Conformal invariance and scalar-tensor theories

A new generalisation of Einstein’s theory is proposed which is invariant under conformal mappings. Two scalar fields are introduced in addition to the metric tensor field, so that two special choices of gauge are available for physical interpretation, the ‘Einstein gauge’ and the ‘atomic gauge’. The theory is not unique but contains two adjustable parameters ζ anda. Witha=1 the theory viewed from the atomic gauge is Brans-Dicke theory (ω=−3/2+ζ/4). Any other choice ofa leads to a creation-field theory. In particular the theory given by the choicea=−3 possesses a cosmological solution satisfying Dirac’s ‘large numbers’ hypothesis.

Constraining gravity using entanglement in AdS/CFT

We investigate constraints imposed by entanglement on gravity in the context of holography. First, by demanding that relative entropy is positive and using the Ryu-Takayanagi entropy functional, we find certain constraints at a nonlinear level for the dual gravity. Second, by considering Gauss-Bonnet gravity, we show that for a class of small perturbations around the vacuum state, the positivity of the two point function of the field theory stress tensor guarantees the positivity of the relative entropy. Further, if we impose that the entangling surface closes off smoothly in the bulk interior, we find restrictions on the coupling constant in Gauss-Bonnet gravity. We also give an example of an anisotropic excited state in an unstable phase with broken conformal invariance which leads to a negative relative entropy.

Gaussian Schell-model beams and general shape invariance

We present two six-parameter families of anisotropic Gaussian Schell-model beams that propagate in a shape-invariant manner, with the intensity distribution continuously twisting about the beam axis. The two families differ in the sense or helicity of this beam twist. The propagation characteristics of these shape-invariant beams are studied, and the restrictions on the beam parameters that arise from the optical uncertainty principle are brought out. Shape invariance is traced to a fundamental dynamical symmetry that underlies these beams. This symmetry is the product of spatial rotation and fractional Fourier transformation.

Analysis of Rectangular Folded-Waveguide Millimeter-Wave Slow-wave Structures using Conformal Transformations

An analysis of rectangular folded-waveguide slow-wave structure was developed using conformal mapping technique through Schwarz's polygon transformation and closed form expressions for the lumped capacitance and inductance per period of the slow-wave structure were derived in terms of the physical dimensions of the structure, incorporating the effects of the beam hole in the lumped parameters. The lumped parameters were subsequently interpreted for obtaining the dispersion and interaction impedance characteristics of the structure. The analysis was benchmarked for two typical millimeter-wave structures, one operating in Ka-band and the other operating in Q-band, against measurement and 3D electromagnetic modeling using MAFIA.

Gauge theory of a group of diffeomorphisms. II. The conformal and de Sitter groups

The extension of Hehl's Poincaré gauge theory to more general groups that include space-time diffeomorphisms is worked out for two particular examples, one corresponding to the action of the conformal group on Minkowski space, and the other to the action of the de Sitter group on de Sitter space, and the effect of these groups on physical fields.

The scale-invariance of spatial patterning in a developing system

Regulating systems, that is, those which exhibit scale-invariant patterns in the adult, are supposed, to do so on account of interactions between cells during development. The nature of these interactions has to be such that the system of positional information (ldquomaprdquo) in the embryo also regulates. To our knowledge, this supposition regarding a regulating map has not been subjected to a direct test in any embryonic system. Here we do so by means of a simple and novel criterion and use it to examine tip regeneration in the mulicellular stage (slug) ofDictyostelium discoideum. When anterior, tip-containing fragments of slugs are amputated, a new tip spontaneously regenerates at the cut surface of the (remaining) posterior fragment. The time needed for regeneration to occur depends on the relative size of the amputated fragment but is independent of the total size of the slug. We conclude from this finding that there is at least one system underlying positional information in the slug which regulates.

Invariance group properties and exact solutions of equations describing time-dependent free surface flows under gravity

Using the method of infinitesimal transformations, a 6-parameter family of exact solutions describing nonlinear sheared flows with a free surface are found. These solutions are a hybrid between the earlier self-propagating simple wave solutions of Freeman, and decaying solutions of Sachdev. Simple wave solutions are also derived via the method of infinitesimal transformations. Incomplete beta functions seem to characterize these (nonlinear) sheared flows in the absence of critical levels.

Gauging the conformal group

It is demonstrated that Kibble’s method of gauging the Poincaré group can be applied to the gauging of the conformal group. The action of the gauge transformations is the action of general spacetime diffeomorphisms (or coordinate transformations) combined with a local action of an 11-parameter subgroup of SO(4,2). Because the translational subgroup is not an invariant subgroup of the conformal group the appropriate generalisation of the derivative of a physical field is not a covariant derivative in the usual sense, but this does not lead to any inconsistencies.

Generalised quaternion methods in conformal geometry

A new approach to Penrose's twistor algebra is given. It is based on the use of a generalised quaternion algebra for the translation of statements in projective five-space into equivalent statements in twistor (conformal spinor) space. The formalism leads toSO(4, 2)-covariant formulations of the Pauli-Kofink and Fierz relations among Dirac bilinears, and generalisations of these relations.

Numerical integration over arbitrary polygonal domains based on Schwarz-Christoffel conformal mapping

This paper presents a new numerical integration technique oil arbitrary polygonal domains. The polygonal domain is mapped conformally to the unit disk using Schwarz-Christoffel mapping and a midpoint quadrature rule defined oil this unit disk is used. This method eliminates the need for a two-level isoparametric mapping Usually required. Moreover, the positivity of the Jacobian is guaranteed. Numerical results presented for a few benchmark problems in the context of polygonal finite elements show that the proposed method yields accurate results.

Poincaré invariance of anomalous gauge theories

We establish the Poincaré invariance of anomalous gauge theories in two dimensions, for both the Abelian and non-Abelian cases, in the canonical Hamiltonian formalism. It is shown that, despite the noncovariant appearance of the constraints of these theories, Poincaré generators can be constructed which obey the correct algebra and yield the correct transformations in the constrained space.

Spectral Invariance of SG Pseudo-Differential Operators on L-P(R-n)

We prove the spectral invariance of SG pseudo-differential operators on L-P(R-n), 1 < p < infinity, by using the equivalence of ellipticity and Fredholmness of SG pseudo-differential operators on L-p(R-n), 1 < p < infinity. A key ingredient in the proof is the spectral invariance of SC pseudo-differential operators on L-2(R-n).

Complete Pick positivity and unitary invariance

The characteristic function for a contraction is a classical complete unitary invariant devised by Sz.-Nagy and Foias. Just as a contraction is related to the Szego kernel k(S) (z, w) = (1 - z (w) over tilde)(-1) for |z|, |w| < 1, by means of (1/k(S))(T,T*) >= 0, we consider an arbitrary open connected domain Omega in C-n, a complete Pick kernel k on Omega and a tuple T = (T-1, ..., T-n) of commuting bounded operators on a complex separable Hilbert space H such that (1/k)(T,T*) >= 0. For a complete Pick kernel the 1/k functional calculus makes sense in a beautiful way. It turns out that the model theory works very well and a characteristic function can be associated with T. Moreover, the characteristic function is then a complete unitary invariant for a suitable class of tuples T.