Quantum corrections to Lorentz invariance violating theories: Fine-tuning problem

It is of general agreement that a quantum gravity theory will most probably mean a breakdown of the standard structure of space–time at the Planck-scale. This has motivated the study of Planck-scale Lorentz Invariance Violating (LIV) theories and the search for its observational signals. Yet, it has been recently shown that, in a simple scalar-spinor Yukawa theory, radiative corrections to tree-level Planck-scale LIV theories can induce large Lorentz violations at low energies, in strong contradiction with experiment, unless an unnatural fine-tuning mechanism is present. In this Letter, we show the calculation of the electron self-energy in the framework given by the Myers–Pospelov model for a Lorentz invariance violating QED. We find a contribution that depends on the preferred frame four-velocity which is not Planck-scale suppressed, showing that this model suffers from the same disease. Comparison with Hughes–Drever experiments requires a fine-tuning of 21 orders of magnitude for this model not to disagree with experiment.     

Parity violating Compton amplitude in unified theories

The leading contribution of O(αG_F) to the parity violating piece of the electron Compton amplitude is calculated in the Weinberg Salam unified theory and is shown to vanish. The reason for this is the nonexistence of parity violating charge and electric dipole moment. In the computation, a problem with the usual gauge fixing terms in the Weinberg model was encountered and resolved; the gauge fixing terms made it impossible to satisfy the electromagnetic Ward identity off mass-shell. The resolution of this problem has led to changed Feynman rules resulting in fewer graphs and gauge dependent vertices. Further features are a renormalization of the parity violating infinities, and a discussion of how the γ₅-algebra and n-dimensional regularization can coexist peacefully.     

Gauge invariance in fractional field theories

The principle of local gauge invariance is applied to fractional wave equations and the interaction term is determined up to order in the coupling constant . Based on the Riemann-Liouville fractional derivative definition, the fractional Zeeman effect is used to reproduce the baryon spectrum accurately. The transformation properties of the non-relativistic fractional Schrödinger-equation under spatial rotations are investigated and an internal fractional spin is deduced.     

Gauge invariance of effective potential in Kaluza-Klein theories

The vacuum energy in Kaluza-Klein theories, as calculated at the extremals of the generating functional of vertices, is gauge independent. The general assertion is illustrated by vacuum-energy calculations in a one-parameter gauge in vector-field theory and of Einstein gravity in the space M₄×T_D–4, where M₄ is Minkowski space and T_D is a D-dimensional torus.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 20–25, December, 1988.     

Density of states in random lattices with translational invariance

We propose a random matrix approach to describe vibrations in disordered systems. The dynamical matrix M is taken in the form M = AA ^T , where A is a real random matrix. It guaranties that M is a positive definite matrix. This is necessary for mechanical stability of the system. We built matrix A on a simple cubic lattice with translational invariance and interaction between nearest neighbors. It was found that for a certain type of disorder acoustical phonons cannot propagate through the lattice and the density of states g(ω) is not zero at ω = 0. The reason is a breakdown of affine assumptions and inapplicability of the macroscopic elasticity theory. Young modulus goes to zero in the thermodynamic limit. It reminds of some properties of a granular matter at the jamming transition point. Most of the vibrations are delocalized and similar to diffusons introduced by Allen, Feldman, et al., Phil. Mag. B 79, 1715 (1999). We show how one can gradually return rigidity and phonons back to the system increasing the width of the so-called phonon gap (the region where g(ω) ∝ ω²). Above the gap the reduced density of states g(ω)/ω² shows a well-defined Boson peak which is a typical feature of glasses. Phonons cease to exist above the Boson peak and diffusons are dominating. It is in excellent agreement with recent theoretical and experimental data.     

Thermodynamic translational invariance in concurrent multiscale simulations of liquids

AdResS multi scale simulations of liquid systems allow for a free exchange of particles between regions, where their interactions are described by different models. The desired “model coexistence” is somewhat reminiscent of phase-coexistence. But while the latter describes heterogeneous systems with position-independent interactions, AdResS is meant to generate homogeneous systems with position-dependent interactions. Here we formulate the bulk equilibrium conditions for model coexistence, discuss the connection between the Hamiltonian H-AdResS scheme and widely used free energy methods based on the Kirkwood coupling parameter method of thermodynamic integration, and point out the relation between thermodynamic corrections in AdResS simulations and tail corrections for truncated long-range potentials. In particular, we use the analogy to derive expressions for the form of the correction profiles in narrow transition zones, which cannot be fully described by the local coupling parameter approximation. Finally, we illustrate how to treat transient mergers of small, diffusing all atom zones attached to reference particles in dynamic AdResS simulations without additional calibrations beyond the initial parameterization of the correction profile for individual all atom zones.     

Topological invariance in supersymmetric theories with a continuous spectrum

We calculate the superpartition function tr(?1)^F_e?βH with a continuous spectrum. This object depends explicitly on β although it is a topological invariant with respect to compact perturbations of the potential. We show that it can be evaluated by reducing the relevant functional integral to an integral over constant configurations. The relationship with the open-space trace theorem of Callias, Bott and Seeley is discussed.     

Modular invariance and anomalies in lower dimensional superstring theories

We show explicitly that the lower dimensional, modular invariant superstrings with Z₂ twisting of the fermionic fields have anomaly free massless sectors.     

Duality and modular invariance in rational conformal field theories

We investigate the polynomial equations which should be satisfied by the duality data for a rational conformal field theory. We show that by these duality data we can construct some vector spaces which are isomorphic to the spaces of conformal blocks. One can construct explicitly the inner product for the former if one deals with a unitary theory. These vector spaces endowed with an inner product are the algebraic reminiscences of the Hilbert spaces in a Chern-Simons theory. As by-products, we show that the polynomial equations involving the modular transformations for the one-point blocks on the torus are not independent. We discuss the solution of structure constants for a physical theory. Making some assumption, we obtain a neat solution. And this solution in turn implies that the quantum groups of the left sector and of the right sector must be the same, although the chiral algebras need not be the same. Some examples are given. Finally, we discuss the reconstruction of the quantum group in a rational conformal theory.     

Modular invariance in N=2 superconformal field theories

The modular invariance properties of two-dimensional N=2 superconformal field theories are studied. It is shown that the character formulae of the central charge c<3 unitary highest weight representation for the untwisted algebras can be written in terms of the string functions and the theta functions of the affine su(2) Kac-Moody algebra. Deriving the modular transformation of the characters we construct the modular invariant partition functions on a torus. The character relation corresponding to the coset space construction of the unitary discrete series in the N=2 algebra is also obtained.     

Discrete translational invariance and mass-renormalization in the two-dimensional sine-Gordon model

We consider the sine-Gordon model as a field-theory in which the interaction has a discrete translational invariance. It is shown that apart from a Gaussian contribution the Helmholtz free-energy exhibits this symmetry as well. Thus in an originally massless theory, the insertion of a mass term can be considered as a rather weak symmetry-breaking. Within the Amit-Goldschmidt-Grinstein renormalization-scheme we show that mass-renormalization is trivial.     

Real-space renormalization group for Langevin dynamics in absence of translational invariance

A novel exact dynamical real-space renormalization group for a Langevin equation derivable from a Euclidean Gaussian action is presented. It is demonstrated rigorously that an algebraic temporal law holds for the Green function on arbitrary structures of infinite extent. In the case of fractals it is shown on specific examples that two different fixed points are found, at variance with periodic structures. Connection with the growth dynamics of interfaces is also discussed.     

Sonically produced heat in a fluid with bulk viscosity and shear viscosity

In a viscous fluid, sound produces heat in a spatial pattern which, in general, depends on the relative magnitudes of the shear viscosity coefficient eta and the bulk viscosity coefficient B'. It is well known that when the particle velocity components ui relative to Cartesian coordinates xi are given for an arbitrary sound field, or any field of flow, the volume rate of heat production qv can be determined from a dissipation function in the form B'T1 + eta T2. Here, T1 and T2 are quadratic functions involving derivatives of the type delta ui/delta xj. In this paper, examples are discussed for continuous monofrequency sound fields, including crossed plane waves, as well as focused and unfocused fields. In these examples, spatial distributions of the time-averaged quantity [qv] for media in which the loss mechanism is primarily bulk viscosity are compared to those for media in which shear viscosity dominates.     

Progress in metric-affine gauge theories of gravity with local scale invariance

Einstein's general relativity theory describes very well the gravitational phenomena in themacroscopic world. In themicroscopic domain of elementary particles, however, it does not exhibit gauge invariance or approximate Bjorken type scaling, properties which are believed to be indispensible for arenormalizable field theory. We argue that thelocal extension of space-time symmetries, such as of Lorentz and scale invariance, provides the clue for improvement. Eventually, this leads to aGL(4, R)-gauge approach to gravity in which the metric and the affine connection acquire the status ofindependent fields. The Yang-Mills type field equations, the Noether identities, and conformal models of gravity are discussed within this framework. After symmetry breaking, Einstein's GR surfaces as an effective low-energy theory.Based on a plenary talk given by one of us (EWM) at the 53rd annual meeting of the Deutsche Physikalische Gesellschaft in Bonn on March 14, 1989.Supported by the German-Israeli Foundation for Scientific Research and Development (GIF), Jerusalem and Munich.Supported by the Deutsche Forschungsgemeinschaft (DFG), Bonn, project He 528/12-1.Supported in part by DOE Grant DE-FG05-85-ER40200.     

Gauge invariance properties of transition amplitudes in gauge theories. I

The functional approach developed earlier for scattering theory in quantum field theory makes it possible to make an explicit and complete study of the gauge invariance properties oftransition amplitudes (not just of the gauge transformations of Green's functions) in covariant and noncovariant gauges. This paper is devoted to the Abelian gauge theory of quantum electrodynamics. Using the powerful technique of functional differentiation and starting from the Coulomb gauge, the gauge invariance property of transition amplitudes,up to gauge-dependent scaling factors, isexplicitly established in arbitrary gauges. The key ingredients in the analysis are the derived exact expression for the vacuum-to-vacuum transition amplitude, introducing in the process arbitrary gauges, and the idea of stimulated emissions by external sources studied earlier.