Transseries for causal diffusive systems

The large proper-time behaviour of expanding boost-invariant fluids has provided many crucial insights into quark-gluon plasma dynamics. Here we formulate and explore the late-time behaviour of nonequilibrium dynamics at the level of linearized perturbations of equilibrium, but without any special symmetry assumptions. We introduce a useful quantitative approximation scheme in which hydrodynamic modes appear as perturbative contributions while transients are nonperturbative. In this way, solutions are naturally organized into transseries as they are in the case of boost-invariant flows. We focus our attention on the ubiquitous telegrapher’s equation, the simplest example of a causal theory with a hydrodynamic sector. In position space we uncover novel transient contributions as well as Stokes phenomena which change the structure of the transseries based on the spacetime region or the choice of initial data.

GCM Procedure

This chapter describes the general covariant modulation (GCM) procedure in detail. It considers an axially symmetric polarized spacetime region R foliated by two functions (u, s) such that: on R, (u, s) defines an outgoing geodesic foliation as in section 2.2.4. The chapter then outlines the elliptic Hodge lemma. It also looks at the deformations of S surfaces, frame transformations, and the existence of GCM spheres. It recalls the transformation formulas recorded in Proposition 2.90, before rewriting a subset of these transformations in a more useful form. In the proof of existence and uniqueness of GCMS, one needs, in addition to the equations derived so far, an equation for the average of α‎. Finally, the chapter discusses the construction of GCM hypersurfaces.

A measure for quantum paths, gravity and spacetime microstructure

The number of classical paths of a given length, connecting any two events in a (pseudo) Riemannian spacetime is, of course, infinite. It is, however, possible to define a useful, finite, measure [Formula: see text] for the effective number of quantum paths [of length [Formula: see text] connecting two events [Formula: see text]] in an arbitrary spacetime. When [Formula: see text], this reduces to [Formula: see text] giving the measure for closed quantum loops of length [Formula: see text] containing an event [Formula: see text]. Both [Formula: see text] and [Formula: see text] are well-defined and depend only on the geometry of the spacetime. Various other physical quantities like, for e.g. the effective Lagrangian, can be expressed in terms of [Formula: see text]. The corresponding measure for the total path length contributed by the closed loops, in a spacetime region [Formula: see text], is given by the integral of [Formula: see text] over [Formula: see text]. Remarkably enough [Formula: see text], the Ricci scalar; i.e. the measure for the total length contributed by infinitesimal closed loops in a region of spacetime gives us the Einstein–Hilbert action. Its variation, when we vary the metric, can provide a new route towards induced/emergent gravity descriptions. In the presence of a background electromagnetic field, the corresponding expressions for [Formula: see text] and [Formula: see text] can be related to the holonomies of the field. The measure [Formula: see text] can also be used to evaluate a wide class of path integrals for which the action and the measure are arbitrary functions of the path length. As an example, I compute a modified path integral which incorporates the zero-point-length in the spacetime. I also describe several other properties of [Formula: see text] and outline a few simple applications.

From Green function to quantum field

A pedagogical introduction to the theory of a Gaussian scalar field which shows firstly, how the whole theory is encapsulated in the Wightman function [Formula: see text] regarded abstractly as a two-index tensor on the vector space of (spacetime) field configurations, and secondly how one can arrive at [Formula: see text] starting from nothing but the retarded Green function [Formula: see text]. Conceiving the theory in this manner seems well suited to curved spacetimes and to causal sets. It makes it possible to provide a general spacetime region with a distinguished “vacuum” or “ground state”, and to recognize some interesting formal relationships, including a general condition on [Formula: see text] expressing zero-entropy or “purity”.

Topological Effects of a Circular Cosmic String

The behaviour of a quantum particle in the spacetime region exterior to a circular cosmic string is studied by constructing a connection one-form in the tetrad formalism. In the weak-field approximation, near the string core, the space exhibits a conical singularity, with an attendant topological phase and distortion of the energy spectrum of a scalar particle determined by the global properties of the spacetime structure of the string loop.

Quantum electrodynamics in intense laser fields

A very intense laser field polarizes the virtual electron-positron pairs that populate the vacuum. This provides for a coupling between different modes of the electromagnetic field, giving rise to effects such as scattering of light by light, a refractive index of the vacuum, vacuum birefringence, etc. Given enough energy in a sufficiently small spacetime region, the virtual pairs can become real, which leads to pair production in the intense field under the action of a third agent. These, as well as related effects, are summarized with respect to their orders of magnitude and conditions under which they might become accessible to experiment. Some other processes that are normally mentioned in this context, such as Thomson (Compton) scattering at high intensities, are considered, too, even though they are unrelated to the vacuum structure of quantum electrodynamics.