Holography and SL(2,\bR) symmetry in 2D Rindler spacetime

It is shown that it is possible to define quantum field theory of a massless scalar free field on the Killing horizon of a 2D-Rindler spacetime. Free quantum field theory on the horizon enjoys diffeomorphism invariance and turns out to be unitarily and algebraically equivalent to the analogous theory of a scalar field propagating inside Rindler spacetime, nomatter the value of the mass of the field in the bulk. More precisely, there exists a unitary transformation that realizes the bulk-boundary correspondence under an appropriate choice for Fock representation spaces. Secondly, the found correspondence is a subcase of an analogous algebraic correspondence described by injective *-homomorphisms of the abstract algebras of observables generated by abstract quantum free-field operators. These field operators are smeared with suitable test functions in the bulk and exact 1-forms on the horizon. In this sense the correspondence is independent from the chosen vacua. It is proven that, under that correspondence the ``hidden'' SL(2,\bR) quantum symmetry found in a previous work gets a clear geometric meaning, it being associated with a group of diffeomorphisms of the horizon itself.Comment: Title changed, further minor changes, references added, accepted for publication in J. Math. Phy

Aspects of hidden and manifest SL(2,R) symmetry in 2D near-horizon black-hole background

The invariance under unitary representations of the conformal group SL(2,R) of a quantum particle is rigorously investigated in two-dimensional spacetimes containing Killing horizons using DFF model. The limit of the near-horizon approximation is considered. If the Killing horizon is bifurcate the conformal symmetry is hidden, i.e. it does not arise from geometrical spacetime isometries, but the whole Hilbert space turns out to be an irreducible unitary representation of SL(2,R) and the time evolution is embodied in the unitary representation. In this case the symmetry does not depend on the mass of the particle and, if the representation is faithful, the conformal observable K shows thermal properties. If the Killing horizon is nonbifurcate the conformal symmetry is manifest, i.e. it arises from geometrical spacetime isometries. The SL(2,R) representation which arises from the geometry selects a hidden conformal representation. Also in that case the Hilbert space is an irreducible representation of SL(2,R) and the group conformal symmetries embodies the time evolution with respect to the local Killing time. However no thermal properties are involved. The conformal observable K gives rise to Killing time evolution of the quantum state with respect to another global Killing time present in the manifold. Mathematical proofs about the developed machinery are supplied and features of the operator H_g = -({d^2}/{dx^2})+ ({g}/{x^2}), with g=-1/4 are discussed. It is proven that a statement, used in the recent literature, about the spectrum of self-adjoint extensions of H_g is incorrect.Comment: 22 pages, 1 figure, latex 2e, some misprint corrected, a reference and a footnote adde

Quantitative estimation of the reinforcing effect of layered silicates in PP nanocomposites

Various polypropylene/layered silicate composites were prepared with different silicate contents. Montmorillonites with and without organophilization as well as three maleinated polypropylenes were used to change the extent of exfoliation and hence the properties of the composites. Structure was characterized by X-ray diffraction (XRD), scanning (SEM) as well as transmission electron microscopy (TEM) and tensile properties were also measured. The analysis of the tensile yield stress values of a large number of composites showed a broad range of variation in mechanical properties. XRD and TEM results do not reflect the differences in properties and they usually do not give quantitative information about the extent of exfoliation either. PP/clay composites containing maleinated PP, which do not exhibit a silicate reflection in XRD, may have very poor mechanical properties indicating small extent of exfoliation. The composition dependence of tensile yield stress of these composites may be described and evaluated quantitatively by a simple model developed earlier for particulate filled polymers. The use of a few simple assumptions most of which are supported by previous results allows us to estimate the extent of exfoliation quantitatively. The tensile yield stress of about 40 composites was analyzed with the model. Some of the composites were prepared by us, while results on others were taken from papers published in the literature. The analysis indicated that the extent of exfoliation is very low in most composites; it reaches maximum 8% of the theoretically possible value in the best case. This result is in agreement with our observation that complete exfoliation can be seldom reached in thermoplastic/clay composites; the structure is complex and hierarchical including large particles and individual silicate layers. The results prove that further efforts must be done to increase the extent of exfoliation in order to achieve reinforcement levels forecasted earlier.\ud \u

Morphology Characterization of PP/Clay Nanocomposites Across the Length Scales of the Structural Architecture

The structure and rheological properties of a large number of layered silicate poly(propylene) nanocomposites were studied with widely varying compositions. Morphology characterization at different length scales was achieved by SEM, TEM, and XRD. Rheological measurements supplied additional information about structure. The results showed that these materials possess a very complex structural architecture. The introduction of functionalized polymer decreased the size of the original clay particles and improved their interaction with the polymer matrix. However, relatively large silicate particles were found also in composite samples yielding XRD traces without silicate reflection. XRD supplies information of limited value if the amount of silicate is low, the gallery distance of the stacks is large or their regularity is limited. On the other hand, XRD indicates intercalation well. Although exfoliated individual layers can be detected by TEM, the method cannot be used to draw general conclusions about the structure of layered silicate nanocomposites because of statistical sampling and bias. A large number of individual layers, i.e. large extent of exfoliation, led to the formation of a silicate network structure, which can be detected very sensitively by Cole-Cole plots of dynamic viscosity. We found that all four morphological entities (particles, intercalated stacks, individual layers, network) may be present simultaneously in the composites. The presence and relative amount must definitely influence composite properties. However, currently used experimental protocols do not supply sufficient information even to estimate the relative influence as well as interplay among different structural features quantitatively. We may conclude, however, that XRD and TEM alone are not sufficient for the characterization of nanocomposites with a complex structure