Forward Light Cone Research Articles

Tachyons in the Milne universe

Superluminal particles (tachyons) are studied in a Robertson-Walker cosmology with linear expansion factor and negatively curved 3-space (Milne universe). This cosmology admits globally geodesic rest frames for uniformly moving observers, isometric copies of the forward lightcone, which can be synchronized by Lorentz boosts. We investigate superluminal wave propagation, a real Proca field with negative mass-square, coupled to subluminal matter in analogy to the electromagnetic field. For photons, the eikonal approximation is exact in Robertson-Walker cosmology, and the Proca field is coupled to the background geometry in such a way that this also holds for tachyons. The spectral decomposition of freely propagating tachyon fields in the Milne universe is derived. We study the wave-particle duality in terms of the spectral elementary waves and their orthogonal ray bundles, in the comoving frame as well as in the individual geodesic rest frames of galactic observers. The spectral energy density of a tachyon background radiation is discussed.

Direct emission of multiple strange baryons in ultrarelativistic heavy-ion collisions from the phase boundary

We discuss a model for the space-time evolution of ultrarelativistic heavy-ion collisions which employs relativistic hydrodynamics within one region of the forward light-cone, and microscopic transport theory (i.e. UrQMD) in the complement. Our initial condition consists of a quark-gluon plasma which expands hydrodynamically and hadronizes. After hadronization the solution eventually changes from expansion in local equilibrium to free streaming, as determined selfconsistently by the interaction rates between the hadrons and the local expansion rate. We show that in such a scenario the inverse slopes of the m T -spectra of multiple strange baryons ( Ξ, Ω) are practically unaffected by the purely hadronic stage of the reaction, while the flow of p's and Λ's increases. Moreover, we find that the rather “soft” transverse expansion at RHIC energies (due to a first-order phase transition) is not washed out by strong rescattering in the hadronic stage. The earlier kinetic freeze-out as compared to SPS-energies results in similar inverse slopes (of the m T -spectra of the hadrons in the final state) at RHIC and SPS energies.

Modular groups of quantum fields in thermal states

For a quantum field in a thermal equilibrium state we discuss the group generated by time translations and the modular action associated with an algebra invariant under half-sided translations. The modular flows corresponding to the algebras of the forward light cone and a space-like wedge admit a simple geometric description in two-dimensional models that factorize in light-cone coordinates. At large distances from the domain boundary compared to the inverse temperature, the flow pattern is essentially the same as time translations, whereas the zero temperature results are approximately reproduced close to the edge of the wedge and the apex of the cone. For each domain there is also a one-parameter group with a positive generator, for which the thermal state is a ground state. Formally, this may be regarded as a certain converse of the Unruh effect.

Inequivalence of Jordan and Einstein Frame: What Is the Low Energy Gravity in String Theory?

It is well-known that any scalar can be promoted to a Jordan-Brans-Dicke type scalar coupling to the Einstein-Hilbert term through a field dependent Weyl transformation of the metric. The Weyl rescaling also transforms mass terms into coupling constants between matter and the scalar. It is pointed out that there exists a distinguished metric where all scalars decouple from an arbitrary fiducial fermion, e.g. the nucleon. If bound states of this fermion are used to define distances and to probe the interior of the forward light cone, it seems reasonable to say that the metric in that particular frame defines the local geometry of space-time at low energies, as probed by experimental gravity and cosmology.

The first-order perturbed SBS equations

Using standard multiscale techniques, a first-order perturbation theory for SBS is developed. In the presence of small damping, we find that there is a stationary solution for a soliton which is a fixed point. The velocity of this soliton is determined by the damping coefficients. In addition, there is also a constant shift in the pump intensity in the region between the front of the backward moving soliton and the forward light cone of the pump.

Causality in the Schrödinger picture: Diagrammatical approach.

The Schroedinger picture of Einstein causality is discussed in the framework of relativistic local field theory. The essential tool is the diagrammatic expansion of the time-evolution kernel which elucidates the propagation of the disturbance at each step of the interaction. At the level of the wave function, the causal character of the theory is rather complicated. However if the spacelike density matrix {rho}{sub {ital s}} is defined by integrating over the timelike field variables, it receives no influence of the signal. Any local disturbance is shown to propagate inside the forward light cone. It is also shown that causality is most clearly seen in the Wigner representation of the density matrix. The result automatically includes a proof of nonrelativistic causality.

Covariant collapse of the state vector and realism

We point out that the assumption that a quantum-mechanical state vector collapses in a Lorentz-covariant way upon measurement is consistent, but that it implies that the state vector, and any reality which it represents, depends on position. We consider the specific example of collapse along the forward light cone of the measurement.

Electromagnetic momenta for systems of charged particles

The electromagnetic linear and angular momenta are defined as an integral of the appropriate density over a forward lightcone. It will be shown how to relate the electromagnetic momenta, for a system of charged point particles, as an integral over finite segments of each world line. Other treatments of this problem, where the momenta is defined as the integral over a spacelike hyperplane, require the integral over infinite segments of the world lines. In general, these integrals diverge for point particles. However, the integrals over the forward lightcones are finite. Several examples are worked.

Hilbert space representations of the Poincaré group for the Landau gauge

The theory of the electromagnetic quantum field in the Landau gauge has always lacked a treatment based on a Hilbert space of C4 -valued functions (the one particle space) on which the Poincaré group acts via a continuous representation realized by bounded operators. In the present paper the gap will be filled and it will be shown, through the mathematical technique of jets on the forward light cone in momentum space, how the theory presented extends the previous treatments of the problem. In order to show that the massless vector field theory in the Landau gauge can be obtained, the equation of motion and the two-point function will be derived in the present model according to the Wightman axioms.

Second Microlocalization and the Mellin Transformation

in an open set GdR such that (t, x, j)=(0, 0, In paper [1] J.M. Bony showed that if for £ Q u is regular outside Dp 2)2, S3 and the forward light cone starting at the point Sx fl S2 fl S3. A basic tool used in the proof of this fact was, among other things, the notion of second microlocalization and in particular of second wave front set. As it is well known the notion of the first wave front set WFu of a distribution u can be described with the help of the Fourier transformation. We show that in the case of the second wave front set an analogous role is played by the Mellin transformation. Moreover we give a new proof of Bony 's theorem on propagation of 2-microlocal singularities.

Nonlocal character of the Rastall model

The discussion of whether quantum theory is compatible with the locality idea that no causal influence can act outside the forward light cone has recently been tightly linked to the corresponding question for a much simpler model theory that enjoys all of the pertinent properties of quantum theory but is much easier to fully comprehend. It is shown here that, contrary to recent claims, the model theory is incompatible with the locality idea mentioned above. The logical structure of the argument, which is similar to the one employed for quantum theory, is discussed.

Conservation of angular momentum for systems of charged particles

It is shown that the interaction part of the electromagnetic angular momentum density of a system of charged particles is integrable over any forward light cone. A covariant definition of the mechanical momentum is then given and it is shown that angular momentum is conserved.

The extended classical charged particle

A theory of the extended classical charged particle is presented. The theory assumes extension along the forward light cone of the particle instead of the usual now-plane. Solutions are given for many of the traditional problems including 4/3, instability, infinite self-energy, and runaway velocity. The Lorentz and Lorentz-Dirac equations are derived from a more general equation of motion.

Conservation of momentum for systems of charged particles

It is shown that the interaction part of the symmetric energy-momentum tensor for a system of charged particles is integrable over any forward light cone. A covariant definition of the mechanical momentum is then given and conservation of momentum is shown.

Differentiable vectors and sharp momentum states of helicity representations of the Poincaré group

This paper discusses some mathematical difficulties in handling sharp momentum eigenvectors for a massless helicity representation of the Poincaré group, related to the non-nuclearity of the space of differentiable vectors, and to the existence of singularities in the Lorentz group generators. A simple characterization of the nuclear space of differentiable vectors of the extension of the representation to a representation of the conformal group is given in terms of functions on the space R4− {0}. Using a fibration of this space over the forward light cone (in momentum space), the singularity in the generators is shown to be related to the fact that the standard presentation of the helicity representations should be reformulated in terms of nontrivial sections over the light cone. The problem is partly identical with the one encountered in the study of an electron in a magnetic monopole field, the generator singularity taking the place of the Dirac string.

The theory of superselection rules. II. Sectors of the free electromagnetic field

In the first part of this study [J. Math. Phys. 19, 1751 (1978)], a wide class of inequivalent irreducible *-representations of the C*-algebra 𝔄 of quasilocal observables of the free electromagnetic field was constructed and analyzed. A *-representation π is called positive if it satisfies the spectrum condition, i.e., if space-time automorphisms are implemented in π by a strongly continuous unitary representation of Minkowski space M4 whose infinitesimal generator has spectrum contained in the forward light cone. Here, we characterize the subclass of the class of *-representations constructed in Ref. 1, which consists of positive *-representations. This then leads to the exhibition of new superselection sectors, i.e., equivalence classes of positive *-representations.

Light-cone quantisation of the Liouville and Toda field theories

The Toda field is a multicomponent field in two space-time dimensions satisfying a generalisation of the Liouville equation ϖ 2 ϕ + exp ϕ = 0. We define the quantum field theory, and solve for the fields in terms of their initial values on a forward light-cone, demonstrating that our solution is regular. We give an explicit result for the Liouville equation which is the quantum version of the well-known classical solution. We also discuss the energy-momentum spectrum, and the conformal properties of the theory.

Local normality properties of some infrared representations

We consider the positive energy representations of the algebra of quasilocal observables for the free massless Majorana field described in preceding papers [1, 2]. We show that by an appropriate choice of the (partially) occupied one particle modes we can find irreducible, type II∞ or IIIλ representations in this class which are unitarily equivalent to the vacuum representation when restricted to any forward light cone and disjoint from it when restricted to any backward light cone, or conversely. We give an elementary explicit proof of local normality of each representation in the above class.

Dispersion relations with finite propagation speed

Considers waves generated at t=0 by prescribing Cauchy data. It is shown that, under weak assumptions the waves propagate inside the forward light cone only if Zj omega j2(k) is a polynomial in k of degree 2.

Matrix-valued special functions and representation theory of the conformal group. I. The generalized gamma function

This article examines in detail the matrix-valued gamma function \[ Γ λ 0 ( α ) = ∫ P e − tr ( r ) λ 0 ( r , r ¯ ) ( det r ) α − 2 d r {\Gamma ^{{\lambda ^0}}}\,(\alpha )\, = \,\int _P {{e^{ - {\text {tr}}(r)}}{\lambda ^0}(r,\,\bar r){{(\det \,r)}^{\alpha - 2}}\,} dr \] associated to the conformal group G = U ( 2 , 2 ) G\, = \,U(2,\,2) . Here, α \alpha is a continuous complex parameter, λ 0 {\lambda ^0} runs through a family of “weights” of K = U ( 2 ) × U ( 2 ) K\, = \,U(2)\, \times \,U(2) , P is the cone of 2 × 2 2\, \times \,2 positive-definite Hermitian matrices, and the integral is well known to converge absolutely for Re ( α ) > 1 {\text {Re}}(\alpha )\, > \,1 . However, until now very little has been known about the analytic continuation for the general weight λ 0 {\lambda ^0} . The results of this paper include the following: The complete analytic continuation of Γ λ 0 {\Gamma ^{{\lambda ^0}}} is determined for all weights λ 0 {\lambda ^0} . In analogy to the case of the classical gamma function it is proved that for any weight λ 0 {\lambda ^0} the mapping α → Γ λ 0 ( α ) − 1 \alpha \, \to \,{\Gamma ^{{\lambda ^0}}}\,{(\alpha )^{ - 1}} is entire. A new integral formula is given for the inverse of the gamma function. An explicit calculation is given for the normalized variant of the gamma matrix that arises in the reproducing kernel for the spaces in which the holomorphic discrete series of G is realized, and one observes that the behavior of the analytic continuation for weights “in general position” is markedly different from the special cases in which the gamma function “is scalar". The full analytic continuation of the holomorphic discrete series for G is determined. The gamma function for the forward light cone (the boundary orbit) is found, and the associated Hardy space of vector-valued holomorphic functions is described. Analogs are given for some of the well-known formulas for the classical gamma function. As an epilogue, applications of the matrix-valued gamma function, such as generalizations to 2 × 2 2\, \times \,2 matrix space of the classical binomial theorem, are announced. These applications require a detailed understanding of the (generalized) Bessel functions associated to the conformal group that will be treated in the sequel to this paper.