Narrow Results

The conventional group of four-dimensional diffeomorphisms is not realizeable as a canonical transformation group in phase space. Yet there is a larger field-dependent symmetry transformation group which does faithfully reproduce 4-D diffeomorphism symmetries. Some properties of this group were first explored by Bergmann and Komar. More recently the group has been analyzed from the perspective of projectability under the Legendre map. Time translation is not a realizeable symmetry, and is therefore distinct from diffeomorphism-induced symmetries. This issue is explored further in this paper. It is shown that time is not "frozen". Indeed, time-like diffeomorphism invariants must be time-dependent. Intrinsic coordinates of the type proposed by Bergmann and Komar are used to construct invariants. Lapse and shift variables are retained as canonical variables in this approach, and therefore will be subject to quantum fluctuations in an eventual quantum theory. Concepts and constructions are illustrated using the relativistic classical and quantum free particle. In this example concrete time-dependent invariants are displayed and fluctuation in proper time is manifest.

A covariant symplectic structure can be identified, in general, starting from conservation laws connected to generic Hamiltonian invariants constructed from covariant vectors, bivectors and tensors. This feature leads to both holonomic and anholonomic formulations of Hamilton equations and Poisson brackets and it seems a deep link common to all interactions including gravity. A key role in the approach is played by affine connections instead of metric.

Spacetime emergence refers to the notion that classical spacetime "emerges" as an approximate macroscopic entity from a non-spatio-temporal structure present in a more complete theory of interacting fundamental constituents. In this paper, we propose a novel mechanism involving the "soldering" of internal and external spaces for the emergence of spacetime and the twin transmutation of general covariance. In the context of string theory, this mechanism points to a critical four-dimensional spacetime background.

Characteristic length scale of the post-Newtonian corrections to the gravitational field of a body is given by its gravitational radius r_g. The role of this scale in quantum domain is discussed in the context of the low-energy effective theory. The question of whether quantum gravity effects appear already at r_g leads to the question of correspondence between classical and quantum theories, which in turn can be unambiguously resolved by considering the issue of general covariance. The O(ℏ⁰) loop contributions turn out to violate the principle of general covariance, thus revealing their essentially quantum nature. The violation is O(1/N), where N is the number of particles in the body. This leads naturally to a macroscopic formulation of the correspondence principle.

We derive a general covariant symplectic structure which leads to holonomic and anholonomic formulations of Hamilton equations of motion related to the hydrodynamic picture of mechanics. Such a structure can be achieved starting from generic bilinear Hamiltonians, constructed by covariant vector, bivector or tensor fields. This feature is gage-free and it realizes a deep link toward canonical quantization for interactions like electromagnetism, gravity and Dirac fields.

The requirement of general covariance of quantum field theory (QFT) naturally leads to quantization based on the manifestly covariant De Donder–Weyl formalism. To recover the standard noncovariant formalism without violating covariance, fields need to depend on time in a specific deterministic manner. This deterministic evolution of quantum fields is recognized as a covariant version of the Bohmian hidden-variable interpretation of QFT.

Since the appearance of General Relativity, its intrinsec general covariance has been very often misinterpreted as implying that physically meaningful quantities (and conclusions extracted from the theory) have to be absolutely independent on observers. This incorrect point of view is sometimes expressed by discarding the very concept of observer in the structure and applications of the theory. As we shall stress in this essay, through some examples, the concept of observer is as essential to General Relativity as it is to any physical theory.

General covariance is the cornerstone of Einstein’s general relativity (GR) and implies that any two metrics related by diffeomorphisms are physically equivalent. There are, however, many examples pointing to the fact that this strict statement of general covariance needs refinement. There are a very special (measure-zero) subset of diffeomorphisms, the residual diffeomorphisms, to which one can associate well-defined conserved charges. This would hence render these diffeomorphic geometries physically distinct. We discuss that these symmetries may be appropriately called “symplectic symmetries”. Existence of residual diffeomorphisms and symplectic symmetries can be a quite general feature and not limited to the examples discussed so far in the literature. We propose that, in the context of black holes, these diffeomorphic, but distinct, geometries may be viewed as “symplectic soft hair” on black holes. We comment on how this may remedy black hole microstate problem, which in this context are dubbed as “horizon fluffs”.

In this paper, the formal structure of Penrose’s gravitationally induced reduction of the wave function mechanism is analyzed. It is shown that pushing Penrose’s argument forward leads to the interpretation of quantum coherence in microscopic systems as an observable signature violation of general covariance. We discuss potential avenues to avoid this conclusion, among them emergent quantum mechanics and super-determinism.