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Nov 1962

Volume 3, Issue 6, pp. 1059-1304

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Molecular Collisions. IV. Nearly Spherical Rigid Body Approximation

George Gioumousis and C. F. Curtiss

J. Math. Phys. 3, 1059 (1962); http://dx.doi.org/10.1063/1.1703818 (14 pages) | Cited 18 times

Online Publication Date: 22 December 2004

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Rotational transitions in the collisions of diatomic molecules are treated by means of a rigid body approximation. The deviation from sphericity is assumed to be sufficiently small that it may be treated as a perturbation. The differential cross section is calculated for the H2‐HD collision in which the hydrogen remains in its ground state while the hydrogen deuteride goes from its ground state to the first excited rotational state.

Quantization of Fields with Infinite‐Dimensional Invariance Groups. III. Generalized Schwinger‐Feynman Theory

Bryce S. DeWitt

J. Math. Phys. 3, 1073 (1962); http://dx.doi.org/10.1063/1.1703819 (21 pages) | Cited 17 times

Online Publication Date: 22 December 2004

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The formal methods of Schwinger and Feynman are applied to nonlinear field theories having elementary vertex functions of arbitrarily high order. In the first half of the paper, familiar theorems are rederived by noncanonical methods. Emphasis is given to purely formal aspects of the theory which may be expected to survive generalization to situations in which standard asymptotic conditions are inapplicable. Since the context in which the field nonlinearities are assumed to appear is that of a non‐Abelian infinite‐dimensional invariance group, detailed attention is given to the question of a group invariant measure for the Feynman functional integral. It is shown that the physically important part of the measure is not determined by the group.
The second half of the paper is devoted to the theory of the propagators and correlation functions which characterize the system when invariant variables are introduced. The existence of a c‐number action functional Γ which contains a complete description of all quantum processes is proved. The second variational derivatives of this functional constitute the wave operator for the one‐particle propagators (including all radiative corrections), and its higher derivatives are the renormalized vertex functions. A description of the renormalization process is easily carried out in terms of Γ. Finally, the implications which its existence has for quantum gravidynamics are discussed. Because it leads to nonlocal covariant equations for a complex metric tensor the way is open to transmutations of topology at the quantum level.

On the Gauge Covariance of Quantum Electrodynamics

I. Bialynicki‐Birula

J. Math. Phys. 3, 1094 (1962); http://dx.doi.org/10.1063/1.1703851 (5 pages) | Cited 15 times

Online Publication Date: 22 December 2004

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The gauge covariant formulation of quantum electrodynamics given by Zumino is further investigated. The method used is that of functional integration. This allows for a slight generalization of Zumino's results and provides a direct relation between the gauge condition in classical and quantum theories. All results are derived without any reference to the canonical formalism. The same procedure can be applied to the Yang‐Mills field with or without a mass, leading in both cases to a renormalizable theory. The Landau gauge is studied in some detail and it is shown that in the perturbation expansion of the propagators no terms violating the gauge invariance appear. Finally, a new interpretation of the generalized Ward identity is proposed.

Modified Mandelstam Representation for Heavy Particles

Jamal N. Islam

J. Math. Phys. 3, 1098 (1962); http://dx.doi.org/10.1063/1.1703852 (9 pages) | Cited 3 times

Online Publication Date: 22 December 2004

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The fourth‐order Feynman amplitude ceases to satisfy the Mandelstam representation when the external masses are sufficiently large. A representation which replaces the Mandelstam one is found in the cases where the four mass invariants are equal in pairs. The physical interpretation is briefly discussed.

Axiomatic Perturbation Theory for Retarded Functions

H. M. Fried

J. Math. Phys. 3, 1107 (1962); http://dx.doi.org/10.1063/1.1703853 (9 pages) | Cited 6 times

Online Publication Date: 22 December 2004

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A construction of the retarded n‐point functions of perturbation theory is given within the Lehmann, Symanzik, and Zimmermann framework and without the specification of an interaction Lagrangian. An intermediate‐state expansion of retarded functionals is employed to define a systematic set of equations representing approximations to the (integral) unitarity conditions; the requirement of symmetry of the n − 1 retarded coordinates of an n‐point retarded function enters in an essential way. The class of solutions to these equations contains the renormalized perturbation theory retarded functions corresponding to local renormalizable Lagrangian interactions, as well as more singular functions corresponding to nonrenormalizable interactions; if the latter are excluded all the n‐point functions may be successively determined to all orders in the renormalized coupling constants. The construction is explicitly performed for the first radiative corrections to the 2‐ and 3‐point functions of a self‐interacting neutral scalar boson field, yielding the finite renormalized results of perturbation theory. Similar but slightly singular results are quoted for the π‐π scattering amplitude.

On the Restricted Lorentz Group and Groups Homomorphically Related to It

A. J. Macfarlane

J. Math. Phys. 3, 1116 (1962); http://dx.doi.org/10.1063/1.1703854 (14 pages) | Cited 28 times

Online Publication Date: 22 December 2004

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A study is made of the real restricted Lorentz group, L, and of its relationship
(a) to the group, SL(2C), of complex unimodular two‐dimensional matrices, and
(b) to the group, O3, of orthogonal transformations in a complex space of three dimensions.
The discussion of case (a) is an improved version of the treatment by Wightman. Its notable features are, firstly, that it gives important formulas in new concise forms and their proofs in an elegant and economical manner, and, secondly, that it deals with the nontrivial matter of proving the internal consistency of the formalism. To illustrate the practical utility of the theory, the product of two nonparallel pure Lorentz transformations is studied. In the discussion of case (b), explicit formulas realizing the isomorphism of O3 and L are obtained. These formulas are new and have been applied, for illustrative purposes, to the derivation of the transformation properties under L of the electromagnetic field vectors, regarded as a complex three‐vector (E + iH). A result analagous to the factorization of the general element of L into a spatial rotation and a pure Lorentz transformation, and to the polar decomposition of the general element of SL(2C), is derived for O3. Insight into the relationship of O3 to L is provided by considering the unimodular matrix description of the complex Lorentz group, and the contrasting specializations of it that lead to the unimodular matrix descriptions of its subgroups, O3 and L.

A Criterion for Singularities in Perturbation Theory

Francis R. Halpern

J. Math. Phys. 3, 1130 (1962); http://dx.doi.org/10.1063/1.1703855 (5 pages) | Cited 1 time

Online Publication Date: 22 December 2004

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A criterion is proposed to distinguish between the singular and nonsingular portions of the Landau surface on the physical sheet. The set of diagrams under consideration are those containing a single loop. A proof of the Mandelstam representation for the four‐point function is given based on this criterion.

On Spinors in n Dimensions

Abraham Pais

J. Math. Phys. 3, 1135 (1962); http://dx.doi.org/10.1063/1.1703856 (5 pages) | Cited 18 times

Online Publication Date: 22 December 2004

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The matrix which enters in the charge conjugation transformation of the usual spinors in 4‐space is an invariant matrix and is skew symmetric. It is shown that there exists such an invariant matrix C for any number of dimensions (and independent of the number of time like dimensions). Its symmetry properties depend on the dimension number n modulo 8. With the help of the C matrix one can construct, for n = 1, 2, 7, 8 mod 8, an n‐dimensional invariant bilinear in the components of a single n‐dimensional spinor. Some examples are given for n = 2, 3, 7. A bilinear baryon invariant is formed for a theory with high symmetry. Its existence is closely related to the triality property of 8‐space.

Integral Representations for Production Amplitudes in Perturbation Theory

Noboru Nakanishi

J. Math. Phys. 3, 1139 (1962); http://dx.doi.org/10.1063/1.1703857 (8 pages) | Cited 5 times

Online Publication Date: 22 December 2004

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The integral representation proposed previously in the two‐particle scattering is extended to the case of one‐particle‐production amplitudes. Five integral representations are found whose main parts consist of only six terms. Their support properties are investigated for the equal‐mass case in every order of perturbation theory. It is shown that lowest‐order graphs are by no means the representatives of analyticity in perturbation theory even in the equal‐mass case.

Atomic Many‐Body Problem. II. The Matrix Components of the Hamiltonian with Respect to Correlated Wave Functions

Levente Szász

J. Math. Phys. 3, 1147 (1962); http://dx.doi.org/10.1063/1.1703858 (14 pages) | Cited 29 times

Online Publication Date: 22 December 2004

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The wave function for an atom with N electrons in arbitrary configuration will be written in the form
math
,where ψ0 is a Slater determinant and f(ij) is the antisymmetrized product of (N − 2) one‐electron spin‐orbitals and one 2‐electron function Φ(ij∕12). The correlation between the two spin‐orbitals ϕi and ϕi can be taken into account by introducing r12 (the interelectronic distance) explicitly into the 2‐electron function Φ. The purpose of the paper is to analyze the structure of the matrix components of the Hamiltonian with respect to the wave function given above. Starting from exact, general formulas for the matrix components it will be shown that, all integrals which occur in the diagonal, as well as in the nondiagonal matrix components can be reduced to six basic integrals which are 2‐ , 3‐ , and 4‐electron integrals, containing interelectronic distances. It will be indicated that, five of the six basic integrals can be calculated in closed form whereas one of them, (an exchange integral) can be given only in the form of an infinite series.

Hard Core Produced by Orthogonality Constraints

S. Tani and D. A. Uhlenbrock

J. Math. Phys. 3, 1161 (1962); http://dx.doi.org/10.1063/1.1703859 (10 pages) | Cited 3 times

Online Publication Date: 22 December 2004

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It is shown that the effect of a hard core as an external force can be reproduced by requirements of orthogonality alone. A fictitious orthonormal set of functions is introduced, which form a complete set in the domain occupied by the hard core. The actual wave function is constructed as the eigenfunction of a modified kinetic energy operator, and is required to be orthogonal to all members of the fictitious set. The limit of infinitely many constraints is carefully discussed. It turns out to be impossible to regard the hard core effect as equivalent to some kind of ``potential'' acting on free waves (for which the Hamiltonian would be well defined). However, the eigenfunction is well defined in the limit of infinitely many constraints, and is independent of the way in which the fictitious set is chosen.

Studies in Perturbation Theory. V. Some Aspects on the Exact Self‐Consistent Field Theory

Per‐Olov Löwdin

J. Math. Phys. 3, 1171 (1962); http://dx.doi.org/10.1063/1.1703860 (14 pages) | Cited 118 times

Online Publication Date: 22 December 2004

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The independent‐particle model in the theory of many‐particle systems is studied by means of the self‐consistent‐field (SCF) idea. After a review of the characteristic features of the Hartree and Hartree‐Fock schemes, the extension of the SCF method developed by Brueckner is further refined by introducing the exact reaction operator containing all correlation effects. This operator is here simply defined by means of the partitioning technique, and, if the SCF potentials are derived from this operator, one obtains a formalism which is completely analogous to the Hartree scheme but which still renders the exact energy and the exact wave function. An elementary derivation of the linked‐cluster theorem is given, and finally the inclusion of various symmetry properties is discussed.

On a Functional Averaging Method in the Classical Ensemble Theory

A. Galindo

J. Math. Phys. 3, 1185 (1962); http://dx.doi.org/10.1063/1.1703861 (6 pages)

Online Publication Date: 22 December 2004

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Albertoni, Bocchieri, and Loinger (ABL) have given a general prescription to calculate the average E(F) of functionals F[ρ] defined on the space L of the initial states (Liouville density functions) ρ(x) of an arbitrary (finite) dynamical system (S, Σ, μ, H). Then they have proved that E(BH) = 0 for any B ∊ Σ, where
math
[ABL theorem]. On the other hand, these authors claim that E gives the same weight to each ρ∊L and therefore, they contend that their theorem is sufficient to justify the classical statistical mechanics.
In this note, however, the following statements are proved: (A) If (S, Σ, μ, H) is ergodic with discrete spectrum, then every probability measure ν in L for which BH[ρ] = 0, ν‐almost everywhere (for any B ∊ Σ) must be concentrated at e [e(x) ≡ μ−1(S) on S]. (B) Given (S, Σ, μ, Ho) with Ho = 0, every (normally defined) functional average A(F) such that A(BHo) = 0 for any B ∊ Σ must define a probability measure ν in L such that ν({e}) = 1. From (B) it will follow that: (1) The ABL average E(F) does not give the same weight to each ρ∊L; actually, E defines a probability measure in L concentrated at e; consequently, no significant nontrivial conclusion can be naturally drawn from the ABL theorem regarding the foundations of the classical statistical mechanics; (2) it is impossible to restore in general the validity of the ABL theorem by choosing a different normal functional averaging method. To illustrate (2) a new (nonconcentrated) functional average E*(F) is introduced and it is proved that E*(BH) ≠ 0 for every nonweakly mixing system (S, Σ, μ, H) and some B ∊ H.

A Brownian‐Motion Model for the Eigenvalues of a Random Matrix

Freeman J. Dyson

J. Math. Phys. 3, 1191 (1962); http://dx.doi.org/10.1063/1.1703862 (8 pages) | Cited 236 times

Online Publication Date: 22 December 2004

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A new type of Coulomb gas is defined, consisting of n point charges executing Brownian motions under the influence of their mutual electrostatic repulsions. It is proved that this gas gives an exact mathematical description of the behavior of the eigenvalues of an (n × n) Hermitian matrix, when the elements of the matrix execute independent Brownian motions without mutual interaction. By a suitable choice of initial conditions, the Brownian motion leads to an ensemble of random matrices which is a good statistical model for the Hamiltonian of a complex system possessing approximate conservation laws. The development with time of the Coulomb gas represents the statistical behavior of the eigenvalues of a complex system as the strength of conservation‐destroying interactions is gradually increased. A ``virial theorem'' is proved for the Brownian‐motion gas, and various properties of the stationary Coulomb gas are deduced as corollaries.

The Threefold Way. Algebraic Structure of Symmetry Groups and Ensembles in Quantum Mechanics

Freeman J. Dyson

J. Math. Phys. 3, 1199 (1962); http://dx.doi.org/10.1063/1.1703863 (17 pages) | Cited 136 times

Online Publication Date: 22 December 2004

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Using mathematical tools developed by Hermann Weyl, the Wigner classification of group‐representations and co‐representations is clarified and extended. The three types of representation, and the three types of co‐representation, are shown to be directly related to the three types of division algebra with real coefficients, namely, the real numbers, complex numbers, and quaternions. The author's theory of matrix ensembles, in which again three possible types were found, is shown to be in exact correspondence with the Wigner classification of co‐representations. In particular, it is proved that the most general kind of matrix ensemble, defined with a symmetry group which may be completely arbitrary, reduces to a direct product of independent irreducible ensembles each of which belongs to one of the three known types.

Evaluation of the Quantum‐Mechanical Ring Sum with Boltzmann Statistics

Hugh E. DeWitt

J. Math. Phys. 3, 1216 (1962); http://dx.doi.org/10.1063/1.1703864 (13 pages) | Cited 45 times

Online Publication Date: 22 December 2004

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A procedure is given for the evaluation of the quantum‐mechanical ring sum at finite temperature. The method is used for the evaluation of the quantum corrections to the classical Debye‐Hückel free energy for an electron gas obeying Boltzmann statistics. The ring sum is shown to be of the form (βe2∕πλD)P(γ), where γ = /λD, = ℏ/(2mkT)1/2, and λD is the Debye screening length. The quantum effects for finite γ are due only to the operation of the uncertainty principle. The function P(γ) decreases monotonically from the classical value π∕3, and the form is shown to be
math
.The coefficients an and bn are evaluated exactly for small n and asymptotically for large n. The two series converge for γ2 < γc2 = 2.042 ….
For γ ≫ γc the function P(γ) is also evaluated as an asymptotic expansion in inverse powers of γ1∕2. Thus the low‐temperature correlation energy of distinguishable electrons is obtained in random phase approximation. At zero temperature, the result is the same as the correlation energy obtained by Foldy for charged bosons. The correlation pressure in this approximation is negative and diverges as ρ1∕4 at high density.

Quantum Statistics and the Boltzmann Equation

Robert M. Lewis

J. Math. Phys. 3, 1229 (1962); http://dx.doi.org/10.1063/1.1703865 (18 pages) | Cited 2 times

Online Publication Date: 22 December 2004

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The system of hierarchy equations for the reduced density operators of quantum statistical mechanics is replaced by a single functional differential equation for a generating functional. A formal solution of the initial value problem for the latter equation is obtained, leading to series expansions of the reduced density operators. These expansions are used to obtain an improved derivation of the quantum‐mechanical Boltzmann equation.

Theory of Transport Coefficients. III. Quantum Statistical Systems

S. Fujita

J. Math. Phys. 3, 1246 (1962); http://dx.doi.org/10.1063/1.1703866 (5 pages) | Cited 6 times

Online Publication Date: 22 December 2004

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The electrical conductivity of an electron‐phonon system is calculated from Kubo's formula using a perturbation method on the assumptions that the coupling is weak, the system is infinitely large, and that the electrons do not interact between them and obey the Fermi‐Dirac statistics and phonons are in thermodynamic equilibrium. The result is identical with that one would obtain from the usually assumed Boltzmann‐Bloch equation. The calculation is a logical extension of the previous treatment I of the present series, where the same problem is treated on the assumption that the electrons obey the Boltzmann statistics. The relation between the present calculation and the derivation of master equation is critically discussed. A brief sketch is given for the calculation of the viscosity coefficient of a dilate quantum statistical gas.

Numerical Estimation of the Partition Function in Quantum Statistics

Lloyd D. Fosdick

J. Math. Phys. 3, 1251 (1962); http://dx.doi.org/10.1063/1.1703867 (14 pages) | Cited 22 times

Online Publication Date: 22 December 2004

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A method for estimating the partition function of a quantum mechanical system is described. The method is based on a technique for evaluating the Wiener integrals in terms of which the partition function may be expressed. This technique involves first, an approximation of the Wiener integral by an n‐dimensional integral and, second, a Monte Carlo estimation of the value of the n‐dimensional integral. Application of the method to a harmonic oscillator and a pair of interacting particles in a box in two dimensions is described.

Dimers on Rectangular Lattices

Tai Tsun Wu

J. Math. Phys. 3, 1265 (1962); http://dx.doi.org/10.1063/1.1703868 (2 pages) | Cited 28 times

Online Publication Date: 22 December 2004

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A simplified calculation of a known result of the dimer problem on a rectangular lattice is presented.

Approximate Solutions of the Modified Bloch Equations for Low Magnetic Fields

Richard J. Runge

J. Math. Phys. 3, 1267 (1962); http://dx.doi.org/10.1063/1.1703869 (11 pages) | Cited 1 time

Online Publication Date: 22 December 2004

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A set of modified Bloch equations proposed by Torrey, Olds, and Codrington for the description of the time dependence of the magnetic moment expectation value in low magnetic fields, is solved using two approximate methods. The equations are solved approximately for the steady state with a constant field and linearly polarized radio‐frequency field applied and an error analysis of the approximate solution is given. A low‐harmonic solution is developed and alternative numerical integration techniques are given. Results are exhibited for the low‐field electron paramagnetic resonance in anthracene negative ion solution and asphaltene. The methods developed are applicable to a wide range of frequencies, applied field strengths, and ratios of transverse to longitudinal relaxation times.

Selection Rules for Integrals of Bloch Functions

J. Zak

J. Math. Phys. 3, 1278 (1962); http://dx.doi.org/10.1063/1.1703870 (2 pages) | Cited 13 times

Online Publication Date: 22 December 2004

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A simple selection rule for integrals of three Bloch functions is derived by using the irreducible representations of the complete space group. In the final formula only the characters of the small representations appear.

An Orthogonality Property of Hydrogenlike Radial Functions

S. Pasternack and R. M. Sternheimer

J. Math. Phys. 3, 1280 (1962); http://dx.doi.org/10.1063/1.1703871 (1 page) | Cited 61 times

Online Publication Date: 22 December 2004

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An integral involving r−s times the product of two hydrogenlike radial functions of the same n and different l is shown to be zero for a number of values of s.

Green's Distributions and the Cauchy Problem for the Multi‐Mass Klein‐Gordon Operator

John Judson Bowman and Joseph David Harris

J. Math. Phys. 3, 1281 (1962); http://dx.doi.org/10.1063/1.1703872 (10 pages) | Cited 1 time

Online Publication Date: 22 December 2004

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Explicit forms of the Green's functions (which are to be regarded as distributions in the sense of Schwartz) for the multi‐mass Klein‐Gordon operator in n‐dimensional spaces are presented. The homogeneous Green's functions GN(x) and GN1(x), defined in the usual way by independent paths of integration in the k0 plane, are investigated in the neighborhood of the light cone. The parameter N indicates the total number of masses involved. The singularities on the light cone reflect the well‐known difference between even‐ and odd‐dimensional wave propagation. It is found that GN(x; odd n) contains a finite jump on the light cone as well as a linear combination of derivatives up to order ☒(n − 2N − 1) of δ(x2); the singular part of GN1(x; odd n) consists of a logarithmic singularity ln (∣x2∣) along with a polynomial in (x2)−1 of degree ☒(n − 2N − 1). For even‐dimensional spaces, the singular part of both Green's functions consists of a polynomial in (x2)−1∕2 of degree n − 2N + 1 vanishing outside the light cone for GN and vanishing inside the light cone for GN1. In all cases no singularities or finite jumps occur when the order 2N of the operator is greater than the number n + 1 of space‐time dimensions. The general solution of the Cauchy problem is given both for the data carrying surface t = 0 and for arbitrary spacelike data surfaces.

Green's Distributions Associated with the Operator [□m − (−μ2)m]l

John Judson Bowman and Joseph David Harris

J. Math. Phys. 3, 1291 (1962); http://dx.doi.org/10.1063/1.1703873 (7 pages) | Cited 1 time

Online Publication Date: 22 December 2004

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The results of a previous paper on homogeneous Green's functions for a multi‐dimensional iterated Klein‐Gordon operator (□ + μ2)l are extended to include homogeneous Green's functions associated with the operator [□m (− μ2)m]l in multi‐dimensional spaces. The Fourier representation of the Green's functions may be expressed, after some angular integrations, as a one‐dimensional infinite integral which does not in general converge. Using the concepts of distribution analysis, it is shown how this improper integral can be evaluated directly to get explicit expressions for the Green's functions. The Green's functions themselves must then be interpreted as distributions in the sense of Schwartz. Several distributions instrumental in this treatment are introduced and their properties studied. Explicit expressions for the singularities of the Green's functions on the light cone are presented. The well‐known difference between even‐ and odd‐dimensional spaces is reflected in the nature of these singularities. The singularities appearing for odd‐dimensional spaces consist of a finite linear combination of derivatives of the Dirac delta function δ(s2), where s is the space‐time distance. The highest derivative appearing is of order ☒(n−2m−l) with n giving the number of space dimensions. The singular part for even‐dimensional spaces consists of a polynomial in 1∕s of degree n − 2ml + 1. No singularities appear when the order of the operator is greater than the number of space dimensions. Finally, a complete set of homogeneous Δ‐function solutions is given along with their initial conditions at zero time. All of these functions would be needed in obtaining the general solution to the Cauchy problem for the operator considered.
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