Matter as Spectrum of Spacetime Representations

Bound and scattering state Schrödinger functions of nonrelativistic quantum mechanics as representation matrix elements of space and time are embedded into residual representations of spacetime as generalizations of Feynman propagators. The representation invariants arise as singularities of rational representation functions in the complex energy and complex momentum plane. The homogeneous space GL(²)U(2) with rank 2, the orientation manifold of the unitary hypercharge-isospin group, is taken as model of nonlinear spacetime. Its representations are characterized by two continuous invariants whose ratio will be related to gauge field coupling constants as residues of the related representation functions. Invariants of product representations define unitary Poincaré group representations with masses for free particles in tangent Minkowski spacetime.     

Gauge Coupling Constants as Residues of Spacetime Representations

The gauge coupling constants in the electroweak standard model can be written as mass ratios, e.g. the coupling constant for isospin interactions with the mass of the charged weak boson and the mass parameter characterizing the ground state degeneracy. A theory is given which relates the two masses in such a ratio to invariants which characterize the representations of a noncompact nonabelian group with real rank 2. The two noncompact abelian subgroups are operations for time and for a hyperbolic position space in a model for spacetime, homogeneous under dilation and Lorentz group action. The representations of the spacetime model embed the bound state representations of hyperbolic position space as seen in the nonrelativistic hydrogen atom. Interactions like Coulomb or Yukawa interactions are described by Lie algebra representation coefficients. A quantitative determination of the ratio of the invariants for position- and time-related operations, determined by the spacetime representation, gives the right order of magnitude for the gauge coupling constants.     

Representations of Spacetime as Unitary Operation Classes; or Against the Monoculture of Particle Fields

Spacetime is modeled as a homogeneous manifoldgiven by the classes of unitary U(2) operations in thegeneral complex operations GL( ). The residual representations of thisnoncompact symmetric space of rank two are characterized by two continuousreal invariants, one invariant interpreted as a particlemass for a positive unitary subgroup and the second onefor an indefinite unitary subgroup related to nonparticle interpretable interaction ranges.Fields represent nonlinear spacetimeGL( )/U(2) by theirquantization and include necessarily nonparticlecontributions in the timelike part of their flat-space Feynman propagator.     

Complexified Spacetime

It is pointed out that the Dirac position coordinates lead to an underlying non-commutative geometry, which again is symptomatic of an underlying double Weiner (Nelsonian) process.     

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 → ∓.     

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.     

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.     

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.     

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.     

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.     

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.     

Mechanics in Six-Dimensional Spacetime

The peculiarities of mechanical motion in Minkovski space with three-dimensional time are considered. A variation principle for deriving equations of motion is defined and the vector nature of energy and conservation laws for six-dimensional energy-momentum vector are discussed. Difficulties connected with vacuum instability and the possibility of anomalous nuclear reactions are removed due to the time irreversibility principle. The motion of a charged particle in a constant electric field is studied as an example of multitime processes. Some results concerning planet motion in the multitime gravitation field are presented.     

Propagation of Photons in Spacetime

A quantum field theory QED formalism is systematically developed to describe photon propagation in spacetime as a time evolution process based on the actual physical process of propagation between emitters and detectors as applied to the reflection of photons. This development, as well as early studies by Feynman, clearly show that a practical, computational and predictive dynamical formalism in spacetime was lacking. The present one generalizes to different experimental situations and other interacting field theories as well emphasizing the practicality of the problem treated here.     

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.     

Spacetime locality of the BRST formalism

The spacetime locality of the BRST formalism is investigated. The analysis covers gauge theories with either closed or open algebras and is undertaken in the explicit context of the antifield formulation of the BRST theory. Under appropriate conditions, the homology of the Koszul-Tate differential modulo the spacetime exterior derivative is shown to be trivial in the space of non-integrated densities with positive antighost and pure ghost numbers. As a result: (i) the solution of the master equation can be taken to be a local functional; (ii) the gauge fixed action is also a local functional provided one takes the gauge fixing fermion to be a local functional as well; and (iii) the BRST transformation is local.     

Quantum Communication in Rindler Spacetime

A state that an inertial observer in Minkowski space perceives to be the vacuum will appear to an accelerating observer to be a thermal bath of radiation. We study the impact of this Davies-Fulling-Unruh noise on communication, particularly quantum communication from an inertial sender to an accelerating observer and private communication between two inertial observers in the presence of an accelerating eavesdropper. In both cases, we establish compact, tractable formulas for the associated communication capacities assuming encodings that allow a single excitation in one of a fixed number of modes per use of the communications channel. Our contributions include a rigorous presentation of the general theory of the private quantum capacity as well as a detailed analysis of the structure of these channels, including their group-theoretic properties and a proof that they are conjugate degradable. Connections between the Unruh channel and optical amplifiers are also discussed.     

New Curved Spacetime Dirac Equations

I propose three new curved spacetime versions of the Dirac Equation. These equations have been developed mainly to try and account in a natural way for the observed anomalous gyromagnetic ratio of Fermions. The derived equations suggest that particles including the Electron which is thought to be a point particle do have a finite spatial size which is the reason for the observed anomalous gyromagnetic ratio. A serendipitous result of the theory, is that, to of the equation exhibits an asymmetry in their positive and negative energy solutions the first suggestion of which is clear that a solution to the problem as to why the Electron and Moun—despite their acute similarities—exhibit an asymmetry in their mass is possible. The Moun is often thought as an Electron in a higher energy state. Another of the consequences of three equations emanating from the asymmetric serendipity of the energy solutions of two of these equations, is that, an explanation as to why Leptons exhibit a three stage mass hierarchy is possible. “The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble.” Paul Adrien Maurice Dirac (1902–1984)