Volume 4, Issue 4, April 1963
Index of content:

Axiomatic Perturbation Theory. II. Partial Sum Formalism
View Description Hide DescriptionA generalization of axiomatic perturbation theory is described in which propagator structure is introduced into the basic formalism, effectively resulting in a systematic rearrangement of the perturbation expressions representing a given process. The exact retarded functional of a simple Boson field theory is decomposed into a sum of approximate functionals, and all the n‐point functions of the first few approximate functionals are determined from modified unitarity equations and a knowledge of the corresponding perturbation functions. The resulting amplitudes differ from the perturbation amplitudes by the replacement of the perturbative spectral functions with related spectral functions depending upon the latter and on the propagator structure. Sets of integral equations for the higher functionals are proposed and solved.

Group Properties of Free‐Field Equations
View Description Hide DescriptionWe show that a set of n free‐field equations and suitable commutation relations relevant to spin 0 or ½ fields, is invariant with respect to a continuous group of transformations isomorphic to the n‐dimensional real orthogonal group. We then find that such a set can always be derived starting from a given particle multiplet. The continuous group found in this way, includes as subgroups the isotopic rotations and the gauge groups related to baryon number and strangeness. Hence this method enables us to reobtain the results reached by other authors in some particular cases, and to extend them to all particle multiplets.

Symmetry in Quantum Theory
View Description Hide DescriptionThe aim of this note is to give a precise definition of symmetry in quantum theory in order to generalize Wigner's representation theorem, in the framework of lattice theory and projective geometry. We do not require the concept of physical state; the results are valid for any field (division ring) used in the realization of the lattice of propositions. Physical systems with the most general superselection rules are included in the theory.

N‐Body Bose System with a Finite Number of States. I. Irreducible Representations
View Description Hide DescriptionFor the purpose of investigating the N‐Body Bose System with a finite number of states, we analyze X_{ij} = A_{i}†A_{j}, i, j ≤ K. (A and A ^{†} are the usual annihilation and creation operators of the second quantization formalism). The Hamiltonian for a fixed number of particles may be expressed in terms of the (finite‐dimensional) irreducible representations of the X_{ij} . A set of fundamental equations is defined for the irreducible representations X_{ij} (N, K), i, j ≤ K and is solved for arbitrary N and K. The analysis of the structure of the X_{ij} (N, K) yields a simple, systematic method for listing all possible ways in which n _{1}+ … n_{i} + … + n_{K} = N may be satisfied for n_{i} positive integral or zero. This leads to a particularly simple method for constructing the X_{ij} (N, K), i, j ≤ K.

Generalization of a Theorem by Goldberg and Sachs
View Description Hide DescriptionA connection is established between algebraic degeneracy of the Weyl tensor, the existence of a null geodesic shear‐free congruence, and certain restrictions on the Ricci tensor which are weaker than the gravitational equations for empty space. The result is roughly, with some important qualifications, that any two of these conditions imply the third. These restrictions on the metric are shown to be invariant under conformal transformations.

Kinematics of the Relativistic Two‐Particle System
View Description Hide DescriptionKnowing the (canonical) forms that the generators of the infinitesimal transformations of P (the restricted Poincaré Group) take when operating within the Hilbert space of states of an arbitrary irreducible representation of P, such as that which affords a kinematic description of a single relativistic particle, we develop the forms in which they appear when operating within the Hilbert space of states of the (reducible) direct product representation of P which describes a system of two non‐interacting relativistic particles. On introducing the Clebsch‐Gordan (C‐G) series of P which expresses the resolution of this latter Hilbert space into a direct integral over the Hilbert spaces wherein operate the irreducible constituents of the direct product representation, we prove that the generators operate in canonical form within each of these subspaces. If we adopt the alternative viewpoint that the C‐G series of P must be written exactly so as to achieve this, our analysis may be regarded as providing a fully explicit and mathematically complete derivation of the formula for the C‐G coefficients of P that appear in the C‐G series. This formula has previously been suggested only on the basis of heuristic physical argument, and its proof is the principal accomplishment of the present work. One important fact which receives further clarification from the explicit nature of our analysis is the following: in order to see how the intrinsic angular momentum of the two‐particle system, in a state of given linear momentum, is compounded from the relative angular momentum and intrinsic spins of the particles, one must view the state from a frame of reference wherein it appears to have zero momentum.

Derivation of the Gell‐Mann‐Okubo Mass Formula
View Description Hide DescriptionA short proof is given for a mathematical identity which is used to derive the Gell‐Mann‐Okubo mass formula.

High‐Energy Behavior in Perturbation Theory
View Description Hide DescriptionThe dominant‐high energy behavior of a wide class of Feynman diagrams is investigated. When the leading contributions are summed they are shown to give a behavior consistent with the Reggepole hypothesis. Series expansions for the trajectory and residue of the dominant Regge pole are obtained in this approximation.

Transition Matrix for Nucleon‐Nucleon Scattering. II
View Description Hide DescriptionThe Fredholm reduction of the singular integral equation satisfied by the reactance matrix, which was developed in a previous paper, is extended so as to constitute a complete and unified Fredholm formalism for the various integral equations which occur in the momentum‐space formulation of two‐body (potential) scattering problems. The essential simplification which permits this unified treatment is the demonstration of the formal similarity of the scatteringintegral equations, whether or not part of the interaction includes a hard core. The principal result consists in an apparatus which is useful in performing those two‐body calculations which occur in investigations of high‐energy nucleon‐nucleus scattering; in particular, the technique is used to obtain the solutions of integral equations satisfied by transition matrices which appear typically in multiple‐scattering theories. The relation between the uniqueness of solutions of the scatteringintegral equations and the validity of the Fredholm formalism is also discussed. Finally, some methods which were previously considered for obtaining approximate solutions of the Fredholm equations are generalized.

Unified Variational Formulation of Classical and Quantum Dynamics. II
View Description Hide DescriptionIn a previous article in this series [E. M. Corson, J. Math. Phys. 4, 42(1963); herein referred to as I], we showed that a new variational principle, of Lagrangian structure, includes Hamilton's principle as a unique derived consequent and encompasses both the classical and quantal definitions of ``state'' and the corresponding Lagrangian and Schrödinger equations of motion, respectively. In this theory, the essential difference between classical and quantal domains was shown to arise solely from the superadded postulates of determinism vs indeterminism; other than this, the two basic postulates may be reduced to (a) Newton's equations and (b) the new variation principle.
It was shown that in the quantal domain, Schrödinger's equations obtain for the transition amplitudes (propagators) between pure eigenstates of homologous variables in configuration space, at different times, thus clearly implying all the results of conventional Hamiltonian‐operator quantum formalism. In the present article we show that our variational formulation, under the familiar classical canonical transformation to momentum space, also automatically leads to the corresponding Schrödinger equations for the transition amplitudes between pure eigenstates of homologous variables in momentum space, at different times, the unitarity of the formalism and the reciprocal Fourier integral relations between coordinate and momentum representations. Thus we bridge a long‐standing gap in theory and derive, uniquely and apparently for the first time, the basic quantum commutators (P.B.'s) which are postulated in the more familiar Hamiltonian theory.

Eigenfunction Form of the Nonrelativistic Coulomb Green's Function
View Description Hide DescriptionBy treating the contour integrals in the momentum plane, the conventional eigenfunction form is deduced from the characteristic form for the nonrelativistic Coulomb Green's function. A straight‐forward extension of the argument employed here leads to the most general form of the eigenfunction form.

Some Analytic Properties of Green Functions and Self‐Energy Parts for a System of Interacting Bosons
View Description Hide DescriptionSome analytic properties of the single‐particle Green functions and proper self‐energy parts for a system of interacting bosons at nonzero temperatures are derived. In particular, it is shown how to obtain the analytical continuations of simple functions of the proper self‐energy parts, which can be obtained from perturbation theory at an infinite set of points along the imaginary energy axis, to the whole of the complex energy plane. This is useful in determining the poles of the Green functions.

Exact Statistical Mechanics of a One‐Dimensional System with Coulomb Forces. III. Statistics of the Electric Field
View Description Hide DescriptionThe statistics of the electric field in a one‐dimensional ``sheet'' plasma is studied. It is shown that the electric field as a function of the space coordinate is a Markov process, whose basic probabilities are related to the Fourier coefficients of certain functions that play an important role in the previously found development of the grand partition function. Some implications regarding the statistics of particle locations are explored. The one‐point probability distribution of the electric field is found to be asymptotically Gaussian in the plasma limit. It is pointed out that the one‐dimensional analog of the Holtsmark calculation leads to an incorrect conclusion because electrostatic shielding is not properly taken into account.

Generalized Master Equation and t‐Matrix Expansion
View Description Hide DescriptionSome mathematical aspects of the generalized master equation are discussed. The resolvent operator is expanded in terms of a two‐body scattering matrix and this result is used to express the quantities in the generalized master equation in terms of expansions in the scattering matrix. The expansions are utilized in studying the density dependence of these quantities.

Nonlinear Coupled Oscillators. I. Perturbation Theory; Ergodic Problem
View Description Hide DescriptionA study is made of some of the problems which arise in determining the long‐time behavior of a system of coupled oscillators. Standard perturbation methods are examined in the light of certain classic results due to Poincaré and Whittaker concerning the construction of constants of motion which are analytic in the coupling constant, λ. These considerations lead to the study of perturbation methods which are not ordered in powers of λ. An examination of the various advantages of these methods leads to a method which removes secular terms in such a way as to mitigate the classic problem of small divisors. The significance of studies which attempt to relate the existence of analytic constants of the motion with the ergodic behavior of a system is examined.

Lyapunov's Stability Criteria for Plasmas
View Description Hide DescriptionThe orbit stability theory of Lyapunov has been adapted to the Vlasov‐Boltzmann equation governing plasmas. Both linear and nonlinear stability are considered. The theory is characterized by a search for Lyapunov functions, whose existence implies stability in analogy with particles trapped in a potential well, as in the energy principle. The most important result is an existence theorem for Lyapunov functions quadratic in perturbations in all linearly stable cases in which perturbations damp asymptotically (sufficiently fast). As a corollary, without damping, the existence of a quadratic Lyapunov function is necessary and sufficient to prevent exponential growth of perturbations. A prescription is given for finding Lyapunov functions which are constants of motion. An example is treated. The implication of nonlinear stability from linear stability with damping is discussed, and Dr. C. S. Gardner's direct proof of nonlinear stability of a Maxwellian plasma by Lyapunov's method is reported.

Partition Function for Certain Simple Lie Algebras
View Description Hide DescriptionThe partition function `which yields the multiplicities of weights in representations' is computed for the following Lie algebras:A _{2}, B _{2}, G _{2}, and A _{3}.

Lie Algebraic Solution of Linear Differential Equations
View Description Hide DescriptionThe solutionU(t) to the linear differential equationdU/dt = h(t)U can be represented by a finite product of exponential operators; In many interesting cases the representation is global. U(t) = exp[g _{1}(t)H _{1}] exp [g _{2}(t)H _{2}] … exp[g_{n} (t)H_{n} ] where g_{i} (t) are scalar functions and H_{i} are constant operators. The number, n, of terms in this expansion is equal to the dimension of the Lie algebra generated by H(t). Each term in this product has time‐independent eigenvectors. Some applications of this solution to physical problems are given.

Erratum: Zero‐Point Energy of an Electron Lattice
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