Conformal geometrodynamics regained: Gravity from duality

There exist several ways of constructing general relativity from ‘first principles’: Einstein’s original derivation, Lovelock’s results concerning the exceptional nature of the Einstein tensor from a mathematical perspective, and Hojman–Kucha?-Teitelboim’s derivation of the Hamiltonian form of the theory from the symmetries of space–time, to name a few. Here I propose a different set of first principles to obtain general relativity in the canonical Hamiltonian framework without presupposing space–time in any way. I first require consistent propagation of scalar spatially covariant constraints (in the Dirac picture of constrained systems). I find that up to a certain order in derivatives (four spatial and two temporal), there are large families of such consistently propagated constraints. Then I look for pairs of such constraints that can gauge-fix each other and form a theory with two dynamical degrees of freedom per space point. This demand singles out the ADM Hamiltonian either in (i) CMC gauge, with arbitrary (finite, non-zero) speed of light, and an extra term linear in York time, or (ii) a gauge where the Hubble parameter is conformally harmonic.     

A commutation theorem and duality for free Bose fields

We first present an elementary commutation theorem for operator algebras on Hilbert space. This theorem is then applied to give an elementary proof of duality for free Bose fields.     

Neutrino in matter and external fields

A technique for describing various processes proceeding in matter and involving neutrinos and electrons is discussed. This technique is based on “the method of exact solutions,” which implies the use of solutions to proper Dirac equations for particle wave functions in matter. Exact solutions for the neutrino and the electron in the cases of uniform nonmoving and rotating matter are discussed. On studying relativistic neutrino motion and associated neutrino-energy quantization in rotating matter, a semiclassical interpretation of particle finite motion is developed. In the general case of neutrino and electron motion in matter with varying parameters, the corresponding effective force acting on the particles is determined. The possibility of electromagnetic-wave radiation by an electron that moves in a dense neutrino flux of varying density and which is accelerated by this kind of force is predicted.     

Neutrinos in matter and external fields

A brief survey of effects generated by the influence of the environment on neutrinos is presented. The issues considered here include flavor and spin oscillations of neutrinos in matter, in electromagnetic fields of various configurations, and in gravitational fields; the electromagnetic properties of the neutrinos and the environment-induced change in these properties; photoneutrino processes generated by the environment; urca processes in magnetic fields; various mechanisms that may be responsible for the asymmetry of neutrino radiation from neutron stars; quantum states of neutrinos in matter and the spin light of neutrinos in matter and external fields; and astrophysical and cosmological applications of the above processes and phenomena. The method that is employed to describe the effect of the environment on neutrinos (as well as on electrons) and which is based on the application of exact solutions to the corresponding modified Dirac equations for particles in matter is briefly discussed.     

Cohomological derivation of the couplings between an abelian gauge field and matter fields

Consistent couplings between an abelian gauge field and a system of matter fields are investigated by means of Lagrangian cohomological arguments.     

N=1 supergravity theories coupled to matter and duality transformations

The most general action for chiral and complex linear superfields coupled to theN=1 old minimal supergravity is given. Scalar potentials for pure complex linear and mixed cases are found. A condition for the breakdown of the duality transformation, which transforms a theory with complex linear superfields to one with chiral scalar superfields, is obtained. When this condition is satisfied, the potentials and couplings cannot be transformed, in general, into a Kähler form; examples are given. Some aspects of vanishing cosmological constant are considered in this context.     

The connexion of duality and causal properties for generalized free fields

It is shown that the time-slice axiom and the diamond property are equivalent for the generalized free field. If, in addition, there is a mass gap, duality is equivalent to either causality requirement. It is further shown that the local rings associated with certain space-time regions are factors in the case of causal generalized free fields with mass gap. Necessary and sufficient conditions for causality and duality and some examples for causal and acausal generalized free fields are also given.     

Calculation of fields on plasma ions by collective coordinates

The probability of finding a given electric field on an ion in a plasma is computed with the aid of the collective coordinates. All the ions are assumed to have the same charge, and the free electrons are replaced by a rigid negative smear. The Jacobian of the transformation to collective coordinates is expanded in a series of Hermite polynomials, and one of the higher terms is computed as an estimate of the error. This error calculation indicates that a lower limit in the temperature probably exists below which present methods may not be used. This lower limit depends upon the form of the short range potential that is chosen. A cutoff in space at a particle separation ofr _c is employed in place of the cutoff in wave number atk _c. An average short range potential as a function of temperature appears. The values of the probability are shown to be surprisingly insensitive to the value ofr _c at infinite temperature and to compare well with the Holtsmark distribution.     

Gravity coupled with matter and the foundation of non-commutative geometry

We first exhibit in the commutative case the simple algebraic relations between the algebra of functions on a manifold and its infinitesimal length elementds. Its unitary representations correspond to Riemannian metrics and Spin structure whileds is the Dirac propagatords=x−x=D ⁻¹, whereD is the Dirac operator. We extend these simple relations to the non-commutative case using Tomita's involutionJ. We then write a spectral action, the trace of a function of the length element, which when applied to the non-commutative geometry of the Standard Model will be shown ([CC]) to give the SM Lagrangian coupled to gravity. The internal fluctuations of the non-commutative geometry are trivial in the commutative case but yield the full bosonic sector of SM with all correct quantum numbers in this slightly non-commutative case. The group of local gauge transformations appears spontaneously as a normal subgroup of the diffeomorphism group.     

Continuous matter fields in Regge calculus

We find that the continuous matter fields are ill-defined in Regge calculus in the physical 4D theory since the corresponding effective action has infinite terms unremovable by the UV renormalisation procedure. These terms are connected with the singular nature of the curvature distribution in Regge calculus, namely, with the presence in d>2 dimensions of the (d−3)-dimensional simplices where the (d−2)-dimensional ones carrying different conical singularities are meeting. Possible resolution of this difficulty is discretisation of matter fields in Regge background.