Diagrammatic proof of the large N melonic dominance in the SYK model

Letters in Mathematical Physics - A crucial result on the celebrated Sachdev–Ye–Kitaev model is that its large N limit is dominated by melonic graphs. In this letter, we offer a...     

Space-time in a simple model of a physical system

A model of space-time is worked out starting from the two primitive concepts of preparticle and of membership relation of set theory. We obtain as derivative concepts those of space-time and inertial reference frame, also those of energy, frequency, momentum, and wavelength of a physical system in a given reference frame. Proportionality relations between energy and frequency, and between momentum and (wavelength)^–1 are shown to be satisfied in our model. The same constant of proportionality intervenes in these two relations, and we interpret it as the Planck constant expressed in a particular system of units. Energy and momentum are conserved in the usual sense, provided we consider sufficiently large regions of the space-time diagram associated to the reference frame under consideration. Lorentz transformations and Heisenberg's inequalities are discussed within the framework of our model.     

Space-time singularities

A set of conditions for the reasonableness of space-time is proposed and investigated. Using these, together with strong causality and an assumption of genericness, it is shown that future timelike or null geodesically incomplete space-times contain either curvature or intermediate singularities, or primordial singularities.     

Space-time groups for the lattice

In the assumption of a lattice theory in which the continuous limit is not taken, the metric of the discrete space-time should be invariant under integral transformations. Based on local isomorphisms between real forms, a method is proposed in order to find the rational and integral elements of the pseudoorthogonal groups. Besides, the rational and integral trigonometric and hyperbolic functions are constructed on the lattice.The ideas in this paper were presented in the VI Seminar on Quantum Theory and the Structure of Space and Time, Tutzing, July 1984.     

Space-time complexity in Hamiltonian dynamics

New notions of the complexity function C(epsilon;t,s) and entropy function S(epsilon;t,s) are introduced to describe systems with nonzero or zero Lyapunov exponents or systems that exhibit strong intermittent behavior with "flights," trappings, weak mixing, etc. The important part of the new notions is the first appearance of epsilon-separation of initially close trajectories. The complexity function is similar to the propagator p(t(0),x(0);t,x) with a replacement of x by the natural lengths s of trajectories, and its introduction does not assume of the space-time independence in the process of evolution of the system. A special stress is done on the choice of variables and the replacement t-->eta=ln t, s-->xi=ln s makes it possible to consider time-algebraic and space-algebraic complexity and some mixed cases. It is shown that for typical cases the entropy function S(epsilon;xi,eta) possesses invariants (alpha,beta) that describe the fractal dimensions of the space-time structures of trajectories. The invariants (alpha,beta) can be linked to the transport properties of the system, from one side, and to the Riemann invariants for simple waves, from the other side. This analog provides a new meaning for the transport exponent mu that can be considered as the speed of a Riemann wave in the log-phase space of the log-space-time variables. Some other applications of new notions are considered and numerical examples are presented.     

Space-time supersymmetry of the covariant superstring

We explicitly derive the ghost oscillator contribution to the gauge covariant fermion emission vertex. This vertex is used to construct the space-time supersymmetry transformation laws which are shown to be an invariance of the free gauge covariant action of the superstring. We develop methods to deal with the quadratic exponentials which appear in the fermion emission vertex, in order to study the closure of the supersymmetry algebra. As a by-product, we complete the proof of the equivalence between the “old” and “new” formulations of the superstring.     

Space-time spaceless-timeless interactions

An argument, based on Lorentz invariance for the number of discrete objects and Lorentz non-invariance for continuous physical quantities, is used to arrive at an uncertainty relation involving dipole moment and mass. Applied to a photon, a virtual dipole moment is defined and the photon itself is described as an electromagnetic wave. The small distance singularity in the Coulomb potential is removed by using a complex number for distance.     

Space-time foliations in gravitation theory

A relationship is established between gravitational fields and space-time foliations on a manifold X⁴; the gravitational singularities are described as singularities of these foliations representing critical points of real functions on X⁴.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 20–23, September, 1982.     

Space-time symmetries in gauge theories

A general definition of symmetries of gauge fields is proposed and a method developed for constructing symmetric fields for an arbitrary gauge group. Scalar fields occur naturally in the formalism and the pure gauge theory reduces to a Higgs model in lower dimensions.Laboratoire propre du C.N.R.S. associé à l'Ecole Normale Supérieure et à l'Université de Paris Sud     

Space-time metrics inducing the radiation connection

It is shown that Lorentzian manifolds giving rise to the radiation connection, i.e. such that the canonical connection induces a unique connection on an isotropic hypersurface, are of type D or conformally flat in Petrov classification, generalizing the Robinson-Bertotti electromagnetic universe with cosmological constant. Under an holonomy hypothesis it is deduced that the underlying space-time manifold is a direct product of two two-dimensional spaces, one of them being space-like, and the other one time-like.     

Space-time is four-dimensional

The quantum state of the universe is described by Hartle and Hawking's ground state which is defined by a path integral over all compact metrics. The most probable classical evolution of the universe can be considered to come from some gravitational instanton by a quantum tunneling. These arguments have been generalized to the case of Kaluza-Klein models. It is found that in d= 11 simple supergravity, with a minisuperspace ansatz, all instantons must have a four dimensional sector. It suggests that this is the main reason why space-time is four-dimensional.This essay received the third award from the Gravity Research Foundation for the year 1985—Ed.     

Space-time foam as the universal regulator

A distribution of virtual black holes in the vacuum will induce modifications in the density of states for small perturbations of gravitational and matter fields. If the virtual black holes fill the volume of a typical spacelike surface then perturbation theory becomes more convergent and may even be finite, depending on how fast the number of virtual black holes increases as their size decreases. For distributions of virtual black holes which are scale invariant the effective dimension of space-time is lowered to a noninteger value less than 4, leading to an interpretation in terms of fractal geometry. In this case general relativity is renormalizable in the 1/N expansion without higher derivative terms. As the Hamiltonian is not modified the theory is stable.This essay received the second award from the Gravity Research Foundation for the year 1985—Ed.