Volume 6, Issue 11, November 1965
Index of content:

On the Dimensionality of the Real World
View Description Hide DescriptionThe ``strength'' of systems of field equations is defined following a procedure due to Einstein. For an arbitrary dimension number, the strength of Einstein's field equations and of Maxwell's equations are calculated, and it is shown that only in four dimensions do the two systems of field equations determine their respective fields equally strongly. In addition, in four dimensions, Weyl's equation exhibits the same strength as the Einstein or Maxwell fields. The results are used to show that a necessary condition for a unified field‐theoretic description of gravity and electromagnetism is that the world be four dimensional. If one also demands that the neutrino field be determined as strongly as the Einstein and Maxwell fields, one must have a two‐component neutrino, rather than a four‐component neutrino.

Invariance and Conservation Laws in Classical Mechanics
View Description Hide DescriptionIt is commonly supposed that, in classical mechanics, invariance of a physical system to coordinate translation implies conservation of linear momentum. ``Invariance'' may be defined in a number of ways. If it is defined to mean invariance of the equation of motion, it is shown that invariance of this equation with respect to coordinate translation does not imply conservation of linear momentum. The effects of scale transformation and coordinate inversion invariance are also investigated. Both the Lagrangian approach and Newton's (second) law approach are considered. It is shown that each of the above invariances implies a condition on the equation of motion, while a combination of these and time‐inversion invariance is needed to obtain ordinary momentum conservation.

Electromagnetic Field inside a Current‐Carrying Region
View Description Hide DescriptionThe improper integrals, appearing in the course of evaluating the vector potential A and the electric fieldE inside a current‐carrying region, are carefully examined. It is found that the integrals exist and have a well‐defined meaning only when the current density function satisfies a Hölder condition. A definite and precise way of evaluating them is derived and A, E are shown to satisfy the usual inhomogeneous equations.

Relativistic Three‐Particle SU _{3} States
View Description Hide DescriptionThe scheme for classifying three‐particle states according to SU _{3} is extended to the relativistic domain. A discussion is also given of the basic group theory lying behind the scheme.

Analytic Properties of the Elastic Unitarity Integral
View Description Hide DescriptionIn this paper, certain results concerning the singularities of holomorphic functions with integral representations are used to investigate the analytic properties of the elastic unitarity integral of the quantum theory of fields. In particular, it is shown that each of the Landau singularities of the scattering amplitude is an actual singularity. It is also shown that if the scattering amplitude is not identically zero it must have a natural boundary on the unphysical sheet.

Crossing Symmetric Regge Representation for the Invariant Scattering Amplitude
View Description Hide DescriptionA crossing symmetric Regge representation for the invariant scattering amplitude is constructed which simultaneously exhibits all Regge poles in the three channels. It is assumed that the amplitude satisfies the Mandelstam representation, and that the usual Mandelstam‐Sommerfeld‐Watson transform exists. To achieve explicit crossing symmetry it is found necessary to work with the Legendre function of the second kind. Except for neglecting the influence of possible angular momentum cuts, the representation is exact for all s, t, and u, with no restriction on the location of the Regge poles. As an illustration of how it might be used in practice, the Chew‐Jones formula for the amplitude of definite signature is derived in the strip approximation.

Correlation Functions and the Coexistence of Phases
View Description Hide DescriptionWe discuss the existence, continuity and other properties of the canonical and grand canonical many‐particle correlation functionsn_{s} (r_{1}, … r_{ s }) in the thermodynamic limit of classical and quantum mechanical systems.
If the pressure of the system for fixed T is constant in the range of specific volume v_{a} to v_{b} , one expects physically to observe the coexistence of two separated phases. In terms of the correlation functions this is expressed by ,where x and x_{b} are the mole fractions of the phases so that v = x_{a}v_{a} + x_{b}v_{b} . With the aid of various lemmas on convex functions we prove that such a ``separation of the phases'' follows rigorously from statistical mechanics provided the correlation functions are ``well defined'' in an appropriate sense.

Alternative Formulations of Scattering in Model Field Theories
View Description Hide DescriptionA comparison is made between the expression for the boson‐fermion scattering amplitude, in a modeltheory, obtained by renormalizing Feynman diagrams, and the one that originates from the dressed‐particle picture. A proof is given of a surmise, previously verified to sixth order, that the two expressions for the transition amplitude are identical, although the former stems from a theory that leads to paradoxes from which the latter is exempt. The proof of the identity of the two representations is extended to transition amplitudes in which a weak interaction, considered to first order, is modified by a strong one considered to all orders. It is shown how the operations in the dressed‐particle picture directly lead to iterations in terms of the physical mass and coupling constant, whereas the expressions obtained from the Feynman diagrams require renormalization before they have this form.

Higher‐Dimensional Periodic Systems
View Description Hide DescriptionThis short note describes a technique which is useful for solving higher‐dimensional periodic systems in which a large amount of homogeneity and symmetry is absent.

Rotationally‐Symmetric Model Field Theories
View Description Hide DescriptionA class of highly symmetric ,nonrelativistic, Euclidean invariant, modelscalar field theories are examined assuming the existence of field and momentum operators that satisfy the canonical commutation relations (CCR). The high degree of symmetry that we assume permits explicit determination of every relevant CCR representation. These consist of a two‐parameter family of unitarily inequivalent representations, some of which are irreducible while the others are reducible. It is demonstrated that only those models that are analogs of the free field can be encompassed within the irreducible representations. Hence every model with interaction—including an analog of the relativistic λφ^{4} theory—requires a reducible CCR representation. For the reducible representations, we determine every relevant Hamiltonian operator possessing the required high degree of symmetry. These Hamiltonians, as well as the generators of space translations, cannot be expressed (solely) as functions of the field and momentum operators, which is characteristic of any system with a unique ground state and reducible CCR representation. Nevertheless, it is demonstrated that these Hamiltonians, as well as the generators of space translations, fulfill the ``weak correspondence principle,'' in which the expectation value of a quantum generator, such as the Hamiltonian operator, in a suitably labeled overcomplete family of states is identified with the associated classical generator, such as the classical Hamiltonian. Our principal results depend on the existence and make extensive use of the countably infinite number of degrees of freedom existing in a field theory. Entirely analogous results apply to related models defined in a finite spacial volume since they still have an infinite number of degrees of freedom.

Matrix Representation of the Angular Momentum Projection Operator
View Description Hide DescriptionFormulas are obtained which give the matrix representation, relative to a product basis, of the projection operator for the total angular momentum of a system. If the individual angular momenta are not too large, the matrix elements depend upon a small number of parameters, independent of the number of angular momenta coupled. Recurrence relations between elements and symmetry properties of elements are derived. These results enable one to perform the vector coupling of a large number of angular momenta in a relatively simple fashion. The connection between the matrix elements and vector coupling coefficients is discussed. Several important special cases are treated in detail.

Solution of an Atomic Integral Containing Three Odd Powers of Interelectronic Separation Coordinates
View Description Hide DescriptionA wavefunction expressed as a linear combination of terms each involving only one interelectronic separation coordinate requires the solution of matrix elements of the Hamiltonian which contain three odd powers of the interelectronic separation coordinates. This paper discusses in detail the integration of .

Contributions to the Quantization Problem in General Relativity
View Description Hide DescriptionIt is possible to quantize most classical field theories by identifying the group of canonical transformations which maintain the covariance properties with a group of unitary transformation in Hilbert space which has the same commutator algebra. The commutators among the canonical field variables are equal to the Dirac delta function times a factor which may be zero. But in the general theory of relativity the classical group of the canonical transformations which maintain the covariance properties of the theory has an invariance subgroup. The ambiguities thus introduced by the usual process of quantization can be avoided by the use of the Dirac quantization procedure for theories with constraints. We establish an analogy between classical Dirac brackets and commutators, and fix an intrinsic coordinate system. This choice of local intrinsic coordinate conditions leads to commutators among the canonical field variables of the general theory of relativity which depend upon the Dirac delta function and its first seven derivatives.

On the Complete Unitarity Equations for Pion‐Pion Scattering
View Description Hide DescriptionThe off‐the‐mass‐shell equations for pion‐pion scattering in the lowest approximation of the complete unitarity formalism of Taylor are discussed. It is shown that if the vertex‐function renormalization constant is taken to be zero, the equations have no (nonzero) solution. These equations are equivalent to certain bootstrap equations, which thus also have no solution. Both systems are discussed in the case where the renormalization constant is not put equal to zero.

Threshold Regge Poles for Coupled Channels
View Description Hide DescriptionThe threshold Regge poles are investigated in a variety of many‐channel problems. The cases considered are a simple j‐independent interaction between two spin‐zero channels, the tensor force problem for two spin‐½ particles and a truncated interaction between a particle and a bound system of two particles.

On the Bound‐State Wavefunctions of a Nonlocal Solvable Potential
View Description Hide DescriptionThe Schrödinger equation is solved in the momentum representation with a nonlocal factorable potential, with at most two point eigenvalues. When the potential is a bilinear form in the Bessel transforms of Yukawa functions, we prove that it is impossible to have two bound states whose asymptotic behavior is determined by the binding energy.

On a Set of Coupled Second‐Order Differential Equations
View Description Hide DescriptionAn alternative method for solving a set of coupled second‐order differential equations, which often appears in theoretical treatments of many‐body problems, is proposed. This method makes use of both mathematical relations derived in matrix theory and physical properties of the potentials provided by the set of equations to be solved.

Singular Logarithmic Potentials in Coordinate Space
View Description Hide DescriptionIn previous works we have studied the problem of the determination of the Jost function for singular repulsive potentials behaving near the origin like inverse powers. The key was to define ``new Jost solutions'' which, still being asymptotically ingoing (or outgoing) waves, tend to constant (Jost functions) near the origin. It was shown from the perturbation expansion in coordinate space of these ``new Jost solutions'' that we can construct the Jost functions by connecting the radial coordinate r and the order of the perturbation expansion p. More precisely, if we introduce an r(p) dependence, the Jost function is the limit of convergent sequences provided r(p) goes to zero less rapidly than a given limiting dependence r_{L} (p). In this paper, working in coordinate space, the same method is extended for two other families of singular potentials: firstly, we consider the case in which the most singular part of the potentials behaves like G ^{2}(log r ^{−1})^{β} r ^{−2n }, (n ≥ 1, β arbitrary) near the origin; secondly, we study exponentially singular repulsive potentials of finite range. It is found that the more singular is supposed to be the potential, the higher becomes the available limiting dependence.

Perturbation Expansion for Lattice Thermal Conductivity
View Description Hide DescriptionA method is given for obtaining a perturbationexpansion for the correlation function formula for thermal conductivity. The problem is reduced to the determination of the matrix elements of the products of resolvent operators X_{ll} ′ and Y_{ll} ′ to different orders in the perturbation λH′. The coefficients in the general equations for X_{ll} ′ and Y_{ll} ′ derived by Van Hove, Janner, and Swenson are iterated, and the resulting approximate equations are used to deduce the formulas for the contributions to the lattice thermal conductivity proportional to λ^{−2} (lowest order) and to λ^{−1}.

Lowest‐Order Contribution to the Lattice Thermal Conductivity
View Description Hide DescriptionIn a previous paper the correlation function formula for the thermal conductivity was used to derive equations for the lowest‐order contribution to the lattice thermal conductivityK^{ij} . An explicit but formal solution of these equations is obtained here, and it is shown how this solution simplifies in the limit of infinite volume. Transportequations equivalent to the familiar Boltzmann equation are derived for perturbations describing both anharmonic forces and lattice imperfections. No approximations are made beyond the restriction to lowest order in the perturbation. It is demonstrated that K^{ij} is symmetric.