Related papers: Absolute versus Relational Debate: a Modern Global…
The algebra of linear and quadratic functions of basic observables on the phase space of either the free particle or the harmonic oscillator possesses a finite-dimensional anomaly. The quantization of these systems outside the critical…
One of the biggest challenges to theoretical physics of our time is to find a background-independent quantum theory of gravity. Today one encounters a profusion of different attempts at quantization, but no fully accepted - or acceptable,…
General relativity poses serious problems for counterfactual propositions peculiar to it as a physical theory. Because these problems arise solely from the dynamical nature of spacetime geometry, they are shared by all schools of thought on…
We revisit the field content and consistency of the New General Relativity family of theories. These theories are constructed in a geometrical framework with a flat and metric-compatible connection, so the affine structure is entirely…
The property of the conformal algebra to contain the Schr\"odinger algebra in one less space dimension is extended to the supersymmetric case. More precisely, we determine the counterpart of any field theory admissible super conformal…
In this article, we consider a second Problem of Time Facet. This started life as Wheeler's Thin Sandwich Problem, within the narrow context of 1) GR-as-Geometrodynamics, in particular its momentum constraint. 2) A Lagrangian variables…
General relativity does not allow one to specify the topology of space, leaving the possibility that space is multiply rather than simply connected. We review the main mathematical properties of multiply connected spaces, and the different…
A new local, covariant ``counter-term'' is used to construct a variational principle for asymptotically flat spacetimes in any spacetime dimension $ d \ge 4$. The new counter-term makes direct contact with more familiar background…
In this research oriented manuscript, foundational aspects of rigid geometry are discussed, putting emphasis on birational side of formal schemes and topological feature of rigid spaces. Besides the rigid geometry itself, topics include the…
One of the main features of covariant theories, in particular general relativity, is that the field equation possesses gauge freedom associated with global diffeomorphisms of the underlying manifold. I shall explain here how the hole…
A complete canonical formulation of general covariance makes it possible to construct new modified theories of gravity that are not of higher-curvature form, as shown here in a spherically symmetric setting. The usual uniqueness theorems…
After reviewing various interpretations of structural realism, I adopt here a definition that allows both relations between things that are already individuated (which I call ``relations between things'') and relations that individuate…
A review is given of recent work on topology changing solutions to the first order form of general relativity. These solutions have metrics which are smooth everywhere, invertible almost everywhere, and have bounded curvature. The…
In the (2+2) formulation of general relativity spacetime is foliated by a two-parameter family of spacelike 2-surfaces (instead of the more usual one-parameter family of spacelike 3-surfaces). In a partially gauge-fixed setting (double-null…
Self-similar potentials generalize the concept of shape-invariance which was originally introduced to explore exactly-solvable potentials in quantum mechanics. In this article it is shown that previously introduced algebraic approach to the…
An enumerative invariant theory in Algebraic Geometry, Differential Geometry, or Representation Theory, is the study of invariants which 'count' $\tau$-(semi)stable objects $E$ with fixed topological invariants $[E]=\alpha$ in some…
Seminar held at JINR, Dubna, May 15, 2012. In General Relativity, spacetime singularities raise a number of problems, both mathematical and physical. One can identify a class of singularities - with smooth but degenerate metric - which,…
The geometric form of standard quantum mechanics is compatible with the two postulates: 1) The laws of physics are invariant under the choice of experimental setup and 2) Every quantum observation or event is intrinsically statistical.…
The new approach to quantize the gravity based on the notion of differential algebra is suggested. It is shown that the differential geometry of this object can not be described in terms of points. The spatialization procedure giving rise…
We consider the possibility that the basic space of physics is not spacetime, but configuration space. We illustrate this on the example with a system of gravitationally interacting point particles. It turns out that such system can be…