Tullio Regge

Role of Surface Integrals in the Hamiltonian Formulation of General Relativity

Abstract It is shown that if the phase space of general relativity is defined so as to contain the trajectories representing solutions of the equations of motion then, for asymptotically flat spaces, the Hamiltonian does not vanish but its value is given rather by a nonzero surface integral. If the deformations of the surface on which the state is defined are restricted so that the surface moves asymptotically parallel to itself in the time direction, then the surface integral gives directly the energy of the system, prior to fixing the coordinates or solving the constraints. Under more general conditions (when asymptotic Poincare transformations are allowed) the surface integrals giving the total momentum and angular momentum also contribute to the Hamiltonian. These quantities are also identified without reference to a particular fixation of the coordinates. When coordinate conditions are imposed the associated reduced Hamiltonian is unambiguously obtained by introducing the solutions of the constraints into the surface integral giving the numerical value of the unreduced Hamiltonian. In the present treatment there are therefore no divergences that cease to be divergences after coordinate conditions are imposed. The procedure of reduction of the Hamiltonian is explicity carried out for two cases: (a) Maximal slicing, (b) ADM coordinate conditions. A Hamiltonian formalism which is manifestly covariant under Poincare transformations at infinity is presented. In such a formalism the ten independent variables describing the asymptotic location of the surface are introduced, together with corresponding conjugate momenta, as new canonical variables in the same footing with the g ij , π ij . In this context one may fix the coordinates in the “interior” but still leave open the possibility of making asymptotic Poincare transformations. In that case all ten generators of the Poincare group are obtained by inserting the solution of the constraints into corresponding surface integrals.

The relativistic spherical top

The classical theory of the free relativistic spherical top is first developed from a Lagrangian viewpoint. Our method allows the invariant mass to be an arbitrary function of the intrinsic spin. A canonical formalism is established following the approach suggested by Dirac for constrained Hamiltonian systems. There is a second arbitrary function in the theory, in addition to the usual one due to reparametrization invariance. The usual Newton-Wigner variables are supplemented by the Euler angles. The quantum theory of the free top is discussed. The classical theory is generalized to included charged tops with magnetic moments.

Gravity and supergravity as gauge theories on a group manifold

Abstract We construct generalizations of gravity, including supergravity, by writing the theory on the group manifold (Poincare for gravity, the graded-Poincare group for supergravity). The action involves forms over the group, restricted to a 4-dimensional submanifold. The equations of motion produce a Lorentz gauge in gravity and supergravity, and an additional anholonomic supersymmetric coordinate transformation which reduces to the “local supersymmetry” of supergravity.

Vortices in He II, current algebras and quantum knots

A canonical quantization scheme is developed for vertices in superfluid He II, using Diracs technique for constrained hamiltonian systems. Quantization introduces in the theory in natural way the structure of the infinite Lie algebra of incompressible flows. We argue that all the topological invariants of the vortex, considered as a knot, can be regarded as observables of the system. Finally unitary representations of measure preserving flows on R3 and current algebra are discussed.

Free boundary of He II and Feynman wave functions

An extension of the standard hydrodynamical equations for superfluid helium atT=0 is discussed. Its relation with Feynman wave functions and the well-known Gross-Pitaevskii treatment is elucidated. Finally, a discussion of the free boundary of He II is carried out and the surface tension computed on the basis of a simple approximation scheme. The result is 0.56 dyne/cm against about 0.35 dyne/cm experimental value.

The monodromy rings of a class of self-energy graphs

The monodromy rings of self-energy graphs, with two vertices and an arbitrary number of connecting lines, are determined.

Improved Hamiltonian for general relativity

Abstract A Hamiltonian formalism for asymptotically flat spaces in general relativity which is manifestly covariant under Poincare transformations at infinity is proposed and some of its implications are briefly discussed.

The monodromy rings of one loop Feynman integrals

The monodromy rings of Feynman integrals for one loop graphs with an arbitrary number of lines are determined.

Band hybridization in rotational motion

Abstract The back-bending behaviour of the moment of inertia as a function of the square of the angular velocity, for the ground-, and s-rotational trajectories in 66 156 Dy 90 , is described in terms of band hybridization.

On the back-bending in the rotational motion

Abstract The anomalous behavior of the moment of inertia, as a function of the angular momentum, for the ground and first excited rotational bands of 154 Gd is accounted for by the mechanism of bands hybridization.