Showing papers in "Journal of Mathematical Physics in 1977"
TL;DR: In this paper, the authors seek a quantum-theoretic expression for the probability that an unstable particle prepared initially in a well defined state ρ will be found to decay sometime during a given interval.
Abstract: We seek a quantum‐theoretic expression for the probability that an unstable particle prepared initially in a well defined state ρ will be found to decay sometime during a given interval. It is argued that probabilities like this which pertain to continuous monitoring possess operational meaning. A simple natural approach to this problem leads to the conclusion that an unstable particle which is continuously observed to see whether it decays will never be found to decay!. Since recording the track of an unstable particle (which can be distinguished from its decay products) approximately realizes such continuous observations, the above conclusion seems to pose a paradox which we call Zeno’s paradox in quantum theory. The relation of this result to that of some previous works and its implications and possible resolutions are briefly discussed. The mathematical transcription of the above‐mentioned conclusion is a structure theorem concerning semigroups. Although special cases of this theorem are known, the ge...
1,822 citations
TL;DR: In this article, conditions on φ and F were given so that, for solutions with nonpositive energy, the following obtains: there exists a finite time T, estimable from above, such that ∥ grad u(t)∥ L 2 (n ) →+∞ as t→T −.
Abstract: Solutions to the Cauchy problem for the equation iu t =Δu+F(|u| 2 )u (x∈ n , t>0), u(x,0)=φ(x), are considered. Conditions on φ and F are given so that, for solutions with nonpositive energy, the following obtains: There exists a finite time T, estimable from above, such that ∥ grad u(t)∥ L 2 ( n ) →+∞ as t→T − . It is also shown that other L q ‐norms of a solution (including q=∞) blow up in finite time.
720 citations
TL;DR: Within the spinorial version of the Cartan structure formulas with the built-in (complex) Einstein vacuum equations some closed semi-Einsteinian substructures are isolated and discussed as discussed by the authors.
Abstract: Within the spinorial version of the Cartan structure formulas with the built‐in (complex) Einstein vacuum equations some closed semi‐Einsteinian substructures are isolated and discussed. Then the idea of semigraviton is introduced, and its relationship to Penrose’s nonlinear graviton is described.
561 citations
TL;DR: In this paper, a general method for finding evolution equations having infinitely many symmetries or flows which preserve them is described, which is applied to the Korteweg-de Vries, modified versions of these equations, Burgers' and sine-Gordon equations.
Abstract: A general method for finding evolution equations having infinitely many symmetries or flows which preserve them is described. This is applied to the Korteweg–de Vries, modified Korteweg–de Vries, Burgers’, and sine–Gordon equations.
556 citations
TL;DR: In this paper, the topology of spacetime is determined by its causal structure, and two corollaries are established: the path topology for the path codes of the topological, differential, and conformal structure of the past and future distinguishing spacetime.
Abstract: The title assertion is proven, and two corollaries are established. First, the topology of every past and future distinguishing spacetime is determined by its causal structure. Second, in e v e r yspacetime the path topology of Hawking, King, and McCarthy codes topological, differential, and conformal structure.
359 citations
TL;DR: In this article, the Lie subalgebras of all real Lie algebra of dimension d?4 are classified into equivalence classes under their groups of inner automorphisms.
Abstract: The Lie subalgebras of all real Lie algebras of dimension d?4 are classified into equivalence classes under their groups of inner automorphisms. Tables of representatives of each conjugacy class are given.
334 citations
TL;DR: In this article, structural axioms are proposed which generate a space SD with dimension D that is not restricted to the positive integers, and integration rules for some classes of functions on SD are derived, and a generalized Laplacian operator is introduced.
Abstract: Five structural axioms are proposed which generate a space SD with ’’dimension’’ D that is not restricted to the positive integers. Four of the axioms are topological; the fifth specifies an integration measure. When D is a positive integer, SD behaves like a conventional Euclidean vector space, but nonvector character otherwise occurs. These SD conform to informal usage of continuously variable D in several recent physical contexts, but surprisingly the number of mutually perpendicular lines in SD can exceed D. Integration rules for some classes of functions on SD are derived, and a generalized Laplacian operator is introduced. Rudiments are outlined for extension of Schrodinger wave mechanics and classical statistical mechanics to noninteger D. Finally, experimental measurement of D for the real world is discussed.
328 citations
TL;DR: In this paper, it was shown that the motion of certain types of helical space curves may be related to the sine-Gordon equation and to the Hirota equation (and consequently to the nonlinear Schrodinger equation and the modified Korteweg-de Vries equation).
Abstract: It is shown that the motion of certain types of helical space curves may be related to the sine–Gordon equation and to the Hirota equation (and consequently to the nonlinear Schrodinger equation and to the modified Korteweg–de Vries equation). The intrinsic equations that govern the motion of space curves are shown to provide the various linear equations that have been introduced to solve these evolution equations by inverse scattering methods.
300 citations
TL;DR: In this paper, the authors illustrate the differences between the properties of the irreducible representations of simple graded Lie algebras and simple simple Lie algesbras.
Abstract: We illustrate through the examples of the osp(2,1) and spl(2,1) algebras the differences between the properties of the irreducible representations of simple graded Lie algebras and simple Lie algebras.
293 citations
TL;DR: In this paper, the scaling limit of the spin-spin correlation function of the two-dimensional Ising model is analyzed for the special case ν = 0 and λ =π−1, where σ and B are given as explicit functions of λ and ν.
Abstract: We explicitly construct the one‐parameter family of solutions, η (ϑ;ν,λ), that remain bounded as ϑ→∞ along the positive real ϑ axis for the Painleve equation of third kind w w′′= (w′)2−ϑ−1 w w′+2νϑ−1(w 3−w) +w 4−1, where, as ϑ→∞, η ∼ 1−λΓ (ν+1/2)2−2νϑ−ν−1/2 e −2ϑ. We further construct a representation for ψ (t;ν,λ) =−ln[η (t/2;ν,λ)], where ψ (t;ν,λ) satisfies the differential equation ψ′′+t −1ψ′= (1/2)sinh(2ψ)+2νt −1 sinh(ψ). The small‐ϑ behavior of η (ϑ;ν,λ) is described for ‖λ‖<π−1 by η (ϑ;ν,λ) ∼ 2σ Bϑσ. The parameters σ and B are given as explicit functions of λ and ν. Finally an identity involving the Painleve transcendent η (ϑ;ν,λ) is proved. These results for the special case ν=0 and λ=π−1 make rigorous the analysis of the scaling limit of the spin–spin correlation function of the two‐dimensional Ising model.
283 citations
TL;DR: In this paper, nonradiating sources, i.e., sources for which the field is identically zero outside a finite region, are introduced, and it is shown that the null space of the Fredholm integral equation is exactly the class of nonradiated sources.
Abstract: A recently developed formulation of the inverse source problem as a Fredholm integral equation of the first kind provides motivation for the development of analytical characterizations of the nonuniqueness in the inverse source problem. Nonradiating sources, i. e., sources for which the field is identically zero outside a finite region, are introduced. It is then shown that the null space of the Fredholm integral equation is exactly the class of nonradiating sources.
TL;DR: In this paper, the problem of wave propagation in a random medium is formulated in terms of Feynman's path integral, which turns out to be a powerful calculational tool.
Abstract: The problem of wave propagation in a random medium is formulated in terms of Feynman’s path integral. It turns out to be a powerful calculational tool. The emphasis is on propagation conditions where the rms (multiple) scattering angle is small but the log‐intensity fluctuations are of order unity—the so‐called saturated regime. It is shown that the intensity distribution is then approximately Rayleigh with calculable corrections. In an isotropic medium, the local or Markov approximation which is commonly used to compute first and second (at arbitrary space–time separation) moments of the wave field is explicitly shown to be valid whenever the rms multiple scattering angle is small. It is then shown that in the saturated regime the third and higher moments can be obtained from the first two by the rules of Gaussian statistics. There are small calculable corrections to the Gaussian law leading to ’’coherence tails.’’ Correlations between waves of different frequencies and the physics of pulse propagation are studied in detail. Finally it is shown that the phenomenon of saturation is physically due to the appearance of many Fermat paths satisfying a perturbed ray equation. For clarity of presentation much of the paper deals with an idealized medium which is statistically homogeneous and isotropic and is characterized by fluctuations of a single typical scale size. However, the extension to inhomogneous, anisotropic, and multiple scale media is given. The main results are summarized at the beginning of the paper.
TL;DR: In this paper, a tensor generalization of the Ernst potential is used to give forms that are manifestly covariant under (i) the external group G of coordinate transformations, (ii) the internal group H of Ehlers transformations, and (iii) the infinite parameter group K of Geroch which combines both.
Abstract: The Einstein equations for stationary axially symmetric gravitational fields are written in several extremely simple forms. Using a tensor generalization of the Ernst potential, we give forms that are manifestly covariant under (i) the external group G of coordinate transformations, (ii) the internal group H of Ehlers transformations and gage transformations, and (iii) the infinite parameter group K of Geroch which combines both. We then show how the same thing can be done to the Einstein–Maxwell equations. The enlarged internal group H′ now includes the Harrison transformations, and is isomorphic to SU(2,1). The enlarged group K′ contains even more parameters, and generates even more potentials and conservation laws.
TL;DR: In this paper, the authors studied the Hamiltonians for nonrelativistic quantum mechanics and for the heat equation in terms of energy forms ∫∇f∇fd'gm, where dμ is a positive, not necessarily finite measure on Rn.
Abstract: We study the Hamiltonians for nonrelativistic quantum mechanics—and for the heat equation—in terms of energy forms ∫∇f∇fd’gm, where dμ is a positive, not necessarily finite measure on Rn. We cover the cases of very singular interactions (e.g., N particles in R3 interacting by two‐body ’’δ potentials’’). We also exhibit, on the other hand, regularity conditions for μ in order that H be realized as a perturbation of the Laplacian by a measurable or generalized functon. The Hamiltonians defined by energy forms alwasy generate Markov semigroups, and the associated processes are symmetric homogeneous strong Markov diffusion Hunt processes with continuous paths realizations. Ergodicity, transiency, and recurrency are also discussed. The associated stochastic differential equaiton is discussed in the situaion were μ is finite but the drift coefficient is only restricted to be l (Rr,dμ). These results provide a large class of examples where solutions of the heat equaion can be expressed by averages with respect t...
TL;DR: In this article, the authors show that the tensor product of two star representations belonging to one class is completely reducible into irreducible representation belonging to the same class.
Abstract: Hermitian representations play a fundamental role in the study of the representations of simple Lie algebras. We show how this concept generalizes for classical simple graded Lie algebras. Star and grade star representations are defined through adjoint and grade adjoint operations. Each algebra admits at most two adjoint and two grade adjoint operations (we list the various possibilities for all classical simple graded Lie algebras). To each adjoint (grade adjoint) operation corresponds a class of star (grade star) representations. The tensor product of two star representations belonging to one class is completely reducible into irreducible representations belonging to the same class. This property is very useful since in general the finite‐dimensional representations of classical simple graded Lie algebras are not completely reducible.
TL;DR: In this article, it was shown that a continuous representation of quantum mechanics exists on a given fuzzy phase space if an only if the corresponding confidence functions for position and momentum measurements satisfy the Heisenberg uncertainty relations.
Abstract: The problem of expressing quantum mechanical expectation values as averages with respect to nonnegative density functions on phase space, by analogy with classical mechanics, is reexamined in the light of some earlier work on fuzzy phase spaces. It is shown that such phase space representations are possible if ordinary phase space is replaced by a so‐called fuzzy phase space, on which the usual marginal distribution functions are redefined to conform to the fact that arbitrarily precise simultaneous measurements on position and momentum are not compatible with quantum mechanics. In the process a generalization of Wigner’s theorem on the nonexistence of phase space representations of quantum mechanics, which also satisfy the standard (classical) marginality conditions in position and momentum, is obtained. It is shown that a (continuous) representation of quantum mechanics exists on a given fuzzy phase space if an only if the corresponding confidence functions for position and momentum measurements satisfy the Heisenberg uncertainty relations.
TL;DR: In this paper, a positive energy argument of Geroch can be modified to rule out a possible class of counterexamples to the cosmic censor hypothesis proposed by Penrose.
Abstract: We show that a positive energy argument of Geroch can be modified to rule out a possible class of counterexamples to the cosmic censor hypothesis proposed by Penrose.
TL;DR: In this article, the Fortuin-Kasteleyn cluster model was used to determine the critical behavior of the classical XY model and the truncated tetrahedron lattice.
Abstract: We construct a class of lattice systems that have effectively nonintegral dimensionality. A reasonable definition of effective dimensionality applicable to lattice systems is proposed and the effective dimensionalities of these lattices are determined. The renormalization procedure is used to determine the critical behavior of the classical XY model and the Fortuin–Kasteleyn cluster model on the truncated tetrahedron lattice which is shown to have the effective dimensionality 2 log3 /log5. It is found that no phase transition occurs at any finite temperature.
TL;DR: In this paper, a discussion of the Kantowski-Sachs cosmological models is given locally as admitting a four-parameter continuous isometry group which acts on spacelike hypersurfaces, and which possesses a threeparameter subgroup whose orbits are 2-surfaces of constant curvature.
Abstract: A discussion is given of the ’’Kantowski–Sachs’’ cosmological models; these are defined locally as admitting a four‐parameter continuous isometry group which acts on spacelike hypersurfaces, and which possesses a three‐parameter subgroup whose orbits are 2‐surfaces of constant curvature (i.e., the models possess spherical symmetry, combined with a translational symmetry, and can thus be regarded as nonempty analogs of part of the extended Schwarzschild manifold). It is shown that all general relativistic models in which the matter content is a perfect fluid satisfying reasonable energy conditions are geodesically incomplete, both to the past and to the future, and that at each resulting singularity the fluid energy density is infinite. In the case where the fluid obeys a barotropic equation of state (which includes all known exact perfect fluid solutions) the field equations are shown to decouple to form a plane autonomous subsystem. This subsystem is examined using qualitative (Poincare–Bendixson) theory, and phase–plane diagrams are drawn depicting the behavior of the fluid’s energy density and shear anisotropy in the course of the models’ evolution. Further diagrams depict the conformal structure of these universes, and a table summarizes the asymptotic properties of all physically relevant variables.
TL;DR: In this article, an algorithm for determining all solutions of the Einstein field equations representing a perfect fluid with metric of the form ds2=dt2−e2αdz2 −e2β(dx2+dy2) and fluid flow vector u=∂/∂t is presented.
Abstract: An algorithm is presented for determining all solutions of the Einstein field equations representing a perfect fluid with metric of the form ds2=dt2−e2αdz2 −e2β(dx2+dy2) and fluid flow vector u=∂/∂t The entire class of solutions is then invariantly characterized These new solutions generalize Szekeres’ inhomogeneous cosmological models containing dust A subclass of these solutions is studied in detail and it is interesting that some of these models approach isotropy but not homogeneity for large cosmological times
TL;DR: A n, B n, C n, D n, and G 2 have been computed in closed terms in this article for simple and classical Lie algebras, and some polynomial identities among infinitesimal generators of these algesbras are derived by means of the same technique.
Abstract: A method of computing eigenvalues of certain types of Casimir invariants has been developed for simple and classical Lie algebras. Especially these eigenvalues for algebrasA n , B n , C n , D n , and G 2 have been computed in closed terms. We also enumerate numbers and functional forms of all linearly independent vector operators in terms of generators in any irreducible representation of these algebras. Some polynomial identities among infinitesimal generators of these algebras are derived by means of the same technique.
TL;DR: In this article, the authors extend the group theoretical analysis of a previous publication to obtain explicitly, as a polynomical in sinγ, cosγ, the function φλμlk(γ) required in the discussion of the quadrupole vibrations of the nucleus.
Abstract: In the present paper we extend the group theoretical analysis of a previous publication to obtain explicitly, as a polynomical in sinγ, cosγ, the function φλμlk(γ) required in the discussion of the quadrupole vibrations of the nucleus. The states appearing in the collective model 〈νλμLV〉=F1λ(β) ΣKφλμLK(γ) DL*MK(φi), l= (ν−λ)/2, are then defined, as Fλl(β), DL*MK(φi) are well known. All matrix elements required in the collective model of the nucleus are related then with the expression (λμL;λ′μ′L′;λ″μ″L″= ∂π0ΣKK′K″ (LL′L″KK′K″) φλμLK(γ) φλ′μ′L′K′ (γ) φλ″μ″L″K″ (γ)sin 3γdγ, which is a reduced 3j‐symbol in the O(5) O(3) chain of groups.
TL;DR: In this article, the modified Fredholm determinant cannot vanish for real k ≥ 0 for noncentral potentials, and the analog of the distinction between zero energy bound states and zero energy resonances for central potentials is found.
Abstract: Fredholm theory is applied to the Lippmann–Schwinger equation for noncentral potentials. For a specified wide class of potentials it is proved that the modified Fredholm determinant cannot vanish for real k≠0. The point k=0 is examined and the analog of the distinction between zero‐energy bound states and zero‐energy resonances for central potentials is found. A generalized Levinson theorem is proved.
TL;DR: In this paper, a simple expression for the normal form of the unitary operator implementing a Bogoliubov transformation on a system of relativistic charged particles is obtained, and necessary and sufficient conditions for the transformation to be unitarily implementable are derived.
Abstract: A simple expression for the normal form of the unitary operator implementing a Bogoliubov transformation on a system of relativistic charged particles is obtained. Necessary and sufficient conditions for the transformation to be unitarily implementable are rederived.
TL;DR: In this paper, the existence of at least one trajectory corresponding to each binary Bernoulli sequence is shown. But the relation between the two trajectories is not investigated, only trajectories in two dimensions with a negative energy (bound states).
Abstract: The anisotropic Kepler problem is investigated in order to establish the one‐to‐one relation between its trajectories and the binary Bernoulli sequences. The Hamiltonian has a quadratic kinetic energy with an anisotropic mass tensor and a spherically symmetric Coulomb energy. Only trajectories in two dimensions with a negative energy (bound states) are discussed. The previous study of this system was based on extensive numerical computations, but the present work uses only analytical arguments. After a review of the earlier results, their relevance to the understanding of the relation between classical and quantum mechanics is emphasized. The main new result is to show the existence of at least one trajectory corresponding to each binary Bernoulli sequence. The proof employs a number of unusual mathematical tools, although they are all elementary. In particular, the virial as a function of the momenta (rather than the action as a function of the position coordinates) plays a crucial role. Also, different ...
TL;DR: In this article, the ground and excited state energies of a system of nucleons interacting through pairing forces are given as a power series in inverse powers of the number of particles, and the expressions are valid for systems with superfluid ground states and either J=0 or L=0 pairing.
Abstract: Expressions for the ground and excited state energies of a system of nucleons interacting through pairing forces are given as a power series in inverse powers of the number of particles. The expressions are valid for systems with superfluid ground states and either J=0 or L=0 pairing. The first three terms in the expansion are given explicitly, and they exhibit excitations with both vibrational and rotational (in isospin space) character. Analytical and numerical results are given for a model system with a two‐level single‐particle spectrum.
TL;DR: In this paper, it was shown that all statistical properties of the generalized Langevin equation with Gaussian fluctuations are determined by a single, two-point correlation function, and that the resulting description corresponds with a stationary, Gaussian, non-Markovian process.
Abstract: It is shown that all statistical properties of the generalized Langevin equation with Gaussian fluctuations are determined by a single, two‐point correlation function. The resulting description corresponds with a stationary, Gaussian, non‐Markovian process. Fokker–Planck‐like equations are discussed, and it is explained how they can lead one to the erroneous conclusion that the process is nonstationary, Gaussian, and Markovian.
TL;DR: In this article, the analytical properties of spin-weighted angular spheroidal functions introduced by Teukolsky are investigated by means of a series involving Jacobi polynomials.
Abstract: The analytic properties of the spin‐weighted angular spheroidal functions introduced by Teukolsky are investigated by means of a series involving Jacobi polynomials. This approach facilitates the numerical determination of eigenvalues, particularly in the case of complex frequencies.
TL;DR: In this article, an algorithm for classifying the closed connected subgroups S of a given Lie group G into conjugacy classes, presented in earlier papers, is further refined so as to provide us with normalized lists of representatives of subalgebra classes.
Abstract: An algorithm for classifying the closed connected subgroups S of a given Lie groupG into conjugacy classes, presented in earlier papers, is further refined so as to provide us with ’’normalized’’ lists of representatives of subalgebra classes. The normalized lists contain the subgroup normalizer Nor G S (Nor G S is the largest subgroup of G for which S is an invariant subgroup) for each subgroup representative. The advantage of having normalized lists is that the problem of merging several different sublists (e.g., the lists of all subgroups of each maximal subgroup of G) into a single overall list becomes greatly simplified. The method is then applied to find all closed connected subgroups of the two de Sitter groups O(3,2) and O(4,1). The classification group in each case is the group of inner automorphisms.
TL;DR: In this paper, it was shown that the H− ion, treated in nonrelativistic approximation with Coulomb interactions only, has only one bound state for the electron to nucleus mass ratio less than 0.21010636.
Abstract: It is rigorously demonstrated that the H− ion, treated in nonrelativistic approximation with Coulomb interactions only, has only one bound state for the electron to nucleus mass ratio less than 0.21010636. This extends earlier work which had proven the result in the fixed (infinite mass) nucleus approximation. The method used can, if desired, also be used to calculate rigorous lower bounds to the energies of those bound states of two electron atomic systems which do exist.