Volume 27, Issue 3, March 1986
Index of content:

On exomorphic types of phase transitions
View Description Hide DescriptionAn algorithmic method is presented to determine the irreducible representations that engender the irreducible representations associated with phase transitions involving a change of symmetry to a subgroup of index n. This method is based on the work of Ascher and Kobayashi [E. Ascher and J. Kobayashi, J. Phys. C 1 0, 1349 (1977)] and the derivation of faithful irreducible representations contained in the permutation representation of transitive subgroups of permutation groups S _{ n }. Character tables of all such irreducible representations, and their epikernels, associated with a change in symmetry to a subgroup of index n=2, 3, 4, 5, and 6 are given explicitly. The relationship to exomorphic types of phase transitions is then discussed. The irreducible representations associated with the phase transitions O ^{1} _{ h } to C ^{1} _{4v } in BaTiO_{3} and D ^{4} _{6h } to D ^{16} _{2h } in β‐K_{2}SO_{4} are derived and it is shown that these two phase transitions belong to the same exomorphic type.

Verma bases for representations of classical simple Lie algebras
View Description Hide DescriptionComplete bases are constructed for all finite‐dimensional irreducible representations of the simple Lie algebras over C of the types A _{ n } (n≥1), B _{ n } and C _{ n } (2≤n≤6), D_{ n } (4≤n≤6), and G _{2}. Each basis vector is given as an explicit sequence of weight‐lowering generators of the algebra acting on the highest weight vector of the representation space. A similar construction (due to D‐N. Verma) for the highest weight representations of all Kac–Moody algebras of rank 2 is presented as well.

Pure Lie algebraic approach to the modified Korteweg–de Vries equation
View Description Hide DescriptionIt is explained, using only Lie algebraic means, how the modified‐KdV‐like equations arise. As an example, the modified‐KdV equation is treated.

Some integrals involving three modified Bessel functions. I
View Description Hide DescriptionThe integrals ∫^{∞} _{0} t ^{1+μ} I _{μ}(a t)K _{ν} (b t)K _{ν}(c t)d t and ∫^{∞} _{0} t ^{1−ν} I _{ν}(a t)K _{ν} (b t)K _{ν}(c t)d t are calculated with the help of the factorization properties of the Appell function F _{4}. Results are given for real parameters a, b, c, both when they are and are not in a triangle configuration.

Some integrals involving three modified Bessel functions. II
View Description Hide DescriptionThe integrals ∫^{∞} _{0} Z _{μ}(a t)K _{ν}(b t) iK_{ρ}(c t)d t, where Z _{μ}=I _{μ}, K _{μ}, are calculated, with the help of the factorization properties of the function F _{4}. Results are given for real parameters a, b, c both when they are and are not in a triangle configuration. Some generalizations using derivation with respect to the parameters are considered.

On the linearization problem for ultraspherical polynomials
View Description Hide DescriptionA direct proof of a formula established by Bressoud in 1981 [D. M. Bressoud, SIAM J. Math. Anal. 1 2, 161 (1981)], equivalent to the linearization formula for the ultraspherical polynomials, is given. Some related results are briefly discussed.

Fractional approximation to the vacuum–vacuum amplitude of a φ^{4}‐potential theory in zero dimensions
View Description Hide DescriptionHere, the vacuum–vacuum amplitude with a φ^{4}‐potential in terms of the fractional approximation to the partition function of a zero‐dimensional field theory is presented. This fractional approximation has been obtained from both the power series and the asymptotic expansion. The power series diverges, nonetheless the fractional approximations are excellent. All the approximations from first to seventh degree are presented, with maximum errors from 0.6% to 1.6×10^{−} ^{5}%, respectively.

On a particular transcendent solution of the Ernst system generalized on n fields
View Description Hide DescriptionA particular solution, a function of a particular form of the fifth Painlevé transcendent, of the Ernst system generalized to n fields is determined, which characterizes both the stationary axially symmetric fields, the solution of the Einstein (n−1) Maxwell equations, and one class of axially symmetric static self‐dual SU(n+1) Yang–Mills fields.

To the complete integrability of long‐wave–short‐wave interaction equations
View Description Hide DescriptionIt is shown that the nonlinear partial differential equations governing the interaction of long and short waves are completely integrable. The methodology used is that of Ablowitz e t a l. [M. J. Ablowitz, A. Ramani, and A. Segur, Lett. Nuovo Cimento 2 3, 333 (1980); M. J. Ablowitz, A. Ramani, and H. Segur, J. Math. Phys. 2 1, 715, 1006 (1980)], though in the last section of our paper the problem also has been discussed in the light of the procedure due to Weiss e t a l. [J. Weiss, M. Tabor, and G. Carnevale, J. Math. Phys. 2 4, 522 (1983)] and a Backlünd transformation has been obtained.

Realizing the Berezin integral as a superspace contour integral
View Description Hide DescriptionIntegration on supermanifolds, using contours and the covariant differential forms of Kostant [B. Kostant, ‘‘Graded manifolds, graded Lie theory and prequantisation,’’ in L e c t u r e N o t e s i n M a t h e m a t i c s, Vol. 570 (Springer, Berlin, 1977)], is described; the good properties that these integrals naturally acquire are considered. It is then shown that the formal process of integration over even and odd variables introduced by Berezin [F. A. Berezin, T h e M e t h o d o f S e c o n d Q u a n t i z a t i o n (Academic, New York, 1966)] using partly covariant and partly contravariant forms can be regarded as a special case of these contour integrals over covariant forms.

Resolving the singularities in the space of Riemannian geometries
View Description Hide DescriptionA method is described for unfolding the singularities of superspace, G=M/D, the space of Riemannian geometries of a manifoldM. This extended, or unfolded superspace, is described by the projection G_{ F(M)}=(M×F(M))/D→M/D=G, where F(M) is the frame bundle of M. The unfolded space G_{ F(M)} is an infinite‐dimensional manifold without singularities. Moreover, as expected, the unfolding of G_{ F(M)} at each geometry [ g _{0}]∈G is parametrized by the isometry group I _{ g 0 }(M) of g _{0}. The construction is completely natural, gives complete control and knowledge of the unfolding at each geometry necessary to make G_{ F(M)} a manifold, and is generally covariant with respect to all coordinate transformations. A similar program is outlined, based on the methods of this paper, of desingularizing the moduli space of connections on a principal fiber bundle.

Effective determinism in a classical field theory with spacelike characteristics
View Description Hide DescriptionFor certain classical field theories, it appears as if spacelike characteristic hypersurfaces can occur in the solutions. In principle, this means that acausal propagation is allowed, and the Cauchy problem breaks down. To investigate if indeed this happens, a class of Kasner‐like solutions of a generalized Einstein–Maxwell theory is examined. It is found that the spacelike characteristics almost never form, and when they do, they cause no trouble.

The structure of the space of homogeneous Yang–Mills fields
View Description Hide DescriptionThe structure of the space of solutions of Yang–Mills equations is examined for solutions that are required to have a specified set of infinitesimal space‐time symmetries. It is shown that when the set consists of Killing vector fields, which are tangent to a compact spacelike Cauchy surface, the space is a smooth ILH manifold near each solution that has only trivial gauge symmetries.

Ermakov and non‐Ermakov systems in quantum dissipative models
View Description Hide DescriptionVia the hydrodynamical formulation of quantum mechanics, a unified protocol to treat the quantum time‐dependent harmonic oscillator with friction is presented, described by two different models: an explicitly time‐dependent, linear Schrödinger equation (Caldirola–Kanai model) and a logarithmic nonlinear Schrödinger equation (Kostin model). For the former model, an Ermakov system that makes it possible to obtain an invariant of Ermakov–Lewis‐type is derived. For the latter model, a non‐Ermakov system is derived instead and it is shown that neither an exact nor an approximate invariant of Ermakov–Lewis‐type exists.

On information gain by quantum measurements of continuous observables
View Description Hide DescriptionA generalization of Shannon’s amount of information into quantum measurements of continuous observables is introduced. A necessary and sufficient condition for measuring processes to have a non‐negative amount of information is obtained. This resolves Groenewold’s conjecture completely including the case of measurements of continuous observables. As an application the approximate position measuring process considered by von Neumann and later by Davies is shown to have a non‐negative amount of information.

Evaluation of a class of integrals that arise in certain path integrals
View Description Hide DescriptionA commonly encountered n‐dimensional integral associated with a relativistic quadratic Lagrangian is explicitly evaluated for arbitrary n. In the limit n→∞, this integral is given by the usual Van Vleck–Morette determinant. The main advantage of the present approach is that it is simple and direct.

Path integration of the time‐dependent forced oscillator with a two‐time quadratic action
View Description Hide DescriptionUsing the prodistribution theory proposed by DeWitt‐Morette [C. DeWitt‐Morette, Commun. Math. Phys. 2 8, 47 (1972); C. DeWitt‐Morette, A. Maheshwari, and B. Nelson, Phys. Rep. 5 0, 257 (1979)], the path integration of a time‐dependent forced harmonic oscillator with a two‐time quadratic action has been given in terms of the solutions of some integrodifferential equations. We then evaluate explicitly both the classical path and the propagator for the specific kernel introduced by Feynman in the polaron problem. Our results include the previous known results as special cases.

Time‐ordering techniques and solution of differential difference equation appearing in quantum optics
View Description Hide DescriptionTime‐ordering techniques based on the Magnus expansion and the Wei–Norman algebraic procedure are discussed and their relevance and usefulness to quantum optics are stressed.

A unified treatment of Wigner D functions, spin‐weighted spherical harmonics, and monopole harmonics
View Description Hide DescriptionA unified, self‐contained treatment of Wigner D functions, spin‐weighted spherical harmonics, and monopole harmonics is given, both in coordinate‐free language and for a particular choice of coordinates.

Screen observables in relativistic and nonrelativistic quantum mechanics
View Description Hide DescriptionScreen observables, which measure the arrival coordinates of free particles at a hyperplane containing a timelike direction, are defined by a covariance property with respect to an irreducible representation of the Poincaŕe or Galilei group. For each representation with m>0 the set of screen observables is constructed explicitly and a unique ideal screen observable of greatest intrinsic accuracy is singled out.