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Volume 10, Issue 11, November 1969

Canonical Transformations and Spectra of Quantum Operators
View Description Hide DescriptionThe claim that unitary transformations in quantum mechanics correspond to the canonical transformations of classical mechanics is not correct. The spectra of operators produced by unitary transformation of Cartesian coordinate and momentum operators (q, k) are necessarily continuous over the entire real domain of their eigenvalues. Operators with spectra which are not everywhere continuous are generated from (q, k) by one‐sided unitary transformations U for which U ^{†} U = 1 but for which UU ^{†} commutes with either q or k (but not both). If UU ^{†} commutes with k, the new coordinates and momenta (r, s) satisfy commutation relations [s_{m}, r_{n} ] = 2πi 1δ_{ m,n }, [s_{m}, s_{n} ] = 0, but [r_{m}, r_{n} ] ≠ 0; (r, s) are canonical only for one‐dimensional systems. The properties of one‐sided unitary transformations are described; they are characterized by φ(K), the eigenvalue of UU ^{†}. The one‐dimensional case for which the one‐sided unitary transformation is canonical is discussed in detail. A prescription is given for obtaining the operator canonically conjugate to any one‐dimensional observable. Generalization to higher dimensions is also discussed.

Canonical Operators for the Simple Harmonic Oscillator
View Description Hide DescriptionAngle and action operators (w, j) for the simple harmonic oscillator are treated as resulting from a canonical transformation of coordinate and momentum operators (q, k) generated by a one‐sided unitary operator U such that U ^{†} U = 1 and UU ^{†} commutes with k but not with q. From the discrete spectrum of the number operator n, eigenvectors η〉 are constructed for every real value of η; the set {η〉} is complete and orthogonal. Another complete set {W〉} is obtained, consisting of the Fourier transforms of the kets in the set {η〉}. The angle operator is The set is not orthogonal; W〉 is not an eigenvector of w. If v is defined as then the creation and destruction operators are given by a = vn ^{½}, a ^{†} = n ^{½} v ^{†}. v is a one‐sided unitary operator such that vv ^{†} = 1, but v ^{†} v = 1 − 0〉 〈0, where 0〉 is the ground state of the oscillator;v and v ^{†} are similar to the operators E _{‐} and E _{+} of Carruthers and Nieto. The Weyl transforms of w and j = 2πℏ(n + ½1) are the classical angle and action variables of the oscillator. The Weyl transform is formulated in terms of the coherent states of the oscillator. A time operator canonical to the Hamiltonian is defined as t = 2πw/ω (ω/2π = frequency). The observables for the oscillator are also given in the Heisenberg picture and their classical limits are considered.

Bounds for Effective Electrical, Thermal, and Magnetic Properties of Heterogeneous Materials
View Description Hide DescriptionDetermining the effective dielectric constant is typical of a broad class of problems that includes effective magnetic permeability, electrical and thermal conductivity, and diffusion. Bounds for these effective properties for statistically isotropic and homogeneous materials have been developed in terms of statistical information, i.e., one‐point and three‐point correlation functions, from variational principles. Aside from the one‐point correlation function, i.e., the volume fraction, this statistical information is difficult or impossible to obtain for real materials. For a broad class of heterogeneous materials (which we shall call cell materials) the functions of the three‐point correlation function that appear in the bounds of effective dielectric constant are simply a number for each phase. Furthermore, this number has a range of values to ⅓ and a simple geometric significance. The number implies a spherical shape, the number ⅓ a cell of platelike shape, and all other cell shapes, no matter how irregular, have a corresponding number between. Each value of this number determines a new set of bounds which are substantially narrower and always within the best bounds in terms of volume fraction alone (i.e., Hashin‐Shtrikman bounds). For dilute suspensions the new bounds are so narrow in most cases as to be essentially an exact solution. There is a substantial improvement over previous bounds for a finite suspension and yet greater improvement for multiphase material where the geometric characteristics of each phase are known.

Bounds for Effective Bulk Modulus of Heterogeneous Materials
View Description Hide DescriptionBounds for the effective bulk modulus for statistically isotropic and homogeneous materials have been developed in terms of statistical information, i.e., one‐point and three‐point correlation function from variational principles. Aside from the one‐point correlation function, i.e., the volume fraction, this statistical information is difficult or impossible to obtain for real materials. For a broad class of heterogeneous materials which we shall call cell materials, the functions of the three‐point correlation function that appear in the bounds of effective bulk modulus are simply a number for each phase. Furthermore, this number has a range of values to ⅓ and a simple geometric significance. The number implies a cell of spherical shape, the number ⅓ a cell of plate‐like shape, and all other cell shapes, no matter how irregular, have a corresponding number between. Each value of this number determines a new set of bounds which are substantially narrower and always within the best bounds for two‐phase media in terms of volume fraction alone (i.e., Hashin‐Shtrikman bounds). For dilute suspensions the new bounds are so narrow in most cases as to be essentially an exact solution. There is a substantial improvement over previous bounds for a finite suspension and yet greater improvement for multiphase materials where the geometric characteristics of each phase are known. The shape factor G is found to have exactly the same range of numerical values and the same geometric significance as was found in the determination of effective dielectric constant bounds. It was found further that under certain conditions the bounds on effective bulk modulus and dielectric constant become numerically identical.

Spectral Representation of the Covariant Two‐Point Function and Infinite‐Component Fields with Arbitrary Mass Spectrum
View Description Hide DescriptionThe general form of a Lorentz covariant two‐point function is written down in momentum space as an expansion in terms of the total spin eigenfunctions. It leads naturally to a local expression for finite‐component fields, but incorporates nonlocal infinite‐component fields. Explicit examples of such fields with increasing mass spectrum are constructed. The two‐point function is shown to fall exponentially for large spacelike separations provided that the lowest mass in the theory is positive.

Perturbation Method for a Nonlinear Wave Modulation. II
View Description Hide DescriptionA perturbation method given in a previous paper of this series is applied to two physical examples, the electron plasma wave and a nonlinear Klein‐Gordon equation. In these systems, and probably in most physical systems, an assumed condition for a mode of l = 0 is not valid. Consequently, the direct application of the method is impossible. In the present paper, we shall illustrate by these examples how this difficulty can be overcome to allow us to use the method. As a result we shall find that, in either case, the original equation can be reduced to the nonlinear Schrödinger equation.

On Canonical SO(4, 1) Transformations of the Dirac Equation
View Description Hide DescriptionCertain matrix transformations of the free‐particle Dirac equation are described as momentum‐dependent SO(4, 1) transformations. Such of these belonging to any one of five subgroups G ^{(α)} (α = 0, 1, 2, 3, 4) are canonical, preserving the Lorentz‐invariant Dirac scalar product in a corresponding one of five modes of expression. The Dirac equation itself is linear in all five components p _{α}[p _{μ} (μ = 0, 1, 2, 3) is the four‐momentum operator, and p _{4} = m] of the ``five‐vector'' p̃, and a transformation in G ^{(β)} has the additional property that the component p _{β} appears linearly also in the transformed equation. The Mendlowitz and the Foldy‐Wouthuysen‐Tani transformation accordingly are in G ^{(0)}, the SO(4) subgroup; and that proposed by Chakrabarti is in G ^{(4)}, the SO(3, 1) subgroup associated with homogeneous Lorentz transformations. For any , obtained from p̃ by a momentum‐dependent SO(4, 1) transformation, there is a corresponding transform of the Dirac equation. Where p _{α} appears in the Dirac equation, appears in the transformed equation. The ambiguities which arise in the specification of the transformation leading to a given such equation are associated with the existence of a ``little group'' for any such .

Internal‐Labeling Problem
View Description Hide DescriptionA method is proposed for labeling the bases of a compact group when reduced with respect to an arbitrary subgroup. The scheme is based on the observation that the heaviest state of a multiplet (subgroup irreducible representation) of an IR (group irreducible representation) can be labeled by a product of heaviest states of simpler ``elementary'' multiplets. Details are worked out for a number of group‐subgroup combinations.

Boson Formalism in Superconductivity
View Description Hide DescriptionAfter a brief summary of the results obtained previously concerning the boson formalism in superconductivity, the formalism is generalized to include finite temperature and Coulomb interactions. Finally, as an illustration, the formalism is used to derive the Landau‐Ginsburg equations and to study the vortices in type II superconductors.

Short‐Range Interactions and Analyticity in Momentum Transfer
View Description Hide DescriptionWe investigate possible characterizations of ``short‐range'' interactions by means of conditions on the total transition probability ∥Tψ∥^{2}. We desire to require that the total transition probability decrease exponentially as the displacement of the wavepacket ψ increases. It is shown that this can only be done for certain types of sequences of wavepackets, and a criterion is developed for selecting these ``proper'' wavepackets. We then show for all such interactions that what is essentially the square of the scattering amplitude is analytic, in the cosine of the scattering angle, in an ellipse which always includes the physical region. We then compare these results with the Schrödinger potential theory and thereby relate the type of the exponential decrease of a potential to the type of the exponential decrease of the corresponding transition probability. Finally, our results are compared with similar results previously obtained by others working on this problem.

Possible Characterization of Short‐Range Interactions
View Description Hide DescriptionIn the preceding paper [J. Math. Phys. 10, 2047 (1969)], we investigated possible characterizations of ``short‐range'' interactions by means of conditions on the total transition probability ∥Tψ∥^{2} which were valid only for certain types of sequences of wavepackets. In this paper we generalize the assumption on the transition probability so that it will apply to an arbitrary wavepacket. We do this by utilizing a measureG[R, ψ] of how much of a wavepacket ψ is within a sphere of radius R about the scattering center at all times. We then characterize a ``short‐range'' interaction by the existence of a positive constant A and positive functions B(p) and C(p) such that for all R ≥ 0,.We then show that for such interactions the square of the scattering amplitude is analytic, in the cosine of the scattering angle, in an ellipse the size of which depends upon B(p). We compare our results with the similar results obtained by the method in the preceding paper and show that these new results much more closely approximate the behavior of the Yukawa interaction.

V‐Particle Decay in the Lee Model
View Description Hide DescriptionThe formalism developed previously for scattering of wavepackets is applied to the N + θ sector of the Lee model. A single analysis suffices to discuss both the stable and unstable case. The rate at which N and θ particles are produced and the number of V particles produced are calculated as a function of time assuming that the incident wave is initially a semi‐infinite, plane‐wave train. An unstable V‐particle state is constructed from the N, θ〉 state by requiring agreement with the scattering analysis. Its dependence on the production process is explicitly shown. The state can be made independent of the production process by requiring normalization. The time dependence of each channel is also calculated for this case. It is shown that both the unstable and stable V‐particle states can be generated from the mathematical V‐particle state, the only difference being the location of the pole which describes the resonance state or the bound state.

Irreducibility of the Ladder Representations of U(2, 2) when Restricted to the Poincaré Subgroup
View Description Hide DescriptionIt is shown that the most degenerate discrete series of unitary irreducible representations of U(2, 2), the so‐called ladder representations, remain irreducible when restricted to representations of the Poincaré subgroup ISL(2, C). They correspond to representations of this subgroup with mass zero and arbitrary integer or half‐integer helicity λ. The basis vectors of the canonical basis are calculated as functions of a lightlike 4‐vector, which is formed by the simultaneous eigenvalues of the generators of the subgroup of translations.

Matrices of Finite Lorentz Transformations in a Noncompact Basis. I. Discrete Series of O(2, 1)
View Description Hide DescriptionWe consider the problem of obtaining the matrices that represent finite group elements in unitary irreducible representations of the group O(2, 1), in a basis in which the ``noncompact'' generator of an O(1, 1) subgroup is diagonal. The discrete series of representations is treated and expressions obtained for the matrix elements of group elements belonging both to the O(2) subgroup and the other O(1, 1) subgroup.

Matrices of Finite Lorentz Transformations in a Noncompact Basis. II. Continuous Series of O(2, 1)
View Description Hide DescriptionThe representation matrices in a noncompact basis for finite elements of the group O(2, 1) are determined in the continuous classes of unitary irreducible representations. Integral as well as half‐integral, and exceptional as well as nonexceptional, representations are treated.

Further Note on Two Binomial Coefficient Identities of Rosenbaum
View Description Hide DescriptionThis note gives additional condition to the finding of Gould on two binomial coefficient identities of Rosenbaum.

On a Generalized Distribution of the Poles of the Unitary Collision Matrix
View Description Hide DescriptionAn expression for the joint distribution of the complex poles of the unitary collision matrix is derived for the single‐channel case, which is valid for all values of the ratio of the width to the spacing. The derivation uses the statistical distribution of the parameters of the real R‐matrix theory. We find that unitarity gives rise to the statistical correlations between the width and the spacing of the collision matrix. It is shown that the distribution of the poles of the unitary collision matrix using Feshbach's unified theory of nuclear reactions is the same as the one obtained using R‐matrix theory, provided we make a particular choice of the arbitrary boundary condition in the latter theory. A remark is made about the use of the random complex orthogonal matrix in the study of the parameters of the statistical collision matrix.

Accidental Degeneracy in the Bethe‐Salpeter Equation
View Description Hide DescriptionThe Bethe‐Salpeter equation for the bound state of the pion‐nucleon system has been studied in the ladder approximation; the propagation time of the exchanged nucleon is neglected. By using the two‐component formalism, the spinor equation is first reduced to a pair of simultaneous integral equations in momentum space. Following Fock, we transform these equations into ones in a four‐dimensional hyperspace and the solutions are obtained in terms of series of O(4) harmonics. As a simple illustration of our method, we have also considered the Bethe‐Salpeter equation for the scalar‐meson system. We find that the pion‐nucleon Bethe‐Salpeter equation shows an accidental degeneracy in the discrete‐energy spectra similar to that in the solutions of the Dirac equation for the hydrogen atom, provided the coupling constant does not exceed a certain critical limit. The scalar problem exhibits at small binding energies a Schrödinger‐type degeneracy. Convergence criteria for the pion‐nucleon Bethe‐Salpeter eigensolutions as series in O(4) harmonics have been discussed; it is found that no solution exists when the coupling constant exceeds a certain critical value. In our approximation scheme, there are no abnormal solutions as are encountered in the fully covariant treatment of the equation.

Theory of Observables
View Description Hide DescriptionAn alternative formulation of a general theory of observables is presented, which contains as special cases the systems proposed by Segal and Mackey. The basic properties are developed, and the exact relations to the aforementioned systems are deduced.

Erratum: Strong‐Coupling Limit in Potential Theory. II
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