Gauge Invariance of a Critical Number of Flavours in QED3

The fermion propagator in an arbitrary covariant gauge can be obtained from the Landau gauge result via a Landau–Khalatnikov–Fradkin transformation. This transformation can be written in a practically useful form in both configuration and momentum space. It is therefore possible to anticipate effects of a gauge transformation on the propagator’s analytic properties. These facts enable one to establish that if a critical number of flavours for chiral symmetry restoration and deconfinement exists in noncompact QED3, then its value is independent of the gauge parameter. This is explicated using simple forms for the fermion–photon vertex and the photon vacuum polarisation. The illustration highlights pitfalls that must be avoided in order to arrive at valid conclusions. Landau gauge is seen to be the covariant gauge in which the propagator avoids modification by a non-dynamical gauge-dependent exponential factor, whose presence can obscure truly observable features of the theory.     

Quantum field theory in curved spacetime,the operator product expansion,and dark energy

To make sense of quantum field theory in an arbitrary (globally hyperbolic) curved spacetime, the theory must be formulated in a local and covariant manner in terms of locally measurable field observables. Since a generic curved spacetime does not possess symmetries or a unique notion of a vacuum state, the theory also must be formulated in a manner that does not require symmetries or a preferred notion of a “vacuum state” and “particles”. We propose such a formulation of quantum field theory, wherein the operator product expansion (OPE) of the quantum fields is elevated to a fundamental status, and the quantum field theory is viewed as being defined by its OPE. Since the OPE coefficients may be better behaved than any quantities having to do with states, we suggest that it may be possible to perturbatively construct the OPE coefficients—and, thus, the quantum field theory. By contrast, ground/vacuum states—in spacetimes, such as Minkowski spacetime, where they may be defined—cannot vary analytically with the parameters of the theory. We argue that this implies that composite fields may acquire nonvanishing vacuum state expectation values due to nonperturbative effects. We speculate that this could account for the existence of a nonvanishing vacuum expectation value of the stress-energy tensor of a quantum field occurring at a scale much smaller than the natural scales of the theory. Fourth Award in the 2008 Essay Competition of the Gravity Research Foundation.     

Quantum cryptography based on homodyne detection (vacuum-state cryptosystem)

A new protocol is proposed for quantum cryptography. The protocol is based on the use of a set of measurements which make it possible to reconstruct completely the density matrix — the information carrier — of a physical system. Such a protocol can be implemented by means of homodyne detection (well known in quantum optics) of an electromagnetic field. An example is given of a quantum cryptosystem in which the vacuum state of the photon field is used as one of two information states. Pis’ma Zh. éksp. Teor. Fiz. 66, No. 1, 64–68 (10 July 1997)     

‘Faster than light’ photons in gravitational fields — Causality,anomalies and horizons

A number of general issues relating to superluminal photon propagation in gravitational fields are explored. The possibility of superluminal, yet causal, photon propagation arises because of Equivalence Principle violating interactions induced by vacuum polarisation in QED in curved spacetime. Two general theorems are presented: first, a polarisation sum rule which relates the polarisation averaged velocity shift to the matter energy-momentum tensor and second, a ‘horizon theorem’ which ensures that the geometric event horizon for black hole spacetimes remains a true horizon for real photon propagation. These results are consequences of an effective action which in QED is valid only for low frequency photons. Their relevance to signal propagation and causality, which are controlled by high frequency propagation, is dependent on the dispersive properties of the modified propagation. This will be discussed elsewhere. A comparison is made with the equivalent results for electromagnetic birefringence and possible connections between superluminal photon propagation, causality and the conformal anomaly are exposed.     

Gauge invariance and the CPT and Lorentz violating induced Chern–Simons-like term in extended QED

The radiative induction of the CPT and Lorentz violating Chern–Simons (CS) term is reassessed. The massless and massive models are studied. Special attention is given to the preservation of gauge symmetry at higher orders in the background vector b _μ when radiative corrections are considered. Both the study of the odd and even parity sectors of the complete vacuum polarization tensor at one-loop order and a non-perturbative analysis show that this symmetry must be preserved by quantum corrections. As a complement we obtain the result that transversality of the polarization tensor does not fix the value of the coefficient of the induced CS term.     

Localizability of tachyonic particles and neutrinoless double beta decay

The quantum field theory of superluminal (tachyonic) particles is plagued by a number of problems, which include the Lorentz non-invariance of the vacuum state, the ambiguous separation of the field operator into creation and annihilation operators under Lorentz transformations, and the necessity of a complex reinterpretation principle for quantum processes. Another unsolved question concerns the treatment of subluminal components of a tachyonic wave packet in the field-theoretical formalism, and the calculation of the time-ordered propagator. After a brief discussion on related problems, we conclude that rather painful choices have to be made in order to incorporate tachyonic spin- \frac12\frac{1}{2} particles into field theory. We argue that the field theory needs to be formulated such as to allow for localizable tachyonic particles, even if that means that a slight unitarity violation is introduced into the S matrix, and we write down field operators with unrestricted momenta. We find that once these choices have been made, the propagator for the neutrino field can be given in a compact form, and the left-handedness of the neutrino as well as the right-handedness of the antineutrino follow naturally. Consequences for neutrinoless double beta decay and superluminal propagation of neutrinos are briefly discussed.     

Photon Productions in a Gravitational Collapsing

We study a possible gravitational vacuum-effect, in which vacuum-energy variation is due to variation of gravitational field, vacuum state gains gravitational energy and releases it by spontaneous photon emissions. Based on the path-integral representation, we present a general formulation of vacuum transition matrix and energy-momentum tensor of a quantum scalar field theory in curved spacetime. Using analytical continuation of dimensionality of the phase space, we calculate the difference of vacuum-energy densities in the presence and absence of gravitational field. Using the dynamical equation of gravitational collapse, we compute the rate of vacuum state gaining gravitational energy. Computing the transition amplitude from initial vacuum state to final vacuum state in gravitational collapsing process, we show the rate and spectrum of spontaneous photon emissions for releasing gravitational energy. We compare our idea with the Schwinger idea for Sonoluminiescence and contrast our scenario with the Hawking effect.     

What can be tested in quantum electrodynamics?

In this paper we examine the theoretical foundations underlying the testing of quantum electrodynamics. We show that for the photon propagator (together with the contiguous vertices) it is not necessary to introduce ad hoc modifications in sufficiently accurate scattering experiments. Energy, momentum transfer, and accuracy determine the tested length in a model-independent way. The situation is quite different with the electron propagator. If gauge invariance is taken for granted, the electron propagator cannot be tested with processes where diagrams with open electron lines are important in the lowest order of perturbation theory. These processes can only give limits for anomalous moment and multiphoton parts of the vertices. On the other hand, processes with closed electron loops (vacuum polarization), such as photon-photon and Delbrück scattering, as well as photon splitting or corresponding low-energy, high-precision experiments can give limits also for the electron propagator. But in these cases only less accurate limits can be obtained, which depend on the modification model. Hence testing of the electron propagator, i.e., roughly speaking, the Dirac equation, is much more difficult than testing of the photon propagator, i.e, Maxwell's equations.Dedicated to the memory of Prof. Wolfgang Yourgrau (1908–1979).Presented at the 1975 International Symposium on Lepton and Photon Interactions at High Energies, Stanford University, Stanford, California.     

On the harmonic-type and linear-type confinement of a relativistic scalar particle yielded by Lorentz symmetry breaking effects

Based on the Standard Model Extension, we investigate relativistic quantum effects on a scalar particle in backgrounds of the Lorentz symmetry violation defined by a tensor field. We show that harmonic-type and linear-type confining potentials can stem from Lorentz symmetry breaking effects, and thus, relativistic bound state solutions can be achieved. We first analyse a possible scenario of the violation of the Lorentz symmetry that gives rise to a harmonic-type potential. In the following, we analyse another possible scenario of the breaking of the Lorentz symmetry that induces both harmonic-type and linear-type confining potentials. In this second case, we also show that not all values of the parameter associated with the intensity of the electric field are permitted in the search for polynomial solutions to the radial equation, where the possible values of this parameter are determined by the quantum numbers of the system and the parameters associated with the violation of the Lorentz symmetry.     

On the Lorentz nature of confinement in heavy—light quarkonia

The Lorentz nature of the effective interquark interaction in a heavy—light quarkonium is studied using the vacuum correlators method and the generalized Nambu—Jona-Lasinio potential quark model. An effective scalar interaction is demonstrated to appear self-consistently owing to chiral-symmetry breaking and to dominate for low-lying states in the bound-state spectrum. The relation between such an effective interquark interaction and the QCD string approach is discussed. On the contrary, the upper part of this spectrum is found to be governed by the spatial Lorentz vector interaction which leads to a degeneracy for the states with opposite parity—the so-called restoration of chiral symmetry for excited hadrons. The text was submitted by the author in English.     

Square-Root Operator Quantization and Nonlocality: A Review

A square-root-operator formalism is developed for quantum systems described with nonrelativistic and relativistic equations of motion. Spectral representation for Green's functions are designed for particles with spin 0, with the implication of its generalization to other spin values. Nonlocal operators suggest that a duality exists between physical particles and dual partners, which are tachyonic mathematical particles. It is shown that nonlocal operators result naturally from square-root operators, with the implication that microcausality holds only asymptotically. Applications help enlighten the formalism in order to envisage realistic situations with Schrödinger equations, Higgs fields, vacuum fluctuations, extra-dimensional methods in the potential theory, and electromagnetic interactions of extended charges and their consequences. It turns out that the innermost structure of these extended charges is associated with nonlocal photon propagators. It is shown that the propagator arisen from the charged torus potential consists of two different parts: a nonlocal photon propagator and a propagator of neutrino-like particles, which is described by square-root-operator equation. We examine the potential of the torus and its propagator as the appearance of superfields in terms of the photon and the massless fermion (photino).     

Is there an observable limit to Lorentz invariance at the Compton wavelength scale?

The possibility of a frame-induced violation of Lorentz invariance due to non-inertial spin-1/2 particle motion is explored in detail for muon decay while in orbit near the event horizon of a microscopic Kerr black hole. It is explicitly shown that kinematic and curvature contributions to the muon’s decay spectrum—in the absence of any unforeseen processes due to quantum gravity—lead to its stabilization at the muon’s Compton wavelength scale. This example is emblematic of the search for unambiguous indicators to critically assess current and future approaches to quantum gravity research.     

Quantum Field Theoretic Treatment of the Casimir Effect at T > 0. Real Time Formalism — Free Field Approximation

The framework of real time quantum field theory at finite temperature is generalized to include boundary conditions for the electromagnetic field strength tensor. In the perturbation theory the usual Feynman rules remain, only the photon propagator is modified. As a first application the Casimir effect is studied in the free field approximation and the known results are rederived with the new method.     

Gravitational anomalies

The effective action for fermions moving in external gravitational and gauge fields is analyzed in terms of the corresponding external field propagator. The central object in our approach is the covariant energy-momentum tensor which is extracted from the regular part of the propagator at short distances. It is shown that the Lorentz anomaly, the conformal anomaly and the gauge anomaly can be expressed in terms of the local polynomials which determine the singular part of the propagator. (There are no coordinate anomalies.) Except for the conformal anomaly, for which we give explicit representations only ind<=4, we consider an arbitrary number of dimensions.     

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.     

The Hadamard Condition for Dirac Fields and Adiabatic States on Robertson–Walker Spacetimes

We characterise the homogeneous and isotropic gauge invariant and quasifree states for free Dirac quantum fields on Robertson–Walker spacetimes. Using this characterisation, we construct adiabatic vacuum states of order n corresponding to some Cauchy surface. It is demonstrated that any two such states (of sufficiently high order) are locally quasi-equivalent. We give a microlocal characterisation of spinor Hadamard states and we show that this agrees with the usual characterisation of such states in terms of the singular behaviour of their associated twopoint functions. The polarisation set of these twopoint functions is determined and found to have a natural geometric form. We finally prove that our adiabatic states of infinite order are Hadamard, and that those of order n correspond, in some sense, to a truncated Hadamard series and therefore allow for a point splitting renormalisation of the expected stress-energy tensor. Received: 30 June 1999 / Accepted: 21 September 2000     

Collective excitations of gluons

Three gauge invariant antisymmetric tensor fields are introduced in the nonabelian gauge theories. They are certain non-linear combinations of the conjugate field tensor and they obey O(3) algebra. An effective chiral lagrangian for these fields is derived. It describes 3 vector and 3 axial mesons with vacuum quantum numbers. The masses are generated by spontaneous restoration of Lorentz invariance.     

Hořava gravity and gravitons at a conformal point

Recently Hořava proposed a renormalizable gravity theory with higher derivatives by abandoning the Lorentz invariance in UV. Here, I construct the Hořava model at λ = 1/3, where a local anisotropic Weyl symmetry exists in the UV limit, in addition to the foliation-preserving diffeomorphism. By considering linear perturbations around Minkowski vacuum for the non-projectable version of the Hořava model, I show that the scalar graviton mode is completely disappeared and only the usual tensor graviton modes remain in the physical spectrum. The existence of the UV conformal symmetry is unique to the theory with the detailed balance and this may explain the importance of the detailed balance condition in quantum gravity.     

The fifth interaction: universal long range force between spins

In this paper we present a review of our investigations on universal long range force between spins mediated by a massless axial vector gauge field which we name as “axial photon”. The invariance of the Lagrangian field theory of particles, possessing spin degrees of freedom, under local Lorentz transformations, necessitates the introduction of such an axial vector gauge field which interacts with spin current of the particles. Classical as well as quantum dynamics of electrons interacting with photon and axial photon are worked out. The new interaction is found to be asymptotically free. It is shown thatqed can be made finite if the coupling strengths of electron to photon and axial photon can be made equal. Experimental consequences of the existence of axial photon are discussed and the strength of the interaction is estimated by comparing predictions of the theory with experiments.