Volume 19, Issue 11, November 1978
Index of content:

A general commutator equation in the algebra of spin operators
View Description Hide DescriptionAn algebraic method of spin operators was developed by Jha and Valatin to solve (a) the Hamiltonian of the isotropic and anisotropicx ymodel in a one‐dimensional lattice of N spin 1/2 particles and (b) the partition function of the Ising model in the absence of magnetic field in two dimensions. The pair of fermion operators used to explain BCS theory in superconductivity were shown to be related to a set of spin operators of Jha and Valatin in a very simple way. Onsager’s Lie algebra for diagonalizing the partition function of the Ising model was found to be included within the said commutator algebra of the spin operators. The structure constants of the algebra are so simple as to allow the entire algebra to be casted in one general commutator equation. In the present paper, the author presents the proof of a general equation from which all sets of commutator relationships existing among the elements of the algebra directly follow.

A nonlinear scalar field theory in isotropic homogeneous space–time
View Description Hide DescriptionA classical nonlinear scalar field theory in isotropic homogeneous space–time of uniform negative curvature is considered which admits a singularity‐freespatially localized dynamically unstable solution. The associated field energy is obtained as a finite positive quantity only for suitably restricted values of a ’’size parameter’’ which measures the degree of spatial localization of the solution. The static flat space–time limit of the present field theory as well as a physically appropriate limitation on the magnitude of the field energy are discussed.

Energetically stable systems
View Description Hide DescriptionFor quantum systems as well as for classical continuous systems energetic stability is defined. It is proved that stability, supplemented with a cluster property, characterizes equilibrium states.

Relativistic quantum kinematics on stochastic phase spaces for massive particles
View Description Hide DescriptionIt is shown that to every Galilei‐covariant nonrelativistic stochastic phase space representation of a system of massive particles, whose generator is rotationally invariant, corresponds a Poincaré‐covariant relativistic representation sharing the same generator. The stochastic phase space probability densities of the two representations overlap in the limit of nonrelativistic velocities in the laboratory frame. The relativistic representations give rise to covariant and conserved probability currents at stochastic space–time points, in complete analogy with their nonrelativistic counterparts. This parallelism extends to the existence of a global representation of the proper Poincaré group in L ^{2}(Γ), which is reduced by each subspace of L ^{2}(Γ) spanned by the set of phase‐space wavefunctionsgenerated by some stochastic phase space representation of all pure states of the system.

Stochastic phase space kinematics of the photon
View Description Hide DescriptionA stochastic phase space description of photon states is obtained by attaching to each circular polarization mode a probability amplitude at stochastic phase space points that are frontally localized. The ensuing formalism gives rise to a probability density at such points that transforms as a scalar under proper Lorentz transformations. There also exist conserved covariant currents associated with these probability densities. The wavefunctions corresponding to all extremal stochastic phase space representations can be embedded in a single Hilbert space, and give rise there to irreducible representations of the proper Poincaré group P^{↑} _{+}.

Representations of the Weyl Lie algebra as models of elementary particles
View Description Hide DescriptionSome irreducible representations of the 11‐parameter Weyl Lie algebra are suggested as models of elementary particles.

An exact solution to Einstein’s equations with a stiff equation of state
View Description Hide DescriptionA solution to the equations of general relativity is given which is spherically‐symmetric and nonstatic with an inhomogeneous density profile ρ and a pressure p given by the stiff equation of statep=ρc ^{2}. The solution may be of use in representing collapsed astrophysical systems or the early stages of an inhomogeneous cosmology.

The conserved densities of the Korteweg–De Vries equation
View Description Hide DescriptionThe conserved densities of the Korteweg–de Vries equation are identified as energy densities associated with higher order equations generated from the KdV equation and governing its solutions.

Bogoliubov transformations, propagators, and the Hawking effect
View Description Hide DescriptionThe thermal spectrum of radiation, seen by suitable observers using ’’Unruh particle detectors,’’ in de Sitter spacetime is recovered using Bogoliubov transformation techniques. Previous attempts by other authors at calculating particle production in de Sitter spacetime, prior to the discovery of thermal radiation using propagator techniques failed. The discrepancy between these previous mode mixing calculations, and the calculations presented here, are traced to ’’de Sitter invariant’’ versus ’’observer dependent’’ formalisms. One consequence is that the initial vacuum state chosen for the quantum field is not unique.

Geometrical spacetime perturbation theory: Regular first‐order structures
View Description Hide DescriptionSpacetimeperturbation theory is formulated in a coordinate independent way by regarding a family of spacetimes as a (4+n) ‐dimensional manifold with a particular standard connection and deriving analogs of the Gauss–Weingarten equations to describe the imbedding of each spacetime in the family.

Causality in homogeneous spaces
View Description Hide DescriptionA simple proof is given of the microcausality of quantum fields defined on certain homogeneous pseudo‐Riemannian spaces. The proof is group theoretic in nature and does not depend on the detailed form of the generalized Pauli–Jordan propagator. As an illustration, applications are given to de Sitter and anti‐de Sitter spaces; in the latter case, it is shown that the commutator of any boson field vanishes for any pair of points in the space.

Coherent states and lattice sums
View Description Hide DescriptionThe expansion of harmonic oscillator states in discrete coherent states on a von Neumann lattice leads to relationships between lattice sums and expansion coefficients of the Weierstrass σ function. It is shown that these relationships can be generalized to arbitrary lattices. Some interesting identities are obtained between infinite sums of different convergence rates.

The Bäcklund problem for the equation ∂^{2} z/∂x ^{1}∂x ^{2}= f (z)
View Description Hide DescriptionThe Bäcklund problem for the equation ∂^{2} z/∂x ^{1}∂x ^{2}=f (z) is discussed for analytic functions f, using the procedure of Estabrook and Wahlquist, starting from a Lagrangian formulation. The condition d ^{2} f/d z ^{2} =k f, k constant, necessary for the existence of nontrivial Bäcklund maps when the space of new dependent variables is R, is shown to be closely related to the structure of the Lie algebra SL(2,R).

Path integrals with a periodic constraint: Entangled strings
View Description Hide DescriptionPath integrals with a periodic constraint =integer) are studied. In particular, the path integral for a string entangled around a singular point in two dimensions is evaluated in polar coordinates. Applications are made for the entangled polymers with and without interactions, the Aharonov–Bohm effect, and the angular momentum projection of a spinning top.

Conservation equations and the gravitational symplectic form
View Description Hide DescriptionBy considering space–times whose metric is given by a perturbation expansion away from a background which admits a Killing vector, conservation equations, based on the energy–momentum tensor, are derived to first and second orders in the perturbation expansion. To the first order, equations are derived independently of the Einstein field equations, and describe secular changes in the energy–momentum distribution of the matter fields. To the second order, a gravitational energy–momentum contribution arises from the conservation equations which may be constructed from the symplectic inner product on the solution space of the linearized Einstein field equations. Considering a similar scheme based on the Bel–Robinson tensor, it is shown that whilst first order conservation equations can be formulated, the lack of a symplectic form for the perturbed Bel–Robinson tensor implies the nonexistence of second order conservation equations, except when the background is flat. The results are applied to perturbations of a stationary black hole, and simple expressions are found for the mass and angular momentum fluxes, through the event horizon, due to a gravitational perturbation. By considering a monochromatic wave, it is seen that the conservation of the gravitational symplectic form reduces, in suitable coordinates, to the Wronskian condition of Teukolsky and Press.

Constructing quantum fields in a Fock space using a new picture of quantum mechanics
View Description Hide DescriptionFor any conventional nonrelativistic quantum theory of a finite number of degrees of freedom, we construct a picture which we call ’’the scattering picture,’’ combining the ’’nice’’ properties of both the interaction and the Heisenberg pictures, and show that, in the absence of bound states, the theory could be formulated in terms of a free Hamiltonian and an effective potential. We generalize the equations thus derived to the relativistic case and show that, given a Poincaré invariant self‐adjoint operator D densely defined on a Fock space, there exists an interacting field which is asymptotically free and has as the scattering matrix the nontrivial operator S=e ^{ i D }, provided that D annihilates the vacuum and the one‐particle states. Crossing relations could easily be imposed on D, but, apart from a few comments, the problem of analyticity of S is left open.

Conditioning of states. II
View Description Hide DescriptionAn alternative axiomatic system describing the concept of conditioning (or preparation) of states in a quantum logic is proposed and its consequences developed. The main difference between this and the system proposed by Pool is our reliance on Mielnik’s idea of transition probabilities, thereby avoiding some ad hoc hypotheses as well as the theory of Baer *‐semigroups.

A unified radon inversion formula
View Description Hide DescriptionA Radon inversion formula which holds in spaces of even or odd dimension n is obtained for functions which admit to a certain general decomposition. The inversion formula which is one member of a Gegenbauer transform pair is used to generate some interesting definite integrals involving special functions. Legendre and Tchebycheff transform pairs are discussed as special cases of the general result.

Electromagnetic and gravitational Hertz potentials
View Description Hide DescriptionWith generality of complex relativity, the classical theory of the electromagnetic Hertz potentials is outlined in terms of spinors and forms. Particularly interesting are D (0,1) and D (1,0) null Hertz potentials. Then, a new spinorial approach to heavens (H spaces) is proposed, which reveals in their structure the presence of the left n u l l gravitational Hertz potential [of the type D (0,2)]. The relevant hints which follow from our results and concern the structure of the most general solutions of the Einstein vacuum equations (type G⊗G), are discussed, in particular on the level of the linearized theory.

Quantum field theory Potts model
View Description Hide DescriptionWe consider a quantum field theory analog to the three states Potts model [R. B. Potts, Proc. Camb. Phil. Soc. 48, 106 (1952)] in two dimensions. Our model can be interpreted as a neutral vector model with discrete gauge symmetry. We prove the existence of the thermodynamic limit by using the lattice approximation and correlation inequalities.