Showing papers in "Journal of Mathematical Physics in 1970"
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TL;DR: In this paper, a new tool, the past and future volume functions, for treating certain global properties of space-times is introduced, which are used to establish two new theorems: (1) a necessary and sufficient condition that a space-time have a Cauchy surface is that it be globally hyperbolic; and (2) the existence of a cauchy surfaces is a stable property of space times.
Abstract: The various properties of the domain of dependence (Cauchy development) which have been found particularly useful in the study of gravitational fields are reviewed. The basic techniques for constructing proofs and counterexamples are described. A new tool—the past and future volume functions—for treating certain global properties of space‐times is introduced. These functions are used to establish two new theorems: (1) a necessary and sufficient condition that a space‐time have a Cauchy surface is that it be globally hyperbolic; and (2) the existence of a Cauchy surface is a stable property of space‐times.
655 citations
TL;DR: In this paper, the authors formulated the quantum theory of linearly polarized light propagating in a 1-dimensional cavity bounded by moving mirrors by utilizing the symplectic structure of the space of solutions of the wave equation satisfied by the Coulombgauge vector potential.
Abstract: The quantum theory of linearly polarized light propagating in a 1‐dimensional cavity bounded by moving mirrors is formulated by utilizing the symplectic structure of the space of solutions of the wave equation satisfied by the Coulomb‐gauge vector potential. The theory possesses no Hamiltonian and no Schrodinger picture. Photons can be created by the exciting effect of the moving mirrors on the zero‐point field energy. A calculation indicates that the number of photons created is immeasurably small for nonrelativistic mirror trajectories and continuous mirror velocities. Automorphic transformations of the wave equation are used to calculate mode functions for the cavity, and adiabatic expansions for these transformations are derived. The electromagnetic field may be coupled to matter by means of a transformation from the interaction picture to the Heisenberg picture; this transformation is generated by an interaction Hamiltonian.
584 citations
TL;DR: In this article, the authors define multipole moments for static, asymptotically flat, source-free solutions of Laplace's equations in flat space, and show that the dependence of these moments on the choice of origin is reflected in the conformal behavior of the P's.
Abstract: Multipole moments are defined for static, asymptotically flat, source‐free solutions of Einstein's equations. The definition is completely coordinate independent. We take one of the 3‐surfaces V, orthogonal to the timelike Killing vector, and add to it a single point Λ at infinity. The resulting space inherits a conformal structure from V. The multipole moments of the solution emerge as a collection of totally symmetric, trace‐free tensors P, Pa, Pab, ⋯ at Λ. These tensors are obtained as certain combinations of the derivatives of the norm of the timelike Killing vector. (For static space‐times, this norm plays the role of a ``Newtonian gravitational potential.'') The formalism is shown to yield the usual multipole moments for a solution of Laplace's equation in flat space, the dependence of these moments on the choice of origin being reflected in the conformal behavior of the P's. As an example, the moments of the Weyl solutions are discussed.
395 citations
TL;DR: In this article, the phase integral approximation for the Green's function is investigated so as to yield an approximate expression for the density of states per unit interval of energy, i.e., the smoothly closed trajectories, unlike the approximate wavefunctions which depend on all possible trajectories.
Abstract: The phase integral approximation for the Green's function is investigated so as to yield an approximate expression for the density of states per unit interval of energy. This quantity is shown for negative energies (bound states) to depend only on the periodic orbits, i.e., the smoothly closed trajectories, unlike the approximate wavefunctions which depend on all possible trajectories. A particle in a periodic box of one, two, and three dimensions is discussed first to demonstrate how the approximate density of states contains a continuous background besides the δ‐function spikes of the discrete spectrum. Then we examine the situation in a spherically symmetric potential where special problems arise because the quasiclassical propagator has to be evaluated at a focal point of the classical trajectory. With the help of the Helmholtz‐Kirchhoff formula of diffraction theory, the amplitude is shown to remain finite at the focus. The orbits which remain entirely in a region of Coulombic potential yield a spectrum of Balmer terms with appropriately reduced degeneracy. However, the orbits which penetrate the screening charge give discrete levels obeying the Bohr‐Sommerfeld conditions with the correct degeneracy. The continuous background in the approximate density of states can be discussed on the basis of the formulas derived in this paper. This is necessary as an introduction to the problem of a particle in a potential where the motion is not multiply periodic.
313 citations
TL;DR: In this article, the Lagrangian of Born and Infeld was applied to nonlinear electrodynamics and the laws of propagation of photons and of charged particles, along with an anisotropic propagation of the wavefronts.
Abstract: After a brief discussion of well‐known classical fields we formulate two principles: When the field equations are hyperbolic, particles move along rays like disturbances of the field; the waves associated with stable particles are exceptional. This means that these waves will not transform into shock waves. Both principles are applied to nonlinear electrodynamics. The starting point of the theory is a Lagrangian which is an arbitrary nonlinear function of the two electromagnetic invariants. We obtain the laws of propagation of photons and of charged particles, along with an anisotropic propagation of the wavefronts. The general ``exceptional'' Lagrangian is found. It reduces to the Lagrangian of Born and Infeld when some constant (probably simply connected with the Planck constant) vanishes. A nonsymmetric tensor is introduced in analogy to the Born‐Infeld theory, and finally, electromagnetic waves are compared with those of Einstein‐Schrodinger theory.
275 citations
TL;DR: In this paper, an inequality for correlation functions in an Ising model with purely ferromagnetic interactions between pairs of spins is established and used to show that the magnetization in such a model is a concave function of external field H for H > 0.
Abstract: An inequality for correlation functions in an Ising model with purely ferromagnetic interactions between pairs of spins is established and used to show that the magnetization in such a model is a concave function of external field H for H > 0. The concavity of magnetization, which holds not only for spin‐½ but also for arbitrary‐spin Ising ferromagnets, provides a basis for certain thermodynamic inequalities near the ferromagnetic critical point, including one involving the ``high temperature'' indices α and γ.
257 citations
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TL;DR: In this paper, the model field theory of the φtt − φxx + m2 sin φ = 0 was studied in detail and a connection with the Kortewegde Vries equation was established.
Abstract: The equation φtt − φxx + m2 sin φ = 0 is presented as a model field theory and studied in detail. It exhibits discrete conserved quantities and extended particle states, with the proper behavior regarding covariance, stability, etc. Those particles do not interact with small oscillations, and we show that this (plus a few reasonable requirements) defines the model uniquely. A connection with the Korteweg‐de Vries equation, which has similar properties, is established.
228 citations
TL;DR: In this article, the Bianchi identity is shown to imply that the Misner-Sharp-Hernandez mass function is an integral of two combinations of Einstein's equations for any energymomentum tensor and that mass energy flow is conservative.
Abstract: The mass‐energy of spherically symmetric distributions of material is studied according to general relativity. An arbitrary orthogonal coordinate system is used whenever invariant properties are discussed. The Bianchi identity is shown to imply that the Misner‐Sharp‐Hernandez mass function is an integral of two combinations of Einstein's equations for any energy‐momentum tensor and that mass‐energy flow is conservative. The two mass equations thus found and the mass function provide a technique for casting Einstein's field equations into alternative forms. This mass‐function technique is applied to the general problem of the motion of a perfect fluid and especially to the examination of negative‐mass shells and their relation to singular behavior. The technique is then specialized to the study of a known class of solutions of Einstein's equations for a perfect fluid and to a brief treatment of uniform model universes and the charged point‐mass solution.
207 citations
TL;DR: In this article, it was shown that neither the mechanical nor the canonical angular momentum is conserved in the presence of a magnetic monopole field, but rather a total angular momentum which incorporates angular momentum resident in the magnetic field.
Abstract: The symmetry of the force field of a magnetic monopole is comparable in its simplicity to that of the hydrogen atom or a harmonic oscillator. Both these latter systems possess a ``hidden'' symmetry which leads to an ``accidental'' degeneracy in the energy spectrum of their Schrodinger equations. Since the monopole field is derived from a vector potential which is not symmetric, but undergoes a gauge transformation under rotation, the concepts of symmetry and constants of the motion must be expressed properly in the presence of velocity‐dependent forces. It is found that neither the mechanical nor the canonical angular momentum is conserved in the presence of a monopole field, but rather a total angular momentum which incorporates angular momentum resident in the magnetic field. The total angular momentum defines a cone, on whose surface the motion takes place, whatever central electrostatic potential may exist. Neither the harmonic oscillator nor the hydrogen atom retain their accidental degeneracy when t...
206 citations
TL;DR: In this paper, the conservation laws derived in an earlier paper for the KortewegdeVries equation are proved to be the only ones of polynomial form, and an algebraic operator formalism is developed to obtain explicit formulas for them.
Abstract: The conservation laws derived in an earlier paper for the Korteweg‐deVries equation are proved to be the only ones of polynomial form. An algebraic operator formalism is developed to obtain explicit formulas for them.
201 citations
TL;DR: In this article, the singularities of the ground state energy of a related ring of interacting spins were found for 2D hydrogen-bonded crystals obeying the ice rule.
Abstract: Models of 2‐dimensional hydrogen bonded crystals obeying the ice rule, which previously have been solved exactly, are generalized by removing the ice rule. Many of the peculiar and unique properties of the solutions for the constrained models are now explained by showing that these models, above critical temperature, are equivalent to new unconstrained models at critical temperature. In addition to locating the critical temperature for the general but unsolved models, we locate the singularities of the ground state energy of a related ring of interacting spins.
TL;DR: In this paper, the existence of a class of smooth solenoidal dynamos, satisfying a no-slip condition on the core boundary, is proved using perturbation theory, where the dynamos are of the form q = q(1) + q(2)+ q(3).
Abstract: The steady‐state kinematic dynamo problem in a homogeneous 3‐dimensional core is studied. The existence of a class of smooth solenoidal dynamos, satisfying a no‐slip condition on the core boundary, is proved using perturbation theory. The dynamos are of the form q = q(1) + q(2) + q(3), where q(1) is spatially periodic on a sufficiently small scale of length, q(2) is zero except near the core boundary, and q(3) is an arbitrary sufficiently small motion. The term q(1) is also a spatially periodic dynamo in an appropriate sense for an infinite core. The last property allows a simple characterization of the bounded dynamos in terms of the admissible q(1).
TL;DR: The number of ways of coloring the bonds of a hexagonal lattice of L sites (L large) with three colors so that no adjacent bonds are colored alike is calculated exactly, giving W = 1.20872.
Abstract: The number of ways WL of coloring the bonds of a hexagonal lattice of L sites (L large) with three colors so that no adjacent bonds are colored alike is calculated exactly, giving W = 1.20872 …. This is equivalent to counting the number of 4‐colorings of the faces of the lattice and can also be regarded as a multiple‐dimer problem. If one introduces activities corresponding to certain vertex configurations, then the system is found to have an infinite‐order phase transition between two ordered states.
TL;DR: In this article, an operator F is constructed which is canonically conjugate, in the Heisenberg sense, to the number operator; and F is used to define a quantum time operator.
Abstract: Apparent difficulties that prevent the definition of canonical conjugates for certain observables, e.g., the number operator, are eliminated by distinguishing between the Heisenberg and Weyl forms of the canonical commutation relations (CCR's). Examples are given for which the uncertainty principle does not follow from the CCR's. An operator F is constructed which is canonically conjugate, in the Heisenberg sense, to the number operator; and F is used to define a quantum time operator.
TL;DR: In this article, a complete, fully explicit, and canonical determination of the matrix elements of all adjoint tensor operators in all U(n) is presented, and it is demonstrated that the canonical resolution of this multiplicity possesses several compatible (or equivalent) properties: classification by null spaces, classification by degree in the Racah invariants and classification by limit properties, and the classification by conjugation parity.
Abstract: A complete, fully explicit, and canonical determination of the matrix elements of all adjoint tensor operators in all U(n) is presented. The class of adjoint tensor operators—those transforming as the IR [1 0 −1]—is the first exhibiting a nontrivial multiplicity. It is demonstrated that the canonical resolution of this multiplicity possesses several compatible (or equivalent) properties: classification by null spaces, classification by degree in the Racah invariants, classification by limit properties, and the classification by conjugation parity. (The concepts in these various classification properties are developed in detail.) A systematic treatment is presented for the coupling of projective (tensor) operators. Six appendices treat in detail the explicit evaluation of all Gel'fand‐invariant operators (Ik), the structural properties of Gram determinants formed of the Ik, the zeros of the norms of the adjoint operators, and the conjugation properties of the canonical adjoint tensor operators.
TL;DR: In this article, a detailed discussion of pair correlations ω2(r) = 〈σ 0σr〉 between spins at lattice sites 0 and r on the axes of anisotropic triangular lattices is given.
Abstract: A detailed discussion of pair correlations ω2(r) = 〈σ0σr〉 between spins at lattice sites 0 and r on the axes of anisotropic triangular lattices is given. The asymptotic behavior of ω2(r) for large spin separation is obtained for ferromagnetic and antiferromagnetic lattices. The axial pair correlation for the ferromagnetic triangular lattice has the same qualitative behavior as that for the ferromagnetic rectangular lattice: There is long‐range order below the Curie point TC and short‐range order above. It is shown that correlations on the anisotropic antiferromagnetic triangular lattice must be given separate treatment in three different temperature ranges. Below the Neel point TN (antiferromagnetic critical point), the completely anisotropic lattice exhibits antiferromagnetic long‐range order along the two lattice axes with the strongest interactions. Spins along the third axis with the weakest interaction are ordered ferromagnetically. Between TN and a uniquely located temperature TD, there is antiferro...
TL;DR: In this paper, the existence of non-invariance groups in the second-quantization picture for fermions distributed in a finite number of states was investigated, and it was shown that the largest such group is a unitary group U(22j+1).
Abstract: We investigate the existence of noninvariance groups in the second‐quantization picture for fermions distributed in a finite number of states. The case of identical fermions in a single shell of angular momentum j is treated in detail. We show that the largest noninvariance group is a unitary group U(22j+1). The explicit form of its generators is given both in the m scheme and in the seniority—angular‐momentum basis. The full set of 0‐, 1‐, 2‐, ⋯, (2j + 1)‐particle states in the j shell is shown to generate a basis for the single irreducible representation [1] of U(22j+1). The notion of complementary subgroups within a given irreducible representation of a larger group is defined, and the complementary groups of all the groups commonly used in classifying the states in the j shell are derived within the irreducible representation [1] of U(22j+1). These concepts are applied to the treatment of many‐body forces, the state‐labeling problem, and the quasiparticle picture. Finally, the generalization to more c...
TL;DR: In this article, the relation between the tensor harmonics given by Regge and Wheeler in 1957 and those given by Jon Mathews in 1962 is analyzed and a convenient orthonormal set of harmonics is given which is useful in studying, for example, gravitational radiation.
Abstract: An analysis is made of the relation between the tensor harmonics given by Regge and Wheeler in 1957 and those given by Jon Mathews in 1962. This makes it possible to use the Regge‐Wheeler harmonics, which are given in terms of derivatives of scalar spherical harmonics, for calculations while using Mathews' form of the harmonics [linear combinations of the elements of the product basis formed from a basis for scalar functions on the 2‐sphere and a basis for symmetric tensors such that the product basis is split into sets which transform under the irreducible representations of SO(3)] to elucidate the properties of tensor harmonics. Thus, a convenient orthonormal set of harmonics is given which is useful in studying, for example, gravitational radiation.
TL;DR: In this article, the conformal group is used to define multipole moments in curved space, and the moments emerge as certain multilinear mappings on the space of conformal Killing vectors.
Abstract: There is an intimate connection between multipole moments and the conformal group. While this connection is not emphasized in the usual formulation of moments, it provides the starting point for a consideration of multipole moments in curved space. As a preliminary step in defining multipole moments in general relativity (a program which will be carried out in a subsequent paper), the moments of a solution of Laplace's equation in flat 3‐space are studied from the standpoint of the conformal group. The moments emerge as certain multilinear mappings on the space of conformal Killing vectors. These mappings are re‐expressed as a collection of tensor fields, which then turn out to be conformal Killing tensors (first integrals of the equation for null geodesics). The standard properties of multipole moments are seen to arise naturally from the algebraic structure of the conformal group.
TL;DR: In this paper, a mathematical conjecture about the distribution of energy levels in complex systems is given, which is much shorter than two that have been published before, and a proof is given.
Abstract: Dyson made a mathematical conjecture in his work on the distribution of energy levels in complex systems. A proof is given, which is much shorter than two that have been published before.
TL;DR: In this article, Wu et al. investigated the asymptotic behavior of the pair correlation ω2(r) = 〈σ 0σr〉 between two spins at sites 0 and r on an axis of an isotropic antiferromagnetic triangular lattice.
Abstract: The asymptotic behavior of the pair correlation ω2(r) = 〈σ0σr〉 between two spins at sites 0 and r on an axis of an isotropic antiferromagnetic triangular lattice is investigated with the aid of the theory of Toeplitz determinants as developed by Wu. The leading terms in the asymptotic expansion are obtained for large spin separation at fixed nonzero temperature. Evidence is presented that the zero‐point behavior of the correlation is of the form ω2(r) ∼ e0r−½ cos ⅔πr, where r = |r| is the spin separation and e0=212(E0T)2=0.632226080…,E0T being the decay amplitude of the pair correlation at the Curie point (critical point) of an isotropic ferromagnetic triangular lattice. A special class of fourth‐order correlations ω4(r) = 〈σ0σδσr σr+δ〉 − 〈σ0σδ〉 〈σrσr+δ〉 between the four spins at sites 0, δ, r, and r + δ on the same lattice axis, where δ is a lattice vector, is reconsidered. The asymptotic form of the correlation for large separation of pairs of spins r = |r| is obtained for all fixed temperatures.
TL;DR: In this article, it was shown that the f −dimensional nonrelativistic Coulomb Green's function and the associated reduced Green's functions can be obtained by differentiation of the corresponding functions in the 1 −dimensional or 2 −dimensional (f even) case.
Abstract: It is shown that the f‐dimensional nonrelativistic Coulomb Green's function and the associated reduced Green's functions can be obtained by differentiation of the corresponding functions in the 1‐dimensional (f odd) or 2‐dimensional (f even) case. A new expansion of the 3‐dimensional coordinate‐space Coulomb Green's function and a new sum formula for a product of two Laguerre polynomials with different arguments are derived.
TL;DR: A complete classification of representations of SO(4, 2) with infinitesimal generators SAB, characterized by the representation relation {SAB, SAC} = −2agBC, and their extension by parity have been determined as mentioned in this paper.
Abstract: A complete classification of representations of SO(4, 2) with infinitesimal generators SAB, characterized by the representation relation {SAB, SAC} = −2agBC, and their extension by parity have been determined. The possible values of a are a = 1 − S2, S = 0, 12, 1, 32, 2, ⋯. These representations have then been reduced according to the two chains SO(4,2)⊃SO(4,1)⊃SO(4)⊃SO(3) and SO(4,2)⊃SO(3,2)⊃SO(3)⊗SO(2). The equivalence of these representations with the oscillator representations is established.
TL;DR: An abstract definition of a general hidden-variables theory is given, and it is shown that such a theory is always possible in the present framework of quantum mechanics and is, in fact, unique in a certain sense as discussed by the authors.
Abstract: An abstract definition of a general hidden‐variables theory is given, and it is shown that such a theory is always possible in the present framework of quantum mechanics and is, in fact, unique in a certain sense. It is noted that the Bohm‐Bub hidden‐variables example is contained in this theory and an attempt is made to clarify the position of this theory with respect to hidden‐variable impossibility proofs. The general definition is used in the consideration of quantum‐mechanical ordering and the measurement process.
TL;DR: In this paper, a general theoretical investigation of 3-wave interaction in the well-defined phase description with special regard to nonlinear explosive instabilities in the presence of linear damping or growth is presented.
Abstract: The paper presents a general theoretical investigation of 3‐wave interaction in the well‐defined phase description with special regard to nonlinear explosive instabilities in the presence of linear damping or growth.
TL;DR: In this article, the stability of time-dependent particlelike solutions of the form ψ=φ(r)e−iωt is examined for the nonlinear field ∇ 2ψ−c−2∂2ψ/∂t2=κ 2 ψ−μ 2 ωψψ*ψ.
Abstract: The stability of time‐dependent particlelike solutions of the form ψ=φ(r)e−iωt is examined for the nonlinear field ∇2ψ−c−2∂2ψ/∂t2=κ2ψ−μ2ψψ*ψ. It is found that such solutions are unstable for all ω.
TL;DR: The work of Finkelstein, Kruskal, Graves and Brill, Carter, and Boyer and Lindquist on the extension and schematic representation of 2-dimensional metrics is systematized and generalized as mentioned in this paper.
Abstract: The work of Finkelstein, Kruskal, Graves and Brill, Carter, and Boyer and Lindquist on the extension and schematic representation of 2‐dimensional metrics is systematized and generalized. As a result, a number of extensions may be found by inspection. Some well‐known examples are given, and the technique is applied to the ``soluzioni oblique'' of Levi‐Civita.
TL;DR: In this paper, the ground state functional of the linearized Einstein theory of gravitation is given as a functional of gauge invariant Ricci tensor, and compared with the corresponding electromagnetic expression.
Abstract: The ground state functional of the linearized Einstein theory of gravitation is given as a functional of the gauge invariant Ricci tensor, and compared with the corresponding electromagnetic expression. The connection of the canonically quantized nonlinear theory of gravitation with the linearized theory is exhibited. Time is treated as a momentum variable rather than as a superspace coordinate, which leads to an ``extrinsic time representation'' h TT ik , hi, t = −½Δ−1πT. The state functional of the linearized theory is shown to be the initial value of the state functional of the canonical theory on a constant extrinsic time hypersurface in the lowest order of a perturbation expansion. By means of the Einstein‐Schrodinger equation, this functional can be integrated off this initial hypersurface.
TL;DR: In this article, a general definition of cosine and sine operators for harmonic oscillator phase is proposed and its consequences examined, and an important feature of the spectral analysis is the ''chain sequence'' condition which ensures that C and S have unit norm.
Abstract: A general definition of ``cosine'' and ``sine'' operators, C and S, for harmonic oscillator phase is proposed and its consequences examined. An important feature of the spectral analysis is the ``chain sequence'' condition which ensures that C and S have unit norm. The (nonunitary) operator U = C + iS is shown to be an annihilation‐type operator whose spectral properties bear a remarkable analogy to those of the standard annihilation operator, although its spectrum fills the unit circle rather than the entire complex plane. Statistical properties of the eigenstates of U are discussed briefly.
TL;DR: A direct connection between spin and conformally weighted functions on the sphere and geometric objects in Minkowski space is established through the isomorphism of the conformal group of the sphere to the restricted Lorentz group as mentioned in this paper.
Abstract: A direct connection between the spin and conformally weighted functions on the sphere and geometric objects in Minkowski space is established through the isomorphism of the conformal group of the sphere to the restricted Lorentz group. It is shown that with the use of these functions one can duplicate all the standard work on the representations of the Lorentz group. It is shown further that these functions can be used to obtain a generalization of the classical equations of motion in which internal degrees of freedom arise naturally.