A short note on extended probability theory
We propose two distinct interpretations of extended probabilities which are realistic for the physical world.
quant-ph↗
arXiv subjects
Johan Noldus
Publications and source records attributed to Johan Noldus.
Free Fermions on causal sets
We construct a Dirac theory on causal sets; a key element in the construction being that the causet must be regarded as emergent in an appropriate sense too. We further notice that mixed norm spaces appear in the construction allowing for negative norm particles and "ghosts".
gr-qc↗
Lorentzian Manifolds and Causal Sets as Partially Ordered Measure Spaces
We consider Lorentzian manifolds as examples of partially ordered measure spaces, sets endowed with compatible partial order relations and measures, in this case given by the causal structure and the volume element defined by each Lorentzian metric. This places the structure normally used to describe spacetime in geometrical theories of gravity in a more general context, which includes the locally finite partially ordered sets of the causal set approach to quantum gravity. We then introduce a function characterizing the closeness between any two partially ordered measure spaces and show that, when restricted to compact spaces satisfying a simple separability condition, it is a distance. In particular, this provides a quantitative, covariant way of describing how close two manifolds with Lorentzian metrics are, or how manifoldlike a causal set is.
gr-qc↗
Should quantum theory change in the light of gravity?
In this paper, we study implications of the geometrical nature of space- time for some of the basic tenets of quantum mechanics. That is, we study two different implications of the principle of general covariance; first we quantize a reparametrization invariant theory, the free particle in Minkowski spacetime and point out in detail where this theory fails (no- tably these comments appear to be missing in the literature). Second we study the covariance of quantum field theory and show how it connects to causality, the outcome of this study is that QFT is what we shall call ultra weakly covariant with respect to the background spacetime. Third, we treat the question of whether evolution in quantum theory (apart from the measurement act) needs to be unitary, it is easily shown that a per- fectly satisfying probabilistic interpretation exists which does not require unitary evolution. Fourth, we speculate on some modifications quantum theory should undergo in order for it to be generally covariant. The results in this paper hint at a profound change of the theory in which causality as a fundamental principle is abandonned.
gr-qc↗
Interpretation of quantal manifolds
In quantum gravity, one looks for alternative structures to spacetime physics than ordinary real manifolds. Here, we propose an alternative universal construction containing the latter as an equilibrium state under the action of the universal diffeomorphism group. Our theory contains many other previous proposals in the literature as special cases. However, the crucial point we make is that those have to be appreciated in the universal context developed here.
gr-qc↗
Foundations of a theory of quantum gravity
After a long technical and consequently philosophical disgression about the necessity of the construction presented in this book, a logically consistent and precise theory of quantum gravity is presented. The construction of this theory goes in several steps; at first we take a fairly conservative point of view and stumble upon some technical difficulties. Consequently, we investigate a new mathematical implication of an old idea to solve these problems; the latter suggest however a completely new way of doing quantum theory. This novel theory automatically incoorporates an extended form of gravity as well as a quantum gauge theory. We compute that the well known free quantum field theories and Newtonian gravity emerge in a suitable limit. The philosophy constructed here is to a high degree Whiteheadian.
physics.gen-ph↗
A new topology on the space of Lorentzian metrics on a fixed manifold
We give a covariant definition of closeness between (time oriented) Lorentzian metrics on a manifold M, using a family of functions which measure the difference in volume form on one hand and the difference in causal structure relative to a volume scale on the other hand. These functions will distinguish two geometric properties of the Alexandrov sets $ A(p,q), \tilde{A} (p,q) $ relative to two space time points $q$ and $p$ and metrics $g$ and $ \tilde{g} $. It will be shown that this family generates uniformities and consequently a topology on the space of Lorentzian metrics which is Hausdorff when restricted to strongly causal metrics. This family of functions will depend on parameters for a volume scale, a length scale (relative to the volume scale) and an index which labels a submanifold with compact closure of the given manifold M.
gr-qc↗
Generalized Gravity I : Kinematical Setting and reformalizing Quantum Field Theory
The first part of this work deals with the development of a natural differential calculus on non-commutative manifolds. The second part extends the covariance and equivalence principle as well studies its kinematical consequences such as the arising of gauge theory. Furthermore, a manifestly causal and covariant formulation of quantum field theory is presented which surpasses the usual Hamiltonian and path integral construction. A particular representation of this theory on the kinematical structure developed in section three is moreover given.
physics.gen-ph↗
Towards a fully consistent relativistic quantum mechanics and a change of perspective on quantum gravity
This paper can be seen as an exercise in how to adapt quantum mechanics from a strict relativistic perspective while being respectful and critical towards the experimental achievements of the contemporary theory. The result is a fully observer independent relativistic quantum mechanics for N particle systems without tachyonic solutions. A remaining worry for the moment is Bell's theorem.
gr-qc↗
A general symmetry preserving reduction scheme and normal form for dynamical systems with a compact symmetry group
We present a generalized Lyapunov Schmidt reduction scheme for diffeomorphisms living on a finite dimensional real vector space V which transform under real one dimensional characters of an arbitrary compact group with linear action V. Moreover we prove a normal form theorem, such that the normal form still has the desirable transformation properties with respect to the character.
math.KT↗
The moduli space of isometry classes of globally hyperbolic spacetimes
This is the last article in a series of three initiated by the second author. We elaborate on the concepts and theorems constructed in the previous articles. In particular, we prove that the GH and the GGH uniformities previously introduced on the moduli space of isometry classes of globally hyperbolic spacetimes are different, but the Cauchy sequences which give rise to well-defined limit spaces coincide. We then examine properties of the strong metric introduced earlier on each spacetime, and answer some questions concerning causality of limit spaces. Progress is made towards a general definition of causality, and it is proven that the GGH limit of a Cauchy sequence of $\mathcal{C}^{\pm}_α$, path metric Lorentz spaces is again a $\mathcal{C}^{\pm}_α$, path metric Lorentz space. Finally, we give a necessary and sufficient condition, similar to the one of Gromov for the Riemannian case, for a class of Lorentz spaces to be precompact.
gr-qc↗
Lorentzian Gromov Hausdorff theory as a tool for quantum gravity kinematics
This thesis start by a review of different approaches to classical and quantum gravity. The main theme is Lorentzian Gromov Hausdorff theory which is an active diffeomorphism invariant theory on the space of Lorentz spaces (think about globally hyperbolic spacetimes). It is argued why such theory might be of significant importance for Lorentzian approaches to quantum gravity such as causal set theory and Lorentzian dynamical triangulations
gr-qc↗
A Lorentzian Gromov-Hausdoff notion of distance
This paper is the first of three in which I study the moduli space of isometry classes of (compact) globally hyperbolic spacetimes (with boundary). I introduce a notion of Gromov-Hausdorff distance which makes this moduli space into a metric space. Further properties of this metric space are studied in the next papers. The importance of the work can be situated in fields such as cosmology, quantum gravity and - for the mathematicians - global Lorentzian geometry.
gr-qc↗
The limit space of a Cauchy sequence of globally hyperbolic spacetimes
In this second paper, I construct a limit space of a Cauchy sequence of globally hyperbolic spacetimes. In the second section, I work gradually towards a construction of the limit space. I prove the limit space is unique up to isometry. I als show that, in general, the limit space has quite complicated causal behaviour. This work prepares the final paper in which I shall study in more detail properties of the limit space and the moduli space of (compact) globally hyperbolic spacetimes (cobordisms). As a fait divers, I give in this paper a suitable definition of dimension of a Lorentz space in agreement with the one given by Gromov in the Riemannian case.
gr-qc↗