A new approach to harmonic analysis on the group SL(2,R)

The Gelfand-Graev horospheric approach to the Fourier transform on SO(n,1) is confined to Class I representations; we extend their method to the regular representation of the group $\text{SL(2,}\text{R}\text{)}$. The role of the cone is played by the manifold of 2×2 real singular matrices, upon which the representation and decomposition theory is much simplified. The discrete series is connected with the appearance of associated homogeneous functions on this manifold; in the inversion formula expressing ?(g) in terms of its components this is related to the presence of double poles in the measure of the continuous series. We examine in some detail the relations between equivalent formulations of the discrete series.     

Fourier analysis on multiplier representations of locally compact Abelian groups

Generalising known results [2] for vector groups, it is shown that, for an arbitrary multiplier ω for an arbitrary locally compact Abelian group G, there is a faithful normal semifinite trace on the von Neuman algebra generated by the regular ω-representation of G which is translation invariant in a certain sense. Analogues of the Fourier transformation, the Plancherel identity and the Fourier inversion formula are obtained in which this trace replaces the Haar measure on the dual group.     

Symmetry breaking at enhanced symmetry points

The influence of world-sheet boundary condensates on the toroidal compactification of bosonic string theories is considered. At the special points in the moduli space at which the closed-string theory possesses an enhanced unbroken G × G symmetry (where G is a semi-simple product of simply laced groups) a scalar boundary condensate parameterizes the coset G × G/G. Fluctuations around this background define an open-string generalization of the corresponding chiral non-linear sigma model. Tree-level scattering amplitudes of on-shell massless states (‘pions’) reduce to the amplitudes of the principal chiral model for the group G in the low-energy limit. Furthermore, the condition for the vanishing of the renormalization group beta function at one loop results in the familiar equation of motion for that model. The quantum corrections to the open-string theory generate a mixing of open and closed strings so that the coset-space pions mix with the closed-string G × G gauge fields, resulting in a Higgs-like breakdown of the symmetry to the diagonal G group. The case of non-oriented strings is also discussed.     

Lifting classes of principal bundles

Given a principal G-bundle P over X, we define a particularly suitable equivalence relation between liftings of P with respect to a group morphism σ:M→G. Supposing that σ has a central kernel C, we obtain a free operation of H¹(X$\text{C}$) (with coefficients in the sheaf of C-valued functions) on the set of lifting classes of P, which is natural under change of groups and base spaces. It is simply transitive, if in addition σ is an epimorphism; otherwise we classify its orbits by sections in the associated bundle P × _G(G/σM).For $\text{C=}\text{Z}{}_{\mathrm{n}}$ we relate the lifting classes to similar classes of n-th roots of associated line bundles. In the differentiable case and for an epimorphism σ with discrete kernel, there is a natural lifting of partial principal connections in each of these lifting classes. Finally, we indicate some applications to geometric quantization.     

Optical computations of cosine transforms

Utilizing the symmetrical properties of cosine transform, a coherent optical Fourier transform system has been modified to compute cosine transforms optically. The modification involves creating a symmetrical pattern at the input. Cosine transform is known to be useful for data compression and image coding especially when the image space-bandwidth product is finite. It is also attractive even when the image size becomes large but is not statistically stationary; in such case it should be processed in (M × M) stationary blocks of finite size.     

Electrical and dielectric properties of Al/HfO2/p-Si MOS device at high temperatures

The temperature dependence of capacitance–voltage (C–V) and conductance–voltage (G/w–V) characteristics of Al/HfO₂/p-Si metal-oxide-semiconductor (MOS) device has been investigated by considering the effect of series resistance (R_s) and interface state density (N_ss) over the temperature range of 300–400 K. The C–V and G/w–V characteristics confirm that the N_ss and R_s of the diode are important parameters that strongly influence the electric parameters in MOS device. It is found that in the presence of series resistance, the forward bias C–V plots exhibits a peak, and its position shifts towards lower voltages with increasing temperature. The density of N_ss, depending on the temperature, was determined from the (C–V) and (G/w–V) data using the Hill–Coleman Method. Also, the temperature dependence of dielectric properties at different fixed frequencies over the temperature range of 300–400 K was investigated. In addition, the electric modulus formalisms were employed to understand the relaxation mechanism of the Al/HfO₂/p-Si structure.     

The mass of the scalar boson beyond the large- N c limit

Within the framework of the 1/N_c expansion of four-fermion interaction models, we analyse the next to leading 1/N_c corrections to the well known large-N _c result M _S = 2M _Q where M _S is the mass of the scalar boson and M _Q is the constituent quark mass. The calculation is performed in the Extended Nambu-Jona Lasinio (ENJL) model which is suitable for describing low energy hadron properties. We treat the model as fully non renormalizable and discuss the comparison with approaches based on the equivalence with renormalizable Yukawa type models. We consider both the G _V = 0 and the G _V ≠ 0 cases with n _f = 2 flavours and study the dependence upon the regularization scheme. We find that pure next-to-leading 1/N _c corrections are large and negative, while a partially resummed treatment can induce positive and smaller corrections. A triplet-singlet states’ splitting is observed.     

Geometry of multidimensional universes

LetG be a compact group of transformation (global symmetry group) of a manifoldE (multidimensional universe) with all orbits of the same type (one stratum). We studyG invariant metrics onE and show that there is one-to-one correspondence between those metrics and triples (g _μv,A _μ ^ä ,h _αβ), whereg _μv is a (pseudo-) Riemannian metric on the space of orbits (space-time),A _μ ^ä is a Yang-Mills field for the gauge groupN|H, whereN is the normalizer of the isotropy groupH inG, andh _αβ are certain scalar fields characterizing geometry of the orbits (internal spaces). The scalar curvature ofE is expressed in terms of the component fields onM. Examples and model building recipes are also given. The results generalize those of non-abelian Kaluza-Klein theories to the case where internal spaces are not necessarily group manifolds.     

The structure of Heesch groups and its relation to material property tensors

The magnetic crystal point groups (Heesch groups) are classified according to their structure with respect to the three inversion operations: space, time, and total inversion. Accordingly the tensors are classified by the irreducible representations of the full inversion group. The groups and tensors are considered under the action of the elements A_i of the group of automorphisms of the full inversion group. The following correspondence theorem is proved: The matrix form of the tensor representation T of the group G coincides with the matrix form of the representation A_iT of the group A_iG. The theorem gives a clear explanation of the so-called “magic numbers” and provides a suitable short cut for the calculation and tabulation of material property tensors.     

GA/GV measured in the decay of polarized neutrons

A measurement of [G_A/G_V] in the decay of polarized neutrons, basically quite similar to our 1968 one, gives a value of 1.258±0.015. The asymmetry coefficient, A, is ?0.113±0.006. These results include the 1968 data.     

The formation and decomposition kinetics of alkylidynes on Pt(111)

The formation and decomposition kinetics of ethylidyne and propylidyne on Pt(111), were studied using static secondary ion mass spectrometry and temperature programmed desorption. For the maximum amounts of dissociatively adsorbed ethylene and propylene formed during adsorption at 200 K and subsequent temperature programmed desorption, the following activation energies (E) and pre-exponential factors (A) are determined: (a) for ethylidyne formation: E = 17±1 kcal mol⁻¹ and A = 1×10^12±1 s⁻¹; (b) ethylidyne decomposition: E = 27 ±2 kcal mole⁻¹ and A = 6×10^11±1 s⁻¹; (c) propylidyne formation: E = 17.5 ± 2 kcal mol⁻¹ and A = 7×10^12±1 s⁻¹; and (d) propylidyne decomposition: E = 22.5±2 kcal mol⁻¹ and A = 4×10^{11 ± 1} s⁻¹.     

Magnetic behaviour of cobalt sponges in cemented carbides

The measurement and analysis of magnetisation curves, including anhysteretic and initial curves of heterogeneous and anisotropic materials (sintered tungsten carbides of constant grain size containing 3–25 wt% Co and Co-rich alloys) are presented. Néel's law of approach to saturation magnetisation for ferromagnetics containing non-magnetic inclusions was compared with the results from an empirical Lamont-Frölich-Kennelly equation and the standard law of approach to saturation magnetisation. Internal demagnetising factors N_if, where f refers to the magnetic fraction obtained from inverse anhysteretic susceptibilities (H ≈ 0) X_af^-1 were compared with those from high-field inverse susceptibilities A_f (LFK equati on) and calculated high-field demagnetising factors a/M_sf (Néel theory); all obtained as a function of volume fraction V of WC. A_f and a/M_sf both increase to a maximum at V ≈ 0.9 and then decrease. A_f is twice the magnitude of a/M_sf at the maximum, having a smaller difference at lower V. A_f is also twice the magnitude of N_if being an increasing function of applied field and is interpreted to be a high-field demagnetising factor. Assuming the Néel equation to account for voids, the LFK equation is interpreted to measure the effects of voids, of demagnetisation by hard directions in hcp Co (at low V) and of demagnetising fields in Bloch walls in very thin Co films at high V. The Néel equation was also fitted directly to the results, showing that the theory derived for isotropic materials is inadequate in dealing with large magnetocrystalline anisotropies. A relationship between H_c, j_r (remanence ratios) and N_if for presintered (pressed powder) and sintered specimens with a range of Co contents and WC grain sizes was established.     

Experimental value of G A/GV from a measurement of both P-odd correlations in free-neutron decay

A new experimental value of the fundamental weak-interaction parameter λ=G _A/G_V (?1.2686±0.0046) is obtained for the first time by an original method that consists in measuring both P-odd correlations in free-neutron decay.     

Loop and Path Spaces and¶Four-Dimensional BF Theories:¶Connections, Holonomies and Observables

We study the differential geometry of principal G-bundles whose base space is the space of free paths (loops) on a manifold M. In particular we consider connections defined in terms of pairs (A,B), where A is a connection for a fixed principal bundle P(M,G) and B is a 2-form on M. The relevant curvatures, parallel transports and holonomies are computed and their expressions in local coordinates are exhibited. When the 2-form B is given by the curvature of A, then the so-called non-abelian Stokes formula follows. For a generic 2-form B, we distinguish the cases when the parallel transport depends on the whole path of paths and when it depends only on the spanned surface. In particular we discuss generalizations of the non-abelian Stokes formula. We study also the invariance properties of the (trace of the) holonomy under suitable transformation groups acting on the pairs (A,B). In this way we are able to define observables for both topological and non-topological quantum field theories of the BF type. In the non-topological case, the surface terms may be relevant for the understanding of the quark-confinement problem. In the topological case the (perturbative) four-dimensional quantum BF-theory is expected to yield invariants of imbedded (or immersed) surfaces in a 4-manifold M. Received: 28 March 1998 / Accepted: 12 September 1998     

Higher order field equations II; Limit properties of green's functions

In a previous paper wave propagation was studied according to a sixth-order partial differential equation involving a complex mass M. The corresponding Yang-Feldman integral equations (indicated as SM-YF-equations), were formulated using modified Green's functions G^M_R(x) and G^M_A(x), which then incorporate the partial differential equation together with certain boundary conditions. In this paper certain limit properties of these modified Green's functions are derived: (a) It is shown that for |M| → ∞ the Green's functions G^M_R(x) and G^M_A(x) approach the Green's functions Δ_R(x) and Δ_A(x) of the corresponding KG-equation (Klein-Gordon equation). (b) It is further shown that the asymptotic behaviour of G^M_A(x) and G^M_A(x) is the same as of Δ_R(x) and Δ_A(x) - and also the same as for D_R(x) and D_A(x) for t→ ± ∞, where D_R and D_A are the Green n's functions for the KG-equation with mass zero. It is essential to take limits in the sense of distribution theory in both cases (a) and (b). The property (b) indicates that the wave propagation properties of the SM-YF-equations, the KG-equation with finite mass and the KG-equation with mass zero are closely related in an asymptotic sense.     

On Certain Kähler Quotients of Quaternionic Kähler Manifolds

We prove that, given a certain isometric action of a two-dimensional Abelian group A on a quaternionic Kähler manifold M which preserves a submanifold N ? M, the quotient M′ = N/A has a natural Kähler structure. We verify that the assumptions on the group action and on the submanifold N ? M are satisfied for a large class of examples obtained from the supergravity c-map. In particular, we find that all quaternionic Kähler manifolds M in the image of the c-map admit an integrable complex structure compatible with the quaternionic structure, such that N ? M is a complex submanifold. Finally, we discuss how the existence of the Kähler structure on M′ is required by the consistency of spontaneous ${\mathcal{N} = 2}$ to ${\mathcal{N} = 1}$ supersymmetry breaking.     

A grand superspace for unified field theories

Agrand superspace is proposed as the phase space for gauge field theories with a fixed structure groupG over a fixed space-time manifoldM. This superspace incorporatesall principal fiber bundles with these data. This phase space is the space of isomorphism classes ofall connections onall G-principal fiber bundles overM (fixedG andM). The justification for choosing this grand superspace for the phase space is that the space-time and the structure group are determinants of the physical theory, but the principal fiber bundle with the givenG andM is not. Grand superspace is studied in terms of a natural universal principal fiber bundle overM, canonically associated withM alone, and with a natural universal connection on this bundle. This bundle and its connection are universal in the sense that all connections on allG-principal fiber bundles (anyG) overM can be recovered from this universal bundle and its universal connection by a canonical construction. WhenG is Abelian, grand superspace is shown to be an Abelian group. Various subspaces of grand superspace consisting of the isomorphism classes of flat connections and of Yang-Mills connections are also discussed.     

Matter fields and metric deformations in multidimensional unified theories

This paper describes a first study of the effects due to including matter fields in generalized Kaluza-Klein (KK) theories with nonabelian compact gauge group G and nontrivial fibres $\text{V}$^K. The approach is based on the first-order Einstein-Cartan (EC) general relativity in (4 + K) dimensions. In the EC theory there are two basic mechanisms which can lead to a spontaneously compactified KK background geometry R⁴ × $\text{V}$^K: (A) a particular kind of energy-momentum density matter condensate in the quantized ground state, or (B) a particular kind of spin-density matter condensate. If (A) or (B) are operating, the inconsistencies usually found between the KK ansatz and the matter-free EC theory are avoided. Mechanism (B) works only when $\text{V}$^K is parallelizable. It is shown that the expansion of matter fields in normal modes on $\text{V}$^K implies that one must include deformations of the Yang-Mills (YM) potentials contained in the usual KK metrics. We discuss and characterize one class of such deformations. As a case study, we consider fibres $\text{V}$^K ～ G′, where G′ is a semisimple compact Lie group. We allow for the “maximal” YM gauge group G_L′ × G_R′. We carry out the harmonic analysis for spinor fields and study the mass spectrum and YM quantum numbers of the normal modes. We rely on mechanism (B) to provide a curvature-free connection (“parallelization”) on $\text{V}$^KG′ by means of a suitable vertical constant torsion. Minimal YM couplings are of size $\text{l}\text{L}\text{≡ g}$, where l is the Planck length and L is the length of the fibre; nonminimal YM couplings are of size L. Nonzero masses are of size L^?1. The massless modes are found and discussed. There would be no massless modes if the parallelizing vertical torsion were absent. This torsion also implies the vanishing of the cosmological constant. When the theory is restricted to massless modes, the YM deformations disappear and the dimensional reduction to four dimensions yields an effective YM theory, which is renormalizable at energies far below L^?1: the effective theory is obtained by letting L → 0 with g ? 1 fixed and by neglecting all masses of order L^?1; g corresponds to the bare YM coupling constant. The surviving effective YM gauge group is G_L′ and the matter fields are in a particular representation of G_L′ × G_R′, corresponding to the zero mass eigenvalue. Explicit examples are discussed for G′ = SU(2) and G′ = SU(3).     

Group actions on quantum bundles

Suppose we are given a group G acting through canonical transformations on a symplectic manifold (M, ω). If there is a quantum bundle over (M, ω), a carrier for wave functions in the geometric quantization theory, then G acts infinitesimally on the bundle in a natural way. We give a necessary and sufficient condition for the infinitesimal G-action to integrate up to a global G-action. This is used for an investigation how the choice of the quantum bundle over (M, ω) influences the integrability of the corresponding infinitesimal G-action. The relationship to group representations is briefly mentioned.