Quantum Correlations,Nuclearity, and Spacetime Curvature

It is proved for a Haag–Araki–Kastler quantum field theory, that gravitation reduces the correlations in the vacuum state. Secondly, we prove Bell's inequalities by nuclearity assumptions. The so-called -content of certain compact mappings restricts the size of the set of measurements which violate Bell's inequalities.     

Spacetime Quantization, Elementary Particles, and Cosmology

Relativistic quantum mechanics is generalized to account for a universally constant quantum of length a. Its value depends on the total convertible energy content of our universe: E_u = hc/2a. The eigenvalues of all (x,y,z,ct) coordinates are integer or half-integer multiples of a in every particular inertial frame. There are thus several spacetime lattices of lattice-constant a: the normal lattice contains the origin of the chosen frame, while inserted lattices are displaced by a/2 along one or several reference axes. States of motion are defined by possible variations of -functions on any one of these lattices. Particle states are defined by their relative phases, specified by four new quantum numbers, u_x, u_y, u_z, u_ct = 0, ±1, ±2,.... They account for all known elementary particles and yield a natural extension of the standard model. Spacetime quantization solves also the EPR paradox and other difficulties that subsisted in the usual continuum theories. It defines inertial frames and is related to cosmology.     

Contracted Representation of Yang's Spacetime Algebra and Buniy-Hsu-Zee's Discrete Spacetime

Motivated by the recent proposition by Buniy, Hsu, and Zee with respect to discrete spacetime and finite spatial degrees of freedom of our physical world with short- and long-distance scales, l _P and L, we reconsider the Lorentz-covariant Yang's quantized spacetime algebra (YSTA), which is intrinsically equipped with two such kinds of scale parameters, λ and R. In accordance with their proposition, we find the so-called contracted representation of YSTA with finite spatial degrees of freedom associated with the ratio R/λ, which gives a possibility of the divergence-free noncommutative field theory on YSTA. The canonical commutation relations familiar in the ordinary quantum mechanics appear as the cooperative Inonu-Wigner's contraction limit of YSTA, λ → 0 and R → ∓.     

Spacetime quantization,generalized relativistic mechanics,and Mach's principle

The introduction of an elementary lengtha representing the ultimate limit for the smallest measurable distance leads to a generalization of Einstein's energy-momentum relation and of the usual Lorentz transformation. The value ofa is left unspecified, but is found to be equal tohc/2E _u, whereE _u is the total energy content of our universe. Particles of zero rest mass can only move at the velocityc of light in vacuum, while material bodies can move slower or faster than light, whena0, without violating the principle of causality. The laws of relativistic mechanics are actually generalized so that they include Mach's principle, since it is found that the universe as a whole can only be in a state of rest for any particular inertial observer.     

Spacetime and Euclidean geometry

Using only the principle of relativity and Euclidean geometry we show in this pedagogical article that the square of proper time or length in a two-dimensional spacetime diagram is proportional to the Euclidean area of the corresponding causal domain. We use this relation to derive the Minkowski line element by two geometric proofs of the spacetime Pythagoras theorem.This article is dedicated to Michael P. Ryan on the occasion of his sixtieth birthday. Mike's passion for, and deft practice of, both geometry and pedagogy is legendary at Maryland. We are pleased with this opportunity to present our pedagogical effort to elucidate the geometry of Minkowski spacetime, the most homogeneous of cosmologies.     

Realism,operationalism, and quantum mechanics

A comprehensive formal system is developed that amalgamates the operational and the realistic approaches to quantum mechanics. In this formalism, for example, a sharp distinction is made between events, operational propositions, and the properties of physical systems.     

Polarized Spacetime Foam

An approximate model of a spacetime foam is presented. It is supposed that in the spacetime foam each quantum handle is like to an electric dipole and therefore the spacetime foam is similar to a dielectric. If we neglect of linear sizes of the quantum handle then it can be described with an operator containing a Grassman number and either a scalar or a spinor field. For both fields the Lagrangian is presented. For the scalar field it is the dilaton gravity + electrodynamics and the dilaton field is a dielectric permeability. The spherically symmetric solution in this case give us the screening of a bare electric charge surrounded by a polarized spacetime foam and the energy of the electric field becomes finite one. In the case of the spinor field the spherically symmetric solution give us a ball of the polarized spacetime foam filled with the confined electric field. It is shown that the full energy of the electric field in the ball can be very big.     

Spacetime vector analysis

Ordinary Gibbs-Heaviside vector algebra is complexified to apply to spacetime. The resulting algebra is isomorphic to both the Pauli algebra, and to the algebra of complex quaternions. Each inertial system is distinguished by a rest frame of real vectors. The rudiments of a spacetime vector analysis are given.     

Mach's Principle and Minkowski Spacetime

It is shown that when the Minkowski metric is approached by a limiting process using two different static, spherically-symmetric, closed cosmological models, that although the energy-stress tensors for the Einstein-Friedmann field equations vanishes, their integral does not. Since part of this integral consists of the mass of the incoherent dust background, which is the same in both models, the Minkowski metric obtained by this limiting process cannot be regarded as anti-Machian, since there is an infinite amount of ponderable matter in the background, albeit at vanishing density. One of the models is the Einstein static universe with its cosmological term. The other model does not employ this term, but instead uses a tensor that has vanishing trace, negative energy density and negative pressure. Gravitational energy is also studied, and it is pointed out that for both models, this energy becomes infinitely negative in the Minkowski limit.     

Rotating Kink Spacetime

A suitably chosen complex parametrization of the 3-sphere is used to construct a (3 + 1)-dimensional spacetime that is homogeneous and satisfies the various standard energy conditions. The spacetime has nonzero vorticity, closed timelike curves and is shown to possess a Finkelstein-Misner kink. Hopf projection from the 3-sphere to a 2-sphere reduces the model to a previously known toy model in lower dimensions.     

Spacetime symplectic extension

It is conjectured that in the origin of spacetime there lies a symplectic rather than metric structure. The complex symplectic symmetry Sp(2l, C), l≥1 instead of the pseudoorthogonal one SO(1, d?1), d≥4 is proposed as the spacetime local structure group. A discrete sequence of the metric spacetimes of the fixed dimensionalities d=(2l)² and signatures, with l(2l?1) timelike and l(2l+1) spacelike directions, defined over the set of Hermitian second-rank spin tensors, is considered as an alternative to the pseudo-Euclidean extra dimensional spacetimes. The basic concepts of the symplectic framework are developed in general, and the ordinary and next-to-ordinary spacetime cases with l=1, 2, respectively, are elaborated in more detail. In particular, the scheme provides the rationale for the four-dimensionality and 1+3 signature of the ordinary spacetime.     

Spacetime Metric and Lightcone Fluctuations

Several aspects of the quantum fluctuations ofspacetime geometry are discussed. A model for lightconefluctuations is described in which a bath of gravitonsleads to metric fluctuations. The operational definitions of such phenomena as lightcone andhorizon fluctuations are examined. The problem ofdescribing fluctuations of a quantum stress tensor isalso discussed. The possibility that one can gain some insights about spacetime geometry fluctuationsfrom studies of the force fluctuations on materialbodies is suggested.     

Spacetime and future quantum theory

Space and time are discussed in connection with the future of quantum theory.     

Building a Quantum Spacetime

A quantum spacetime is constructed from the freedata given at past null infinity. One starts with afield equation for a scalar function Z on the initialsurface and then shows that the solution depends on four constants of integration. Theseconstants become the spacetime points and the levelsurfaces of the scalar function, i.e., Z = const, becomenull hypersurfaces on the derived spacetime. A phase space together with a complex structure areconstructed on past null infinity. This Hilbert space ofincoming gravitons possesses a natural foliation whichdefines superselection sectors on the space of asymptotic quantum states. The dynamics of nullsurface quantization provides spacetime-valued quantumoperators on the superselection sectors. It is shownthat the spacetime points themselves become operators with nonvanishing commutationrelations.     

A Note on Algebraically Special,Asymptotically Simple and Empty Spacetime

It is proved that subject to certain physically reasonable restrictions on the curvature of spacetime, an algebraically special, asymptotically simple and empty spacetime is necessarily a copy of the Minkowski space. This imposes further requirements a nontrivial asymptotically simple and empty spacetime has to satisfy.     

Spacetime Holography and the HOPF Fibration

No Heading Quantum mechanics and general relativity share an equivalence with respect to the holographic principle. Large-scale fluctuations predicted by the holographic principle may be derived from the quantum mechanics of spin. As holographic theories, quantum mechanics and general relativity in quaternionic bases are formally similar. Gravitation may not be properly quantized and unified with quantum fields in the usual manner, but rather gravitation and Dirac quantum fields as two separate spinor fields that form pairs which define octonions.     

Gravity as Spacetime Curvature

Einstein's general theory of relativity conceives the phenomena of gravity as manifestations of the curvature of the spacetime manifold in which physical events take place. I sketch the line of thought that led Einstein to this conception, and I briefly discuss proposals by Jeffreys and Feynman for retaining Einstein's gravitational field equations while discarding their purportedly geometrical meaning.