Einstein Causality Research Articles

Sign of the refractive index in a gain medium with negative permittivity and permeability

We show how the sign of the refractive index in any medium may be derived using a rigorous analysis based on Einstein causality. In particular, we consider left-handed materials, i.e., media that have negative permittivities and permeabilities at the frequency of interest. We find that the consideration of gain in such media can give rise to a positive refractive index.

Negative Refraction at Optical Frequencies in Nonmagnetic Two-Component Molecular Media

There is significant motivation to develop media with negative refractive indices at optical frequencies, but efforts in this direction are hampered by the weakness of the magnetic response at such frequencies. We show theoretically that a nonmagnetic medium with two atomic or molecular constituents can exhibit a negative refractive index. A negative index is possible even when the real parts of both the permittivity and permeability are positive. This surprising result provides a route to isotropic negative-index media at optical frequencies.

Representations of the Poincaré Semigroup and Relativistic Causality

This paper provides the mathematical tools for addressing issues of two kinds of causality in relativistic scattering theory: general causality, i.e., an effect can only be measured after its cause, and Einstein causality, i.e., no propagation of probability outside of the forward light cone. Starting from Wigner's unitary irreducible representations of the Poincare group for noninteracting, one particle states, we describe the mathematical tools necessary to describe scattering states, the Lippmann-Schwinger Dirace kets, and to describe resonances and decaying states, the relativistic Gamow ket. An important step for their derivations is the Hardy space hypothesis. Investigating the transformation properties of scattering and resonance states under the dynamical Poincare semigroup reveals that both kinds of causality result from this hypothesis about nature of the spaces of states and observables.

Causality and the Skyrme model

It is shown that despite the possibility of a breakdown of hyperbolicity, the SU(2) Skyrme model can never exhibit bulk violations of Einstein causality because its energy momentum tensor satisfies the dominant energy condition. It also satisfies the strong energy condition. The Born–Infeld–Skyrme model also satisfies both the dominant and strong energy conditions.

Electrodynamic interaction between two atoms in near-field contact.

A theoretical study of elements of the quantum electrodynamic interaction between two single-electron atoms in near-field contact is presented. The framework of the study is an electromagnetic propagator formalism that allows one to describe the near-field space-time interaction in such a manner that the Einstein causality and lack of photon localizability are manifest. First we set up the atom-field Hamiltonian in the so-called G-gauge, starting from the Coulomb Hamiltonian, and thereafter we show that this leads to a correct propagator description for the retarded part of the transverse electromagnetic field (operator). In the G-gauge approach, renormalization of the two-particle energy level structure stemming from the transverse self-field occurs. The intraparticle renormalization is calculated for a three-level atom (it is trivial for a two-level atom), and the interparticle renormalization, which depends on the atomic separation, is determined for two two-level atoms. The magnitude of the energy renormalization is always finite in our theory because we do not consider the atoms to be point-like objects from an electromagnetic point of view. Throughout, we relate our G-gauge formalism to the multipole theory often used in studies of interatomic electrodynamics.

Time asymmetric quantum theory – I. Modifying an axiom of quantum physics

AbstractA slight modification of one axiom of quantum theory changes a reversible theory into a time asymmetric theory. Whereas the standard Hilbert space axiom does not distinguish mathematically between the space of states (in‐states of scattering theory) and the space of observables (out‐“states” of scattering theory) the new axiom associates states and observables to two different Hardy subspaces which are dense in the same Hilbert space and analytic in the lower and upper complex energy plane, respectively. As a consequence of this new axiom the dynamical equations (Schrödinger or Heisenberg) integrate to a semigroup evolution. Extending this new Hardy space axiom to a relativistic theory provides a relativistic theory of resonance scattering and decay with Born probablilities that fulfill Einstein causality and the exponential decay law.

Abnormal wave propagation in passive media

Abnormal velocities in passive structures such as one-dimensional (1-D) photonic crystals and a slab having a negative index of refraction are discussed. In the case of 1-D photonic crystal, the frequency- and time-domain experiments for waves tuned to the bandgap of the photonic crystal demonstrate a positive group velocity exceeding the speed of light in vacuum (superluminal). In the case of a medium with negative index of refraction, our theoretical studies show that such a medium can support positive group and negative phase velocities (backward waves), as well as negative group and negative phase velocities. The meaning of superluminal group velocity and negative group velocity, or equally, positive superluminal group delay and negative group delay, are discussed. It is shown that despite their counterintuitive meaning there are no contradictions with the requirements of relativistic causality (Einstein causality). To clearly demonstrate this, the important subject of the front is reintroduced.

On Einstein causality and time asymmetry in quantum physics

A theorem of Hegerfeldt shows that if the spectrum of the Hamiltonian is bounded from below, then the propagation speed of certain probabilities does not have an upper bound. We prove a theorem analogous to Hegerfeldt's that appertains to asymmetric time evolutions given by a semigroup of operators. As an application, we consider a characterization of relativistic quasistable states by irreducible representations of the causal Poincare semigroup and study the implications of the new theorem for this special case.

SEARCH FOR SUPERLUMINAL PROPAGATION IN HIGH MAGNETIC FIELDS

We investigate the possibility of superluminal propagation of massive particles in high magnetic fields. Formal arguments suggest that this could occur in the quantum Hall regime near zero temperature, since here a few conducting states in the center of a broadened Landau band carry a macroscopic Hall current. We investigate the microscopic processes which lead to the high velocities of conducting states in the quantum Hall effest and ask what are the highest possible velocities which can be generated in that way. Our analysis shows that in typical quantum Hall systems at the most a tiny fraction of a carrier wave function can describe superluminal behaviour but that the center of mass velocity is far below the velocity of light. On the other hand, on pure theoretical grounds we can construct special systems (with ≪exotic≫ parameters) where the Schrödinger equation does admit solutions with superluminal center of mass velocities. The described propagation has no analogy in classical physics. Einstein's causality is not violated. The time evolution of the corresponding wave function is similar to that of a recently observed superluminal light pulse.1 We discuss analogies and differences between the two phenomena.

Relativity without relativity

We give a derivation of general relativity (GR) and the gauge principle that is novel in presupposing neither spacetime nor the relativity principle. We consider a class of actions defined on superspace (the space of Riemannian 3-geometries on a given bare manifold). It has two key properties. The first is symmetry under 3-diffeomorphisms. This is the only postulated symmetry, and it leads to a constraint linear in the canonical momenta. The second property is that the Lagrangian is constructed from a ‘local’ square root of an expression quadratic in the velocities. The square root is ‘local’ because it is taken before integration over 3-space. It gives rise to quadratic constraints that do not correspond to any symmetry and are not, in general, propagated by the Euler–Lagrange equations. Therefore these actions are internally inconsistent. However, one action of this form is well behaved: the Baierlein–Sharp–Wheeler (Baierlein R F, Sharp D and Wheeler J A 1962 Phys. Rev. 126 1864) reparametrization-invariant action for GR. From this viewpoint, spacetime symmetry is emergent. It appears as a ‘hidden’ symmetry in the (underdetermined) solutions of the Euler–Lagrange equations, without being manifestly coded into the action itself. In addition, propagation of the linear diffeomorphism constraint together with the quadratic square-root constraint acts as a striking selection mechanism beyond pure gravity. If a scalar field is included in the configuration space, it must have the same characteristic speed as gravity. Thus Einstein causality emerges. Finally, self-consistency requires that any 3-vector field must satisfy Einstein causality, the equivalence principle and, in addition, the Gauss constraint. Therefore we recover the standard (massless) Maxwell equations.

Superluminality and spatial field confinement in optical tunneling

The fact that the source domain of an electromagnetic field emitted from an electric-dipole active atom extends over the entire near-field zone of the particle is used to argue that superluminality always appears in optical tunneling processes. The superluminal response does not violate the Einstein causality because it originates solely in our inability to confine photons completely in space and thus not in field propagation (with the vacuum speed of light). The time delay in the tunneling process is shown to depend in a crucial manner on the detection process. Superluminality originating in lack of spatial field confinement in an experiment may look as if photon propagation with a speed larger than c 0 has occurred. We emphasize this by presenting for a single point dipole and an array of dipoles calculations of the apparent velocity of field propagation, and show that this velocity depends on the detector threshold level and the source detector distance.

On the quantum physical relation between photon tunnelling and near-field optics.

The standard theory of optical tunnelling based on a mathematical analogy between the time-independent one-particle Schrödinger equation describing electron tunnelling and the Helmholtz equation for the macroscopic electric field is criticized. In a classical perspective photons are related to the dynamics of the transverse part of the electromagnetic field, and with the assumption that the electrons in a coupled photon-atom system are driven in a linear fashion by the field, a rigorous integro-differential equation for the transverse field is established. In the near-field zone of matter a transverse self-field exists and this may cause the appearance of superluminality in optical tunnelling. In the wake of a brief review of the space-time dynamics of free photons, a first-quantized description of the birth process of a single polychromatic photon in the near-field region of an active atom (molecule, mesoscopic particle) is presented, and afterwards the link between photon localizability, Einstein causality and near-field photon tunnelling is discussed. On the basis of a new one-photon quantum theory of near-field scattering from a microscopic (or mesoscopic) object, it is shown that photon tunnelling always appears in photon-atom scattering. A first-order Born approximation cannot capture the phenomenon, however. The emergence of the energy wave function of the scattered photon is followed in space and time.

Search for superluminal velocities in quantum Hall and related systems

We investigate the possibility of superluminal propagation of massive particles in high magnetic fields. Formal arguments suggest that this may occur in the quantum Hall effect. Our microscopic analysis shows that in typical quantum Hall systems at the most a tiny fraction of a carrier wave function can describe superluminal behaviour but that the centre of mass velocity is far below c. On the other hand, in special (“exotic”) systems the Schrödinger equation does admit solutions with superluminal centre of mass velocities. The described propagation has no analogy in classical physics. Einstein's causality is not violated.

Propagation speed of evanescent modes

The group velocity of evanescent waves (in undersized waveguides, for instance) was theoretically predicted, and has been experimentally verified, to be superluminal (v(g)>c). By contrast, it is known that the precursor speed in vacuum cannot be larger than c. In this paper, by computer simulations based on Maxwell equations only, we show the existence of both phenomena. In other words, we verify the actual possibility of superluminal group velocities, without violating the so-called (naive) Einstein causality.

Time-domain detection of superluminal group velocity for single microwave pulses

Single microwave pulses centered at 9.68 GHz with 100-MHz (full width at half maximum) bandwidth are used to evanescently tunnel through a one-dimensional photonic crystal. In a direct time-domain measurement, it is observed that the peak of the tunneling wave packets arrives (440+/-20) ps earlier than the companion free space (air) wave packets. Despite this superluminal behavior, Einstein causality is not violated since the earliest parts of the signal, also known as the Sommerfeld forerunner, remain exactly luminal. The frequency of oscillations and the functional form of the Sommerfeld forerunner for any causal medium are derived.

Exact relativistic time evolution for a step potential barrier

We derive an exact analytic solution to a Klein-Gordon equation for a step potential barrier with cutoff plane wave initial conditions, in order to explore wave evolution in a classical forbidden region. We find that the relativistic solution rapidly evanesces within a depth $2x_p$ inside the potential, where $x_p$ is the penetration length of the stationary solution. Beyond the characteristic distance $2x_p$, a Sommerfeld-type precursor travels along the potential at the speed of light, $c$. However, no spatial propagation of a main wavefront along the structure is observed. We also find a non-causal time evolution of the wavefront peak. The effect is only an apparent violation of Einstein causality.

SPECIAL RELATIVITY AND SUPERLUMINAL MOTIONS: A DISCUSSION OF SOME RECENT EXPERIMENTS

Some experiments, performed at Berkeley, Cologne, Florence, Vienna, Orsay and Rennes led to the claim that something seems to travel with a group velocity larger than the speed c of light in vacuum. Various other experimental results seem to point in the same direction: For instance, localized wavelet-type solutions of Maxwell equations have been found, both theoretically and experimentally, that travel with Superluminal speed. Even muonic and electronic neutrinos — it has been proposed — might be "tachyons," since their square mass appears to be negative. With regard to the first-mentioned experiments, it was very recently claimed by Guenter Nimtz that those results with evanescent waves or "tunneling photons" — implying Superluminal signal and impulse transmission — violate Einstein causality. In this note, on the contrary, we want to stress that all such results do not place relativistic causality in jeopardy, even if they refer to actual tachyonic motions: In fact, special relativity can cope even with Superluminal objects and waves. For instance, it is possible (at least in microphysics) to solve also the known causal paradoxes, devised for "faster than light" motion, even if this is not widely recognized. Here we show, in detail and rigorously, how to solve the oldest causal paradox, originally proposed by Tolman, which is the kernel of many further tachyon paradoxes. The key to the solution is a careful application of tachyon mechanics, as it unambiguously follows from special relativity.

New concepts in particle physics from the solution of an old problem

Recent ideas on modular localization in local quantum physics are used to clarify the relation between on- and off-shell quantities in particle physics; in particular the relation between on-shell crossing symmetry and off-shell Einstein causality. Among the collateral results of this new nonperturbative approach are profound relations between crossing symmetry of particle physics and Hawking-Unruh like thermal aspects (KMS property, entropy attached to horizons) of quantum matter behind causal horizons, aspects which hitherto were exclusively related with Killing horizons in curved spacetime rather than with localization aspects in Minkowski space particle physics. The scope of this modular framework is amazingly wide and ranges from providing a conceptual basis for the d=1+1 bootstrap-formfactor program for factorizable d=1+1 models to a decomposition theory of QFT's in terms of a finite collection of unitarily equivalent chiral conformal theories placed a specified relative position within a common Hilbert space (in d=1+1 a holographic relation and in higher dimensions more like a scanning). The new framework gives a spacetime interpretation to the Zamolodchikov-Faddeev algebra and explains its thermal aspects.

Causality, delocalization, and positivity of energy

In a series of interesting papers G. C. Hegerfeldt has shown that quantum systems with positive energy initially localized in a finite region, immediately develop infinite tails. In our paper Hegerfeldt's theorem is analysed using quantum and classical wave packets. We show that Hegerfeldt's conclusion remains valid in classical physics. No violation of Einstein's causality is ever involved. Using only positive frequencies, complex wave packets are constructed which at $t = 0$ are real and finitely localized and which, furthemore, are superpositions of two nonlocal wave packets. The nonlocality is initially cancelled by destructive interference. However this cancellation becomes incomplete at arbitrary times immediately afterwards. In agreement with relativity the two nonlocal wave packets move with the velocity of light, in opposite directions.

Time diffraction of evanescent waves.

The concept of time diffraction of evanescent electromagnetic waves is proposed. Transient propagation of electromagnetic waves is studied in two forms: propagating and evanescent waves. Differences and similarities between quantum particle tunneling and photon tunneling are clearly demonstrated. Traversal time cannot be accurately defined due to diffraction in time. Nevertheless, a delay time is defined as the difference between the time of flight and the peak time of the transient light. The delay time is found to increase linearly and slowly with the traversal distance. Superluminal tunneling is found possible only for evanescent waves, and Einstein causality is not violated.