Finitary Spacetime Sheaves

The notion of finitary spacetime sheaves is introduced based on locally finiteapproximations of the continuous topology of a bounded region of a spacetimemanifold. Finitary spacetime sheaves are seen to be sound mathematical modelsof approximations of continuous spacetime observables.     

Algebraic Quantization of Causal Sets

A scheme for an algebraic quantization of the causal sets of Sorkin et al. ispresented. The suggested scenario is along the lines of a similar algebraizationand quantum interpretation of finitary topological spaces due to Zapatrin and thisauthor. To be able to apply the latter procedure to causal sets Sorkin's 'semanticswitch' from 'partially ordered sets as finitary topological spaces' to 'partiallyordered sets as locally finite causal sets' is employed. The result is the definition of'quantum causal sets'. Such a procedure and its resulting definition are physicallyjustified by a property of quantum causal sets that meets Finkelstein's requirementfor 'quantum causality' to be an immediate, as well as an algebraically represented,relation between events for discrete locality's sake. The quantum causal setsintroduced here are shown to have this property by direct use of a result fromthe algebraization of finitary topological spaces due to Breslav, Parfionov, andZapatrin.     

Quantum Geometry on Quantum Spacetime: Distance,Area and Volume Operators

We develop the first steps towards an analysis of geometry on the quantum spacetime proposed in Doplicher et al. (Commun Math Phys 172:187–220, 1995). The homogeneous elements of the universal differential algebra are naturally identified with operators living in tensor powers of Quantum Spacetime; this allows us to compute their spectra. In particular, we consider operators that can be interpreted as distances, areas, 3- and 4-volumes.     

The Spinning and Curving of Spacetime: The Electromagnetic and Gravitational Fields in the Evans Field Theory

The unification of the gravitational and electromagnetic fields achieved geometrically in the generally covariant unified field theory of Evans implies that electromagnetism is the spinning of spacetime and gravitation is the curving of spacetime. The homogeneous unified field equation of Evans is a balance of spacetime spin and curvature and governs the influence of electromagnetism on gravitation using the first Bianchi identity of differential geometry. The second Bianchi identity of differential geometry is shown to lead to the conservation law of the Evans unified field, and also to a generalization of the Einstein field equation for the unified field. Rigorous mathematical proofs are given in appendices of the four equations of differential geometry which are the cornerstones of the Evans unified field theory: the first and second Maurer-Cartan structure relations and the first and second Bianchi identities. As an example of the theory, the origin of wavenumber and frequency is traced to elements of the torsion tensor of spinning spacetime.     

Propagation of Relativistic Quantum Particles in Spacetime in Quantum Field Theory

The paper deals with a recent systematic study of the propagation of relativistic quantum particles in spacetime. This study was a reaction to the overwhelming number of experiments dealing with the localization of not only massive but also of photons by detectors. The method of study is based on a configuration unitarity expansion of the vacuum-to-vacuum transition amplitudes as, massive and massless, particles propagate between emitters and detectors. Topics treated are the amplitudes of propagation from one time-space coordinate to another, limiting velocities of particles and their reconciliations with relativity, emergence of particles into cones in detection regions versus the direction of their moments, stimulated emissions by external sources in spacetime, scattering theory in quantum field theory in configuration space, and finally a spacetime for mulation of closed-time path for multi-particle states.     

Lessons from Quantum Field Theory: Hopf Algebras and Spacetime Geometries

We discuss the prominence of Hopf algebras in recent progress in Quantum Field Theory. In particular, we will consider the Hopf algebra of renormalization, whose antipode turned out to be the key to a conceptual understanding of the subtraction procedure. We shall then describe several occurrences of this, or closely related Hopf algebras, in other mathematical domains, such as foliations, Runge-Kutta methods, iterated integrals and multiple zeta values. We emphasize the unifying role which the Butcher group, discovered in the study of numerical integration of ordinary differential equations, plays in QFT.     

Quantum Charges and Spacetime Topology: The Emergence of New Superselection Sectors

A new form of superselection sectors of topological origin is developed. By that it is meant a new investigation that includes several extensions of the traditional framework of Doplicher, Haag and Roberts in local quantum theories. At first we generalize the notion of representations of nets of C*–algebras, then we provide a brand new view on selection criteria by adopting one with a strong topological flavour. We prove that it is coherent with the older point of view, hence a clue to a genuine extension. In this light, we extend Roberts’ cohomological analysis to the case where 1–cocycles bear non-trivial unitary representations of the fundamental group of the spacetime, equivalently of its Cauchy surface in the case of global hyperbolicity. A crucial tool is a notion of group von Neumann algebras generated by the 1–cocycles evaluated on loops over fixed regions. One proves that these group von Neumann algebras are localized at the bounded region where loops start and end and to be factorial of finite type I. All that amounts to a new invariant, in a topological sense, which can be defined as the dimension of the factor. We prove that any 1–cocycle can be factorized into a part that contains only the charge content and another where only the topological information is stored. This second part much resembles what in literature is known as geometric phases. Indeed, by the very geometrical origin of the 1–cocycles that we discuss in the paper, they are essential tools in the theory of net bundles, and the topological part is related to their holonomy content. At the end we prove the existence of net representations. Dedicated to Klaus Fredenhagen on the occasion of his sixtieth birthday     

The Consistency of Causal Quantum Geometrodynamics and Quantum Field Theory

We consider quantum geometrodynamics and parametrized quantum field theories in the framework of the Bohm-de Broglie interpretation. In the first case, and following the lines of our previous work [1], where a hamiltonian formalism for the bohmian trajectories was constructed, we show the consistency of the theory for any quantum potential, completing the scenarios for canonical quantum cosmology presented there. In the latter case, we prove the consistency of scalar field theory in Minkowski spacetime for any quantum potential, and we show, using this alternative hamiltonian method, a concrete example where Lorentz invariance of individual events is broken.     

The Phase of a Quantum Mechanical Particle in Curved Spacetime

We investigate the quantum mechanical wave equations for free particles of spin 0, 1/2, 1 in the background of an arbitrary static gravitational field in order to explicitly determine if the phase of the wavefunction is S/ = p dx /, as is often quoted in the literature. We work in isotropic coordinates where the wave equations have a simple manageable form and do not make a weak gravitational field approximation. We interpret these wave equations in terms of a quantum mechanical particle moving in medium with a spatially varying effective index of refraction. Due to the first order spatial derivative structure of the Dirac equation in curved spacetime, only the spin 1/2 particle has exactly the quantum mechanical phase as indicated above. The second order spatial derivative structure of the spin 0 and spin 1 wave equations yield the above phase only to lowest order in . We develop a WKB approximation for the solution of the spin 0 and spin 1 wave equations and explore amplitude and phase corrections beyond the lowest order in . For the spin 1/2 particle we calculate the phase appropriate for neutrino flavor oscillations.     

On Explaining Quantum Correlations: Causal vs. Non-Causal

At the basis of the problem of explaining non-local quantum correlations lies the tension between two factors: on the one hand, the natural interpretation of correlations as the manifestation of a causal relation; on the other, the resistance on the part of the physics underlying said correlations to adjust to the most essential features of a pre-theoretic notion of causation. In this paper, I argue for the rejection of the first horn of the dilemma, i.e., the assumption that quantum correlations call for a causal explanation. The paper is divided into two parts. The first, destructive, part provides a critical overview of the enterprise of causally interpreting non-local quantum correlations, with the aim of warning against the temptation of an account of causation claiming to cover such correlations ‘for free’. The second, constructive, part introduces the so-called structural explanation (a variety of non-causal explanation that shows how the explanandum is the manifestation of a fundamental structure of the world) and argues that quantum correlations might be explained structurally in the context of an information-theoretic approach to QT.     

Quantum Kinematic Theory of the Poincare Group in Two-Dimensional Spacetime

Non-Abelian quantum kinematics is applied to thePoincare group P ₊ (1, 1),as an example of the quantization-through-the-symmetryapproach to quantum mechanics. Upon quantizing thegroup, generalized Heisenberg commutation relations are obtained, and aclosed Heisenberg–Weyl algebra follows. Then,according to the general theory, the three basicquantum-kinematic invariant operators are calculated;these afford the superselection rules for diagonalizing theincoherent rigged Hilbert space H(P ₊ ) of the regularrepresentation. This paper examines only one of thesediagonalization schemes, while introducing a irreducible spacetime representation carried by isotopicplane-wave eigenvectors of two compatible superselectionoperators (which define a Poincare-invariant linear2-momentum). Thereafter, the principle of microcausality produces massive 2-spinor isotopic states in 1+ 1 Minkowski space. The Dirac equation is thus deducedwithin the quantum kinematic formalism, and the familiarJordan–Pauli propagation kernel in 2-dimensional spacetime is also obtained as a Hurwitzinvariant integral over the group manifold. The maininterest of this approach lies in the adoptedgroup-quantization technique, which is a strictlydeductive method and uses exclusively the assumed Poincaresymmetry.     

Implications of Local Friendliness Violation for Quantum Causality

We provide a new formulation of the Local Friendliness no-go theorem of Bong et al. [Nat. Phys. 16, 1199 (2020)] from fundamental causal principles, providing another perspective on how it puts strictly stronger bounds on quantum reality than Bell’s theorem. In particular, quantum causal models have been proposed as a way to maintain a peaceful coexistence between quantum mechanics and relativistic causality while respecting Leibniz’s methodological principle. This works for Bell’s theorem but does not work for the Local Friendliness no-go theorem, which considers an extended Wigner’s Friend scenario. More radical conceptual renewal is required; we suggest that cleaving to Leibniz’s principle requires extending relativity to events themselves.     

A Local-Realistic Model of Quantum Mechanics Based on a Discrete Spacetime

This paper presents a realistic, stochastic, and local model that reproduces nonrelativistic quantum mechanics (QM) results without using its mathematical formulation. The proposed model only uses integer-valued quantities and operations on probabilities, in particular assuming a discrete spacetime under the form of a Euclidean lattice. Individual (spinless) particle trajectories are described as random walks. Transition probabilities are simple functions of a few quantities that are either randomly associated to the particles during their preparation, or stored in the lattice nodes they visit during the walk. QM predictions are retrieved as probability distributions of similarly-prepared ensembles of particles. The scenarios considered to assess the model comprise of free particle, constant external force, harmonic oscillator, particle in a box, the Delta potential, particle on a ring, particle on a sphere and include quantization of energy levels and angular momentum, as well as momentum entanglement.     

Classic and Quantum Motions of Scalar Particle in Beltrami-de Sitter Spacetime

From a five-dimensional Minkowski view the five-dimensional angular momentum of a free spin-O particle moving in de Sifter spacefime is conservative, by which its fimelike geodesics can be labeled completely and uniquely. Based on that observation and working in Belframi coordinate for de Sifter spacefime, solutions and their asymptotic behavior to the Belframi-de Sifter-Klein-Gordon equation are given.     

Measures Defined on Quantum Logics of Sets

We study families formed with subsets of any set X which are quantum logics but which are not Boolean algebras. We consider sequences of measures defined on a sets quantum logics and valued on an effect algebra and obtain a sufficient condition for a sequences of such measures to be uniformly strongly additive with respect to order topology of effect algebras.