Mathematical problems of stochastic quantum mechanics: Harmonic analysis on phase space and quantum geometry

In this paper we review the mathematical methods and problems that are specific to the programme of stochastic quantum mechanics and quantum spacetime. The physical origin of these problems is explained, and then the mathematical models are developed. Three notions emerge as central to the programme: positive operator-valued (POV) measures on a Hilbert space, reproducing kernel Hilbert spaces, and fibre bundle formulations of quantum geometries. A close connection between the first two notions is shown to exist, which provides a natural setting for introducing a fibration on the associated overcomplete family of vectors. The introduction of group covariance leads to an extended version of harmonic analysis on phase space. It also yields a theory of induced group representations, which extends the results of Mackey on imprimitivity systems for locally compact groups to the more general case of systems of covariance. Quantum geometries emerge as fibre bundles whose base spaces are manifolds of mean stochastic locations for quantum test particles (i.e., spacetime excitons) that display a phase space structure, and whose fibres and structure groups contain, respectively, the aforementioned overcomplete families of vectors and unitary group representations of phase space systems of covariance.Work supported in part by the Natural Science and Engineering Research Council of Canada (NSERC) grants.     

Feynman path integrals for polynomially growing potentials

A general class of infinite dimensional oscillatory integrals with polynomially growing phase functions is studied. A representation formula of the Parseval type is proved, as well as a formula giving the integrals in terms of analytically continued absolutely convergent integrals. These results are applied to provide a rigorous Feynman path integral representation for the solution of the time-dependent Schrödinger equation with a quartic anharmonic potential. The Borel summability of the asymptotic expansion of the solution in power series of the coupling constant is also proved.     

Symmetric boundary values for the Dirichlet problem for harmonic maps from the disc into the 2-sphere

Let Aut(D) denote the group of biholomorphic diffeormorphisms from the unit disc D onto itself and O(3) the group of orthogonal transformations of the unit sphere S ². The existence of multiple solutions to the Dirichlet problem for harmonic maps from D into S ² is related to the symmetries (if any) of the boundary value γ : ∂D → S ², by invariance of the Dirichlet energy under the action of Aut(D) × O(3). In this paper, we classify the stabilizers in Aut(D) × O(3) of boundary values in H ^1/2(S ¹, S ²) and . We give two applications to the Dirichlet problem for harmonic maps. This work was partially supported by the CMLA, Ecole Normale Supérieure de Cachan, Cachan, France.     

Lie group action and stability analysis of stationary solutions for a free boundary problem modelling tumor growth

In this paper we study asymptotic behavior of solutions for a free boundary problem modelling tumor growth. We first establish a general result for differential equations in Banach spaces possessing a Lie group action which maps a solution into new solutions. We prove that a center manifold exists under certain assumptions on the spectrum of the linearized operator without assuming that the space in which the equation is defined is of either DA(θ) or DA(θ,∞) type. By using this general result and making delicate analysis of the spectrum of the linearization of the stationary free boundary problem, we prove that if the surface tension coefficient γ is larger than a threshold value γ^* then the unique stationary solution is asymptotically stable modulo translations, provided the constant c is sufficiently small, whereas if γ<γ^* then this stationary solution is unstable.     

Finite-dimensional algebraic representations of the infinite phylon group

The concept of phylon is introduced as a generalisation of derivative strings, differential strings and new tensors. The behaviour of phyla under change of coordinates is given by finite-dimensional algebraic representations of a very large group, the infinite phylon group. These representations are studied from both the general and the matrix points of view. Various examples of phyla are given, mainly from a statistical context. The basic structure of these representations is given.     

Harmonic analysis and systems of covariance for phase space representation of the poincaré group

Continuing some earlier work on the Galilei group, the spectral resolution of phase space representations of the Poincaré group is achieved by deriving all possible decompositions into irreducible representations corresponding to reproducing, kernel Hilbert spaces. Systems of covariance related to quantum measurements performed with extended test particles are analyzed, and questions of global unitarity discussed.Supported in part by NSERC Research Grants.     

Polynomial representations of the symplectic groups

The irreducible finite dimensional representations of the symplectic groups are realized as polynomials on the irreducible representation spaces of the corresponding general linear groups. It is shown that the number of times an irreducible representation of a maximal symplectic subgroup occurs in a given representation of a symplectic group, is related to the betweenness conditions of representations of the corresponding general linear groups. Using this relation, it is shown how to construct polynomial bases for the irreducible representation spaces of the symplectic groups in which the basis labels come from the representations of the symplectic subgroup chain, and the multiplicity labels come from representations of the odd dimensional general linear groups, as well as from subgroups. The irreducible representations of Sp(4) are worked out completely, and several examples from Sp(6) are given.     

Polynomial representations of the orthogonal groups

This paper is concerned with realizations of the irreducible representations of the orthogonal group and construction of specific bases for the representation spaces. As is well known, Weyl's branching theorem for the orthogonal group provides a labeling for such bases, called Gelfand-Žetlin labels. However, it is a difficult problem to realize these representations in a way that gives explicit orthogonal bases indexed by these Gelfand-–etlin labels. Thus, in this paper the irreducible representations of the orthogonal group are realized in spaces of polynomial functions over the general linear groups and equipped with an invariant differentiation inner product, and the Gelfand-Žetlin bases in these spaces are constructed explicitly. The algorithm for computing these polynomial bases is illustrated by a number of examples. Partially supported by a grant from the Department of Energy. Partially supported by NSF grant No. MCS81-02345.     

Spectral and phase space analysis of the linearized non-cutoff Kac collision operator

The non-cutoff Kac operator is a kinetic model for the non-cutoff radially symmetric Boltzmann operator. For Maxwellian molecules, the linearization of the non-cutoff Kac operator around a Maxwellian distribution is shown to be a function of the harmonic oscillator, to be diagonal in the Hermite basis and to be essentially a fractional power of the harmonic oscillator. This linearized operator is a pseudodifferential operator, and we provide a complete asymptotic expansion for its symbol in a class enjoying a nice symbolic calculus. Related results for the linearized non-cutoff radially symmetric Boltzmann operator are also proven.     

Semilinear representations of PGL

Let L be the function field of a projective space over an algebraically closed field k of characteristic zero, and H be the group of projective transformations. An H-sheaf on is a collection of isomorphisms for each g ∈ H satisfying the chain rule. We construct, for any n > 1, a fully faithful functor from the category of finite-dimensional L-semilinear representations of H extendable to the semigroup End(L/k) to the category of coherent H-sheaves on The paper is motivated by a study of admissible representations of the automorphism group G of an algebraically closed extension of k of countable transcendence degree undertaken in [4]. The semigroup End(L/k) is considered as a subquotient of G, hence the condition on extendability. In the appendix it is shown that, if is either H, or a bigger subgroup in the Cremona group (generated by H and a certain pair of involutions), then any semilinear of degree one is an integral L-tensor power of It is also shown that this bigger subgroup has no non-trivial representations of finite degree if n > 1.     

Every group is the automorphism group of a rank-3 matroid

Every group is the automorphism group of a rank-3 extension of a rank-3 Dowling geometry.Partially supported by The George Washington University UFF grant.Partially supported by the National Security Agency under grant MDA904-91-H-0030.     

Zeros of Racah coefficients and the pell equation

The interface between Racah coefficients and mathematics is reviewed and several unsolved problems pointed out. The specific goal of this investigation is to determine zeros of these coefficients. The general polynomial is given whose set of zeros contains all nontrivial zeros of Racah (6j) coefficients [this polynomial is also given for the Wigner-Clebsch-Gordan (3j) coefficients]. Zeros of weight 1 3j- and 6j-coefficients are known to be related to the solutions of classic Diophantine equations. Here it is shown how solutions of the quadratic Diophantine equation known as Pell's equation are related to weight 2 zeros of 3j- and 6j-coefficients. This relation involves transformations of quadratic forms over the integers, the orbit classification of zeros of Pell's equation, and an algorithm for determining numerically the fundamental solutions of Pell's equation. The symbol manipulation program MACSYMA was used extensively to effect various factorings and transformations and to give a proof.The results of this paper were presented in an invited talk by one of us (JDL) at the NSF-CBMS Regional Conference on Special Functions, Physics and Computer Algebra, May 20–24, 1985, Arizona State University, Tempe, AZ.     

Multivariate lag-windows and group representations

Symmetries of the auto-cumulant function (a generalization of the auto-covariance function) of a kth-order stationary time series are derived through a connection with the symmetric group of degree k. Using the theory of group representations, symmetries of the auto-cumulant function are demystified and lag-window functions are symmetrized to satisfy these symmetries. A generalized Gabr–Rao optimal kernel is also derived through the developed theory.     

Quasiregular representations of the infinite-dimensional Borel group

The notion of quasiregular (Representation of Lie groups, Nauka, Moscow, 1983) or geometric (Grundlehren der Mathematischen Wissenschaften, Band 220, Springer, Berlin, New York, 1976; Encyclopaedia of Mathematical Science, Vol. 22, Springer, Berlin, 1994, pp. 1-156) representation is well known for locally compact groups. In the present work an analog of the quasiregular representation for the solvable infinite-dimensional Borel group is constructed and a criterion of irreducibility of the constructed representations is presented. This construction uses G-quasi-invariant Gaussian measures on some G-spaces X and extends the method used in Kosyak (Funktsional. Anal. i Prilo?hen 37 (2003) 78-81) for the construction of the quasiregular representations as applied to the nilpotent infinite-dimensional group .     

Stationary phase method in infinite dimensions by finite dimensional approximations: Applications to the Schrödinger equation

We develop an approach by finite dimensional approximations for the study of infinite dimensional oscillatory integrals and the relative method of stationary phase. We provide detailed asymptotic expansions in the nondegenerate as well as in the degenerate case. We also give applications to the derivation of detailed asymptotic expansions in Planck's constant for the Schrödinger equation.     

Relations between the Clausen and Kazhdan-Lusztig representations of the symmetric group

We use Kazhdan-Lusztig polynomials and subspaces of the polynomial ring C[x_1,1,…,x_n,n] to give a new construction of the Kazhdan-Lusztig representations of S_n. This construction produces exactly the same modules as those which Clausen constructed using a different basis in [M. Clausen, Multivariate polynomials, standard tableaux, and representations of symmetric groups, J. Symbolic Comput. (11), 5-6 (1991) 483-522. Invariant-theoretic algorithms in geometry (Minneapolis, MN, 1987)], and does not employ the Kazhdan-Lusztig preorders. We show that the two resulting matrix representations are related by a unitriangular transition matrix. This provides a C[x_1,1,…,x_n,n]-analog of results due to Garsia and McLarnan, and McDonough and Pallikaros, who related the Kazhdan-Lusztig representations to Young’s natural representations.     

A comparison result for elliptic equations in the infinite dimensional Gauss space

We obtain a comparison result for a class of Dirichlet problems for the operator in an infinite dimensional separable Hilbert space X with the Gauss measure γ and a suitable differentiable structure.     

The regularity problem for generalized harmonic maps into homogeneous spaces

Let be a Riemannian surface and be a standard sphere, or more generally a Riemannian manifold on which a Lie group,, acts transitively by isometries. We define generalized harmonic maps by extending the notion of weakly harmonic maps in a natural way (motivated by Noether's Theorem), to mapsu W _loc ^1,1 (, ). We prove that, under some slight technical restrictions, for 1 <-p < 2, there are generalized harmonic mapsu W ^1,p(, ) that are everywhere discontinuous (in particular, this solves an open problem proposed by F. Bethuel, H. Brezis and F. Hélein, in [BBH]). We also show that the natural -regularity condition for such maps is to require <u to belong to the Lorentz space L(2, ). To prove this -regularity result we extend a compensated compactness result of R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes, proved in [CLMS], to the case of Lorentz spaces in duality.     

Killing helices on a symmetric space of rank one

In this paper, we study helices which are orbits of one parameter families of isometries on a symmetric space of rank one. We introduce the notion of structure torsion fields for helices, show the necessary and sufficient condition that they are generated by some Killing vector fields, and study their moduli space. The author is partially supported by Grant-in-Aid for Scientific Research (C)(No. 14540075), Ministry of Education, Science, Sports, Culture and Technology.