Exact combinatorics of Bern-Kosower-type amplitudes for two-loop φ3 theory

Counting the contribution rate of a world-line formula to Feynman diagrams in (φ³ theory, we explain the idea of how to determine precise combinatorics of Bern-Kosower-like amplitudes derived from a bosonic string theory for N-point two-loop Feynman amplitudes. In this connection we also present a method to derive simple and compact world-line forms for the effective action.     

Kerr-Schild metric cannot have an accelerating source

We give a simple proof that if the source of a vacuum electro-magnetic or purely gravitational Kerr-Schild metric is a world-line in the background Minkowski space-time then the 4-acceleration of the world-line must vanish.     

Particles,twistors and the division algebras

We study twistorial mechanics of particles and super-particles in six dimensions. To this end we formulate (in a general division algebra framework) a twistor theory in D = 6 based on quaternionic numbers, and prove the equivalence between this version of particle dynamics and the ordinary one. The super-twistors define a covariant and gauge invariant concept of a super world-line and allow us to write an action for the supersymmetric particle that is not plagued by the content of second class constraints that prevents a covariant quantization in the space-time picture. The notion and geometry of projectile twistor space, and its connection to Minkowski space, are examined and shown to directly generalize the results in D = 3, 4. Quantization is performed and analytic quaternionic eigenfunctions and integrations are discussed. We also draw some conclusions on the possible generalization to ten dimensions.     

Soft-gluon resummation in quark-vector boson vertex at high energy

Resummation of the soft-gluon radiative corrections for the quark-vector boson vertex is performed within the path-integral (world-line) approach. The leading-order expression for the vacuum-averaged Wilson integral for an arbitrary gauge field is found in n-dimensional space-time. The cusp anomalous dimension of the color non-singlet Sudakov form factor of on-mass-shell quark is calculated in an arbitrary covariant gauge in the one-loop order, and the leading double-logarithmic asymptotical behavior is obtained from the corresponding evolution equation.     

Relativistic and separable classical hamiltonian particle dynamics

We show within the Hamiltonian formalism the existence of classical relativistic mechanics of N scalar particles interacting at a distance which satisfies the requirements of Poincaré invariance, separability, world-line invariance and Einstein causality. The line of approach which is adopted here uses the methods of the theory of systems with constraints applied to manifestly covariant systems of particles. The study is limited to the case of scalar interactions remaining weak in the whole phase space and vanishing at large space-like separation distances of the particles. Poincaré invariance requires the inclusion of many-body, up to N-body, potentials. Separability requires the use of individual or two-body variables and the construction of the total interaction from basic two-body interactions. Position variables of the particles are constructed in terms of the canonical variables of the theory according to the world-line invariance condition and the subsidiary conditions of the non-relativistic limit and separability. Positivity constraints on the interaction masses squared of the particles ensure that the velocities of the latter remain always smaller than the velocity of light.     

Relativistic rigid particles: Classical tachyons and quantum anomalies

Causal rigid particles whose action includes anarbitrary dependence on the world-line extrinsic curvature are considered. General classes of solutions are constructed, includingcausal tachyonic ones. The Hamiltonian formulation is developed in detail except for one degenerate situation for which only partial results are given and requiring a separate analysis. However, for otherwise generic rigid particles, the precise specification of Hamiltonian gauge symmetries is obtained with in particular the identification of the Teichmüller and modular spaces for these systems. Finally, canonical quantisation of the generic case is performed paying special attention to the phase space restriction due to causal propagation. A mixed Lorenz-gravitational anomaly is found in the commutator of Lorentz boosts with world-line reparametrisations. The subspace of gauge invariant physical states is therefore not invariant under Lorentz transformations. Consequences for rigid strings and membranes are also discussed.     

H-space and Robinson-Trautman solutions

It is shown that in theH-space of any Robinson-Trautman type II solution there is a uniquely defined world-line and that, under suitable regularity conditions, this world-line is a geodesic.     

MÄRZKE-WHEELER COORDINATES FOR ACCELERATED OBSERVERS IN SPECIAL RELATIVITY

In special relativity, the definition of coordinate systems adapted to generic accelerated observers is a long-standing problem, which has found unequivocal solutions only for the simplest motions. We show that the Märzke-Wheeler construction, an extension of the Einstein synchronization convention, produces accelerated systems of coordinates with desirable properties: (a) they reduce to Lorentz coordinates in a neighborhood of the observers' world-lines; (b) they index continuously and completely the causal envelope of the world-line (that is, the intersection of its causal past and its causal future: for well-behaved world-lines, the entire space-time). In particular, Märzke-Wheeler coordinates provide a smooth and consistent foliation of the causal envelope of any accelerated observer into space-like surfaces.We compare the Märzke-Wheeler procedure with other definitions of accelerated coordinates; we examine it in the special case of stationary motions, and we provide explicit coordinate transformations for uniformly accelerated and uniformly rotating observers. Finally, we employ the notion of Märzke-Wheeler simultaneity to clarify the relativistic paradox of the twins, by pinpointing the local origin of differential aging.     

An Ontology of Nature with Local Causality,Parallel Lives,and Many Relative Worlds

Parallel lives (PL) is an ontological model of nature in which quantum mechanics and special relativity are unified in a single universe with a single space-time. Point-like objects called lives are the only fundamental objects in this space-time, and they propagate at or below c, and interact with one another only locally at point-like events in space-time, very much like classical point particles. Lives are not alive in any sense, nor do they possess consciousness or any agency to make decisions—they are simply point objects which encode memory at events in space-time. The only causes and effects in the universe occur when lives meet locally, and thus the causal structure of interaction events in space-time is Lorentz invariant. Each life traces a continuous world-line through space-time, and experiences its own relative world, fully defined by the outcomes of past events along its world-line (never superpositions), which are encoded in its external memory. A quantum field comprises a continuum of lives throughout space-time, and familiar physical systems like particles each comprise a sub-continuum of the lives of the field. Each life carries a hidden internal memory containing a local relative wavefunction, which is a local piece of a pure universal wavefunction, but it is the relative wavefunctions in the local memories throughout space-time which are physically real in PL, and not the universal wavefunction in configuration space. Furthermore, while the universal wavefunction tracks the average behavior of the lives of a system, it fails to track their individual dynamics and trajectories. There is always a preferred separable basis, and for an irreducible physical system, each orthogonal term in this basis is a different relative world—each containing some fraction of the lives of the system. The relative wavefunctions in the lives’ internal memories govern which lives of different systems can meet during future local interactions, and thereby enforce entanglement correlations—including Bell inequality violations. These, and many other details, are explored here, but several aspects of this framework are not yet fleshed out, and work is ongoing.     

The Z=Z(t/z)-Type Plane Wave Solutions of the Field Equations of General Relativity in a Plane Symmetric Space-Time

Taub (Ann. Math., 53:472?C490, 1951) has studied plane symmetry in Riemannian space-time by considering empty space-time admitting a three parameter group of motions. In this paper, we have deduced the line element of such a space-time for Z=Z(t/z)-type plane gravitational waves using suitable transformations following the concept of Takeno (Sci. Rep. Inst. Theor. Phys. Hiroshima Univ. 1, 1961), Lal and Ali (Tensor 20:281?C302, 1969). Furthermore it has been shown that the deduced space-time admit plane wave solutions of the field equations of general relativity containing electromagnetic terms. Also we have studied electromagnetic field except gauge transformation with particular cases with respect to ??.     

Ramond-Neveu-Schwarz string with new boundary conditions

For the Ramond-Neveu-Schwarz string in D space-time dimensions we seek boundary conditions which preserve Poincaré invariance in d dimensions, d<D. We obtain twisted closed and twisted open strings preserving Gervais-Sakita supersymmetry. Covariant BRST quantization yields D=10. For some boundary conditions, partition functions exhibit space-time supersymmetry.     

The Cauchy problem for the O(N), CP(N − 1), and GC(N,p) models

We study the Cauchy problem for the O(N), $\text{C}$P(N ? 1) and G$\text{C}$(N, p) models in n + 1 dimensional space-time. We prove the existence and uniqueness of solutions for small time intervals and for any n. In space-time dimension two, the previous solutions can be extended to all times by the method of a priori estimates. In space-time dimensions three and four, our estimates yield only partial results on the global existence problem. In all cases the solutions are required only to belong to local spaces, which means that they satisfy local regularity conditions but have no restrictions on their behaviour at infinity in space.     

A background-dependent approach to the theory of gravitation II. Relativistic gravitational equations

On the basis of the results of Paper I and guided by a Machian view of nature, we find new gravitational equations which are background dependent. Such equations describe a purely geometrical theory of gravitation, and their dependence on the background structure is through the total energy-momentum tensor on the past sheet of the light cone of each space-time pointx [θ^μν x, say], i.e., through the integral on the past sheet of the light cone ofx of the parallel transport of the energy-momentum tensor from the space-time point in which it is defined tox along the geodesic connecting the two space-time points. Following Gürsey, we assume that the source of the De Sitter metric is not the cosmological term, but, rather, the energy-momentum tensor of a “uniform distribution of mass scintillations” [T ^μν x, say].T ^μν x, indeed, turns out to be equal to the metric tensor times a constant factor. As a consequence, in any local inhomogeneity A of a space-time whose background structure is determined by the Perfect Cosmological Principle,θ ^μν turns out to be approximately equal to the metric tensor times a constant factor, providedT=g _αβ T ^αβ is sufficiently small and the structure of the past sheet of the light cones of the space-time points belonging to Λ is not too much perturbed by the local gravitational field. As a consequence, in Λ the new equations approximately reduce to Einstein's equations. If one considers a “superuniverse model” in which our universe is considered as a local inhomogeneity in a De Sitter background, then from the above result there follows a fortiori the agreement of the new gravitational equations with the classical tests of gravitation. Furthermore, the dependence on the background structure is such that the new equations (i) incorporate the idea that the frame has to be fixeddirectly in connection with cosmological observations, and (ii) are singular in the absence of matter in the whole space-time. Moreover, (iii) the coupling constant turns out to be dimensionless in natural units (c=1=?), and (iv) a local inertial frame in a De Sitter background is determined by the condition that with respect to it the background structure is homogeneous in space and in time and is Lorentz invariant.     

Topological invariants and the dynamics of an axial vector torsion field

A generalized theory of gravitation is discussed which is based on a Riemann-Cartan space-time,U ₄, with an axial vector torsion field. Besides Einstein's equations determining the metric of theU ₄, a system of nonlinear field equations is established coupling an axial vector source current to the axial vector torsion field. The properties of the solutions of these equations are discussed assuming a London-type condition relating the axial current and torsion field. To characterize the solutions use is made of the Euler and Pontrjagin forms and the associated quadratic curvature invariants for theU ₄ space-time. It is found that there exists for a Riemann-Cartan space-time a relation between the zeros of the axial vector torsion field and the singularities of the Pontrjagin invariant, which is analogous to the well-known Hopf relation between the zeros of vector fields and the Euler characteristic.     

Another gravitational instability of flat space-time

Witten's demonstration of the instability of the five dimensional Kaluza-Klein ground state can be reduced to four, physical dimensions. One can then conclude that flat space-time at zero temperature with one of the spatial directions wrapped up is unstable. The decay rate is of order exp?L²/16πL²_p where L is the spatial periodicity and L_p is the Planck length. The post tunnelling evolution of the space-time is also discussed.     

Space-time geometry and symmetry breaking

We look for solutions of the Einstein-Yang-Mills equations in a 4 + D dimensional space-time. We find solutions where the first 4 dimensions are a flat Minkowskian space-time, while the D others are a compact, space-like manifold of small size. Such solutions can be obtained for an arbitrary compact gauge group K and are invariant under a sub-group G of K related to the space-time geometry. This shows that 4 + D dimensional gravity can give a mechanism for the super-strong symmetry breaking needed in grand unified field theories without introducing Higgs scalars.     

Nonperturbative quark dynamics in a baryon

The field-correlator method is used to calculate nonperturbative dynamics of quarks in a baryon. The general expression for the 3q Green’s function is obtained using the Fock-Feynman-Schwinger (world-line) path-integral formalism, where all dynamics is contained in the 3q Wilson loop with spin-field insertions. Using the lowest cumulant contribution for the Wilson loop, one obtains a Y-shaped string potential vanishing at the string-junction position. Using the einbein formalism for the quark kinetic terms, one automatically obtains constituent quark masses, calculable through the string tension. The resulting effective action for 3q plus Y-shaped strings is quantized in the path-integral formalism to produce two versions of Hamiltonian: one is in the c.m. and the other is in the light-cone system. The hyperspherical formalism is used to calculate masses and wave functions. Simple estimates in the lowest approximation yield baryon masses in good agreement with experiment without fitting parameters.     

Extracting muon momentum scale corrections for hadron collider experiments

The quark condensate is calculated within the world-line effective-action formalism, by using for the Wilson loop an ansatz provided by the stochastic vacuum model. Starting with the relation between the quark and the gluon condensates in the heavy-quark limit, we diminish the current quark mass down to the value of the inverse vacuum correlation length, finding in this way a 64?% decrease in the absolute value of the quark condensate. In particular, we find that the conventional formula for the heavy-quark condensate cannot be applied to the c-quark, and that the corrections to this formula can reach 23?% even in the case of the b-quark. We also demonstrate that, for an exponential parametrization of the two-point correlation function of gluonic field strengths, the quark condensate does not depend on the non-confining non-perturbative interactions of the stochastic background Yang?CMills fields.     

Spinor methods in conformal Killing transport

The equations of conformal Killing transport are discussed using tensor and spinor methods. It is shown that, in Minkowski space-time, the equations for a null conformal Killing vector ξ ^a are completely determined by the corresponding spinor ω ^A and its covariant derivative, which defines a spinor π _A′ . In conformally flat space-time, the covariant derivative of π _A′ is also involved. Some applications to twistor theory are briefly mentioned.