Showing papers in "Journal of Mathematical Physics in 1975"

Nonlinear differential−difference equations

Abstract: A method is presented which enables one to obtain and solve certain classes of nonlinear differential−difference equations. The introduction of a new discrete eigenvalue problem allows the exact solution of the self−dual network equations to be found by inverse scattering. The eigenvalue problem has as its singular limit the continuous eigenvalue equations of Zakharov and Shabat. Some interesting differences arise both in the scattering analysis and in the time dependence from previous work.

On the duality condition for a Hermitian scalar field

Abstract: A general Hermitian scalar field, assumed to be an operator−valued tempered distribution, is considered. A theorem which relates certain complex Lorentz transformations to the TCP transformation is stated and proved. With reference to this theorem, duality conditions are considered, and it is shown that such conditions hold under various physically reasonable assumptions about the field. A theorem analogous to Borchers’ theorem on relatively local fields is stated and proved. Local internal symmetries are discussed, and it is shown that any such symmetry commutes with the Poincare

Some solutions of complex Einstein equations

Abstract: Complex V4’s are investigated where ΓȦḂ =0 and therefore a fortiori equations Gab=0 are fulfilled. A general theory of spaces of this type is outlined and examples of nontrivial solutions of all degenerate algebraic types are provided.

Prolongation structures of nonlinear evolution equations

Abstract: The prolongation structure of a closed ideal of exterior differential forms is further discussed, and its use illustrated by application to an ideal (in six dimensions) representing the cubically nonlinear Schrodinger equation. The prolongation structure in this case is explicitly given, and recurrence relations derived which support the conjecture that the structure is open—i.e., does not terminate as a set of structure relations of a finite‐dimensional Lie group. We introduce the use of multiple pseudopotentials to generate multiple Backlund transformation, and derive the double Backlund transformation. This symmetric transformation concisely expresses the (usually conjectured) theorem of permutability, which must consequently apply to all solutions irrespective of asymptotic constraints.

Higher - Dimensional Unifications of Gravitation and Gauge Theories

Abstract: We give a comprehensive geometric treatment of Kaluza–Klein type unifications of non‐Abelian gauge theories with gravitation. The appearance of a cosmological term is noted.

Topology of Higgs fields

Abstract: It is shown that the conserved magnetic charge discovered by ’t Hooft in non−Abelian gauge theories with spontaneous symmetry breaking is not associated with the invariance of the action under a symmetry group. Rather, it is a topological characteristic of an isotriplet of Higgs fields in a three−dimensional space: the Brouwer degree of the mapping between a large sphere in configuration space and the unit sphere in field space provided by the normalized Higgs field ? a = φ a (φ b φ b )−1/2. The use of topological methods in determining magnetic charge configurations is outlined. A peculiar interplay between Dirac strings and zeros of the Higgs field under gauge transformations is pointed out. The monopole−antimonopole system is studied.

On the quantum mechanical treatment of dissipative systems

Abstract: Two types of Hamiltonians are investigated which describe quantum mechanically a particle moving subject to a linear viscous force under the influence of a conservative force: the conventional explicitly time‐dependent one and an alternative class of nonlinear Hamiltonians. In the latter group we propose a new form. By Ehrenfest’s theorem the expectation values of the operators of physical observables correspond to the classical quantities. For all Schrodinger equations we derive and discuss wavepacket, wave, stationary, and pseudostationary solutions of force free motion, free fall, and harmonic oscillator.

Spacetime symmetries and linearization stability of the Einstein equations. I

Abstract: In a previous paper we began a study of the Fischer–Marsden conditions for the linearization stability of vacuum space–times with compact, Cauchy hypersurfaces. We showed that a space–time of this class is linearization stable if and only if it admits no global Killing vector fields. In this paper we derive the general nonlinear constraints upon the perturbations which are necessary, whenever Killing symmetries occur, to exclude spurious perturbation solutions. We establish the hypersurface independence of these constraints by relating them to the conserved integrals of the perturbation equations associated with the Killing symmetries of the background. As a corollary of this result, we also establish the gauge invariance of the nonlinear constraints. We briefly discuss the noncompact case and mention a possible application of our results to the study of the Hawking process of quantum mechanical particle production by black holes.

Lorentz covariant treatment of the Kerr--Schild geometry

Abstract: It is shown that a Lorentz covariant coordinate system can be chosen in the case of the Kerr–Schild geometry which leads to the vanishing of the pseudo energy–momentum tensor and hence to the linearity of the Einstein equations. The retarded time and the retarded distance are introduced and the Lienard–Wiechert potentials are generalized to gravitation in the case of world‐line singularities to derive solutions of the type of Bonnor and Vaidya. An accelerated version of the de Sitter metric is also obtained. Because of the linearity, complex translations can be performed on these solutions, resulting in a special relativistic version of the Trautman–Newman technique and Lorentz covariant solutions for spinning systems can be derived, including a new anisotropic interior metric that matches to the Kerr metric on an oblate spheroid.

Continuous subgroups of the fundamental groups of physics. I. General method and the Poincaré group

Abstract: We present a general method for reducing the problem of finding all continuous subgroups of a given Lie groupG with a nontrivial invariant subgroup N, to that of classifying the subgroups of N and the subgroups of the factor group G/N. The method is applied to classify all continuous subgroups of the Poincare group (PG) and of the Lorentz group extended by dilatations [the homogeneous similitude group (HSG)]. Lists of representatives of each conjugacy class of subalgebras of the Lie algebras of the groups PG and HSG are given in the form of tables.

Observables, operators, and complex numbers in the Dirac theory

Abstract: The geometrical formulation of the Dirac theory with spacetime algebra is shown to be equivalent to the usual matrix formalism. Imaginary numbers in the Dirac theory are shown to be related to the spin tensor. The relation of observables to operators and the wavefunction is analyzed in detail and compared with some purportedly general principles of quantum mechanics. An exact formulation of Larmor and Thomas precessions in the Dirac theory is given for the first time. Finally, some basic relations among local observables in the nonrelativistic limit are determined.

Quantum theory of anharmonic oscillators. I. Energy levels of oscillators with positive quartic anharmonicity

Abstract: This is an investigation of the energy levels of an anharmonic oscillator characterized by the potential (1/2) x2+λx4. Two regions of λ and n are distinguishable (n being the quantum number of the energy level) one in which the harmonic oscillator levels En=n+1/2 are only slightly distorted and the other in which the purely quartic oscillator form En?cλ1/3(n+1/2)4/3 (c being a constant) is only slightly distorted. Rapidly converging algorithms have been developed, using the Bargmann representation, from which energy levels in any (λ,n) (with λ≳0) regime can easily be computed. Simple formulas are also derived which give excellent approximations to the energy levels in various (λ,n) regimes.

J-Matrix Method: Extensions to Arbitrary Angular Momentum and to Coulomb Scattering

Abstract: The J-matrix method introduced previously for s-wave scattering is extended to treat the lth partial wave kinetic energy and Coulomb Hamiltonians within the context of square integrable (L2), Laguerre (Slater), and oscillator (Gaussian) basis sets The determination of the expansion coefficients of the continuum eigenfunctions in terms of the L2 basis set is shown to be equivalent to the solution of a linear second order differential equation with appropriate boundary conditions, and complete solutions are presented Physical scattering problems are approximated by a well-defined model which is then solved exactly In this manner, the generalization presented here treats the scattering of particles by neutral and charged systems The appropriate formalism for treating many channel problems where target states of differing angular momentum are coupled is spelled out in detail The method involves the evaluation of only L2 matrix elements and finite matrix operations, yielding elastic and inelastic scattering information over a continuous range of energies

Exact recursive evaluation of 3j- and 6j-coefficients for quantum- mechanical coupling of angular momenta

Abstract: Algorithms are developed for the exact evaluation of the 3j‐coefficients of Wigner and the 6j‐coefficients of Racah. These coefficients arise in the quantum theory of coupling of angular momenta. The method is based on the exact solution of recursion relations in a particular order designed to guarantee numerical stability even for large quantum numbers. The algorithm is more efficient and accurate than those based on explicit summations, particularly in the commonly arising case in which a whole set of related coefficients is needed.

Global existence of solutions to the Cauchy problem for time-dependent Hartree equations

Abstract: The existence of global solutions to the Cauchy problem for time‐dependent Hartree equations for N electrons is established. The solution is shown to have a uniformly bounded H1(R3) norm and to satisfy an estimate of the form ∥ ψ (t) ∥H2 ⩽ c exp(kt). It is shown that ’’negative energy’’ solutions do not converge uniformly to zero as t → ∞.

Semiclassical approximations to 3j‐ and 6j‐coefficients for quantum‐mechanical coupling of angular momenta

Abstract: The coupling of angular momenta is studied using quantum mechanics in the limit of large quantum numbers (semiclassical limit). Uniformly valid semiclassical expressions are derived for the 3j (Wigner) coefficients coupling two angular momenta, and for the 6j (Racah) coefficients coupling three angular momenta. In three limiting cases our new expressions reduce to those conjectured by Ponzano and Regge. The derivation involves solving the recursion relations satisfied by these coefficients, by a discrete analog of the WKB method. Terms of the order of the inverse square of the quantum numbers are neglected in the derivation, so that the results should be increasingly accurate for larger angular momenta. Numerical results confirm this asymptotic convergence. Moreover, the results are of a useful accuracy even at small quantum numbers.

Uniform asymptotic expansions in transport theory with small mean free paths, and the diffusion approximation

Abstract: We consider initial−boundary value and boundary value problems for transport equations in inhomogeneous media. We consider the case when the mean free path is small compared to typical lengths in the domain (e.g., the size of a reactor). Employing the boundary layer technique of matched asymptotic expansions, we derive a uniform asymptotic expansion of the solution of the problem. In so doing we find that in the interior of the domain, i.e., away from boundaries and away from the initial line, the leading term of the expansion satisfies a diffusion equation which is the basis of most computational work in reactor design. We also derive boundary conditions appropriate to the diffusion equation. Comparisons with existing results such as the asymptotic and P1 diffusion theories, the PN approximation, and the extrapolated end point condition for these approximations, are made. Finally the uniform validity of our expansions is proved, thus yielding the desired error estimates.

Semisimple Graded Lie Algebras

Abstract: The concept of metric is introduced for graded Lie algebras. Semisimple graded Lie algebras are defined in terms of metric conditions of nonsingularity. It is shown that for this class of algebras the metric tensor generates a quadratic Casimir operator. Also for this class, the grading representation is irreducible and its weights are related to the roots of the Lie algebra (’’root‐weight theorem’’). The problem is solved to find all semisimple graded Lie algebras. For SU(N), N≳2, for O(N), N≳5, and for all exceptional groups there are none. For all other semisimple Lie algebras there is one and only one. These are explicity constructed in terms of a convenient realization of Sp(2N) matrices. SU(2) is discussed in some detail and a new group [GSU(2)] is found which leaves a mixed c‐number/q‐number quadratic form invariant. We also define irreducible tensor operators for this group. SU(N), N≳2, provides examples of nonsemisimple gradings.

Monotonic converging variational approximations to the functional integrals in quantum statistical mechanics

Abstract: We show, by making use of the functional integral technique, that, for a large class of useful quantum statistical systems, the partition function is, with respect to the coupling constant, the Laplace transform of a positive measure. As a consequence, we derive an infinite set of monotonicly converging upper and lower bounds to it. In particular, the lowest approximation appears to be identical to the Gibbs–Bogolioubov variational bound, while the next approximations, for which we give explicit formulas for the first few ones, lead to improve the previous bound. The monotonic character of the variational successive approximations allows a new approach towards the thermodynamical limit.

A note on coherent state representations of Lie groups

Abstract: The analyticity properties of coherent states for a semisimple Lie group are discussed. It is shown that they lead naturally to a classical ’’phase space realization’’ of the group.

The Weyl correspondence and path integrals

Abstract: The method of Weyl transforms is used to rigorously derive path integral forms for position and momentum transition amplitudes from the time‐dependent Schrodinger equation for arbitrary Hermitian Hamiltonians. It is found that all paths in phase space contribute equally in magnitude, but that each path has a different phase, equal to 1/h/ times an ’’effective action’’ taken along it. The latter is the time integral of p⋅q−h (p,q), h (p,q) being the Weyl transform of the Hamiltonian operator H, which differs from the classical Hamiltonian function by terms of order h/2, vanishing in the classical limit. These terms, which can be explicitly computed, are zero for relatively simple Hamiltonians, such as (1/2M)[P−eA (Q)]2+V (Q), but appear when the coupling of the position and momentum operators is stronger, such as for a relativistic spinless particle in an electromagnetic field, or when configuration space is curved. They are always zero if one opts for Weyl’s rule for forming the quantum operator corresponding to a given classical Hamiltonian. The transition amplitude between two position states is found to be expressible as a path integral in configuration space alone only in very special cases, such as when the Hamiltonian is quadratic in the momenta.

Resonantly coupled nonlinear evolution equations

Abstract: A differential matrix eigenvalue problem is used to generate systems of nonlinear evolution equations. They model triad, multitriad, self‐modal, and quartet wave interactions. A nonlinear string equation is also recovered as a special case. A continuum limit of the eigenvalue problem and associated evolution equations are discussed. The initial value solution requires an investigation of the corresponding inverse‐scattering problem.

Exact propagator for a time−dependent harmonic oscillator with and without a singular perturbation

Abstract: By using Feynman’s definition of a path integral, exact propagators for a time−dependent harmonic oscillator with and without an inverse quadratic potential have been evaluated. It is shown that these propagators depend only on the solutions of the classical unperturbed oscillator. The relations between these propagators, the invariants, and the Schrodinger equation are also discussed.

Motion of a body in general relativity

Abstract: A simple theorem, whose physical interpretation is that an isolated, gravitating body in general relativity moves approximately along a geodesic, is obtained.

Singularities in nonsimply connected space--times

Abstract: Space–times with asymptotically flat nonsimply connected spacelike slices are shown to possess enough intrinsic geometric structure to guarantee the existence of singularities under conditions usually considered insufficient. In particular, it is shown that if the normal geodesics to the spacelike slice are converging on a suitable compact set, and the space–time satisfies a standard energy condition, then it is timelike geodesically incomplete. A similar result holds if the space–time satisfies the chronology and generic conditions.

A general setting for Casimir invariants

Abstract: This paper contains a general description of the theory of invariants under the adjoint action of a given finite‐dimensional complex Lie algebra G, with special emphasis on polynomial and rational invariants. The familiar ’’Casimir’’ invariants are identified with the polynomial invariants in the enveloping algebra U (G). More general structures (quotient fields) are required in order to investigate rational invariants. Some useful criteria for G having only polynomial or rational invariants are given. Moreover, in most of the physically relevant Lie algebras the exact computation of the maximal number of algebraically independent invariants turns out to be very easy. It reduces to finding the rank of a finite matrix. We apply the general method to some typical examples.

Continuous subgroups of the fundamental groups of physics. II. The similitude group

Abstract: All subalgebras of the similitude algebra (the algebra of the Poincare group extended by dilatations) are classified into conjugacy classes under transformations of the similitude group. Use is made of the classification of all subalgebras of the Poincare algebra, carried out in a previous article. The results are presented in tables listing representatives of each class and their basic properties.

Neutron transport and diffusion in inhomogeneous media. I

Abstract: The asymptotic solution of the neutron transport equation is obtained for large near‐critical domains D which possess a cellular, nearly periodic structure. A typical mean free path in D is taken to be of the same order of magnitude as a cell diameter, and these are taken to be small (of order e) compared to a typical diameter of D. The solution is asymptotic with respect to the small parameter e. It is a product of two functions, one determined by a detailed cell calculation and the other obtained as the solution of a time dependent diffusion equation. The diffusion equation contains precursor (delayed neutron) densities, equations for which are derived. The coefficients in the diffusion equation, which are determined using the results of the cell calculation, differ from those now used in engineering applications. The initial condition for the diffusion equation is derived, and the problem of determining the boundary condition is discussed.

The variational approach to the two−body density matrix

Abstract: A variational method for the two−body density matrix is developed for practical calculations of the properties of many−fermion systems with two−body interactions. In this method the energy E = JHijkl ρijkl is minimized using the two−body density matrix elements ρijkl = 〈ψ‖a+ja+iakal‖ψ〉 as variational parameters. The approximation consists in satisfying only a subset of necessary conditions—the nonnegativity of the following matrices: the two−body density matrix, the ’’two−hole matrix’’ Qijkl = 〈Ψ‖ajaia+ka+l‖Ψ〉 and the particle−hole matrix Gijkl = 〈Ψ‖ (a+iaj−ρij)+ (a+kal−ρk) ‖Ψ〉. The idea of the method was introduced earlier; here some further physical interpretation is given and a numerical procedure for calculations within a small single−particle model space is described. The method is illustrated on the ground state of Be atom using 1s, 2s, 2p orbitals.

Consistency in the formulation of the Dirac, Pauli, and Schrödinger theories

Abstract: Properties of observables in the Pauli and Schrodinger theories and first order relativistic approximations to them are derived from the Dirac theory. They are found to be inconsistent with customary interpretations in many respects. For example, failure to identify the ’’Darwin term’’ as the s−state spin−orbit energy in conventional treatments of the hydrogen atom is traced to a failure to distinguish between charge and momentum flow in the theory. Consistency with the Dirac theory is shown to imply that the Schrodinger equation describes not a spinless particle as universally assumed, but a particle in a spin eigenstate. The bearing of spin on the interpretation of the Schrodinger theory is discussed. Conservation laws of the Dirac theory are formulated in terms of relative variables, and used to derive virial theorems and the corresponding conservation laws in the Pauli−Schrodinger theory.