Showing papers in "Journal of Mathematical Physics in 1967"

Exponential Operators and Parameter Differentiation in Quantum Physics

Abstract: Elementary parameter‐differentiation techniques are developed to systematically derive a wide variety of operator identities, expansions, and solutions to differential equations of interest to quantum physics. The treatment is largely centered around a general closed formula for the derivative of an exponential operator with respect to a parameter. Derivations are given of the Baker‐Campbell‐Hausdorff formula and its dual, the Zassenhaus formula. The continuous analogs of these formulas which solve the differential equationdY(t)/dt = A(t) Y(t), the solutions of Magnus and Fer, respectively, are similarly derived in a recursive manner which manifestly displays the general repeated‐commutator nature of these expansions and which is quite suitable for computer programming. An expansion recently obtained by Kumar and another new expansion are shown to be derivable from the Fer and Magnus solutions, respectively, in the same way. Useful similarity transformations involving linear combinations of elements of a Lie algebra are obtained. Some cases where the product eAeB can be written as a closed‐form single exponential are considered which generalize results of Sack and of Weiss and Maradudin. Closed‐form single‐exponential solutions to the differential equationdY(t)/dt = A(t) Y(t) are obtained for two cases and compared with the corresponding multiple‐exponential solutions of Wei and Norman. Normal ordering of operators is also treated and derivations, corollaries, or generalization of a number of known results are efficiently obtained. Higher derivatives of exponential and general operators are discussed by means of a formula due to Poincare which is the operator analog of the Cauchy integral formula of complex variable theory. It is shown how results obtained by Aizu for matrix elements and traces of derivatives may be readily derived from the Poincare formula. Some applications of the results of this paper to quantum statistics and to the Weyl prescription for converting a classical function to a quantum operator are given. A corollary to a theorem of Bloch is obtained which permits one to obtain harmonic‐oscillator canonical‐ensemble averages of general operators defined by the Weyl prescription. Solutions of the density‐matrix equation are also discussed. It is shown that an initially canonical ensemble behaves as though its temperature remains constant with a ``canonical distribution'' determined by a certain fictitious Hamiltonian.

Maximal analytic extension of the Kerr metric

Abstract: Kruskal's transformation of the Schwarzschild metric is generalized to apply to the stationary, axially symmetric vacuum solution of Kerr, and is used to construct a maximal analytic extension of the latter. In the low angular momentum case, a2 < m2, this extension consists of an infinite sequence Einstein‐Rosen bridges joined in time by successive pairs of horizons. The number of distinct asymptotically flat sheets in the extended space can be reduced to four by making suitable identifications. Several properties of the Kerr metric, including the behavior of geodesics lying in the equatorial plane, are examined in some detail. Completeness is demonstrated explicitly for a special class of geodesics, and inferred for all those that do not strike the ring singularity.

Spin‐s Spherical Harmonics and ð

Abstract: Recent work on the Bondi‐Metzner‐Sachs group introduced a class of functions sYlm (θ, φ) defined on the sphere and a related differential operator ð. In this paper the sYlm are related to the representation matrices of the rotation group R 3 and the properties of ð are derived from its relationship to an angular‐momentum raising operator. The relationship of the sTlm (θ, φ) to the spherical harmonics of R 4 is also indicated. Finally using the relationship of the Lorentz group to the conformal group of the sphere, the behavior of the sTlm under this latter group is shown to realize a representation of the Lorentz group.

Phase-Integral Approximation in Momentum Space and the Bound States of an Atom

Abstract: The phase integral approximation of the Green's function in momentum space is investigated for an electron of negative energy (corresponding to a bound state) which moves in a spherically symmetric potential. If the propagator rather than the wavefunction is considered, all classical orbits enter into the formulas, rather than only the ones which satisfy certain quantum conditions, and the separation of variables can be avoided. The distinction between classically accessible and classically inaccessible regions does not arise in momentum space, because any two momenta can be connected by a classical trajectory of given negative energy for a typical atomic potential. Three approaches are discussed: the Fourier transform of the phase integral approximation in coordinate space, the approximate solution of Schrodinger's equation in momentum space by a WKB ansatz, and taking the limit of small Planck's quantum in the Feynman‐type functional integral which was recently proposed by Garrod for the energy‐momentum representation. In particular, the last procedure is used to obtain the phase jumps of π/2 which occur every time neighboring classical trajectories cross one another. These extra phase factors are directly related to the signature of the second variation for the action function, and provide a physical application of Morse's calculus of variation in the large. The phase integral approximation in momentum space is then applied to the Coulomb potential. The location of the poles on the negative energy axis gives the Bohr formula for the bound‐state energies, and the residues of the approximate Green's function are shown to yield all the exact wavefunctions for the bound states of the hydrogen atom.

Properties of a Harmonic Crystal in a Stationary Nonequilibrium State

Abstract: The stationary nonequilibrium Gibbsian ensemble representing a harmonic crystal in contact with several idealized heat reservoirs at different temperatures is shown to have a Gaussian r space distribution for the case where the stochastic interaction between the system and heat reservoirs may be represented by Fokker—Planck-type operators. The covariance matrix of this Gaussian is found explicitly for a linear chain with nearest-neighbor forces in contact at its ends with heat reservoirs at temperatures T 1 and T N , N being the number of oscillators. We also find explicitly the covariance matrix, but not the distribution, for the case where the interaction between the system and the reservoirs is represented by very “hard” collisions. This matrix differs from that for the previous case only by a trivial factor. The heat flux in the stationary state is found, as expected, to be proportional to the temperature difference (T 1 − T N ) rather than to the temperature gradient (T 1 − T N )/N. The kinetic temperature of the jth oscillator T(j) behaves, however, in an unexpected fashion. T(j) is essentially constant in the interior of the chain decreasing exponentially in the direction of the hotter reservoir rising only at the end oscillator in contact with that reservoir (with corresponding behavior at the other end of the chain). No explanation is offered for this paradoxical result.

Correlations in Ising Ferromagnets. I

Abstract: An inequality relating binary correlation functions for an Ising model with purely ferromagnetic interactions is derived by elementary arguments and used to show that such a ferromagnet cannot exhibit a spontaneous magnetization at temperatures above the mean-field approximation to the Curie or “critical” point. (As a consequence, the corresponding “lattice gas” cannot undergo a first order phase transition in density (condensation) above this temperature.) The mean-field susceptibility in zero magnetic field at high temperatures is shown to be an upper bound for the exact result.

Dynamics of Pressure‐Free Matter in General Relativity

Abstract: An orthonormal tetrad system and associated coordinate system is obtained, which may be used to locally describe any dust‐filled space‐time. This is used to study dust‐filled spaces in which there exist multiply transitive groups of motions; all such spaces are classified in detail. Spaces containing shear‐free dust are also considered; it is shown that σ = 0 ⇒ ωΘ = 0. Three classes of solution with σ = 0, ω ≠ 0 are studied. Several new solutions of the field equations are contained in these results.

Topology in general relativity

Abstract: A number of theorems and definitions which are useful in the global analysis of relativistic world models are presented. It is shown in particular that, under certain conditions, changes in the topology of spacelike sections can occur if and only if the model is acausal. Two new covering manifolds, embodying certain properties of the universal covering manifold, are defined, and their application to general relativity is discussed.

Stability of Matter. II

Abstract: The stability of a system of charged point particles is proved under the assumption that all negatively charged particles are fermions. A lower bound on the energy is found to be −Aq⅔Nme4ℏ−2, where q is the number of distinct negative species, N the total number of negative particles, m an upper bound for their mass, e an upper bound for the absolute value of the charge on both negative and positive particles, and A is a numerical constant.

Wigner Method in Quantum Statistical Mechanics

Abstract: The Wigner method of transforming quantum‐mechanical operators into their phase‐space analogs is reviewed with applications to scattering theory, as well as to descriptions of the equilibrium and dynamical states of many‐particle systems. Inclusion of exchange effects is discussed.

Absence of Ordering in Certain Classical Systems

Abstract: A classical inequality giving lower bounds for fluctuations about ordered states is derived. The inequality, analogous to a quantum result due to Bogoliubov, is established by a purely classical argument which makes explicit the nature of the surface boundary conditions required, a point which is rather obscure in the quantum derivations. As in the quantum case the inequality is useful in excluding certain kinds of phase transitions in one‐ and two‐dimensional systems. This is illustrated for several kinds of classical spin systems.

Correlations in Ising Ferromagnets. II. External Magnetic Fields

Abstract: Results of a previous paper showing that 〈σkσl〉 and [〈σkσlσm σn〉 − 〈σkσl〉 〈σmσn〉] are always positive for a system of Ising spins σi = ±1 coupled by a purely ferromagnetic interaction (〈 〉 denotes a thermal average) are extended to the cases where (i) certain spins are constrained to have the value +1 or (ii) the system is placed in an external (``parallel'') magnetic field H. The theorems thus obtained provide a simple proof of the existence of ``bulk'' values for 〈σkσl〉 and for 〈σk〉; the latter is identical with the usual bulk magnetization per spin. The correlation functions 〈σkσl〉 are monotone nondecreasing in |H| for fixed temperature T. Both 〈σkσl〉 and 〈σk〉 (and thus the bulk magnetization) are monotone nonincreasing in T for fixed H ≥ 0.

Real spinor fields.

Abstract: The Dirac equation is expressed entirely in terms of geometrical quantities by providing a geometrical interpretation for the (−1)½ which appears explicitly in the Dirac equation. In the modification of the Dirac electron theory which ensues, the (−1)½ appears as the generator of rotations in the spacelike plane orthogonal to the plane containing the electron current and spin vectors. This amounts to a further ``relativistic'' constraint on the spinor theory and so may be expected to have physical consequences. It does not, however, conflict with well‐substantiated features of the Dirac theory.

Deformation and Contraction of Lie Algebras

Abstract: This paper deals with the theory of deformation of Lie algebras. A connection is established with the usual contraction theory, which leads to some ``more singular'' contractions. As a consequence it is shown that the only groups which can be contracted in the Poincare group are SO(4, 1) and SO(3, 2).

n‐Representability Problem for Reduced Density Matrices

Abstract: In this paper we prove some theorems about the n‐representability problem for reduced density operators. The first theorem (Theorem 6) sharpens a theorem proved by Garrod and Percus. Let Tnp be the set of all n‐representable p‐density operators. Then a density operator Dp belongs to Tnp¯ (the bar indicates the closure with respect to a certain topology) if and only if Tr (DpBp) ≥ 0 for all bounded self‐adjoint p‐particle operators Bp, such that their n‐expansion (pn)ΓpnBp≡ ∑ i1<…

Multiple Scattering of Electromagnetic Waves by Arbitrary Configurations

Abstract: This paper extends to three‐dimensional vector electromagnetic scattering problems our previous development of the scalar problems. We introduce a vector‐dyadic formalism that facilitates exploiting the previous results, and derive analogous integral equations which specify the multiple‐scattering amplitudes for many objects in terms of the corresponding functions for isolated scatterers. One representation is in terms of the dyadic analog of Beltrami's operator. For arbitrary configurations, the multi‐scattered amplitudes are developed as series in inverse powers of the separations of scatterers (with coefficients in terms of isolated scatterer amplitudes and their derivatives); for two scatterers, we derive a corresponding closed form in terms of a differential operator. Another representation is a system of algebraic equations for the many‐body multipole coefficients in terms of the isolated scatterer values. Explicit closed forms are derived for two arbitrarily spaced elementary scatterers (electric dipoles, magnetic dipoles, etc.) both by separations of variables, and by working with elementary dyadic fields.

Ground‐State Energy of a Finite System of Charged Particles

Abstract: A trial wavefunction of superconducting type is postulated for the ground state of a system of N positive and N negative charges with Coulomb interactions in the absence of any exclusion principle. The ground‐state binding energy is rigorously proved to be greater than AN75 Ry, where A is an absolute constant. Results of earlier perturbation‐theoretic calculations for an infinite system are confirmed. The author, with A. Lenard, has previously proved that the exclusion principle, holding for particles with one sign of charge only, is a sufficient condition for the stability of matter; the present paper shows that the exclusion principle is also necessary for stability.

Discrete series for the universal covering group of the 3 + 2 de sitter group.

Abstract: A classification is given of the irreducible unitary representations of the universal covering group of the 3 + 2 de Sitter group which contract to the usual physical representations of the Poincare group. These representations include the discrete series for the 3 + 2 de Sitter group. The classification problem is reduced from one for the group to the corresponding one for the Lie algebra. The method used by Thomas for the representations of the 4 + 1 de Sitter group is then followed, except that a representation is reduced out with respect to the irreducible unitary representations of a noncompact 2 + 2 subalgebra. It is conjectured that each representation of this subalgebra occurs at most once. The action on the representation spaces of a basis for the Lie algebra is given. The contractions of the representations to those of the Poincare, oscillator and the Galilei groups are briefly considered.

Solution of the Dimer Problem by the Transfer Matrix Method

Abstract: It is shown how the monomer‐dimer problem can be formulated in terms of a transfer matrix, and hence in terms of simple spin operators as was originally done for the Ising problem. Thus, we rederive the solution to the pure dimer problem without using Pfaffians. The solution is extremely simple once one sees how to formulate the transfer matrix.

Difficulties in the Kinetic Theory of Dense Gases

Abstract: For the determination of the transport coefficients of a dense gas, the long‐time behavior of the pair distribution function F2 for small intermolecular distances is obtained from a density expansion in terms of the first distribution function F1. On the basis of the dynamics of small groups of particles, it is shown that this expansion contains divergences so that it cannot be used for (a) the computation of the long‐time behavior of F2 beyond O(n); (b) the demonstration of the decay of the initial state beyond O(n2). Similar divergences are encountered in the computation of the transport coefficients from time‐correlation functions. The nature of the divergences suggests (a) there is no kinetic stage in the approach of a dense gas to equilibrium, in the sense of Bogoliubov; (b) a weak logarithmic density dependence of the transport coefficients.

Canonical Definition of Wigner Coefficients in Un

Abstract: Two general results applicable to the problem of a canonical definition of the Wigner coefficient in Un are demonstrated: (1) the existence of a canonical imbedding of Un × Un into Un2 and (2) a general factorization lemma for operators defined in the boson calculus. Using these results, a resolution of the multiplicity problem for U3 is demonstrated, in which all degenerate operators are shown to split completely upon projection into U2.

Decomposition of the Unitary Irreducible Representations of the Group SL(2C) Restricted to the Subgroup SU(1, 1)

Abstract: The unitary irreducible representations of the group SL(2C) belonging to the principal series restricted to the subgroup SU(1, 1) are decomposed into a direct integral of unitary irreducible representations of SU(1, 1) The matrix elements of the unitary operator which performs the decomposition are given explicitly and used to obtain a relation between the matrix elements of the unitary irreducible representations of the groups SL(2C) and SU(1, 1) Similar identities between the matrix elements of nonunitary representations of these groups are obtained by means of analytic continuation The relevance of these results to the theory of complex angular momentum and of high energy nearly forwardscattering is pointed out

One‐Dimensional Ising Model with General Spin

Abstract: The one‐dimensional Ising model with general spin S has been formulated as an eigenvalue problem of order 2S + 1. Two methods to reduce the order to [S + 1] have been developed for calculating the energy and the susceptibility at zero external field. Exact solutions for S = 32 and S = 1 have been obtained. Numerical calculations of S = 32, 1, and ½ have been compared.

Remarks on Relativistic Statistical Mechanics. I

Abstract: The modifications introduced by the specific forms of relativistic dynamics of many‐particle systems are shown to give rise to a different (with respect to the nonrelativistic case) manner to set the problems involved in a tentative construction of relativistic statistical mechanics. Although the difficult problems of relativistic dynamics are not solved, it is possible to define relativistic generalizations of phase space, distribution functions, Gibbs ensembles, and average values. In particular, phase space is chosen for convenience and is no longer related (as is usually the case) to the ``initial data,'' whose nature is yet unknown. As a consequence, only those observables which depend on the variables characterizing phase space give rise to easily computed average values. However, it is possible to enlarge at will the basic phase space and to define subsequent densities from which average values may be calculated. [Example: The calculation of average values of observables A(…xiμ,uiμ…) needs only den...

Lowering and Raising Operators for the Orthogonal Group in the Chain O(n) ⊃ O(n − 1) ⊃ … , and their Graphs

Abstract: Normalized lowering and raising operators are constructed for the orthogonal group in the canonical group chain O(n) ⊃ O(n − 1) ⊃ … ⊃ O(2) with the aid of graphs which simplify their construction By successive application of such lowering operators for O(n), O(n − 1), … on the highest weight states for each step of the chain, an explicit construction is given for the normalized basis vectors To illustrate the usefulness of the construction, a derivation is given of the Gel'fand‐Zetlin matrix elements of the infinitesimal generators of O(n)

Existence and Bifurcation Theorems for the Ginzburg‐Landau Equations

Abstract: The second‐order transition of a superconducting material from normal to superconducting state according to the Ginzburg‐Landau theory is rigorously discussed. The bifurcation of a superconducting state is proved for both the Abrikosov mixed state and the case of a film in a parallel magnetic field when the flux or external field is slightly less than critical. The existence of a mixed state for all values of flux below the critical value is also proved.

Statistical Mechanics of Dimers on a Quadratic Lattice

Abstract: The partition function for the square lattice completely filled with dimers is analyzed for a finite n × m rectangular lattice with edges and for the corresponding lattice with periodic boundary conditions. The total free energy is calculated asymptotically for fixed ξ = n/m up to terms o(1/n2−δ) for any δ > 0. The bulk terms proportional to nm, the surface terms proportional to (n + m) which vanish with periodic boundary conditions, and the constant terms which reveal a parity and shape dependence are expressed explicitly using dilogarithms and elliptic theta functions.

Stress‐Tensor Commutators and Schwinger Terms

Abstract: We investigate, in local field theory, general properties of commutators involving Poincare generators or stress‐tensor components, particularly those of local commutators among the latter. The spectral representation of the vacuum stress commutator is given, and shown to require the existence of singular ``Schwinger terms'' at equal times, similar to those present in current commutators. These terms are analyzed and related to the metric dependence of the stress tensor in the presence of a prescribed of a prescribed gravitational field and some general results concerning this dependence presented. The resolution of the Schwinger paradox for the Tμν commutators is discussed together with some of its implications, such as ``nonclassical'' metric dependence of Tμν. A further paradox concerning the vacuum self‐stress—whether the stress tensor or its vacuum‐subtracted value should enter in the commutators—is related to the covariance of the theory, and partially resolved within this framework.

Description of extended bodies by multipole moments in special relativity.

Abstract: The description of an extended charged body in a given electromagnetic field in flat space‐time is considered, and it is shown that such a body may be completely specified by a certain set of multipole moments of the energy‐momentum tensor Tαβ and the charge‐current vector Jα. These moments include the momentum vector, spin tensor, and total charge of the body, and they completely determine Tαβ and Jα. It is shown that the only relations between the moments due to the ``generalized conservation equations'' ∂βTαβ = −FαβJβ and ∂αJα = 0 are the constancy of the total charge and equations of motion for the momentum vector and spin tensor, in contrast to previous descriptions by moments, such as that of Mathisson, which have an infinite number of such relations. The equations of motion are given exactly, as infinite series in the moments, without assuming the applied electromagnetic field to be analytic, and an approximation procedure is developed, based on the smallness of the body compared with a typical len...

Results on Certain Non‐Hermitian Hamiltonians

Abstract: We present a few results on the spectral properties of a class of physically reasonable non‐Hermitian Hamiltonians. These theorems relate the spectral properties of a non‐self‐adjoint operator (of the aforementioned class) in terms of that of a self‐adjoint operator. These theorems can be specialized to yield conditions under which the perturbed eigenvalues (of the above class of operators) vary continuously from the eigenvalues of the unperturbed operators. If the Schrodinger equation has to be solved numerically, a knowledge of the spectral properties of the non‐Hermitian Hamiltonian would insure when the eigensolutions exist.