shown that a class of Schwartz distributions on the real axis can be continued to holomorphic functions in the upper and lower complex half‐planes such that the ``jump'' on the real axis represents the distribution. Many operations with distributions can be reduced to operations with the associated holomorphic functions which is of particular interest for the convolution product and for Fourier transforms. By means of the continuations several kinds of multiplications for distributions are being defined, which is of interest for quantum field theory.
TL;DR: The connection of spin and commutation relations for different fields is studied in this paper, where it is proved on the basis of the Lorentz invariance and spectrum conditions that any regular weak locality is equivalent to the normal weak locality plus a set of evenoddness conservation laws.
Abstract: The connection of spin and commutation relations for different fields is studied. The normal locality is defined as the property that two Boson fields as well as a Boson field and a Fermion field commute, while two Fermion fields anticommute with each other at a spacelike distance. A regular locality is defined as any combination of commutativity and anticommutativity between various pairs of fields at spacelike distance, where the kinematically related fields are assumed to obey the same type of commutation relations. The normal and regular weak locality is defined in a corresponding way. It is proved on the basis of the Lorentz invariance and spectrum conditions that any regular locality is equivalent to the normal locality plus a set of even‐oddness conservation laws. It is further shown, under the assumption of the normal weak locality between pairs of the same field, that any regular weak locality is equivalent to the normal weak locality plus a set of even‐oddness conservation laws.
TL;DR: In this paper, a set of coupled integral equations describing the collisions of diatomic molecules is developed by exploitation of the properties of the irreducible representations of the three-dimensional rotation group.
Abstract: A set of coupled integral equations describing the collisions of diatomic molecules is developed by exploitation of the properties of the irreducible representations of the three‐dimensional rotation group. An expansion of the cross section in spherical harmonics is described, and its virtues argued.
TL;DR: In this article, it was shown that the Landau curve for a reduced sixth-order diagram can acquire acnodes and real cusps as the masses are varied, and the problem of obtaining general criteria for acnode and Cusps is discussed.
Abstract: It is shown that the Landau curve for a reduced sixth‐order diagram can acquire acnodes and real cusps as the masses are varied. They are associated with complex singularities that under certain conditions are in the physical sheet and cause a breakdown of the Mandelstam representation. The problem of obtaining general criteria for acnodes and cusps is discussed.