Volume 7, Issue 1, January 1966
Index of content:

Scattering by the Singular Potential r ^{−4}
View Description Hide DescriptionThe potential 1/r ^{4} is the only known potential more singular than the centrifugal term, for which the Schrödinger equation can be solved exactly. In the present investigation, we consider the potential in more detail than has been done so far. In particular the physical S‐matrix is obtained, shown to be unitary, and compared with expressions of other derivations given in the literature. The eigenvalues of the Mathieu equation are finally discussed, and the behavior of the Regge trajectories is indicated.

Local Approximation in Renormalizable and Nonrenormalizable Theories. I
View Description Hide DescriptionWe present a framework of quantum field theory which is wide enough to incorporate renormalizable, as well as nonrenormalizable theories. A universal high‐energy bound for matrix elements of fields is derived. The ``string'' approximation of the two‐point function for various couplings is studied.

Local Approximations in Renormalizable and Nonrenormalizable Theories. II
View Description Hide DescriptionThe investigations of a previous paper are generalized to two‐point matrix elements. A principle is formulated, which yields unique finite Feynman rules in the renormalizable case, i.e., permits a unique separation of counterterms. For nonrenormalizable theories this principle yields uniqueness up to a ``scaling'' parameter. The results are generalized to a large class of Feynman graphs. For this subset of graphs, field‐theoretical principles do not determine this scaling parameter.

On Series Expansion of a Determinant and Solution of the Secular Equation
View Description Hide DescriptionA determinant and its cofactors are expanded in terms of the cyclic products of its elements. With the aid of this expansion, an implicit equation for the eigenvalue and an explicit equation for amplitudes of the corresponding eigenfunction are obtained, respectively, in terms of a ratio of two simple series expansions. Comparisons with Feenberg's perturbation formula and with Sasakawa's perturbation method are also discussed.

Distribution of Zeros of the Grand Partition Function
View Description Hide DescriptionThe distribution of zeros of the grand partition function is calculated in the thermodynamic limit for a class of one‐dimensional gas models in two ways: (1) from the equation of state and (2) directly from the partition function. In this way one obtains (for these cases) a verification of the assumptions we had to make in order to associate a unique distribution of zeros with a given equation of state. In the Appendix we present some numerical evidence for the validity of these assumptions also in the case of the van der Waals gas.

Approximation by Sequences of Padé Approximants in Regions of Meromorphy
View Description Hide DescriptionIt is shown that sequences {P_{mn} (z)} of Padé approximants with m → ∞ and n → ∞ approximate a function f(z) inside a certain circle of meromorphy with centre at z = 0. Extensions of this result are discussed.

Complex Space‐Time and Classical Field Theory. I
View Description Hide DescriptionThis is the first of the series of three papers which introduces complex space‐time to describe physical phenomena. The objective of this generalization is twofold: firstly, to geometrize gauge transformations and electromagnetic fields, and secondly, to quantize space‐time in order to remove serious divergences from the field theory. In this paper classical fields are discussed in complex space‐time with a view of subsequent generalization to quantum field theory in quantized space‐time.

The Quantized Complex Space‐Time and Quantum Theory of Free Fields. II
View Description Hide DescriptionIn this paper a covariant quantization of complex space‐time is proposed. As a consequence of this quantization each of the four real coordinates can take discrete values n ^{½} l, and furthermore, measurements of these coordinates are noninterfering with each other. Next the general theory of quantized free fields is developed in the background of quantized space‐time. As an example the case of complex scalar field has been dealt with and it is found that the resulting Green's functions are nonsingular.

Complex Space‐Time and Geometrization of Electromagnetism. III
View Description Hide DescriptionIt is shown here that if we demand the covariance under general circular transformations (rotations in complex planes), the electromagnetic potentials can be geometrized as the affine connections necessary for such covariance. The geodesic hypothesis yields the Lorentzequation of motion with an interesting correction term.

Asymptotic Behavior of the Scattering Amplitude in the Left Half‐Plane of Complex Angular Momentum
View Description Hide DescriptionIt is shown, for a certain class of analytic potentials, that the previously reported result,where the sign ± is chosen such that ‐ π/2 ≤ arg ± (λ/k) ≤ π/2, is valid also for Re λ < 0 except for a discrete set of points on the negative real axis. The result holds for arbitrary complex k ≠ 0, finite or infinite.

Mathematical Aspects of the Weyl Correspondence
View Description Hide DescriptionThe Weyl correspondence between classical and quantum observables is rigorously formulated for a linear mechanical system with a finite number of degrees of freedom. A multiplication of functions and a *‐operation are introduced to make the Hilbert space of Lebesgue square‐integrable complex‐valued functions on phase space into a H*‐algebra. The Weyl correspondence is realized as a *‐isomorphism f → W(f) of this H*‐algebra onto the H*‐algebra of Hilbert‐Schmidt operators on the Hilbert space of Lebesgue square‐integrable complex‐valued functions on configuration space. Moreover, the kernel of W(f) is exhibited in terms of a Fourier‐Plancherel transform of f. Elementary properties of the Wigner quasiprobability density function and its characteristic function are deduced and used to obtain these results.

Physical Region Decoupling Procedure for Coupled Channel Scattering Amplitudes
View Description Hide DescriptionA method for expressing unitary, coupled channel, scattering amplitudes in terms of amplitudes satisfying ``uncoupled'' or ``elastic'' unitarity equations is presented. The method is given a physical interpretation by relating the equations to equations of the Heitler type. It is shown how continuous channels may be incorporated into the formalism, in some circumstances, without requiring the solution of integral equations. The influence of inelastic channels on pure elasticscattering is mentioned briefly and a pseudoelastic form of the exact unitarity equation is discussed. No applications of the method are undertaken here.

Relation between the Onsager and Pfaffian Methods for Solving the Ising Problem. II. The General Lattice
View Description Hide DescriptionThe considerations of a previous paper are extended to the case of a general lattice. It is shown algebraically why the Pfaffian and Onsager methods of solution of the Ising problem coincide only when the lattice is planar, and that the problem is then a linear one. When the lattice is nonplanar the Pfaffian method breaks down due to the appearance of unwanted negative signs, and it is shown how the Onsager method compensates for this at the expense of making the problem nonlinear.

Asymptotic Behavior of Phase Shift in λ for Strongly Singular Potentials
View Description Hide DescriptionThe asymptotic form of the phase shift is derived for strongly singular potentials for large complex λ, arg λ < ½π, and real k. This result is valid also for usual (regular) potentials. It is shown also that the S‐matrix for strongly singular potentials must have an infinite number of poles in λ‐plane accumulating asymptotically in a narrow region along the imaginary axis in the first and third quadrants.

Rigorous Treatment of the Van Der Waals‐Maxwell Theory of the Liquid‐Vapor Transition
View Description Hide DescriptionRigorous upper and lower bounds are obtained for the thermodynamic free‐energy density a(ρ, γ) of a classical system of particles with two‐body interaction potential q(r) + γ^{ν}φ(γr) where ν is the number of space dimensions and ρ the density, in terms of the free‐energy density a ^{0}(ρ) for the corresponding system with φ(x) ≡ 0. When φ(x) belongs to a class of functions, which includes those which are nonpositive and those whose ν‐dimensional Fourier transforms are nonnegative, the upper and lower bounds coincide in the limit γ → 0 and lim_{γ → 0} a(ρ, γ) is the maximal convex function of ρ not exceeding a ^{0}(ρ) + ½αρ^{2}, where α ≡ ∫ φ(x) dx. The corresponding equation of state is given by Maxwell's equal‐area rule applied to the function p ^{0}(ρ) + ½αρ^{2} where p ^{0}(ρ) is the pressure for φ(x) ≡ 0. If a ^{0}(ρ) + ½αρ^{2} is not convex the behavior of the limiting free energy indicates a first‐order phase transition. These results are easily generalized to lattice gases and thus apply also to Ising spin systems.
The two‐body distribution function is found, in the limit γ → 0, to be normally identical with that for φ(x) ≡ 0, but if the system has a phase transition it has the form appropriate to a two‐phase system.
Some of the upper and lower bounds on a(ρ, γ) are simple enough to be useful for finite γ. Also, some of our results remain valid for quantum systems.

Representation of Discrete Symmetry Operators
View Description Hide DescriptionRepresentations of discrete symmetry operators (DSO's) connected with space (P), time (T), and generalized charge (C) are considered. It is shown that if one writes a DSO as exp (iπΩ) × a phase transformation, then (under certain conditions on Ωs) to each DSO there corresponds a set of Ωs which is closed with respect a Lie algebra, which is isomorphic to the Lie algebra of generators of rotation in an n‐dimensional Euclidean space; where n is the number of commuting observables that changes sign under this DSO in a given representation (e.g. linear momentum representation). In the particular case of the (TCP) operation, there are six Ωs, of which two are diagonal, viz. the generalized charge Q, and spin projection along the z axis Sz; corresponding Euclidean group is four‐dimensional. For the sake of completeness, the representations are also given for the following cases: (i) nonrelativistic quantum mechanics, (ii) quantum theory of free fields, in terms of field operators.

Interacting Fermions in One Dimension. II. Attractive Potential
View Description Hide DescriptionThe exact energies and wavefunctions for a system of (N − 1) one‐dimensional fermions all of the same spin and one fermion of the opposite spin are calculated in the large volume, finite density limit, when the particles interact via an attractive delta function potential. It is found that the attractive potential gives rise to a bound state, but, in spite of the presence of this bound state, all of the physical properties which are calculated (ground‐state energy,effective mass of a certain class of excitations, etc.) are analytic continuations in the coupling constant of the corresponding results in the repulsive case. In addition, it is possible to have eigenstates which do not have the bound state present. These excited states are also discussed and are found to exhibit a negative effective mass and to modify the particle density at very large distances from the different particle.

Some Theorems on the Unimodular Complex Degree of Optical Coherence
View Description Hide DescriptionThe complex degree of coherence γ(r_{1}, r_{2}, t _{1} − t _{2}) of a stationary optical field is defined as the normalized cross‐correlation function (γ ≤ 1) of the light disturbances at two space‐time points (r_{1}, t _{1}), (r_{2}, t _{2}), the disturbances being represented by means of Gabor's analytic signals.
In the present paper the general form of γ is examined under the condition that γ takes on the extreme value unity for all possible time differences τ = t _{1} − t _{2} (− ∞ < τ < ∞). Several cases are distinguished, depending on whether this condition is satisfied for some points or for all points in some fixed domain of space. It is pointed out, that the previously published derivations of the relevant theorems contain serious errors. The methods employed in the present paper make use of the property of nonnegative definiteness which the complex degree of coherence is shown to obey. The results have a bearing on the important but as yet unsolved ``phase problem'' of optical coherencetheory.

Application of Houston's Method to the Sum of Plane Waves over the Brillouin Zone. I. Simple‐Cubic and Face‐Centered‐Cubic Lattices
View Description Hide DescriptionIn certain problems in solid‐state physics, the radial functions g_{j} (r) in the expansion χ(r) = Σ_{ j=0} ^{∞} g_{j} (r)K_{j} (θ, φ), where χ(r) is a known function and the K_{j} 's are Kubic Harmonics, are of interest. This paper deals with the functions χ(r) ≡ N ^{−1} Σ_{k} e ^{ ik·r}, where the sum runs over the first Brillouin Zone of a crystal. In particular, the functions χ(r) for simple cubic and face‐centered cubic lattices are expanded into series of Kubic Harmonics and the radial functions g_{j} (r) for several values of j are found using Houston's method, in which the expansion into series of Kubic Harmonics contains only a finite number of terms with lowest j's. g _{0}(r) is calculated using 3, 6, and 9‐term expansion, g _{2}(r) and g _{3}(r) using only 3 and 6‐term expansion. Comparing g_{j} (r) obtained from the formulas with different numbers of terms it is established that for r in the region 〈0, 2a〉, where a is the lattice constant, the 6‐term approximation is very good. In practice, the functions g_{j} (r) usually occur in integrands, together with atomic orbitals, and the tabulated results are expected to be particularly useful in the study of Wannier functions in the OPW scheme.

On a Quantum Field Theory with Degenerate Vacuum and a Special Type of Symmetry
View Description Hide DescriptionWe investigate a frame of a model of quantum field theory with a degenerate vacuum. At least one vacuum is cyclic. The field is covariant under a ``gauge'' transformation. We show the existence of non‐gauge‐invariant vacuum states. We define ``observables'' as gauge‐invariant operators; the algebra of these does not coincide with the algebra of the field operator; the reduction of the former algebra reflects a superselection rule.