Showing papers in "Journal of Mathematical Physics in 1990"

Local symmetries and constraints

Abstract: The general relationship between local symmetries occurring in a Lagrangian formulation of a field theory and the corresponding constraints present in a phase space formulation are studied. First, a prescription—applicable to an arbitrary Lagrangian field theory—for the construction of phase space from the manifold of field configurations on space‐time is given. Next, a general definition of the notion of local symmetries on the manifold of field configurations is given that encompasses, as special cases, the usual gauge transformations of Yang–Mills theory and general relativity. Local symmetries on phase space are then defined via projection from field configuration space. It is proved that associated to each local symmetry which suitably projects to phase space is a corresponding equivalence class of constraint functions on phase space. Moreover, the constraints thereby obtained are always first class, and the Poisson bracket algebra of the constraint functions is isomorphic to the Lie bracket algebra of the local symmetries on the constraint submanifold of phase space. The differences that occur in the structure of constraints in Yang–Mills theory and general relativity are fully accounted for by the manner in which the local symmetries project to phase space: In Yang–Mills theory all the ‘‘field‐independent’’ local symmetries project to all of phase space, whereas in general relativity the nonspatial diffeomorphisms do not project to all of phase space and the ones that suitably project to the constraint submanifold are ‘‘field dependent.’’ As by‐products of the present work, definitions are given of the symplectic potential current density and the symplectic current density in the context of an arbitrary Lagrangian field theory, and the Noether current density associated with an arbitrary local symmetry. A number of properties of these currents are established and some relationships between them are obtained.

Noncommutative differential geometry of matrix algebras

Abstract: The noncommutative differential geometry of the algebra Mn (C) of complex n×n matrices is investigated. The role of the algebra of differential forms is played by the graded differential algebra C(sl(n,C),Mn (C))=Mn (C)⊗Λsl(n,C)*,sl(n,C) acting by inner derivations on Mn (C). A canonical symplectic structure is exhibited for Mn (C) for which the Poisson bracket is, to within a factor i, the commutator. Also, a canonical Riemannian structure is described for Mn (C). Finally, the analog of the Maxwell potential is constructed and it is pointed out that there is a potential with a vanishing curvature that is not a pure gauge.

Spherically symmetric static SU(2) Einstein–Yang–Mills fields

Abstract: The discrete family of global solutions of the static spherically symmetric SU(2) Einstein–Yang–Mills equations that were recently numerically obtained by Bartnik and McKinnon [Phys. Rev. Lett. 6 1, 141 (1988)] is studied in greater detail, both numerically and analytically. A similar discrete sequence of numerical solutions outside a regular event horizon is shown to exist for every radius of the horizon.

Noncommutative differential geometry and new models of gauge theory

Abstract: The noncommutative differential geometry of the algebra C∞(V)⊗Mn(C) of smooth Mn(C)‐valued functions on a manifold V is investigated. For n≥2, the analog of Maxwell’s theory is constructed and interpreted as a field theory on V. It describes a U(n)–Yang–Mills field minimally coupled to a set of fields with values in the adjoint representation that interact among themselves through a quartic polynomial potential. The Euclidean action, which is positive, vanishes on exactly two distinct gauge orbits, which are interpreted as two vacua of the theory. In one of the corresponding vacuum sectors, the SU(n) part of the Yang–Mills field is massive. For the case n=2, analogies with the standard model of electroweak theory are pointed out. Finally, a brief description is provided of what happens if one starts from the analog of a general Yang–Mills theory instead of Maxwell’s theory, which is a particular case.

Infinite Dimensional Algebras and a Trigonometric Basis for the Classical Lie Algebras

Abstract: This paper explores features of the infinite‐dimensional algebras that have been previously introduced. In particular, it is shown that the classical simple Lie algebras (AN, BN, CN, DN) may be expressed in an ‘‘egalitarian’’ basis with trigonometric structure constants. The transformation to the standard Cartan–Weyl basis, and the particularly transparent N→∞ limit that this formulation allows is provided.

On identically closed forms locally constructed from a field

Abstract: Let M be an n‐dimensional manifold with derivative operator ∇a and let B(M) be an arbitrary vector bundle over M, equipped with a connection. A cross section of B defines a field φ on M. Let α be a p‐form on M (with p

Covariant quantization of gauge theories in the framework of extended BRST symmetry

Abstract: The quantization rules for gauge theories in the Lagrangian formalism are formulated on the basis of the requirement of an extended BRST symmetry. The independence of the S matrix to the choice of a gauge is proved. The Ward identities are derived, and the existence theorem for the solutions of the generating equations within the given formalism is proved. Rank 1 gauge theories are considered as an example.

Integral bounds for radar ambiguity functions and Wigner distributions

Abstract: An upper bound is proved for the L p norm of Woodward’s ambiguity function in radar signal analysis and of the Wigner distribution in quantum mechanics when p >2. A lower bound is proved for 1 ≤p < 2. In addition, a lower bound is proved for the entropy. These bounds set limits to the sharpness of the peaking of the ambiguity function or Wigner distribution. The bounds are best possible and equality is achieved in the L P bounds if and only if the functions/ and g that enter the definition are both Gaussians.

Three-dimensional quantum groups from contractions of SU(2) q

Abstract: Contractions of Lie algebras and of their representations are generalized to define new quantum groups. An explicit and complete exposition is made for the one‐dimensional Heisenberg H(1)q and the two‐dimensional Euclidean quantum group E(2)q obtained by contracting SU(2)q.

Solvable Lie algebras of dimension six

Abstract: Six‐dimensional solvable Lie algebras over the field of real numbers that possess nilradicals of dimension four are classified into equivalence classes. This completes Mubarakzyanov’s classification of the real six‐dimensional solvable Lie algebras.

Tight frames of compactly supported affine wavelets

Abstract: This paper extends the class of orthonormal bases of compactly supported wavelets recently constructed by Daubechies [Commun. Pure Appl. Math. 41, 909 (1988)]. For each integer N≥1, a family of wavelet functions ψ having support [0,2N−1] is constructed such that {ψjk(x)=2j/2ψ(2jx−k) kj,k∈Z} is a tight frame of L2(R), i.e., for every f∈L2(R), f=c∑jk 〈ψjk‖f〉ψjk for some c>0. This family is parametrized by an algebraic subset VN of R4N. Furthermore, for N≥2, a proper algebraic subset WN of VN is specified such that all points in VN outside of WN yield orthonormal bases. The relationship between these tight frames and the theory of group representations and coherent states is discussed.

Exact model for a relativistic star

Abstract: Assuming that the physical three‐space in a relativistic superdense star has the geometry of a three‐spheroid, a static spherically symmetric model based on an analytic closed‐form solution of Einstein’s field equations is presented. Assuming the density of the order of 2×1014 g cm−3, estimates of the total mass and size of the stars of the model are obtained for various values of a density‐variation parameter that is suitably defined. The total mass and the boundary radius of each of these models are of the order of the mass and size of a neutron star.

Character formulas for irreducible modules of the Lie superalgebras sl(m/n)

Abstract: Kac distinguished between typical and atypical finite‐dimensional irreducible representations of the Lie superalgebras sl(m/n) and provided an explicit character formula appropriate to all the typical representations. Here, the range of validity of some character formulas for atypical representations that have been proposed are discussed. Several of them are of the Kac–Weyl type, but then it is proved that all formulas of this type fail to correctly give the character of one particular atypical representation of sl(3/4). Having ruled out, therefore, all such formulas, a completely new extension of the Kac–Weyl character formula is proposed. The validity of this formula in the case of all covariant tensor irreducible representations is proved, and some evidence in support of the conjecture that it covers all irreducible representations of sl(m/n) is presented.

The potential group approach and hypergeometric differential equations

Abstract: This paper proposes a generalized realization of the potential groups SO(2,1) and SO(2,2) to describe the confluent hypergeometric and the hypergeometric equations, respectively. It implies that the classes of Schrodinger equations with solvable potentials whose analytical solutions are related to the confluent hypergeometric and the hypergeometric functions can be realized in terms of the above group structure.

Quantum mechanics as an infinite‐dimensional Hamiltonian system with uncertainty structure: Part II

Abstract: Making reference to the formalism developed in Part I to formulate Schrodinger quantum mechanics, the properties of Kahlerian functions in general, almost Kahlerian manifolds, are studied.

Extended BRST quantization of gauge theories in the generalized canonical formalism

Abstract: The rules of canonical quantization of gauge theories are formulated on the basis of the extended BRST symmetry principle. The existence of solutions of the generating equations of the gauge algebra is proved. Equivalence between the extended BRST quantization and the standard method of generalized canonical quantization is established. Ward identities corresponding to invariance of a theory under the extended BRST symmetry are obtained.

Reliability of perturbation theory in general relativity

Abstract: The relation between perturbation theory and exact solutions in general relativity is tackled by investigating the existence and properties of smooth one‐parameter families of solutions. On the one hand, the coefficients of the Taylor expansion (in the parameter) of any given smooth family of solutions necessarily satisfy the hierarchy of perturbation equations. On the other hand, it is the converse question (does any solution of the perturbation equations come from Taylor expanding some family of exact solutions ?) which is of importance for the mathematical justification of the use of perturbation theory. This converse question is called the one of the ‘‘reliability’’ of perturbation theory. Using, and completing, recent results on the characteristic initial value problem, the local reliability of perturbation theory for general relativity in vacuum is proven very generally. This result is then generalized to the Einstein–Yang–Mills equations (and therefore, in particular, to the Einstein–Maxwell ones). These local results are then partially extended to global ones by: (i) proving the existence of semiglobal vacuum space‐times (respectively, Einstein–Yang–Mills solutions) which are stationary before some retarded time u0, and radiative after u0, and which admit a smooth conformal structure at future null infinity; and (ii) constructing smooth one‐parameter families of such solutions whose Taylor expansions are of the ‘‘multipolar post‐Minkowskian’’ type which has been recently used in perturbation analyses of radiative space‐times.

Some aspects of quantum groups and supergroups

Abstract: Some features of Manin’s construction of quantum groups are developed and extended to supergroups.

Physical constraints on the coefficients of Fourier expansions in cylindrical coordinates

Abstract: It is demonstrated that (i) the postulate of infinite differentiability in Cartesian coordinates and (ii) the physical assumption of regularity on the axis of a cylindrical coordinate system provide significant simplifying constraints on the coefficients of Fourier expansions in cylindrical coordinates. These constraints are independent of any governing equations. The simplification can provide considerable practical benefit for the analysis (especially numerical) of actual physical problems. Of equal importance, these constraints demonstrate that if A is any arbitrary physical vector, then the only finite Fourier terms of A_r and A_θ are those with m=1 symmetry. In the Appendix, it is further shown that postulate (i) may be inferred from a more primitive assumption, namely, the arbitrariness of the location of the cylindrical axis of the coordinate system.

A posterior Schrödinger equation for continuous nondemolition measurement

Abstract: A continuous model for a nondemolition observation of an atom is given. An equation for the corresponding instrument is found and a stochastic dissipative Schrodinger equation for the unnormalized posterior wave function of the atom is derived. It is shown that the continuously observed isolated atom relaxes to the ground state without mixing.

The gravitational field of plane symmetric thick domain walls

Abstract: Exact solutions of Einstein’s equations for a scalar field with a potential V(Φ) =V0 cos2(1−n) (Φ/f(n)) (0

Conformal transformations and the effective action in the presence of boundaries

Abstract: The conformal properties of the heat kernel expansion are used to determine the local form of the coefficients in a manifold with boundary The conformal transformation of the effective action is obtained A novel derivation of the boundary term in the Gauss–Bonnet–Chern theorem is detailed

Singular anharmonicities and the analytic continued fractions. II. The potentials V(r)=ar2+br−4+cr−6

Abstract: The c=0 results of Paper I [J. Math. Phys. 30, 23 (1989)] are extended. In spite of the presence of an additional coupling constant, the Laurent series solutions of the Schrodinger equation that are obtained remain similar to Mathieu functions. Indeed, the recurrences for coefficients preserve their three‐term character, their analytic continued fraction solutions still converge, etc. The formulas become even slightly simpler for c≠0 due to a certain symmetry of the equations to be solved. An acceleration of convergence is better understood and a few numerical illustrations of efficiency are also delivered.

Conformally symmetric static fluid spheres

Abstract: Solutions of the Einstein–Maxwell equations for static spheres of charged imperfect fluids are investigated, where the space‐time geometry is assumed to admit a conformal symmetry. Previous work is generalized by considering a nonstatic conformal symmetry. This allows the possibility of solutions that are nonsingular at the center, unlike the previous solutions based on a static conformal symmetry. Two such regular solutions are presented for charged spheres. The further generalization necessary to find stable exact stellar models with a conformal symmetry is indicated.

The Duistermaat-Heckman integration formula on flag manifolds

Abstract: An exposition is given of various geometrical properties of flag manifolds and of the Duistermaat–Heckman integration formula as applied to flag manifolds.

Variational and perturbative schemes for a spiked harmonic oscillator

Abstract: A variational analysis of the spiked harmonic oscillator Hamiltonian operator −d2/ dx2+x2+l(l+1)/x2+λ‖x‖−α , where α is a real positive parameter, is reported in this work. The formalism makes use of the functional space spanned by the solutions of the Schrodinger equation for the linear harmonic oscillator Hamiltonian supplemented by a Dirichlet boundary condition, and a standard procedure for diagonalizing symmetric matrices. The eigenvalues obtained by increasing the dimension of the basis set provide accurate approximations for the ground state energy of the model system, valid for positive and relatively large values of the coupling parameter λ. Additionally, a large coupling perturbative expansion is carried out and the contributions up to fourth‐order to the ground state energy are explicitly evaluated. Numerical results are compared for the special case α=5/2.

The free Dirac operator on compact and noncompact graphs

Abstract: The free Dirac operator defined on composite one‐dimensional structures consisting of finitely many half‐lines and intervals is investigated. The influence of the connection points between the constituents is modeled by transition conditions for the wave functions or equivalently by different self‐adjoint extensions of the Dirac operator. General relations between the parameters of the extensions and the eigenvalues resp. the scattering coefficients are derived and then applied to the cases of a bundle of half‐lines, a point defect, a branching line, and an eye‐shaped structure.

Space‐time geometry of relativistic particles

Abstract: A three‐dimensional space‐time geometry of relativistic particles is constructed within the framework of the little groups of the Poincare group. Since the little group for a massive particle is the three‐dimensional rotation group, its relevant geometry is a sphere. For massless particles and massive particles in the infinite‐momentum limit, it is shown that the geometry is that of a cylinder and a two‐dimensional plane. The geometry of a massive particle continuously becomes that of a massless particle as the momentum/mass becomes large. The geometry of relativistic extended particles is also considered. It is shown that the cylindrical geometry leads to the concept of gauge transformations, while the two‐dimensional Euclidean geometry leads to a deeper understanding of the Lorentz condition.

An Sp(2)‐covariant version of generalized canonical quantization of dynamical systems with linearly dependent constraints

Abstract: An Sp(2)‐covariant version of the method of generalized canonical quantization of dynamical systems with linearly dependent first‐class constraints is proposed. The existence theorem for solutions of generating equations of a gauge algebra is proved and the natural arbitrariness in these solutions is described. The scheme proposed is shown to be equivalent to the standard version of generalized canonical quantization.

Conserved quantities and symmetries of KP hierarchy

Abstract: Conserved quantities and symmetries of the KP equation from the point of view of the Sato theory that provides a unifying approach to soliton equations is studied. Conserved quantities are derived from the generalized Lax equations. Some reductions of the KP hierarchy such as KdV, Boussinesq, a coupled KdV, and Sawada–Kotera equation are also considered. By expansion of the squared eigenfunctions of the Lax equations in terms of the τ function, symmetries of the KP equations are obtained. The relationship of this procedure to the two‐dimensional recursion operator newly found by Fokas and Santini is discussed.