Conformally invariant σ models in 2n dimensions

The conformally invariant σ-model of Felsager and Leinaas in four dimensions (n=2) is generalised to 2n dimensions, and the unit-charge instantons are presented. This hierarchy of σ-models starts with the O(3) σ-model (n=1). The relationship of these models with the generalised YM systems for even n=2p is pointed out.     

Does a varying speed of light solve the cosmological problems?

We propose a new generalisation of general relativity which incorporates a variation in both the speed of light in vacuum (c) and the gravitational constant (G) and which is both covariant and Lorentz invariant. We solve the generalised Einstein equations for Friedmann universes and show that arbitrary time-variations of c and G never lead to a solution to the flatness, horizon or Λ problems for a theory satisfying the strong energy condition. In order to do so, one needs to construct a theory which does not reduce to the standard one for any choice of time, length and energy units. This can be achieved by breaking a number of invariance principles such as covariance and Lorentz invariance.     

Poisson-Lie T-duality and Bianchi type algebras

All Bianchi bialgebras have been obtained. By introducing a non-degenerate adjoint invariant inner product over these bialgebras the associated Drinfeld doubles have been constructed, then by calculating the coupling matrices for these bialgebras several σ-models with Poisson-Lie symmetry have been obtained. Two simple examples as prototypes of Poisson-Lie dual models have been given.     

Relativistic decay theory: Corrections to the Weisskopf-Wigner approximation

An analysis is undertaken of the corrections to the Wigner-Weisskopf approximation for relativistic decay processes. It is shown that difficulties with relativistic invariance are encountered. A proposal is made for a Lorentz invariant approximation, which preserves as much of the original expression as possible. The effects of measurements are discussed, and it is noted that, indeed, a watched pot cannot boil. Curves are presented for a hypothetical σ→ππ decay process, and the deviations from exponentiality are discussed when measurements both are and are not being made.     

Pathologies of the large-N limit for RPN−1, CPN−1, QPN−1 and mixed isovector/isotensor σ-models

We compute the phase diagram in the N→∞ limit for lattice RP^N−1, CP^N−1 and QP^N−1 σ-models with the quartic action, and more generally for mixed isovector/isotensor models. We show that the N=∞ limit exhibits phase transitions that are forbidden for any finite N. We clarify the origin of these pathologies by examining the exact solution of the one-dimensional model: we find that there are complex zeros of the partition function that tend to the real axis as N→∞. We conjecture the correct phase diagram for finite N as a function of the spatial dimension d. Along the way, we prove some new correlation inequalities for a class of N-component σ-models, and we obtain some new results concerning the complex zeros of confluent hypergeometric functions.     

An arena for model building in the Cohen—Glashow very special relativity

The Cohen—Glashow Very Special Relativity (VSR) algebra is defined as the part of the Lorentz algebra which upon addition of CP or T invariance enhances to the full Lorentz group, plus the space—time translations. We show that noncommutative space—time, in particular noncommutative Moyal plane, with light- like noncommutativity provides a robust mathematical setting for quantum field theories which are VSR invariant and hence set the stage for building VSR invariant particle physics models. In our setting the VSR invariant theories are specified with a single deformation parameter, the noncommutativity scale ╕_NC. Preliminary analysis with the available data leads to ╕_NC ≳ 1–10 TeV.     

High spin limits and non-abelian T-duality

The action of the non-abelian T-dual of the WZW model is related to an appropriate gauged WZW action via a limiting procedure. We extend this type of equivalence to other σ-models with non-abelian isometries and their non-abelian T-duals, focusing on Principal Chiral models. We reinforce and refine this equivalence by arguing that the non-abelian T-duals are the effective backgrounds describing states of an appropriate parent theory corresponding to divergently large highest weight representations. The proof involves carrying out a subtle limiting procedure in the group representations and relating them to appropriate limits in the corresponding backgrounds. We illustrate the general method by providing several non-trivial examples.     

Derivation of an effective Lagrangian from a generalized scalar curvature

A spinor Lagrangian invariant under global coordinate, local Lorentz and local chiral SU(n) × SU(n) gauge transformations is presented. The invariance requirement necessitates the introduction of boson fields, and a theory for these fields is then developed by relating them to generalizations of the vector connections in general relativity and utilizing an expanded scalar curvature as a boson Lagrangian. In implementing this plan, the local Lorentz group is found to greatly facilitate the correlation of the boson fields occurring in the spinor Lagrangian with the generalized vector connections.The independent boson fields of the theory are assumed to be the inhomogeneously transforming irreducible parts of the connections. It turns out that no homogeneously transforming parts are necessary to reproduce the chiral Lagrangian usually used as a basis for phenomenological field theories. The Lagrangian in question appears when the gravitational interaction is turned off. It includes pseudoscalar, spinor, vector, and axial vector fields, and the vector fields carry mass in spite of the fact that the theory is locally gauge invariant.     

Reconsidering a Scientific Revolution: The Case of Einstein <Emphasis Type="Italic">versus</Emphasis> Lorentz

The relationship between Albert Einstein's special theory of relativity and Hendrik A. Lorentz's ether theory is best understood in terms of competing interpretations of Lorentz invariance. In the 1890s, Lorentz proved and exploited the Lorentz invariance of Maxwell's equations, the laws governing electromagnetic fields in the ether, with what he called the theorem of corresponding states. To account for the negative results of attempts to detect the earth's motion through the ether, Lorentz, in effect, had to assume that the laws governing the matter interacting with the fields are Lorentz invariant as well. This additional assumption can be seen as a generalization of the well-known contraction hypothesis. In Lorentz's theory, it remained an unexplained coincidence that both the laws governing fields and the laws governing matter should be Lorentz invariant. In special relativity, by contrast, the Lorentz invariance of all physical laws directly reflects the Minkowski space-time structure posited by the theory. One can use this observation to produce a common-cause argument to show that the relativistic interpretation of Lorentz invariance is preferable to Lorentz's interpretation.     

Quasirelativistische Urfermionen

In recent papers we have reformulated the quantum theory of fermions in a lattice space with a finite number of points, sayZ ³. Of course, Lorentz invariance holds only in the limitZ→∞. So far a lattice space withZ ³ points provides us only with a convenient method to define the order of some limiting processes. — Here we discuss a more fundamental question. According to general relativity Lorentz invariance can not hold all over the world and must also be broken at tiny distances of the centers of particles. If the space curvature is not essential for the theory of elementary particles, as generally is assumed, we must only take into account the topological structure of the deviations from Lorentz invariance. In this respect the lattice space provides us with a rather rough but sufficient picture of a space, being approximately Lorentz invariant only for intermediate distances. — In a lattice withZ ³ points the Hilbert space has a finite number of dimensions. It seems to us important that this assumption will hold, even if we proceed to a more refined space model.     

Lorentz invariant exact solution of the anomalous chiral Schwinger model

We present an exact solution of the anomalous chiral Schwinger model using Fermionic variables. We implement infrared regularization by considering the model on a spatial circleS ¹. Quantum effects modify the gauge constraints through the appearance of Schwinger terms in the gauge algebra. We perform a careful analysis of the resulting second class gauge constraints by implementing Dirac's method at the quantum level and obtain the spectrum of the theory. We get a consistent unitary Lorentz invariant theory for particular values of the counterterms. We find that when we regulate the fermionic sector of the model without reference to the gauge fields Lorentz invariance requires that we add both Lorentz variant and gauge variant counterterms.     

Conformal invariance in quantum gravity

Trace anomalies in a conformal invariant theory do not arise when its conformal invariance in four dimensions is extended to an arbitrary number n of space-time dimensions: the theory can be made finite in any order of perturbation theory by conformal invariant counterterms in n dimensions. Such an extension of conformal invariance is possible provided one works in the framework of spontaneously broken conformal invariance. This is shown explicitly by working out several examples at the one-loop level and by examining the Ward identities which lead to a general proof.We speculate upon possible consequences of these results on the nature of gravitation and other fundamental interactions.     

Gauge invariant and gauge fixed actions for various higher-spin fields from string field theory

We propose a systematic procedure for extracting gauge invariant and gauge fixed actions for various higher-spin gauge field theories from covariant bosonic open string field theory. By identifying minimal gauge invariant part for the original free string field theory action, we explicitly construct a class of covariantly gauge fixed actions with BRST and anti-BRST invariance. By expanding the actions with respect to the level N   of string states, the actions for various massive fields including higher-spin fields are systematically obtained. As illustrating examples, we explicitly investigate the level N?3$N?3$ part and obtain the consistent actions for massive graviton field, massive 3rd rank symmetric tensor field, or anti-symmetric field. We also investigate the tensionless limit of the actions and explicitly derive the gauge invariant and gauge fixed actions for general rank n symmetric and anti-symmetric tensor fields.     

CPT violation does not lead to violation of Lorentz invariance and vice versa

We present a class of interacting nonlocal quantum field theories, in which the CPT invariance is violated while the Lorentz invariance is present. This result rules out a previous claim in the literature that the CPT violation implies the violation of Lorentz invariance. Furthermore, there exists the reciprocal of this theorem, namely that the violation of Lorentz invariance does not lead to the CPT violation, provided that the residual symmetry of Lorentz invariance admits the proper representation theory for the particles. The latter occurs in the case of quantum field theories on a noncommutative space–time, which in place of the broken Lorentz symmetry possesses the twisted Poincaré invariance. With such a CPT-violating interaction and the addition of a C-violating (e.g., electroweak) interaction, the quantum corrections due to the combined interactions could lead to different properties for the particle and antiparticle, including their masses.     

Invariant length scale in relativistic kinematics: lessons from Dirichlet branes

Dirac–Born–Infeld theory is shown to possess a hidden invariance associated with its maximal electric field strength. The local Lorentz symmetry O(1,n) on a Dirichlet-n-brane is thereby enhanced to an O(1,n)×O(1,n) gauge group, encoding both an invariant velocity and acceleration (or length) scale. The presence of this enlarged gauge group predicts consequences for the kinematics of observers on Dirichlet branes, with admissible accelerations being bounded from above. An important lesson is that the introduction of a fundamental length scale into relativistic kinematics does not enforce a deformation of Lorentz boosts, as one might assume naively. The exhibited structures further show that Moffat's non-symmetric gravitational theory qualifies as a candidate for a consistent Born–Infeld type gravity with regulated solutions.     

The Geometry of Antiferromagnetic Spin Chains

It has been a conjecture of F.D.M.Haldane that long-wavelength excitations around antiferromagnetic vacua of certain spin chains are related to relativistic two-dimensional σ-models. We construct a class of such spin chains and associated σ-models, using symplectic geometry as a main tool. The target space of the σ-model can be an arbitrary complex flag manifold, and we find universal expressions for the metric and θ-term. The derivation relies on the fact that the flag manifold is a Lagrangian submanifold in a direct product of Grassmannians associated (via geometric quantization) to the representations of spins at consecutive sites of the spin chain. The true goal is not to reach the uttermost limits, but to discover a completeness that knows no boundaries. Rabindranath Tagore      

Constraints and actions in two and three dimensionalN=1 conformal supergravity

We investigate closure of the gauge algebra and constraints inN=1 conformal supergravity in 2 and 3 dimensions. In the 2 dimensional case, contrary to 3 or higher dimensions, some parts of the gauge fields are algebraically unsolvable in the constraint equations on group curvatures. It will be shown that these unsolvable parts are decoupled from the transformation law as well as from the kinetic multiplets. Hence they are absent in the invariant action for matter multiplets coupled to conformal supergravity which is relevant to the old superstrings. Explicit construction of the invariant actions are illustrated for the case of spinning strings and locally supersymmetric σ-models with the Wess-Zumino term.