Quantum optics: Dedication

The Cauchy Problem

We will pick up the concepts of partial and complete observables introduced by Rovelli in Conceptional Problems in Quantum Gravity, Birkhauser, Boston (1991); Class Quant Grav, 8:1895 (1991); Phys Rev, D65:124013 (2002); Quantum Gravity, Cambridge University Press, Cambridge (2007) in order to construct Dirac observables in gauge systems. We will generalize these ideas to an arbitrary number of gauge degrees of freedom. Different methods to calculate such Dirac observables are developed. For background independent field theories we will show that partial and complete observables can be related to Kuchař’s Bubble-Time Formalism (J Math Phys, 13:768, 1972). Moreover one can define a non-trivial gauge action on the space of complete observables and also state the Poisson brackets of these functions. Additionally we will investigate, whether it is possible to calculate Dirac observables starting with partially invariant partial observables, for instance functions, which are invariant under the spatial diffeomorphism group.

Partial and Complete Observables for Hamiltonian Constrained Systems

Constraints on dark matter density and axion mass from the large-scale structure of spacetime

In this work, we consider different forms of relativistic perfect fluid Lagrangian densities that yield the same gravitational field equations in general relativity (GR). A particularly intriguing example is the case with couplings of the form [1+f{sub 2}(R)]L{sub m}, where R is the scalar curvature, which induces an extra force that depends on the form of the Lagrangian density. It has been found that, considering the Lagrangian density L{sub m}=p, where p is the pressure, the extra-force vanishes. We argue that this is not the unique choice for the matter Lagrangian density, and that more natural forms for L{sub m} do not imply the vanishing of the extra force. Particular attention is paid to the impact on the classical equivalence between different Lagrangian descriptions of a perfect fluid.

Nonminimal coupling of perfect fluids to curvature

In a special class of globally hyperbolic, topologically trivial, asymptotically flat at spatial infinity spacetimes selected by the requirement of absence of supertranslations (compatible with Christodoulou-Klainermann spacetimes) it is possible to define the rest-frame instant form of ADM canonical gravity by using Dirac's strategy of adding ten extra variables at spatial infinity and ten extra first class constraints implying the gauge nature of these variables. The final canonical Hamiltonian is the weak ADM energy and a discussion of the Hamiltonian gauge transformations generated by the eight first class ADM constraints is given. When there is matter and the Newton constant is switched off, one recovers the description of the matter on the Wigner hyperplanes of the rest-frame instant form of dynamics in Minkowski spacetime.

REVIEW: The Rest-Frame Instant Form of Metric Gravity

The intrinsic covariant 1-time description (rest-frame instant form) for N relativistic scalar particles is defined. The system of N charged scalar particles plus the electromagnetic field is described in this way: the study of its Dirac observables allows the extraction of the Coulomb potential from field theory and the regularization of the classical self-energy by using Grassmann-valued electric charges. The 1-time covariant relativistic statistical mechanics is defined.

N- and 1-time Classical Description of N-body Relativistic Kinematics and the Electromagnetic Interaction

Newton's standard theory of gravitation is reformulated in terms of a generally Galilei-covariant action principle as a gauge theory of the extended Galilei group. A suitable modification of Utiyama's method for gauging the projective realization of the Galilei group associated with the free mass-point, together with the connections between the consequent 11 external gauge fields and known facts about Galilean and Newtonian geometrical structures, are discussed from a unified point of view. Then the problem of the existence of an action principle for the dynamical evolution of the gauge fields is analysed. Since it is not known how to extend Utiyama's method from the case of `invariance' to the case of `quasi-invariance, modulo the equations of motion', which turns out to be the key factor in the Galilean case, an action principle is derived, starting from a suitable power expansion of the 4-metric tensor, as a contraction, for , of the ADM-De-Witt action of general relativity. The Galilean action depends on 27 fields (i.e. it contains 16 auxiliary fields besides the 11 gauge fields) and is indeed quasi-invariant, modulo the equations of motion under general Galilean coordinate transformations. The physical equivalence of this theory and Newton's theory of gravity is shown explicitly, by analysing its first- and second-class constraints. Finally, we discuss the feasibility of a symplectic reduction of the 27-fields theory to a minimal theory depending on only the 11 gauge fields, in the sense of Utiyama's method.

Standard and generalized Newtonian gravities as `gauge' theories of the extended Galilei group: I. The standard theory

A new version of tetrad gravity in globally hyperbolic, asymptotically flat at spatial infinity spacetimes with Cauchy surfaces diffeomorphic to R
3 is obtained by using a new parametrization of arbitrary cotetrads to define a set of configurational variables to be used in the ADM metric action. Seven of the fourteen first class constraints have the form of the vanishing of canonical momenta. A comparison is made with other models of tetrad gravity and with the ADM canonical formalism for metric gravity.

A New Parametrization for Tetrad Gravity

A relativistic kinematics for the N-body problem which solves all the problems raised until now on this topic is constructed by exploting the Wigner-covariant rest-frame instant form of dynamics in the context of parametrized Minkowski theories. The Wigner hyperplanes, orthogonal to the total timelike four-momentum of any N-body configuration, define the intrinsic rest frame and realize the separation of the center-of-mass motion. The point chosen as origin in each Wigner hyperplane can be made to coincide with the covariant noncanonical Fokker–Pryce center of inertia. As is well known, the latter is distinct from the canonical pseudo- four-vector describing the decoupled motion of the center of mass (which possess the same Euclidean covariance as the quantum Newton–Wigner three-position operator) and from the noncanonical pseudo-four-vector known as Moller’s center of energy. Our approach leads to the splitting of the notion of center of mass into an external one, defined in terms of an external Poincare group realization, and an internal one defined in terms of an internal unfaithful realization of the group inside the Wigner hyperplane. Because of the first class constraints defining the rest frame (vanishing of the internal three-momentum), the latter three internal concepts of center of mass weakly coincide. The resulting unique internal center of mass is thereby a gauge variable which, by a suitable gauge fixing, can be localized at the origin of the Wigner hyperplane. An adapted canonical basis of relative variables is constructed by means of the classical counterpart of the Gartenhaus–Schwartz transformation. The invariant mass of the N-body configuration is the Hamiltonian for the relative motions. Within this general framework, the rotational kinematics can be developed in terms of the same dynamical body frames, orientation-shape variables, spin frame, and canonical spin bases already introduced in the case of the nonrelativistic N-body problem.

Luca Lusanna

Papers

REVIEW: The Rest-Frame Instant Form of Metric Gravity

N- and 1-time Classical Description of N-body Relativistic Kinematics and the Electromagnetic Interaction

Standard and generalized Newtonian gravities as `gauge' theories of the extended Galilei group: I. The standard theory

A New Parametrization for Tetrad Gravity

Centers of mass and rotational kinematics for the relativistic N-body problem in the rest-frame instant form