Volume 5, Issue 10, October 1964
Index of content:
5(1964); http://dx.doi.org/10.1063/1.1704069View Description Hide Description
A manifestly Lorentz invariant Hamiltonian formalism for multichannel scattering and production processes is developed by making two simple and natural extensions of the ordinary quantum mechanical formalism. The first is the asymptotic covariance postulate of Fong and Sucher which is essentially a necessary and sufficient condition for Lorentz invariance of the scattering amplitudes. The second is a Lorentz invariant extension of the asymptotic condition. It is shown that the latter is in fact no extension at all in a case where the total momentum operators of the asymptotic (unperturbed) systems are the same as the total momentum operators of the interacting system. In such a case the ordinary multichannel scattering formalism is completely Lorentz invariant whenever asymptotic covariance is satisfied.
5(1964); http://dx.doi.org/10.1063/1.1704070View Description Hide Description
From an investigation of the two‐point function in nonrenormalizable field theories, it is shown that at least in certain approximations, a nonrenormalizable field is nonlocalizable. This is intimately connected with the occurrence of essential singularities on the light cone in the Wightman functions of the field. Green's functions cannot be defined and the observables, in particular the scattering matrix elements, have to be expressed directly in terms of the unordered expectations values.
5(1964); http://dx.doi.org/10.1063/1.1704072View Description Hide Description
The dispersion relation is obtained for the ``hose'' instability in a modulated beam of charged particles traveling through a finite ohmic plasma channel. For a relativistic beam, the low‐frequency mode obeys a dispersion relation of the same form as for the unmodulated beam, but it involves a constant which will have to be obtained by machine computation. For large enough modulation frequency the dispersion relation is identical to that for an unmodulated beam.
5(1964); http://dx.doi.org/10.1063/1.1704073View Description Hide Description
The magnetic configuration of a cylinder with infinite conductivity is computed numerically for different values of the height 2h. The calculation indicates that the current densityj(z) diverges at the edges as I(h)/(h 2−z 2)½ and permits the evaluation of the function I(h). The regular part of j(z) seems to be approximated by an elliptical profile. The case of a cut cylinder is also discussed.
Complex Angular Momenta and Many‐Particle States. I. Properties of Local Representations of the Rotation Group5(1964); http://dx.doi.org/10.1063/1.1704074View Description Hide Description
The properties of the ``local representations'' of the rotation group corresponding to complex angular momentum are further developed. Completeness and bi‐orthogonality relations are derived and a reduction of products is carried out, giving a generalization of the Clebsch‐Gordan reduction. The connection with the representation theory of the group SL(2,R) is considered and a generalization of Regge's use of the Sommerfeld‐Watson transform is made to the case where three momentum transfer variables occur in the description of scattering amplitudes.
5(1964); http://dx.doi.org/10.1063/1.1704075View Description Hide Description
The arrangement of atoms in solid solutions and alloys, prepared at high temperatures and cooled nonadiabatically, is not the one which is thermodynamically most stable. In establishing theories of phenomena related to the internal degrees of freedom of such a system, such as magnetism, one must be careful to account for this nonequilibrium distribution of atoms. In this paper, systems are treated with the aid of a fictitious equilibrium system. This fictitious system is constructed such that its thermal equilibrium properties are the same as the properties of the non‐thermal‐equilibrium system. Thus one can treat nonequilibrium systems by applying well known thermal equilibrium techniques to the fictitious system.
The method is illustrated via the example of a magnetically dilute alloy. Brout's result for a very dilute Ising system is obtained with the aid of the theory of classical fluids, without collecting diagrams. A method for applying the higher approximations developed for classical fluids to the present problem is suggested; calculations and discussions of which are retained for a forthcoming paper.
5(1964); http://dx.doi.org/10.1063/1.1704076View Description Hide Description
The Mandelstam representation is a statement about the region of analyticity and asymptotic behavior (polynomial boundedness) of a scattering amplitude. In virtue of the unitarity condition, however, these two are not completely independent. Some physical consequences, e.g., uniqueness, polynomial boundedness of the total cross section, etc., which have been already derived from the Mandelstam representation, are shown to be preserved, even if the polynomial boundedness is replaced by a somewhat weaker assumption. By making use of unitarity, analyticity, and crossing symmetry, the following type of scattering amplitude F = E + M, where E is an entire function in both variables s and t, while M denotes a Mandelstam‐type function with finite number of subtraction, is shown to be ruled out. Similarly, F = EM is also ruled out, if one imposes the additional restriction that E should increase less fast than an exponential in one variable while the other is finite.
5(1964); http://dx.doi.org/10.1063/1.1704077View Description Hide Description
Scalar multiple scattering effects due to a random distribution of spheres are considered in detail. Transformation from a volume to a surface integral allows one to take account of the ``hole corrections'' involved in the equation of multiple scattering, and yields a secular equation for the propagation constant K of the composite medium. In the low‐frequency limit a result is given which appears to be exact over the entire range 0 ≤ δ ≤ 1, where δ is the fractional volume occupied by scatterers. Also in this limit, the boundary conditions appropriate to the boundary of the composite medium are established from examination of the total transmitted and reflected fields.
5(1964); http://dx.doi.org/10.1063/1.1704078View Description Hide Description
The solution of the Fredholm homogeneous equation,whereand x = (1 − ξ2)½ is found to be the associated Legendre function even, and the characteristic numbers of this kernel are obtained. The solution of the corresponding equation of the second kind is also found. The kernel of the homogeneous equation whose solution is , n + m odd, is obtained.
5(1964); http://dx.doi.org/10.1063/1.1704079View Description Hide Description
Various types of gyroscopic precessional motion are discussed by the application of operational methods to the vector Euler equation describing the motion of a rigid body about a fixed point. Discussed herein are free precession and forced precessions due to external torques both fixed in the body and fixed in space.
5(1964); http://dx.doi.org/10.1063/1.1704080View Description Hide Description
An iterative scheme is presented for solving the N/Dequations in the case where the left‐hand cut consists entirely as a sum of poles. At no step is recourse to a matrix inversion of an algebraic system required. A scheme for approximating arbitrary cuts by sequences of poles is also presented.
5(1964); http://dx.doi.org/10.1063/1.1704081View Description Hide Description
It is shown that the postulate of indistinguishability of the Maxwell field tensor from its dual leads to the concept of the electromagnetic field tensor as a spinor component in dual space. The demand for algebraic consistency dictates a unique connection with the gravitational field. The Maxwell field must be viewed as a set of potentials, and the necessity for a duality gauge condition excludes the existence of magnetic monopoles.
Representations of the Inhomogeneous Lorentz Group in Terms of an Angular Momentum Basis: Derivation for the Cases of Nonzero Mass and Zero Mass, Discrete Spin5(1964); http://dx.doi.org/10.1063/1.1704082View Description Hide Description
In a previous paper the authors showed how the infinitesimal generators of the proper, orthochronous, inhomogeneous Lorentz group acted in a basis in which the square of the angular momentum, the z component of the angular momentum, the helicity and the energy were diagonal for the irreducible representations which correspond to the cases of nonzero and zero mass, discrete spin. In that paper no derivation of the results were given. It was possible, however, to verify them directly. In the present paper we carry out the derivation.
5(1964); http://dx.doi.org/10.1063/1.1704083View Description Hide Description
Asymptotic behavior and subtraction problem of the perturbation‐theoretical integral representation (PTIR) are investigated in detail. Six theorems are rigorously proved in this connection. It is shown that a function represented by an unsubtracted PTIR may asymptotically increase in particular directions. The relation between the asymptotic behavior and the subtraction number is clarified for the subtracted PTIR. As a by‐product one obtains a consistent definition of a finite part of the integral involving x −1θ(x).
5(1964); http://dx.doi.org/10.1063/1.1704084View Description Hide Description
A method of generating orthogonal polar coordinate systems in N‐dimensional space is given. Commuting angular‐momentum operators are easily found in the coordinate systems generated; these operators are of a single form which depends on two parameters. A short table of coordinate systems and the resulting structure of quantum numbers and eigenfunctions is given.
Inequalities Relating the Nearest‐Neighbor Spin Correlation and the Magnetization for the Heisenberg Hamiltonian5(1964); http://dx.doi.org/10.1063/1.1704085View Description Hide Description