arXiv subjects
Publications and source records attributed to Alejandro Corichi.
The authors present concepts and mathematical developments which give rise to the Hamiltonian formulation of Einstein's general relativity, first introduced by Arnowitt, Deser, and Misner. All the geometrical quantities needed for the construction are explicitly obtained, and examples of some of the application of the formalism are given.
In canonical quantum gravity the physical states solve the constraints but are typically not normalizable in the kinematical Hilbert space. Group averaging constructs them from normalizable kinematical seeds. We describe the geometry of this seed-to-state map in the language of geometric quantum mechanics. When averaging is an orthogonal projection, the admissible kinematical rays form a holomorphic vector bundle over the physical ray space. The embedded exact solutions form its zero section, and its fibers are strictly larger than gauge orbits. Observables descend to the base exactly when the Hessian of the squared constraint vanishes along their flow. For distributional physical states, group averaging on an explicit Sobolev test space yields an affine bundle of representatives. Its sections are embeddings of the physical ray space whose images contain no exact solutions. The test-space inner product singles out one of them, and narrowing the seed profile gives Weyl sequences of approximate solutions. We apply this to solvable loop quantum cosmology. Group averaging reproduces the Klein--Gordon physical inner product on an explicit test space. Because the scalar-field momentum enters the constraint quadratically, the constraint spectrum has an edge near the shell. The canonical section is then mode-dependent and not Kähler isometric, and the canonical seeds of small universes are biased off the shell. The edge also separates the Page--Wootters state from the slice of the Klein--Gordon solution. An ideal clock shows none of these effects. The results separate the canonical structures, namely physical rays, physical geometry and Dirac observables, from those that depend on a choice of seeds, test space or clock.
Exact normalizable solutions of quantum constraints form a symplectic submanifold of the space of rays, whereas classical first-class constraints define a surface on which the symplectic form is degenerate. We ask how the degenerate directions can be recovered when quantum theory is regarded as fundamental and the constraints are imposed through a single positive master operator. For a nonnegative master function whose zero set is a Morse--Bott critical manifold, the image of the linearized Hamiltonian vector field is the symplectic orthogonal of the zero set, and coisotropy is equivalent to this linearization having square zero. The image is fixed by the zero set, but the theorem identifies it with the response of the master dynamics, which makes it accessible to quantum states. Three realizations follow. The exact quantum physical space is the opposite, rotational case of the same dichotomy. For quadratic masters, the bottom of the spectrum is $(\hbar/2)\Tr_+L$, so first-class zero sets carry no offset of order $\hbar$; within the harmonic approximation this explains the $\hbar$ versus $\hbar^2$ offsets observed in tests of the master-constraint programme. Finally, the second time derivative of the limiting covariance of a semiclassical packet, which can be determined by bounded probes, has the gauge distribution as its image. Two oscillators at fixed total energy give exact master evolution, an exact realization of this covariance shear, and the reduced symplectic limit. For the non-Abelian SL(2,$\R$) model we compute the coherent and Weyl master symbols, prove essential self-adjointness and a finite-time propagation estimate, and derive the spectral bottom $\hbar^2/2$.
Geometric methods have proved useful in the Hamiltonian description of classical constrained systems. We develop the corresponding geometry for quantum constrained systems, using the formulation of quantum mechanics in which the space of rays $\Ps$ is a Kähler manifold and unitary transformations are its canonical transformations. The Dirac conditions $\hat C_iΨ=0$ define the zero set of a single function $\Cq=\ev{\Mop}$, the expectation value of the master constraint operator. Our central identity is $\Cq=|μ|^2+\tfrac12\|X_μ\|^2$, where $μ=(\ev{\hat C_i})$ is the moment map of the gauge action on $\Ps$ and $X_μ$ its fundamental vector fields: the quantum condition imposes the classical-type condition $μ=0$ \emph{and} the collapse of the gauge orbits to points. This explains why no second reduction is needed. The physical state space is a totally geodesic Kähler submanifold. $\Cq$ is a Morse--Bott function which gradient flow is heat-kernel-smeared group averaging, converges to the nearest-point projection onto physical states, and its fibres are invariant under the complexified gauge group. Physical observables are characterised to all orders by the double-commutator condition, which is equivalent to $[\hat O,\hat P_0]=0$, with $\hat P_0$ the projector onto physical states. When zero lies in the continuous spectrum, the gradient flow converges to Dirac states in the sense of Grundling and Hurst, and its large-time asymptotics reproduce the rigging map of refined algebraic quantization. Finally, pulled back along coherent states, $\Cq$ equals the classical master constraint plus an $O(\hbar)$ term given by the squared speed of the classical gauge orbits. Together these results connect the Dirac procedure, the master constraint programme, refined algebraic quantization and symplectic reduction within one geometric picture.
If one takes seriously the postulate of quantum mechanics in which physical states are rays in the standard Hilbert space of the theory, one is naturally lead to a geometric formulation of the theory. Within this formulation of quantum mechanics, the resulting description is very elegant from the geometrical viewpoint, since it allows to cast the main postulates of the theory in terms of two geometric structures, namely a symplectic structure and a Riemannian metric. However, the usual superposition principle of quantum mechanics is not naturally incorporated, since the quantum state space is non-linear. In this note we offer some steps to incorporate the superposition principle within the geometric description. In this respect, we argue that it is necessary to make the distinction between a 'projective superposition principle' and a 'decomposition principle' that extend the standard superposition principle. We illustrate our proposal with two very well known examples, namely the spin 1/2 system and the two slit experiment, where the distinction is clear from the physical perspective. We show that the two principles have also a different mathematical origin within the geometrical formulation of the theory.
The Generalized First Law (GFL) and the Generalized Second Law (GSL) of thermodynamics are studied for cosmological scenarios with spatial curvature through an apparent horizon. We focus on effective and alternative cosmic systems motivated by quantum cosmological models, where the entropy is considered a function of the apparent area, transforming the effective cosmological model into the standard form in cosmology. The general conditions for the validity of the GSL are analyzed for entropy as a general function of area and logarithmic corrections to the usual Black Hole entropy. The Weak Energy Condition (WEC) and the Strong Energy Condition (SEC) are implemented for the matter entropy part. In particular, we study the GFL and the regions where the GSL is valid for effective Loop Quantum Cosmology (LQC) models with spatial curvature $k=0,\pm 1$, taking every possible value of the logarithmic contributing factor for the entropy analysis. In addition, in order to solve some violations of the GSL, we explore the possibility of admitting negative absolute temperatures (NAT) in our system, where the validity conditions for an extended generalized second law (EGSL) are studied, and the time arrow from the second law is discussed for the LQC models.
The second law of thermodynamics constitutes a fundamental principle of physics, precluding the existence of perpetual motion machines and providing a natural definition of the arrow of time. Its scope extends across virtually all areas of physical theory. Nonetheless, certain systems are known to admit negative absolute temperatures under well-defined conditions, a phenomenon that has been experimentally observed. In this work, we formulate an extended version of the first and second laws, which recovers the conventional statement for positive temperatures and extends its applicability to the negative-temperature domain. Illustrative examples are discussed in the contexts of quantum cosmology and Onsager's vortices.
The Hamiltonian description of classical gauge theories is a well studied subject. The two best known approaches, namely the covariant and canonical Hamiltonian formalisms have received a lot of attention in the literature. However, in our opinion, a full understanding of the relation between them is not available, specially for gauge theories that are defined over regions with boundaries. Here we consider this issue, for spacetimes with isolated horizons as inner boundaries (representing black holes in equilibrium), and assess whether their corresponding descriptions can be seen as equivalent. First, we shall review and reanalyze both formalisms from anew, focusing on some subtleties not considered before. Next we show that, if we compare them at face value, there are subtle differences between both formalisms, for instance in the derivation of horizon energy, a physical observable of the theory, as well as in the number of boundary degrees of freedom. At first sight, it seems that the canonical formalism introduces some additional ambiguities that make more challenging the comparison between both formalisms. We analyze different possible interpretations in both approaches and show how one can obtain complete correspondence between results within covariant and canonical formalisms. We also show that the correspondence can be achieved in the original phase spaces, with no boundary degrees of freedom, or by appropriately extending them by adding horizon degrees of freedom. Along the way, we shed light on the role of boundary terms in the symplectic structure in relation to boundary degrees of freedom.
The Entropic Uncertainty Relations (EUR) result from inequalities that are intrinsic to the Hilbert space and its dual with no direct connection to the Canonical Commutation Relations. Bialynicky-Mielcisnky obtained them in \cite{bialynicki1975uncertainty} attending Hilbert spaces with a Lebesgue measure. The analysis of these EUR in the context of singular Hilbert spaces has not been addressed. Singular Hilbert spaces are widely used in scenarios where some discretization of the space (or spacetime) is considered, e.g., loop quantum gravity, loop quantum cosmology and polymer quantum mechanics. In this work, we present an overview of the essential literature background and the road map we plan to follow to obtain the EUR in polymer quantum mechanics.
The Hamiltonian description of classical gauge theories is a very well studied subject. The two best known approaches, namely the covariant and canonical Hamiltonian formalisms have received a lot of attention in the literature. However, a full understanding of the relation between them is not available, specially when the gauge theories are defined over regions with boundaries. Here we consider this issue, by first making precise what we mean by equivalence between the two formalisms. Then we explore several first order gauge theories, and assess whether their corresponding descriptions satisfy the notion of equivalence. We shall show that, even when in several cases the two formalisms are indeed equivalent, there are counterexamples that signal that this is not always the case. Thus, non-equivalence is a generic feature for gauge field theories. These results call for a deeper understanding of the subject.
The notion of Isolated Horizons has played an important role in gravitational physics, being useful from the characterization of the endpoint of black hole mergers to (quantum) black hole entropy. In particular, the definition of {\it weakly} isolated horizons (WIHs) as quasilocal generalizations of event horizons is purely geometrical, and is independent of the variables used in describing the gravitational field. Here we consider a canonical decomposition of general relativity in terms of connection and vierbein variables starting from a first order action. Within this approach, the information about the existence of a (weakly) isolated horizon is obtained through a set of boundary conditions on an internal boundary of the spacetime region under consideration. We employ, for the self-dual action, a generalization of the Dirac algorithm for regions with boundary. While the formalism for treating gauge theories with boundaries is unambiguous, the choice of dynamical variables on the boundary is not. We explore this freedom and consider different canonical formulations for non-rotating black holes as defined by WIHs. We show that both the notion of horizon degrees of freedom and energy associated to the horizon is not unique, even when the descriptions might be self-consistent. This represents a generalization of previous work on isolated horizons both in the exploration of this freedom and in the type of horizons considered. We comment on previous results found in the literature.
We investigate three issues that have been discussed in the context of inflation: Fading of the importance of quantum non-commutativity; the phenomenon of quantum squeezing; and the ability to approximate the quantum state by a distribution function on the classical phase space. In the standard treatments, these features arise from properties of mode functions of quantum fields in (near) de Sitter space-time. Therefore, the three notions are often assumed to be essentially equivalent, representing different facets of the same phenomenon. We analyze them in general Friedmann-Lemaitre- Robertson-Walker space-times, through the lens of geometric structures on the classical phase space. The analysis shows that: (i) inflation does not play an essential role; classical behavior can emerge much more generally; (ii) the three notions are conceptually distinct; classicality can emerge in one sense but not in another; and, (iii) the third notion is realized in a surprisingly strong sense; there is exact equality between completely general $n$-point functions in the classical theory and those in the quantum theory, provided the quantum operators are Weyl ordered. These features arise already for linear cosmological perturbations by themselves: considerations such as mode-mode coupling, decoherence, and measurement theory --although important in their own right-- are not needed for emergence of classical behavior in any of the three senses discussed. Generality of the results stems from the fact that they can be traced back to geometrical structures on the classical phase space, available in a wide class of systems. Therefore, this approach may also be useful in other contexts.
We study the canonical Hamiltonian analysis of gauge theories in the presence of boundaries. While the implementation of Dirac's program in the presence of boundaries, as put forward by Regge and Teitelboim, is not new, there are some instances in which this formalism is incomplete. Here we propose an extension to the Dirac formalism --together with the Regge-Teitelboim strategy,-- that includes generic cases of field theories. We see that there are two possible scenarios, one where there is no contribution from the boundary to the symplectic structure and the other case in which there is one, depending on the dynamical details of the starting action principle. As a concrete system that exemplifies both cases, we consider a theory that can be seen both as defined on a four dimensional spacetime region with boundaries --the bulk theory--, or as a theory defined both on the bulk and the boundary of the region --the mixed theory--. The bulk theory is given by the 4-dimensional Maxwell + $U(1)$ Pontryagin action while the mixed one is defined by the 4-dimensional Maxwell + 3-dimensional $U(1)$ Chern-Simons action on the boundary. Finally, we show how these two descriptions of the same system are connected through a canonical transformation that provides a third description. The focus here is in defining a consistent formulation of all three descriptions, for which we rely on the geometric formulation of constrained systems, together with the extension of the Dirac-Regge-Teitelboim (DRT) formalism put forward in the manuscript.
We perform the canonical Hamiltonian analysis of a topological gauge theory, that can be seen both as a theory defined on a four dimensional spacetime region with boundaries --the bulk theory--, or as a theory defined on the boundary of the region --the boundary theory--. In our case the bulk theory is given by the 4-dimensional $U(1)$ Pontryagin action and the boundary one is defined by the $U(1)$ Chern-Simons action. We analyse the conditions that need to be imposed on the bulk theory so that the total Hamiltonian, smeared constraints and generators of gauge transformations be well defined (differentiable) for generic boundary conditions. We pay special attention to the interplay between the constraints and boundary conditions in the bulk theory on the one side, and the constraints in the boundary theory, on the other side. We illustrate how both theories are equivalent, despite the different canonical variables and constraint structure, by explicitly showing that they both have the same symmetries, degrees of freedom and observables.
We study numerically the solutions to the effective equations of Bianchi IX spacetimes within Loop Quantum Cosmology. We consider Bianchi IX models with and without inverse triad corrections whose matter content is a scalar field without mass. The solutions are classified using the classical observables. We show that both effective theories --with lapse N=V and N=1-- solve the big bang singularity and reproduce the classical dynamics far from the bounce. Moreover, due to the spatial compactness, there is an infinity number of bounces and recollapses. We study the limit of large volume and show that both effective theories reproduce the same dynamics, thus recovering general relativity. We implement a procedure to identify amongst the Bianchi IX solutions, those that behave like k=0,1 FLRW as well as Bianchi I, II, and VII_0 models. The effective solutions exhibit Bianchi I phases with Bianchi II transitions and also Bianchi VII_0 phases, which had not been studied before, at the quantum nor effective level. We comment on the possible implications of these results for a quantum modification to the classical BKL behaviour.
Isolated Horizons have played an important role in gravitational physics, from characterization of the endpoint of black hole mergers to black hole entropy. With an eye towards a canonical formulation we consider general relativity in first order form. We focus on two issues: i) The role of the internal gauge freedom in consistent formulations of the action principle, and ii) the role a 3+1 decomposition has in the allowed internal gauge. We clarify how the requirement of well posed variational principles compatible with general weakly isolated horizons (WIHs) does lead to a partial gauge fixing in the first order descriptions used previously in the literature. We consider the Palatini action together with the Holst extension, with and without boundary terms at the horizon. We show that, for the complete configuration space --with no gauge fixing--, the Palatini action is differentiable without additional surface terms at the WIH boundary, but the more general Holst action is not. A surface term at the horizon --that renders the action for asymptotically flat configurations differentiable-- makes the Holst action differentiable, but only if one restricts configuration space and partially reduces the internal Lorentz gauge. For the second issue, we show that upon performing a 3+1 decomposition and imposing the time gauge, there is a further gauge reduction of the Hamiltonian theory in terms of Ashtekar-Barbero variables to a $U(1)$-gauge theory on the horizon. We show that even when the Holst action is differentiable without additional surface terms or any gauge fixing for Type I spherically symmetric (strongly) isolated horizons --and a preferred foliation--, this result does not go through for more general isolated or weakly isolated horizons. Our results represent the first comprehensive study of these issues and clarify some contradictory statements found in the literature.
We study here a complete quantization of a Callan-Giddings-Harvey-Strominger (CGHS) vacuum model following loop quantum gravity techniques. Concretely, we adopt a formulation of the model in terms of a set of new variables that resemble the ones commonly employed in spherically symmetric loop quantum gravity. The classical theory consists of two pairs of canonical variables plus a scalar and diffeomorphism (first class) constraints. We consider a suitable redefinition of the Hamiltonian constraint such that the new constraint algebra (with structure constants) is well adapted to the Dirac quantization approach. For it, we adopt a polymeric representation for both the geometry and the dilaton field. On the one hand, we find a suitable invariant domain of the scalar constraint operator, and we construct explicitly its solution space. There, the eigenvalues of the dilaton and the metric operators cannot vanish locally, allowing us to conclude that singular geometries are ruled out in the quantum theory. On the other hand, the physical Hilbert space is constructed out of them, after group averaging the previous states with the diffeomorphism constraint. In turn, we identify the standard observable corresponding to the mass of the black hole at the boundary, in agreement with the classical theory. We also construct an additional observable on the bulk associated with the square of the dilaton field, with no direct classical analog.
We consider the loop quantization of the (diagonal) Bianchi type IX cosmological model. We explore different quantization prescriptions that extend the work of Wilson-Ewing and Singh. In particular, we study two different ways of implementing the so-called inverse triad corrections. We construct the corresponding Hamiltonian constraint operators and show that the singularity is formally resolved. We find the effective equations associated with the different quantization prescriptions, and study the relation with the isotropic $k$=1 model that, classically, is contained within the Bianchi IX model. We use geometrically defined scalar observables to explore the physical implications of each of these theories. This is the first part in a series of papers analyzing different aspects of the Bianchi IX model, with inverse corrections, within loop quantum cosmology.