In search of relativistic time

This paper explores the status of some notions which are usually associated to time, like datations, chronology, durations, causality, cosmic time and time functions in the Einsteinian relativistic theories. It shows how, even if some of these notions do exist in the theory or for some particular solution of it, they appear usually in mutual conflict: they cannot be synthesized coherently, and this is interpreted as the impossibility to construct a common entity which could be called time. This contrasts with the case in Newtonian physics where such a synthesis precisely constitutes Newtonian time. After an illustration by comparing the status of time in Einsteinian physics with that of the vertical direction in Newtonian physics, I will conclude that there is no pertinent notion of time in Einsteinian theories.Comment: to appear in Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physic

A new look at Lorentz-Covariant Loop Quantum Gravity

In this work, we study the classical and quantum properties of the unique commutative Lorentz-covariant connection for loop quantum gravity. This connection has been found after solving the second-class constraints inherited from the canonical analysis of the Holst action without the time-gauge. We show that it has the property of lying in the conjugacy class of a pure \su(2) connection, a result which enables one to construct the kinematical Hilbert space of the Lorentz-covariant theory in terms of the usual \SU(2) spin-network states. Furthermore, we show that there is a unique Lorentz-covariant electric field, up to trivial and natural equivalence relations. The Lorentz-covariant electric field transforms under the adjoint action of the Lorentz group, and the associated Casimir operators are shown to be proportional to the area density. This gives a very interesting algebraic interpretation of the area. Finally, we show that the action of the surface operator on the Lorentz-covariant holonomies reproduces exactly the usual discrete \SU(2) spectrum of time-gauge loop quantum gravity. In other words, the use of the time-gauge does not introduce anomalies in the quantum theory.Comment: 28 pages. Revised version taking into account referee's comment

A Lorentz-Covariant Connection for Canonical Gravity

We construct a Lorentz-covariant connection in the context of first order canonical gravity with non-vanishing Barbero-Immirzi parameter. To do so, we start with the phase space formulation derived from the canonical analysis of the Holst action in which the second class constraints have been solved explicitly. This allows us to avoid the use of Dirac brackets. In this context, we show that there is a "unique" Lorentz-covariant connection which is commutative in the sense of the Poisson bracket, and which furthermore agrees with the connection found by Alexandrov using the Dirac bracket. This result opens a new way toward the understanding of Lorentz-covariant loop quantum gravity

On three quantization methods for particle on hyperboloid

We compare the respective efficiencies of three quantization methods (group theoretical, coherent state and geometric) by quantizing the dynamics of a free massive particle in two-dimensional de Sitter space. For each case we consider the realization of the principal series representation of $SO_0(1,2)$ group and its two-fold covering SU(1,1). We demonstrate that standard technique for finding an irreducible representation within the geometric quantization scheme fails. For consistency we recall our earlier results concerning the other two methods, make some improvements and generalizations.Comment: 22 pages, no figures, revte

Historical Lagrangian Dynamics

research paperThis paper presents (in its Lagrangian version) a very general " historical" formalism for dynamical systems, including time-dynamics and field theories. It is based on the universal notion of history. Its condensed and universal formulation provides a synthesis and a generalization different approaches of dynamics. It is in our sense closer to its real essence. The formalism is by construction explicitely covariant and does not require the introduction of time, or of a time function in relativistic theories. It considers space-time (in field theories) exactly in the same manner than time in usual dynamics, with the only difference that it has 4 dimensions. Both time and space-time are considered as particular cases of the general notion of an evolution domain. In addition, the formalism encompasses the cases where histories are not functions (e.g., of time or of space-time), but forms. This applies to electromag-netism and to first order general relativity (that we treat explicitely). It has both Lagrangian and Hamiltonian versions. An interesting result is the existence of a covariant generalized symplectic form, which generalizes the usual symplectic or the multisymplectic form, and the symplectic currents. Its conservation on shell provides a genuine symplectic form on the space of solutions