Jose Zapata

Followers

Following

Co-authors

Public Views

Interests

Uploads

Papers by Jose Zapata

Discrete models and General Boundary Field Theory

Download

Spin Foam Model for 3D Gravity in the Continuum

Topics in Mathematical Physics, General Relativity and Cosmology in Honor of Jerzy Plebanski, 2006

Download

Gauge from holography

arXiv: General Relativity and Quantum Cosmology, 2017

In a spacetime divided into two regions $U_1$ and $U_2$ by a hypersurface $\Sigma$, a perturbatio... more In a spacetime divided into two regions $U_1$ and $U_2$ by a hypersurface $\Sigma$, a perturbation of the field in $U_1$ is coupled to perturbations in $U_2$ by means of the first-order holographic imprint that it leaves on $\Sigma$. The linearized glueing field equation constrains perturbations on the two sides of a dividing hypersurface. This linear operator may have a nontrivial null space; a nontrivial perturbation of the field leaving a holographic imprint on a dividing hypersurface which does not affect perturbations on the other side should be considered physically irrelevant. This condition, together with a locality requirement, leads to the notion of gauge equivalence in first-order Lagrangian field theory.

Download

Topological aspects of fields modeled using extended lattice fields: a first study

Lattice formulations of physical systems encounter difficulties in capturing their topological pr... more Lattice formulations of physical systems encounter difficulties in capturing their topological properties when the fields describing them are separated into disconnected topological sectors. In this work we show that augmenting the usual lattice fields to include extra variables describing local topological information, the topology of the field space in the continuum is faithfully reproduced. We apply this extended lattice formulation to some simple models with non trivial topological charges, and we study their properties both analytically and via Monte Carlo simulations.

Download

Homotopy data as part of the lattice field: A first study

International Journal of Modern Physics C, 2021

Fields exhibit a variety of topological properties, like different topological charges, when fiel... more Fields exhibit a variety of topological properties, like different topological charges, when field space in the continuum is composed by more than one topological sector. Lattice treatments usually encounter difficulties describing those properties. In this work, we show that by augmenting the usual lattice fields to include extra variables describing local topological information (more precisely, regarding homotopy), the topology of the space of fields in the continuum is faithfully reproduced in the lattice. We apply this extended lattice formulation to some simple models with nontrivial topological charges, and we study their properties both analytically and via Monte Carlo simulations.

Download

Homotopy classes of gauge fields and the lattice

Advances in Theoretical and Mathematical Physics, 2019

Given a smooth manifold M and a Lie group G, we consider parallel transport maps-semigroup homomo... more Given a smooth manifold M and a Lie group G, we consider parallel transport maps-semigroup homomorphisms from a path groupoid in M to Gas an alternative description of principal G-bundles with smooth connections on them. Using a cellular decomposition C of M , and a system of paths associated to C , we define a homotopical equivalence relation of parallel transport maps, leading to the concept of an extended lattice gauge (ELG) field. A lattice gauge field, as used in Lattice Gauge Theory, is part of the data contained in an ELG field, but the latter contains additional topological information of local nature, sufficient to reconstruct a principal G-bundle up to equivalence, in the spirit of Barrett [3]. Following a theorem of Pachner [14], we give a criterion to determine when two ELG fields over different cell decompositions C and C ′ define equivalent bundles. As a first concrete physical application, we define a simple operation that captures the geometric essence of the 't Hooft loop operator in the theory of quantum gauge fields, and study its relation with the bundle structure of an ELG field. Contents 37 Appendix B. The equivalence theorem 39 References 42

Download

Macroscopic observables from the comparison of local reference systems

Classical and Quantum Gravity, 2019

Parallel transport as dictated by a gauge field determines a collection of local reference system... more Parallel transport as dictated by a gauge field determines a collection of local reference systems. Comparing local reference systems in overlapping regions leads to an ensemble of algebras of relational kinematical observables for gauge theories including general relativity. Using an auxiliary cellular decomposition, we propose a discretization of the gauge field based on a decimation of the mentioned ensemble of kinematical observables. The outcome is a discrete ensemble of local subalgebras of "macroscopic observables" characterizing a measuring scale. A set of evaluations of those macroscopic observables is called an extended lattice gauge field because it determines a G-bundle over M (and over submanifolds of M that inherit a cellular decomposition) together with a lattice gauge field over an embedded lattice. A physical observable in our algebra of macroscopic observables is constructed. An initial study of aspects of regularization and coarse graining, which are special to this description of gauge fields over a combinatorial base, is presented. The physical relevance of this extension of ordinary lattice gauge fields is discussed in the context of quantum gravity.

Download

Gauge from Holography and Holographic Gravitational Observables

Advances in High Energy Physics, 2019

In a spacetime divided into two regions U1 and U2 by a hypersurface Σ, a perturbation of the fiel... more In a spacetime divided into two regions U1 and U2 by a hypersurface Σ, a perturbation of the field in U1 is coupled to perturbations in U2 by means of the holographic imprint that it leaves on Σ. The linearized gluing field equation constrains perturbations on the two sides of a dividing hypersurface, and this linear operator may have a nontrivial null space. A nontrivial perturbation of the field leaving a holographic imprint on a dividing hypersurface which does not affect perturbations on the other side should be considered physically irrelevant. This consideration, together with a locality requirement, leads to the notion of gauge equivalence in Lagrangian field theory over confined spacetime domains. Physical observables in a spacetime domain U can be calculated integrating (possibly nonlocal) gauge invariant conserved currents on hypersurfaces such that ∂Σ⊂∂U. The set of observables of this type is sufficient to distinguish gauge inequivalent solutions. The integral of a conse...

Download

On a Continuum Limit for Loop Quantum Cosmology

AIP Conference Proceedings, 2008

The use of non-regular representations of the Heisenberg-Weyl commutation relations has proved to... more The use of non-regular representations of the Heisenberg-Weyl commutation relations has proved to be useful for studying conceptual and technical issues in quantum gravity. Of particular relevance is the study of Loop Quantum Cosmology (LQC), a symmetry reduced theory that is related to Loop Quantum Gravity, and that is based on a non-regular, polymeric representation. Recently, a soluble model was used by Ashtekar, Corichi and Singh to study the relation between Loop Quantum Cosmology and the standard Wheeler-DeWitt theory and in particular the passage to the limit in which the auxiliary parameter (interpreted as "quantum geometry discreetness") is sent to zero in hope to get rid of this 'regulator ambiguity' in the LQC dynamics. In this note we outline the first steps toward reformulating this question within the program developed by the authors for studying the continuum limit of polymeric theories, which was successfully applied to simple systems such as a Simple Harmonic Oscillator and the Free Particle.

Download

On the Classical Limit of Loop Quantized Theories

Most of the work presented was done in collaboration with Madhavan Varadarajan1. We study the cla... more Most of the work presented was done in collaboration with Madhavan Varadarajan1. We study the classical limit of kinematic loop quantum gravity. We observe that there are no states on which, both, the geometric operators and the holonomy operators behave quasi-classically. Thus, we propose a new set of connection measuring operators. These are "magnetic flux" type operators based on a physical lattice specified by the quasi-classical states themselves. Our aim is to approximate classical data of the kinematical phase space of connection dynamic approach to quantum gravity in the framework of loop quantization2. The proposed "magnetic flux" operators come along with an ambitious plan to study other quantum gauge fields. That is, we hope that our proposal helps the loop picture of quantum geometry to promote lattice gauge theory (LGT) to a physical arena for the study of renormalizable field theories that is free of ultraviolet divergences.

Polymer quantum mechanics and its continuum limit

Physical Review D, 2007

A rather non-standard quantum representation of the canonical commutation relations of quantum me... more A rather non-standard quantum representation of the canonical commutation relations of quantum mechanics systems, known as the polymer representation has gained some attention in recent years, due to its possible relation with Planck scale physics. In particular, this approach has been followed in a symmetric sector of loop quantum gravity known as loop quantum cosmology. Here we explore different aspects of the relation between the ordinary Schrödinger theory and the polymer description. The paper has two parts. In the first one, we derive the polymer quantum mechanics starting from the ordinary Schrödinger theory and show that the polymer description arises as an appropriate limit. In the second part we consider the continuum limit of this theory, namely, the reverse process in which one starts from the discrete theory and tries to recover back the ordinary Schrödinger quantum mechanics. We consider several examples of interest, including the harmonic oscillator, the free particle and a simple cosmological model.

Download

Local gauge theory and coarse graining

Journal of Physics: Conference Series, 2012

Within the discrete gauge theory which is the basis of spin foam models, the problem of macroscop... more Within the discrete gauge theory which is the basis of spin foam models, the problem of macroscopically faithful coarse graining is studied. Macroscopic data is identified; it contains the holonomy evaluation along a discrete set of loops and the homotopy classes of certain maps. When two configurations share this data they are related by a local deformation. The interpretation is that such configurations differ by "microscopic details". In many cases the homotopy type of the relevant maps is trivial for every connection; two important cases in which the homotopy data is composed by a set of integer numbers are: (i) a two dimensional base manifold and structure group U (1), (ii) a four dimensional base manifold and structure group SU (2). These cases are relevant for spin foam models of two dimensional gravity and four dimensional gravity respectively. This result suggests that if spin foam models for twodimensional and four-dimensional gravity are modified to include all the relevant macroscopic degrees of freedom-the complete collection of macroscopic variables necessary to ensure faithful coarse graining-, then they could provide appropriate effective theories at a given scale.

Download

Loop quantization from a lattice gauge theory perspective

Classical and Quantum Gravity, 2004

We present an interpretation of loop quantization in the framework of lattice gauge theory. Withi... more We present an interpretation of loop quantization in the framework of lattice gauge theory. Within this context the lack of appropriate notions of effective theories and renormalization group flow exhibit loop quantization as an incomplete framework. This interpretation includes a construction of embedded spin foam models which does not rely on the choice of any auxiliary structure (e.g. triangulation) and has the following straightforward consequences: The values of the coupling constants need to be those of an UV-attractive fixed point. The kinematics of canonical loop quantization and embedded spin foam models are compatible. The weights assigned to embedded spin foams are independent of the 2-polyhedron used to regularize the path integral, |J| x = |J| x′. An area spectrum with edge contributions proportional to l PL 2(j + 1/2) is not compatible with embedded spin foam models and/or canonical loop quantization.

Download

Quantum theory of geometry: III. Non-commutativity of Riemannian structures

Classical and Quantum Gravity, 1998

The basic framework for a systematic construction of a quantum theory of Riemannian geometry was ... more The basic framework for a systematic construction of a quantum theory of Riemannian geometry was introduced recently. The quantum versions of Riemannian structures-such as triad and area operators-exhibit a noncommutativity. At first sight, this feature is surprising because it implies that the framework does not admit a triad representation. To better understand this property and to reconcile it with intuition, we analyze its origin in detail. In particular, a careful study of the underlying phase space is made and the feature is traced back to the classical theory; there is no anomaly associated with quantization. We also indicate why the uncertainties associated with this non-commutativity become negligible in the semi-classical regime.

Download

Hamiltonian and physical Hilbert space in polymer quantum mechanics

Classical and Quantum Gravity, 2007

In this paper, a version of polymer quantum mechanics, which is inspired by loop quantum gravity,... more In this paper, a version of polymer quantum mechanics, which is inspired by loop quantum gravity, is considered and shown to be equivalent, in a precise sense, to the standard, experimentally tested, Schrödinger quantum mechanics. The kinematical cornerstone of our framework is the so called polymer representation of the Heisenberg-Weyl (H-W) algebra, which is the starting point of the construction. The dynamics is constructed as a continuum limit of effective theories characterized by a scale, and requires a renormalization of the inner product. The result is a physical Hilbert space in which the continuum Hamiltonian can be represented and that is unitarily equivalent to the Schrödinger representation of quantum mechanics. As a concrete implementation of our formalism, the simple harmonic oscillator is fully developed.

Download

Translation symmetry in 2+1 Regge calculus

Classical and Quantum Gravity, 1993

... Translation symmetry in 2 + 1 Regge calculus Henri Waelbroeck and Josh Antonio Zapata lnstitu... more

(2 + 1) covariant lattice theory and 't Hooft's formulation

Classical and Quantum Gravity, 1996

We show that 't Hooft's representation of (2+1)-dimensional gravity in terms of flat polygonal ti... more We show that 't Hooft's representation of (2+1)-dimensional gravity in terms of flat polygonal tiles is closely related to a gauge-fixed version of the covariant Hamiltonian lattice theory. 't Hooft's gauge is remarkable in that it leads to a Hamiltonian which is a linear sum of vertex Hamiltonians, each of which is defined modulo 2π. A cyclic Hamiltonian implies that "time" is quantized. However, it turns out that this Hamiltonian is constrained. If one chooses an internal time and solves this constraint for the "physical Hamiltonian", the result is not a cyclic function. Even if one quantizes a la Dirac, the "internal time" observable does not acquire a discrete spectrum. We also show that in Euclidean 3-d lattice gravity, "space" can be either discrete or continuous depending on the choice of quantization. Finally, we propose a generalization of 't Hooft's gauge for Hamiltonian lattice formulations of topological gravity dimension 4.

Download

Observable currents in lattice field theories

Observable currents are spacetime local objects that induce physical observables when integrated ... more Observable currents are spacetime local objects that induce physical observables when integrated on an auxiliary codimension one surface. Since the resulting observables are independent of local deformations of the integration surface, the currents themselves carry most of the information about the induced physical observables. I introduce observable currents in a multisymplectic framework for Lagrangian field theory over discrete spacetime. One family of examples is composed by Noether currents. A much larger family of examples is composed by currents, spacetime local objects, that encode the symplectic product between two arbitrary vectors tangent to the space of solutions. A weak version of observable currents, which in general are nonlocal, is also introduced. Weak observable currents can be used to separate points in the space of physically distinct solutions. It is shown that a large class of weak observable currents can be "improved" to become local. A Poisson brack...

Download

Effective theories of connections and curvature: abelian case

We introduce a notion of measuring scales for quantum abelian gauge systems. At each measuring sc... more We introduce a notion of measuring scales for quantum abelian gauge systems. At each measuring scale a finite dimensional affine space stores information about the evaluation of the curvature on a discrete family of surfaces. Affine maps from the spaces assigned to finer scales to those assigned to coarser scales play the role of coarse graining maps. This structure induces

Download

A Hamiltonian Formulation of Topological Gravity

Topological gravity is the reduction of Einstein's theory to spacetimes with vanishing curvature,... more Topological gravity is the reduction of Einstein's theory to spacetimes with vanishing curvature, but with global degrees of freedom related to the topology of the universe. We present an exact Hamiltonian lattice theory for topological gravity, which admits translations of the lattice sites as a gauge symmetry. There are additional symmetries, not present in Einstein's theory, which kill the local degrees of freedom. We show that these symmetries can be fixed by choosing a gauge where the torsion is equal to zero. In this gauge, the theory describes flat space-times. We propose two methods to advance towards the holy grail of lattice gravity: A Hamiltonian lattice theory for curved space-times, with first-class translation constraints.

Download

Jose Zapata

Uploads

Papers by Jose Zapata

Log In