Volume 3, Issue 3, May 1962

A Convergent Perturbation Expansion in First‐Quantized Electrodynamics
View Description Hide DescriptionIt is shown that a real physical problem exists which, when calculated in first‐quantized electrodynamics, possesses a convergent perturbation expansion. The result is demonstrated by proving the analyticity in a region of nonzero radius about the origin in the complex coupling‐constant plane, of the transition probability for pair creation by two electromagnetic fields. Some singularities in the complex plane are located, which limit the radius of convergence only for a discrete set of values for the energies of the electromagnetic fields which define the problem.

Green's Distributions and the Cauchy Problem for the Iterated Klein‐Gordon Operator
View Description Hide DescriptionAn explicit form of the homogeneous Green's function for the multi‐dimensional iterated Klein‐Gordon operator is obtained. By a direct calculation from its Fourier representation, the Green's function is expressed as a one‐dimensional, infinite integral of the Sonine type. Although this integral is classically divergent when the order of the operator is less than the number of space dimensions, it can be treated rigorously under these conditions using the concepts of distribution analysis. A generalized Sonine integral is developed and the result applied to obtaining an explicit expression for the Green's function, which is now to be regarded as a distribution in the sense of Schwartz. Using a distribution introduced for this purpose, the Green's function is written in a form which explicitly displays its singularities on the light cone. The well‐known difference between even‐ and odd‐dimensional spaces is reflected in the nature of these singularities. The singularities appearing for an odd number of space dimensions consist of a finite linear combination of derivatives of the Dirac delta function δ(s ^{2}) where s is the space‐time distance. The highest derivative appearing is of order ½(n − 2l − 1) with n giving the number of space dimensions and 2l giving the order of the operator. The singular part for even‐dimensional spaces consists of a polynomial in 1/s of degree n − 2l + 1. No singularities appear when the order of the operator is greater than the number of dimensions. The general solution of Cauchy's problem for the iterated Klein‐Gordon operator is obtained in convolution form. An explicit solution for the ordinary Klein‐Gordon equation is presented in a form which exhibits separately the contributions due to the singular part and the regular part of the Green's function.

Simple Realizations of the Infinitesimal Generators of the Proper Orthochronous Inhomogeneous Lorentz Group for Mass Zero
View Description Hide DescriptionA realization of the infinitesimal generators for the mass‐zero case of the proper orthochronous inhomogeneous Lorentz group is given explicitly for both continuous and discrete spin cases in terms of a uniform notation. The realization for the discrete spin case is unitarily equivalent to that given by Shirokov.
For the sake of completeness the infinitesimal generators for the case of nonzero mass, derived by Foldy, are also given. Hence the present paper contains realizations for all irreducible unitary representations of the inhomogeneous Lorentz group of physical interest.
Since the irreducible representations of the two‐dimensional Euclidean group play an important role in the massless case, simple realizations of the irreducible representations of the generators of this group are also given.

Properties of the Irreducible Angular Momentum Tensors
View Description Hide DescriptionThe angular momentum operators define a set of irreducible tensors which are unique except for a normalization constant. The normalization is conveniently defined in terms of statistical tensors which describe oriented states. The properties of the tensors discussed here include: (1) the trace of products of components of such tensors, (2) symmetry properties of the traces, and (3) the expansion of products of components of these tensors into a sum of irreducible tensors. The corresponding expansion of commutators and anticommutators of these components is also discussed briefly.

The Quantum‐Mechanical Scattering Problem
View Description Hide DescriptionThe purpose of this paper is threefold: to provide a mathematically rigorous formulation of the quantum‐mechanical scattering problem from the time‐independent point of view as has been done by Jauch from the time‐dependent point of view, to establish a union between the two formulations, and to investigate the necessity of the asymptotic condition which occurs as a postulate in the time‐dependent formulation. The formulation of the problem depends only on the ``total'' and ``free'' Hamiltonian operators. Under the conditions necessary for the time‐dependent formulation, the wave operators defined by the asymptotic limits provide a unique solution of this problem. The possibility that solutions can exist when the asymptotic conditions are not valid is investigated by defining wave operators by an integral representation. The conditions sufficient for these to provide a unique solution are shown to be possibly weaker than the asymptotic conditions; there may be a class of Hamiltonian operators for which such solutions exist but for which the asymptotic limits do not. An explicit characterization of such a set of Hamiltonian operators is not achieved, but this question of the necessity of the asymptotic condition has been reduced to a specific mathematical problem. It is hoped that this paper will find a reader who is able to carry the mathematical investigation further.

The Quantum‐Mechanical Scattering Problem. II. Multi‐Channel Scattering
View Description Hide DescriptionThe study reported in an earlier paper of the single channel scattering problem is extended to include the multi‐channel case.

Theory of High‐Energy Potential Scattering
View Description Hide DescriptionThe exact amplitude for scattering of a Schrödinger or Dirac particle by a static potential is rewritten in a two‐potential form by splitting the potential into two parts, one of which contributes only to exactly forward scattering. Replacement of the exact wave function by a modified plane wave gives a high‐energy approximation that is shown to be equivalent to the Saxon‐Schiff approximation in the Schrödinger case. Corrections to the approximation are obtained in principle from a simplified series expansion of the exact wave function having the modified plane wave as leading term. The approximate amplitude reduces at small scattering angles to a well‐known result; at large angles, it reduces to Schiff's stationary‐phase approximation in the Dirac case but not, as shown by the example of a Gaussian potential, in the Schrödinger case.

Perturbation Theory and Lie Algebras
View Description Hide DescriptionAn approach to perturbation theory is considered based on the formula exp (—iB) A exp (iB) = A exp (iB]), where [AB] = AB − BA is the commutator. The operators A constitute a linear space and the operators B considered are such that B] take into itself. The present discussion considers the case where is finite dimensional with coordinates c _{1}, ···, c_{n} ; i.e., is isomorphic with the set of n‐dimensional vectors c. Under these circumstances sufficient conditions for the basic formula are given in terms of the ``analytic vectors'' of Nelson. The set, , of B's available can be considered closed under the processes of taking linear combinations and forming the commutator. Thus is a Lie algebra. Exponentiation leads to a Lie group of operators U, and A and A′ are said to be B equivalent if A′ = U ^{−1} AU. For finite dimensional each B is associated with an n × n matrix, b, which specifies the operation B] relative to the vectors c. Effectively then, is finite dimensional. The matrices b form a Lie algebra with a corresponding Lie group of matrices u such that A and A′ are B equivalent if and only if the corresponding two vectors c and c′ are u images; i.e., c′ = u c. Computationally, therefore, the set of A′ equivalent to a given A is obtained by considering the orbit of a given vector c under the Lie group; i.e., the set of u c. A neighborhood of a given A consists of those operators A′ in the form A + δA where δA is arbitrary except for a restriction on the size of its coordinates, c_{i}. Given A, a neighborhood can be found for which one can obtain by a well‐known construction on the orbits a set of functionally complete and functionally independent invariants for B equivalence. Computationally global and rational invariants are desirable and these can be obtained in the form of similarity invariants, provided that a can be mapped onto a set of n × n matrices a in such a way that if c corresponds to a, then b c corresponds to a′ = [ab]. If the sets and are identical such a mapping is immediately available and if, in addition, the corresponding Lie algebra is semisimple, in general, given A, the A's in some neighborhood are each equivalent to an A″ which is a function of A. This corresponds to a case in which a very simple perturbation of A levels occurs. Two examples are discussed.

A Procedure for Estimating Eigenvalues
View Description Hide DescriptionA new procedure is given for calculation of lower bounds to the eigenvalues of self‐adjoint operators. Computation of the lower bounds is reduced to the solution of linear algebraic problems.

A Note on Perturbation Theory in Nearly Periodic Systems
View Description Hide DescriptionIt is shown that for a particle in periodic or nearly periodic motion, an integrated form of the equations of motion may be the better starting point for an approximate calculation of the orbit. The motion of a charged particle in a static, inhomogeneous magnetic field is used to illustrate how this approach avoids the difficulty of the spurious secular terms to all orders of the approximation.

Stochastic Models for Many‐Body Systems. I. Infinite Systems in Thermal Equilibrium
View Description Hide DescriptionSome model Hamiltonians are proposed for quantum‐mechanical many‐body systems with pair forces. In the case of an infinite system in thermal equilibrium, they lead to temperature‐domain propagator expansions which are expressible by closed, formally exact equations. The expansions are identical with infinite subclasses of terms from the propagator expansion for the true many‐body problem. The two principal models introduced correspond, respectively, to ring and ladder summations from the true propagator expansion, but augmented by infinite classes of self‐energy corrections. The latter are expected to yield damping of single‐particle excitations. The eigenvalues of the ring and ladder model Hamiltonians are real, and they are bounded from below if the pair potential obeys certain conditions. The models are formulated for fermions, bosons, and distinguishable particles. In addition to the ring and ladder models, two simpler types are discussed, one of which yields the Hartree‐Fock approximation to the true problem. A novel feature of all the model Hamiltonians (except the Hartree‐Fock) is that they contain an infinite number of parameters whose phases are fixed by random choices. Explicit closed expressions are obtained for the Helmholtz free energy of all the models in the classical limit.

Stochastic Models for Many‐Body Systems. II. Finite Systems and Statistical Nonequilibrium
View Description Hide DescriptionIn a preceding paper [J. Math. Phys. 3, 475 (1962)], some model Hamiltonians were proposed for quantum‐mechanical many‐body systems with pair forces. For infinite systems in thermal equilibrium, they led to temperature‐domain propagator expansions which were formally summable and expressible by closed equations. These expansions were identical with infinite subclasses of terms from the propagator expansion for the true many‐body problem. The two principal models corresponded to ring‐ and ladder‐diagram summations from the true propagator expansion, augmented by infinite classes of self‐energy corrections. The model Hamiltonians were called stochastic because they contained parameters whose phases were fixed by random choices. In the present paper, more general models are formulated which yield formally summable propagator expansions for finite systems. The analysis is extended to correlation and Green's functions defined for nonequilibrium ensembles. The nonequilibrium treatment is developed in the Heisenberg representation in such a way that unlinked diagrams do not arise. A basic convergence question associated with the formal closed equations for the model propagators and correlation functions is examined by means of finite‐difference integration of the Heisenberg equations of motion. This procedure appears to converge independently of whether the perturbation expansions for the propagators and correlation functions converge. It yields substantial support for the validity of the formal closed modelequations.

Asymptotic Expansion of the Bardeen‐Cooper‐Schrieffer Partition Function by Means of the Functional Method
View Description Hide DescriptionThe canonical operator exp [−β(H − _{μ} N)] associated with the Bardeen‐Cooper‐Schrieffer (BCS)model Hamiltonian of superconductivity is represented as a functional integral by the use of Feynman's ordering parameter. General properties of the partition function in this representation are discussed. Taking the inverse volume of the system as an expansion parameter, it is possible to calculate the thermodynamic potential including terms independent of the volume. This yields a new proof that the BCS variational value is asymptotically exact. The behavior of the canonical operator for large volume is described and related to the state of free quasiparticles. A study of the terms of the thermodynamic potential which are of smaller order in the volume in the low‐temperature limit, shows that the ground state energy is nondegenerate and belongs to a number eigenstate.

Antisymmetric Functions and Slater Determinants
View Description Hide DescriptionGiven a function w completely antisymmetric in n variables, there may exist a set of nfunctions of one variable such that the given function is a Slater determinant in the latter. The first problem considered is that of obtaining a criterion for this to be the case for a given function. This problem is solved by considering the function w as a mapping of the space of functions in n − 1 variables onto the space of functions of one variable. A necessary and sufficient condition for the initial function to be a Slater determinant is then shown to be that the image space be n dimensional. This criterion is converted into practical algorithms which can be employed for the determination. The application of one of these yields the theorem that an arbitrary linear combination of the n + 1 Slater determinants in n variables formed from n + 1 one‐variable functions can always be written as a single Slater determinant. It is further proved that if the image space of the mapping is m(>n) dimensional, the original function can be expressed as a linear combination of m!/(m − n)!n! Slater determinants in n variables formed from m one‐variable functions. Playing an important role in the analysis is the product of the mapping described above by its adjoint (the product is simply related to Dirac's density matrix for a quantum mechanical system of identical particles) as well as the eigenvectors and eigenvalues of this Hermitian positive semidefinite mapping. The latter form a basis for a systematic approximation procedure for representing a given function by a single Slater determinant or by sums of Slater determinants formed from a particular number of one‐variable functions, which yields results obtained previously by Löwdin. Problems of simultaneous approximation of sets of antisymmetric functions and possible physical applications to many‐fermion systems are briefly discussed.

Excitation Spectrum of a Fermion System
View Description Hide DescriptionBy the method of ``modes'' introduced by Sawada, the excitation spectra of a fermion system with singular interactions have been obtained in three cases; (1) with one additional particle above and and one hole below the Fermi surface, (2) with one additional particle above the Fermi surface, and (3) with two particles above the Fermi surface. The argument holds not only for the system of low density (nuclear matter with a hard‐core repulsion), but also for the case of the interacting electron gas in the high‐density region. The results have been compared with the Brueckner‐Goldstone perturbation‐expansion formulas, by using diagrams in the second‐ and third‐order of the expansion in interaction strength. We show that in the approximation of our treatment, the occupation probability function at temperature T has the same form as the Fermi distribution function.

Locally Maxwellian Solutions for a ``Positronium'' Plasma
View Description Hide DescriptionExact locally Maxwelliansolutions of a neutral ``positronium'' plasma are explicitly presented. They correspond to a flow which is a superposition of a parallel flow, a radial expansion, and a rigidbody rotation.

Studies in Nonequilibrium Rate Processes. V. The Relaxation of Moments Derived from a Master Equation
View Description Hide DescriptionA study has been made of the relaxation of the moments of probability distributions whose time evolution are governed by a master equation. The necessary and sufficient condition for the first moment, M _{1}(t), to undergo a simple exponential relaxation is found to bewhere A_{nm} is the transition probability per unit time for transitions from state m to n, and where β and γ are constants. The necessary and sufficient condition under which the first k moments, M _{1}(t), M _{2}(t), ···, M_{k} (t), satisfy a closed system of linear equations is found to beNear equilibrium, i.e., as t → ∞, all the moments M_{r} (t) obey, to a good approximation, a simple exponential relaxation law irrespective of the form of the A_{nm} .
For systems described by the Fokker‐Planck equationthe necessary and sufficient condition that the first moment M _{1}(t) undergo a simple exponential relaxation is found to be b _{1}(x) = βx + γ and the necessary and sufficient condition for the 2nd moment, M _{2}(t) to have a simple exponential relaxation is . It is shown that these conditions are equivalent to the conditions on the A_{nm} stated above.

Double‐Tensor Operators for Configurations of Equivalent Electrons
View Description Hide DescriptionTo every irreducible representation W of the rotation group in 2l + 1 dimensions that is used to classify states of the electronic configurations l^{n}, there correspond two couples (v, S), where v and S stand for the seniority number and total spin, respectively. Determinantal product states are introduced to examine this correspondence in detail. It is shown that for two double tensorsW ^{(κk)} and W ^{(κ′k)}, the set of reduced matrix elementsfor fixed n, v _{1}, v _{1}′, W, and W′, is proportional to the setwhere ξ and ξ′ are additional labels that may be required to define the states uniquely, provided (a) the two couples (v _{1}, S _{1}) and (v _{2}, S _{2}) are distinct, (b) the two couples (v _{1}′, S _{1}′) and (v _{2}′, S _{2}′) are distinct, and (c) the sum κ + κ′ + k is odd. The amplitudes of the double tensors are chosen so that the constant of proportionality is equal to the ratio of two 3‐j symbols, multiplied by a phase factor. An explicit expression for this factor is given for f electrons, and a number of applications are discussed.

Transient Response of a Dipole Antenna
View Description Hide DescriptionThe transient current in a long dipole antenna excited by a step‐function voltage across an infinitesimal center gap has recently been calculated by Wu. A simpler derivation of Wu's result is given using double Fourier transformation.

An Approach to Gravitational Radiation by a Method of Spin Coefficients
View Description Hide DescriptionA new approach to general relativity by means of a tetrad or spinor formalism is presented. The essential feature of this approach is the consistent use of certain complex linear combinations of Ricci rotation coefficients which give, in effect, the spinor affine connection. It is applied to two problems in radiationtheory; a concise proof of a theorem of Goldberg and Sachs and a description of the asymptotic behavior of the Riemann tensor and metric tensor, for outgoing gravitational radiation.