Quantum General Invariance and Loop Gravity

A quantum physical projector is proposed for generally covariant theories which are derivable from a Lagrangian. The projector is the quantum analogue of the integral over the generators of finite one-parameter subgroups of the gauge symmetry transformations which are connected to the identity. Gauge variables are retained in this formalism, thus permitting the construction of spacetime area and volume operators in a tentative spacetime loop formulation of quantum general relativity.     

Dirac’s Equation in R-Minkowski Spacetime

We recently constructed the R-Poincaré algebra from an appropriate deformed Poisson brackets which reproduce the Fock coordinate transformation. We showed then that the spacetime of this transformation is the de Sitter one. In this paper, we derive in the R-Minkowski spacetime the Dirac equation and show that this is none other than the Dirac equation in the de Sitter spacetime given by its conformally flat metric. Furthermore, we propose a new approach for solving Dirac’s equation in the de Sitter spacetime using the Schrödinger picture.     

Clifford代数中的双曲相位变换群及其在四维相对论时空中的应用

将Clifford代数所定义的双曲复空间R_H和作用在双曲复空间R_H上的双曲相位变换群U₄(H)赋予了明确的物理意义. 双曲复空间R_H同构于四维Minkowski时空，而其上的双曲相位变换群U₄(H)就是四维相对论时空中的洛仑兹(Lorentz)变换群. 进一步，利用U₄(H)群的复合变换性质，自然导出了四维Minkowski时空中Lorentz变换和速度变换的一般表达式. 由此，将狭义相对论中的特殊Lorentz变换作为特例包含其中. 关键词： 双曲复数 双曲相位变换 Minkowski时空 Clifford代数     

Gravitational Redshift Induces Quantum Interference

Quantum field theory in curved spacetime is used to show that gravitational redshift induces a unitary transformation on the quantum state of propagating photons. It is found that the transformation is a mode-mixing operation, and a protocol that exploits gravity to induce a Hong–Ou–Mandel-like interference effect on the state of two photons is devised. It is discussed how the results of this work can provide a demonstration of quantum field theory in curved spacetime.     

Conformally covariant vector–spinor field in de Sitter space

Unlike the Lorentz transformation which replaces the Galilean transformation among inertial frames at high relative velocities, there seems to be no such a consensus in the case of coordinate transformation between inertial frames and uniformly rotating ones. There have been some attempts to generalize the Galilean rotational transformation to high rotational velocities. Here we introduce a modified version of one of these transformations proposed by Philip Franklin in 1922. The modified version is shown to resolve some of the drawbacks of the Franklin transformation, specially with respect to the corresponding spacetime metric in the rotating frame. This new transformation introduces non-inertial eccentric observers on a uniformly rotating disk and the corresponding metric in the rotating frame is shown to be consistent with the one obtained through Galilean rotational transformation for points close to the rotation axis. Employing the threading formulation of spacetime decomposition, spatial distances and time intervals in the spacetime metric of a rotating observer’s frame are also discussed.     

Cloaks,editors, and bubbles: applications of spacetime transformation theory

Spacetime or ‘event’ cloaking was recently introduced as a concept, and the theoretical design for such a cloak was presented for illumination by electromagnetic waves [McCall et al., J. Opt. 2011]. Here it is described how event cloaks can be designed for simple wave systems, using either an approximate ‘speed cloak’ method, or an exact full‐wave one. Further, details of many of the implications of spacetime transformation devices are discussed, including their (usually) directional nature, spacetime distortions (as opposed to cloaks), and how leaky cloaks manifest themselves. More exotic concepts are also addressed, in particular concepts that follow naturally on from considerations of simple spacetime transformation devices, such as spacetime modeling and causality editors. A proposal for implementing an interrupt‐without‐interrupt concept is described. Finally, the design for a time‐dependent ‘bubbleverse’ is presented, based on temporally modulated Maxwell's Fisheye transformation device (T‐device) in a flat background spacetime.     

Electron paths,tunnelling, and diffraction in the spacetime algebra

This paper employs the ideas of geometric algebra to investigate the physical content of Dirac's electron theory. The basis is Hestenes' discovery of the geometric significance of the Dirac spinor, which now represents a Lorentz transformation in spacetime. This transformation specifies a definite velocity, which might be interpreted as that of a real electron. Taken literally, this velocity yields predictions of tunnelling times through potential barriers, and defines streamlines in spacetime that would correspond to electron paths. We also present a general, first-order diffraction theory for electromagnetic and Dirac waves. We conclude with a critical appraisal of the Dirac theory.Supported by a SERC studentship.     

Spherical symmetry and the fluid of multidimensional objects

The subject of the paper is the formulation and analysis of the notion of spherical symmetry for a spacetime with matter of multidimensional objects (also called extended particles) i.e. matter, whose elementary constituents are extensive objects. The paper includes also eduction of a subclass of vacuum, static, spherically-symmetric solutions of the Einstein equation in the spacetime mentioned above. In particular a close analogue of the Schwarzschild solution is obtained.     

Achieving acoustic cloak by using compressible background flow

We propose a scheme of acoustic spherical cloaking by means of background irrotational flow in compressible fluid.The background flow forms a virtual curved spacetime and directs the sound waves to bypass the cloaked objects. To satisfy the laws of real fluid, we show that spatially distributed mass source and momentum source are necessary to supply. The propagation of sound waves in this system is studied via both geometric acoustics approximation and full wave approach.The analytic solution of sound fields is obtained for plane wave incidence. The results reveal the effect of phase retardation(or lead) in comparison with the ordinary transformation-acoustic cloak. In addition, the ability of cloaking is also evaluated for unideal background flows by analyzing the scattering cross section.     

Path integrals and quantum interference effects in a five-dimensional Kaluza-Klein theory

The Aharonov-Bohm (AB) effect, the gravitational analogue to the AB effect, and the topological shifts in an AB set-up are derived by evaluating the propagator for a particle in a five-dimensional Kaluza-Klein spacetime via path integrals.     

Conformal transformations and conformal invariance in gravitation

Conformal transformations are frequently used tools in order to study relations between various theories of gravity and Einstein's general relativity theory. In this paper we discuss the rules of these transformations for geometric quantities as well as for the matter energy‐momentum tensor. We show the subtlety of the matter energy‐momentum conservation law which refers to the fact that the conformal transformation “creates” an extra matter term composed of the conformal factor which enters the conservation law. In an extreme case of the flat original spacetime the matter is “created” due to work done by the conformal transformation to bend the spacetime which was originally flat. We discuss how to construct the conformally invariant gravity theories and also find the conformal transformation rules for the curvature invariants R², R_abR^ab, R_abcdR^abcd and the Gauss‐Bonnet invariant in a spacetime of an arbitrary dimension. Finally, we present the conformal transformation rules in the fashion of the duality transformations of the superstring theory. In such a case the transitions between conformal frames reduce to a simple change of the sign of a redefined conformal factor.     

Finite field-dependent symmetries in perturbative quantum gravity

In this paper we discuss the absolutely anticommuting nilpotent symmetries for perturbative quantum gravity in general curved spacetime in linear and non-linear gauges. Further, we analyze the finite field-dependent BRST (FFBRST) transformation for perturbative quantum gravity in general curved spacetime. The FFBRST transformation changes the gauge-fixing and ghost parts of the perturbative quantum gravity within functional integration. However, the operation of such symmetry transformation on the generating functional of perturbative quantum gravity does not affect the theory on physical ground. The FFBRST transformation with appropriate choices of finite BRST parameter connects non-linear Curci–Ferrari and Landau gauges of perturbative quantum gravity. The validity of the results is also established at quantum level using Batalin–Vilkovisky (BV) formulation.     

Inertial Frame Dragging in an Acoustic Analogue Spacetime

We report an incipient exploration of the Lense‐Thirring precession effect in a rotating acoustic analogue black hole spacetime. An exact formula is deduced for the precession frequency of a gyroscope due to inertial frame dragging, close to the ergosphere of a ‘Draining Bathtub’ acoustic spacetime which has been studied extensively for acoustic Hawking radiation of phonons and also for ‘superresonance’. The formula is verified by embedding the two dimensional spatial (acoustic) geometry into a three dimensional one where the similarity with standard Lense‐Thirring precession results within a strong gravity framework is well known. Prospects of experimental detection of this new ‘fixed‐metric’ effect in acoustic geometries, are briefly discussed.     

Relativistic Anandan quantum phase and the Aharonov–Casher effect under Lorentz symmetry breaking effects in the cosmic string spacetime

From the modified Maxwell theory coupled to gravity, we establish a possible scenario of the violation of the Lorentz symmetry and write an effective metric for the cosmic string spacetime. Then, we investigate the arising of an analogue of the Anandan quantum phase for a relativistic Dirac neutral particle with a permanent magnetic dipole moment in the cosmic string spacetime under Lorentz symmetry breaking effects. Besides, we analyse the influence of the effects of the Lorentz symmetry violation and the topology of the defect on the Aharonov–Casher geometric quantum phase in the nonrelativistic limit.     

On General Møller Transformation

We give a study on the general Møller transformation and emphatically introduce its differential form. In this paper, a definition of acceleration is given in spacetime language and the inertial reference frame is also settled. With a discussion of thegeodesic equations of motion, the differential form of the general Møller transformation at arbitrary direction is presented.     

Graded manifold theory as the geometry of supersymmetry

Building upon Kostant's graded manifold theory, we present a new way of introducing spinors into the spacetime manifold, by expanding the algebra of functions on spacetime to a graded algebra. The elements of differential geometry are generalized to accomodate the expanded algebra of functions and in this enriched geometry we find the elements of supersymmetry and of supergravity theory. The geometrical role of the supergravity fields is discussed and a derivation of their transformation rules is given.Research supported in part by the National Science Foundation under Grant Nos. PHY77-22864 and PHY77-05299     

Microscopic and macroscopic entropy of extremal black holes in string theory

It has been observed on a number of occasions that complex transformations, of real solutions of the field equations to other real solutions, often preserve certain properties of the Weyl tensor. That is, the Petrov type and/or gravito-electromagnetic (GEM) properties of the Weyl tensor are preserved. In this context, we present an outstanding example of a complex windmill transformation of a static (non-physical) anisotropic fluid spacetime of Petrov type $I(M^+)$ that maps to a purely magnetic (PM) spacetime of Petrov type $I(M^{\infty })$ . The PM spacetime is analyzed and compared to the Arianrhod–Lun–McIntosh–Perjés spacetime. It is shown that these spacetimes, although similar in some aspects, are distinct solutions. The main distinction is that the generated PM spacetime satisfies all the standard energy-conditions. This intriguing but purely mathematical scenario may have implications in the area of GEM duality.     

Particle Production in Expanding Spacetime

The complex time WKB (CWKB) approximation has been an effective technique to study particle production in expanding space time. The success of the approximation technique both in time and space dependent gauge has motivated us to study the method in relation to the time dependent approximation. In this work we try to understand the adiabatic and non-adiabatic transition within the framework of complex time WKB approximation. We find that the emergence of thermal radiation is due to some topological characteristics of cosmological spacetime that separates the spacetime into Euclidean and non-Euclidean region. This applies also to blackhole spacetime. The complex WKB trajectory approach shows that the Euclidean vacuum fluctuation is root cause of thermal particle production and is basically a Hawking effect. We also study here the sensitivity of particle production on the rise of scale factor at early times. It is found that the tunneling paths are responsible for the origin of thermal radiation whereas the slope of the scale factor determines the magnitude of the temperature of the thermal particle production. We also substantiate Hu's assertion in this connection.