Self-consistent solutions for fermions in constant SU(2) gauge potentials

We derive self-consistent solutions for $\text{isospin}\text{-}\text{1}\text{2}$ Dirac particles coupled to constant SU(2) vector potentials. Such solutions can be identified as classical approximations to non-topological fermionic solitons. We discuss the stability of these states under quantum fluctuations.     

On the existence of non-one-dimensional soliton-like solutions for some field theories

Some general conditions of the existence of non-dimensional spherically symmetric solitons are discussed. As a particular case, the condition of the existence of instanton-like solutions is obtained which coincides in an amasing way with the well-known formula $\text{p =}\text{2D}\text{(D?2)}$ the condition for theory to be renormalizable. (D is the space time dimension, p is the degree of nonlinearity.)     

Quantum effects in the Bertotti-Robinson metric

Scalar and spin $\text{1}\text{2}$ fields have been studied in the Bertotti-Robinson space-time and analytical solutions have been obtained for the Klein-Gordon equation and the Dirac equation. For large e particle creation takes place as in the pseudoeuclidean metric under the influence of a constant external field.     

Soliton excitations in a commensurability 2 mixed peierls system

We consider a one-dimensional half-filled band system with the electrons coupled to both internal mode and bond modulation. For weak and intermediate coupling strength the site dimerization are mutually exclusive in the uniform ground states and inside neutral solitons. They coexist however inside charged solitons. The solitons carry charge $\text{±}\text{e}\text{2}$ per spin. Creation energy decreases as the two couplings become less different. The position of the gap state depends on the occupation number.     

Monopoles,duality, triality

The existence of magnetic charges could be a raison d'être not only for the quantization of electricity in units $\text{1}\text{3}\text{e}$ but also for the confinement of the quarks at the end of “observable”, “electric” Dirac strings (quarks have no magnetic charge in this scheme). We first review the Dirac quantization condition, using a “sum over histories” approach, and get a more general result: A string attached to the dyon (e₁, g₂) is observable by a dyon (e₂, g₂) unless (1?x) e₂g₁ ? xe₁g₂ = nh, where x is an arbitrary parameter which reflects an ambiguity in the action principle. The Dirac and Schwinger-Zwanziger quantization rules are special cases, with $\text{x = 0}\text{and}\text{1}\text{2}$, respectively. Then, we look for the values of x and of the magnetic charges to “explain” that (i) the electric charge is quantized in units $\text{1}\text{3}\text{e}$, (ii) the string attached to an electron is unobservable, (iii) the string attached to a quark is observable. We find a denumerable set of solutions. In most cases, the magnetic charges also are connected with observable strings.     

Linear relativistic hamiltonians

A 2(2J + 1)-component relativistic Hamiltonian H that describes free particles of mass m and spin J is said to be linear if it has the form $\text{H = h}\text{x}\text{?}\text{·}\text{p}\text{+ gm}$, where $\text{x}\text{\_.}\text{= i[H,}\text{x}\text{]}{}_{\mathrm{?}}$, h is a numerical factor, and g commutes with x and p. All such Hamiltonians are found, provided that the metric is either the unit matrix or ?₃ and provided that the theory is invariant under the discrete symmetries. If the operator Γ in the generator $\text{K}\text{=}\text{1}\text{2}\text{[}\text{x}\text{, H]}{}_{\mathrm{+}}\text{+ Γ}$ of Lorentz boosts is required to be local, there are only two possibilities; either Γ = 0, which generalizes the Dirac spin-$\text{1}\text{2}$ theory, or $\text{Γ = ??}{}_3\text{(}\text{1}\text{2m}\text{)}\text{S × p}$, which generalizes the Sakata-Taketani spin-0 and spin-1 theories. The relationship to linear manifestly covariant equations and its significance is discussed.     

Relativistic hamiltonian equations for any spin

We discuss 2(2J + 1)-component Poincaré-invariant Hamiltonian theories that describe free particles of definite mass and spin and that are subject to the conditions (a) every observable $\text{O}$ is either Hermitian or pseudo-Hermitian (i.e., O = ?₃O⁺?₃) and (b) the theory is invariant under the discrete symmetries. Our treatment is based on the Heisenberg equations of motion and on the Lie algebra of the Poincaré group. Explicit formulas are found for the generators of this algebra, including the Hamiltonian H, and all relations between the operators Γ and H that are both necessary and sufficient for $\text{K =}\text{1}\text{2}\text{[x}\text{, H]}{}_{\mathrm{+}}\text{+ Γ}$ to generate Lorentz boosts are found. To illustrate the utility of our results, we apply them to obtain explicit generalizations of the Dirac equation to any spin, by requiring that Γ = 0, and of the Sakata-Taketani spin-0 and spin-1 equations to any spin, by requiring that $\text{Γ = ??}{}_3\text{(}\text{1}\text{2m}\text{)}\text{S × p}$.     

Dynamical model of spin-1 and spin-0 particles (II)

Equations are proposed that completely describe $\text{s = 0,}\text{1}\text{2}\text{, 1}$ particles, i.e. specify 2s+2 limiting states in the E → ∞ limit. A set of states closed with respect to the E → ∞ limiting procedure is considered, whence follows the necessity for the introduction of a λ-state corresponding to one-dimensional unitary representation of the Lorentz group. It is shown further, that a cosmological term proportional to the square of the λ-state strength should be introduced in gravitation theory. On the basis of the Lorentz group, a table of fundamental interactions is proposed and a dual transformation for the interacting system is introduced. Appendices deal with a generalisation of the Duffin-Kemmer formalism and a new method for the introduction of interactions.     

Dirac dynamics on stochastic phase spaces for spin 1/2 particles

The Foldy-Wouthuysen representation of the dynamics of a free spin $\text{1}\text{2}$ particle is formulated in a Hilbert space $\text{H}$(Γ) of spinor-valued functions over Γ-space. $\text{H}$(Γ) carries a reducible Wigner-type representation of the Poincaré group. The transition to the Dirac representation in a new bispinor Hilbert space $\text{K}$(Γ) is effected by means of a generalized inverse Foldy-Wouthuysen transformation. Explicit expressions are derived for the resolution generators η of invariant subspaces $\text{K}$±(Γ_η) carrying irreducible representations of the resulting representations of the Poincaré group. The formalism is recast in a manifestly covariant form and the Dirac equation on $\text{H}$(Γ_s) with minimal coupling to a four-potential is examined. It is shown that the resulting external field theory is gauge-invariant and relativistically covariant.     

A positive energy dynamics and scattering theory for directly interacting relativistic particles

Starting from the tensor product of N irreducible positive energy representations of the Poincaré group describing N free relativistic particles with arbitrary spins and positive masses, we construct an interacting positive energy representation by modifying the total 4-momentum operator. We first make a transformation to a Hilbert space on which the free total 4-momentum operator equals the product of a dimensionless center-of-mass 4-vector ($\text{(|k|}{}^2\text{+ 1)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, k) and a free “reduced Hamiltonian” H_r⁰, which is a positive operator acting only on internal variables, and then replace H_r⁰ by an interacting reduced Hamiltonian H_r = H_r⁰ + V, where V commutes with the Lorentz group and is such that H_r is a positive operator. The resulting product form is shown to imply that the wave operators interwine the free and interacting representations so that the S-operator is Lorentz invariant. From a physical point of view the scheme is related to the framework first introduced by Bakamjian and Thomas, in which the Hamiltonian and boost generators are modified, but the above procedure makes a mathematically rigorous discussion much simpler. In the spin-zero case we introduce a natural generalization of the pair potentials of nonrelativistic N-particle Schrödinger theory to the present relativistic setting, study its scattering theory, and point out some problems that do not have analogs at the nonrelativistic level. In the $\text{spin}\text{-}\text{1}\text{2}$ case we propose, inspired by the Dirac equation, explicit reduced Hamiltonians to describe atomic energy levels and present arguments making plausible that their eigenvalues are in closer agreement with the experimental data than their nonrelativistic counterparts. We also consider extensions to arbitrary spin and, in the $\text{spin}\text{-}\text{1}\text{2}$ case, coupling of a quantized radiation field. In view of eventual applications to “completely integrable” one-dimensional field theories the case of one space dimension is studied as well, both in quantum mechanics and in classical mechanics.     

On the geometrical aspects of electromagnetism,I

The trajectory of a charged test particle under a Lorentz force is obtained as the geodesic of a riemannian four dimensional manifold. Originally, the geodesic equation is nonlinear in some vector field A′_μ. The nonlinearity is traded in for the correct characteristic $\text{e}\text{m}$ of the test particle through a gauge condition, imposed upon A′_μ, which turns the geodesic into the fully covariant linear and gauge invariant Lorentz equation. Fitting the $\text{e}\text{m}$ ratio inside the gauge leaves F′_μν independent of $\text{e}\text{m}$ and allows its identification with the E.-M. tensor F_μν. This four dimensional approach allows the identification of the fifth coordinate used in Kaluza's geometrization |1,2|. The gauge function appears as the sum of Hamilton-Jacobi function plus an additional term, related to the “length” of the trajectory. It is this latter term which guarantees the correct “normalisation” of the $\text{e}\text{m}$ ratio.     

Algebraic study of a class of relativistic wave equations

First-order relativistic wave equations are considered whose irreducible matrix coefficients satisfy the simplest (except for the Dirac algebra) unique mass condition, (β · p)³ = p²(β · p), which is also sufficient to guarantee causality in a minimally coupled external electromagnetic field. All of the associated representations of SL(2, ©) are classified and studied up to and including those which are the direct sum of four irreducible components, (n, m), with either n or m less than two. A large number of families of representations are found which permit the algebraic condition to be satisfied. These are tabulated according to whether a Hermitian choice for β⁰ is possible and their spin content is given. If a unique spin is described, then the only possible representations are

$\text{(1) (}\text{n}\text{,0) ⊕ (}\text{n}\text{? 1/2, 1/2)}$

$\text{(2) (}\text{n}\text{,0) ⊕ (}\text{n}\text{+ 1/2, 1/2)}$

$\text{(3) (}\text{n}\text{+ 1/2, 1/2) ⊕ (}\text{n}\text{,0) ⊕ (}\text{n}\text{? 1/2, 1/2)}$

$\text{(4) (1,0) ⊕ (1/2, 1/2) ⊕ (0,1)}$

and their conjugates. If, in addition, the representation is assumed to be self-conjugate, then only the Dirac and Petiau-Duffin-Kemmer equations survive.     

A slice theorem for the space of solutions of Einstein's equations

A slice for the action of a group G on a manifold X at a point x ? X is, roughly speaking, a submanifold S_x which is transverse to the orbits of G near x. Ebin and Palais proved the existence of a slice for the diffeomorphism group of a compact manifold acting on the space of all Riemannian metrics. We prove a slice theorem for the group $\text{D}$ of diffeomorphisms of spacetime acting on the space $\text{E}$ of spatially compact, globally hyperbolic solutions of Einstein's equations. New difficulties beyond those encountered by Ebin and Palais arise because of the Lorentz signature of the spacetime metrics in $\text{E}$ and because $\text{E}$ is not a smooth manifold- it is known to have conical singularities at each spacetime metric with symmetries. These difficulties are overcome through the use of the dynamic formulation of general relativity as an infinite dimensional Hamiltonian system (ADM formalism) and through the use of constant mean curvature foliations of the spacetimes in $\text{E}$. (We devote considerable space to a review and extension of some special properties of constant mean curvature surfaces and foliations that we need.) The conical singularity structure of $\text{E}$, the sympletic aspects of the ADM formalism, and the uniqueness of constant mean curvature foliations play key roles in the proof of the slice theorem for the action of $\text{D}$ on $\text{E}$. As a consequence of this slice theorem, we find that the space $\text{D}$ = $\text{E}$/$\text{D}$ of gravitational degrees of freedom is a stratified manifold with each stratum being a sympletic manifold. The spaces for homogeneous cosmologies of particular Bianchi types give rise to special finite dimensional symplectic strata in this space $\text{G}$. Our results should extend to such coupled field theories as the Einstein-Yang-Mills equations, since the Yang-Mills system in a given background spacetime admits a slice theorem for the action of the gauge transformation group on the space of Yang-Mills solutions, since there is a satisfactory Hamiltonian treatment of the Einstein-Yang-Mills system, and since the singularity structure of the solution set is known.     

Classical electrodynamics of a nonlinear Dirac field free solutions

In previous papers it has been shown that adding a positive scalar self-interaction ($\text{ψ}$ψ)² to the Dirac field Lagrangian provides a reasonably satisfactory model to describe the barions. In this work, we analyze other solutions of the same nonlinear Dirac equation, making progress in the direction of a systematic analysis. These solutions could provide the ground states for more elaborate interacting schemes of the real particles. Unfortunately the new solutions appear to have energies consistently higher than the ones analyzed in previous papers. Also, the more complicated solutions, whose energy seems to be much higher than the simplest one, leave us little hope for a low minimum energy state.     

A new series expansion in lattice statistics

The total energy W(a₀) lost by a charged particle in crossing a plate of thickness a₀ is calculated. Intermediate transition to the Heaviside transformation $\text{W}\text{?}\text{(a=}\text{1}\text{p}\text{)}$ is used. Several limiting cases of the Lorentz factor of particle and plate thickness are analyzed. Theory is compared with experiment.     

Variation of the local field through the liquid-vapour interface

We derive an integral equation for the self-consistent local field E^loc(z) within an inhomogeneous non-polar fluid, with particular application to the liquid-vapour interface. Approximate solutions are given for the cases of induced atomic dipoles oriented perpendicular and parallel to the interface. For the perpendicular case we relate the average field to the local field and thus obtain an equation for the static dielectric constant ?(z) in terms of the density profile n(z). The departures of the local field from Lorentz form $\text{E}{}^{\mathrm{<mtext>ext</mtext>}}\text{/(1 + (}\text{8}\text{3}\text{)παn(z))}$ and of the dielectric constant from the Clausius-Mossotti form ($\text{1 + (}\text{8}\text{3}\text{)πan(z))/(1 ? (}\text{4}\text{3}\text{)παn(z))}$ are shown to be small. For the parallel case we discuss fringing of the external field and show that the dipoles align themselves with the average field, not the external field. The departure of the local field from $\text{E}{}^{\mathrm{<mtext>ave</mtext>}}\text{/(1 ? (}\text{4}\text{3}\text{παn(z))}$ is shown to be small.     

The Dyson instability and asymptotics of the perturbation series in QED

The probability of the ground state decay at e² < 0 is calculated by the steepest descent method in fermionic electrodynamics. The saddle points are the solution of some equations which have been obtained by calculating the asymptotic of the Dirac operator determinant in a very strong external field. The SO(3) × O(2) solutions are found explicitly. The main contribution from them to the Nth coefficient of perturbation theory is proportional to $\text{(?α)}{}^{\mathrm{<mtext>N</mtext><mtext>2</mtext>}}\text{Γ(}\text{N}\text{2}\text{)}$, where $\text{S = 2}{}^2\text{3}{}^{\mathrm{<mtext>?3</mtext><mtext>2</mtext>}}\text{π}{}^3$.