Two-point functions and renormalized observables

The coefficient of the Rmu nu Rmu nu -1/3R2 part of the conformal anomaly of the stress tensor for arbitrary spin Weinberg-type fields in a curved background spacetime is calculated. The method used involves obtaining terms in the De-Witt-Schwinger expansion for the two-point function by explicit calculation of the complete two-point function in de Sitter space.

In the upper and lower quadrants of flat spacetime, (\ifmmode\pm\else\textpm\fi{}T>\ensuremath{\Vert}X\ensuremath{\Vert}) it is possible to introduce a system of coordinates in which the metric is explicitly time dependent. The conventional interpretation of quantum field theory, based on a Fock basis, cannot be implemented in a natural fashion in such coordinates. This difficulty suggests that quantum theory is covariant only for a subset of the coordinate transformations which are allowed in the classical theory.

The stress tensor is expressed in terms of an arbitrary symmetric tensor field of second rank and the couple-stress tensor. The stress and couple-stress tensors are represented by arbitrary tensor fields satisfying the homogeneous equilibrium equations. These tensors are also given in the form of the expressions satisfying the inhomogeneous equilibrium equations used in the microcontinuum theory of elasticity. The stress tensor functions are considered.

Traditional concept of force fields is based on Newton's mechanics and defines the dynamics of various physical systems. It is shown that it is equivalent to the use of the corresponding metric of an anisotropic space. This is a base of the geometric approach describing the motions of a physical system. Such approach makes it possible to get rid of several known paradoxes; it could be also used for the further development of the theory. Examples from classical mechanics, hydrodynamics, electrodynamics, quantum mechanics and gravitation theory are given.

We discuss the slow relaxation phenomenon in glassy systems by means of replicas by constructing a static field theory approach to the problem. At the mean field level we study how criticality in the four point correlation functions arises because of the presence of soft modes and we derive an effective replica field theory for these critical fluctuations. By using this at the gaussian level we obtain many physical quantities: the correlation length, the exponent parameter that controls the mode-coupling dynamical exponents for the two-point correlation functions, and the prefactor of the critical part of the four point correlation functions. Moreover, we perform a one-loop computation in order to identify the region in which the mean field gaussian approximation is valid. The result is a Ginzburg criterion for the glass transition. We define and compute in this way a proper Ginzburg number. Finally, we present numerical values of all these quantities obtained from the hypernetted chain approximation for the replicated liquid theory.

Using the counterterm subtraction technique we calculatehe stress-energy tensor, action, and other physical quantities for Kerr-AdS black holes in various dimensions. For Kerr-AdS_5 with both rotation parameters non-zero, we demonstrate that stress-energy tensor, in the zero mass parameter limit, is equal to the stress tensor of the weakly coupled four dimensional dual field theory. As a result, the total energy of the generalKerr-AdS_5 black hole at zero mass parameter, exactly matches the Casimir energy of the dual field theory. We show that at high temperature, the general Kerr-AdS_5 and perturbative field theory stress-energy tensors are equal, up to the usual factor of 3/4. We also use the counterterm technique to calculate the stress tensors and actions for Kerr-AdS_6, and Kerr-AdS_7 black holes, with one rotation parameter, and we display the results. We discuss the conformal anomalies of the field theories dual to the Kerr-AdS_5 and Kerr-AdS_7 spacetimes. In these two field theories, we show that the rotation parameters break conformal invariance but not scale invariance, a novel result for a non-trivial field theory. For Kerr-AdS_7 the conformal anomalies calculated on the gravity side and the dual (0,2) tensor multiplet theory are equal up to 4/7 factor. We expect that the Casimir energy of the free field theory is the same as the energy of the Kerr-AdS_7 black hole (with zero mass parameter), up to that factor.

Studies the topology of 2nd-order symmetric tensor fields. Degenerate points are basic constituents of tensor fields. From the set of degenerate points, an experienced researcher can reconstruct a whole tensor field. We address the conditions for the existence of degenerate points and, based on these conditions, we predict the distribution of degenerate points inside the field. Every tensor can be decomposed into a deviator and an isotropic tensor. A deviator determines the properties of a tensor field, while the isotropic part provides a uniform bias. Deviators can be 3D or locally 2D. The triple-degenerate points of a tensor field are associated with the singular points of its deviator and the double-degenerate points of a tensor field have singular local 2D deviators. This provides insights into the similarity of topological structure between 1st-order (or vectors) and 2nd-order tensors. Control functions are in charge of the occurrences of a singularity of a deviator. These singularities can further be linked to important physical properties of the underlying physical phenomena. For a deformation tensor in a stationary flow, the singularities of its deviator actually represent the area of the vortex core in the field; for a stress tensor, the singularities represent the area with no stress; for a Newtonian flow, compressible flow and incompressible flow as well as stress and deformation tensors share similar topological features due to the similarity of their deviators; for a viscous flow, removing the large, isotropic pressure contribution dramatically enhances the anisotropy due to viscosity.

Recently it has been shown that it is possible to retain the Lorentz-invariant interpretation of the non-commutative field theory.1,2,3 This was achieved by the means of the twisted action of the Poincaré group on the tensor product of the fields. We investigate the consequences of this approach for the quantized fields.

We generalize, to any space-time dimension, the unitarity bounds of highest weight UIR's of the conformal groups with Lie algebras $so(2,d)$. We classify gauge theories invariant under $so(2,d)$, both integral and half-integral spins. A similar analysis is carried out for the algebras $so^*(2n)$. We study new unitary modules of the conformal algebra in $d>4$, that have no analogue for $d\leq 4$ as they cannot be obtained by "squaring" singletons. This may suggest the interpretation of higher dimensional non-trivial conformal field theories as theories of "tensionless" $p$-branes of which tensionless strings in $d=6$ are just particular examples.

The modal interpretation is actually a family of interpretations sharing the feature that a system’s density operator constrains the possibilities for assigning definite values to its observables (Healey 1989, Dieks & Vermaas 1998). We start with a brief review of the basic idea behind these interpretations from an algebraic point of view (Zimba & Clifton 1998, Halvorson & Clifton 1999). Consider a ‘universe’ comprised of a quantum system U, with finitely many degrees of freedom, represented by a Hilbert space Hu = ®i Hi. At any given time, U will occupy a pure vector state x E Hu that determines a reduced density operator Ds on the Hilbert space Hi of any subsystem S U. Let B(Hs) de­

Mathai–Quillen Formalism

Reordering of baryon chiral perturbation theory

We define a pure state associated with every sufficiently regular state of linear quantum field theory. For a scalar field on a globally hyperbolic spacetime, the energy relative to a timelike vector field defines a quasifree state, whose associated pure state is a natural vacuum of the scalar field. If the energy state is defined relative to a globally timelike complete Killing field, its associated pure state is the well known `frequency splitting' stationary vacuum state. It is proved that such vacua corresponding to two commuting Killing fields are equal. It is conjectured that the vacua are equal even if the Killing fields do not commute. Implications for the particle interpretation of quantum field theory in curved spacetimes are discussed.

Metaphor’s peculiar property to yield cognitive insight — often in otherwise false sentences — has been the focus of contemporary studies of metaphor. In Metaphor: Its Linguistic Structure and Cognitive Force, Kittay (1987) attempts to understand the cognitive force of metaphor by utilizing semantic field theory — a theory that assimilates conceptual structure to a semantic structure underlying the utterances we produce. According to semantic field theory, the meaning of a word is, in part, a function of its relation to words occupying the same conceptual terrain. For example, “red” takes its meaning, in part, from the other color terms in the language, terms with which it has the semantic relation of co-hyponym. Similarly, the meaning of “hot,” is determined, in part, in contrast to other terms in a gradable antonymy, that is, in contrast to “cold,” “warm,” and “cool.” The meaning of terms is also a function, in part, of the words with which they can collocate. For example, “fisherman” stands in the relation of AGENT to the verb “fish.” That is, at least part of what it means to be a fisherman is to be one who engages in the action of fishing. Furthermore, semantic field theory proposes that the lexicon is organized along conceptual affinities and contrasts reflected in the semantic relations that structure fields.

We present a visualization method for the exploration of three-dimensional tensor fields. The representation of the tensor field on a one-parameter family of two-dimensional surfaces as stretched, compressed and bent piece of fabric reflects the physical properties of stress and strain tensor fields. The texture parameters as the fiber density and fiber direction are controlled by tensor field. The surface family is defined as a set of isosurfaces extracted from an additional scalar field. This field can be a “connected” scalar field, for example, pressure or a scalar field representing some symmetry or inherent structure of the dataset. The texture generation consists basically of three steps. The first is the transformation of the tensor field into a positive definite metric. In the second step, we generate a spot noise texture as input for the final fabric generation. Shape and density of the spots are controlled by the eigenvalues of the tensor field. This spot image incorporates the entire information defined by the three eigenvalue fields. In the third step we use line integral convolution (LIC) to provide a contin- uous representation that enhances the visibility of the eigendirections. This method supports an intuitive distinction between positive and negative eigenvalues and supports the additional visualization of a connected scalar field.