arXiv subjects

José A. Zapata

Publications and source records attributed to José A. Zapata.

15 recordsLinked to original sources

arXiv2026

FALSE: The EPRL amplitude is supported on flat connections

There is an error in our recent preprint ``The EPRL amplitude is supported on flat connections''. The error is in Section 3. Here we leave the original text unchanged, but add a note in Section 3 pointing out exactly what the error is. We apologize for the false alarm. After one month, this preprint will be withdrawn from the arxiv. {\bf Original Abstract:} For the version of the EPRL model based on the original vertex amplitude and the face amplitude selected by its gluing properties, we prove that the EPRL amplitude of any region with the topology of a 4-ball is supported on flat connections. We state immediate consequences of this result, comment on some applications, and discuss physical implications. The results hold in general; they do not rely on a semiclassical analysis.

gr-qc↗

arXiv2026

Homotopy lattice gauge fields 1: The fields and their properties

We introduce homotopy lattice gauge fields (HLGFs), a version of gauge fields over a discretized base, based on a notion of higher parallel transport that enriches the usual parallel transport along paths on a lattice to also consider higher dimensional paths. Higher dimensional data keeps information about the parallel transport along homotopies of curves. With this data, a HLGF on a base space of dimension two or three determines a principal bundle over the base manifold. This data is also responsible for our formulas for the topological charge on two-dimensional bases. Our framework is an application of a nonabelian algebraic topology framework developed to solve the local to global problem in higher dimensional homotopy. No previous knowledge of higher category theory is assumed. The second part will be devoted to the space of fields as an arena for doing Quantum Field Theory, and to give the first examples of how our framework refines standard lattice gauge theory.

math-ph↗

arXiv2025

Towards relational foundations for spacetime quantum physics

Rovelli's relational interpretation of quantum mechanics tells us that the description of a system in the formalism of quantum mechanics is not an absolute, but it is relative to the observer itself. The interpretation goes further and proposes a set of axioms. In standard non relational language, one of them states that an observer can only retrieve finite amount information from a system by means of measurement. Our contribution starts with the observation that quantum mechanics, i.e. quantum field theory (QFT) in dimension 1, radically differs from QFT in higher dimensions. In higher dimensions boundary data (or initial data) cannot be specified by means of finitely many measurements. This calls for a notion of measuring scale, which we provide. At a given measuring scale the observer has partial information about the system. Our notion of measuring scale generalizes the one implicitly used in Wilsonian QFT, where at each measuring scale there are effective theories, which may be corrected, and if the theory turns out to be renormalizable the mentioned corrections converge to determine a completely corrected (or renormalized) theory at the given measuring scale. The notion of a measuring scale is the cornerstone of Wilsonian QFT. This notion tells us that we are not describing a system from an absolute perspective. An effective theory at that scale describes the system with respect to the observer, which may retrieve information from the system by means of measurement in a specific way determined by our notion of measuring scale. We claim that a relational interpretation of quantum physics for spacetimes of dimension greater than 1 is Wilsonian.

quant-ph↗

arXiv2023

A discretization of Holst's action for general relativity

We present a simplicial model for gravity written in terms of a discretized Lorentz connection and a discretized tetrad field. The continuum limit of its action is Holst's action for general relativity. With the intention of using it to construct spin foam modes for quantum gravity, we write two other equivalent models written in terms of a discretized and constrained $B$ field. The differences between our model and existing models are most likely inessential in the sense that a quantization would lead to equivalent quantum theories in the Wilsonian continuum limit. Nevertheless, we mention two features leading to possible advantages: Curvature degrees of freedom are described at the level of each 4-simplex. Our model offers a picture of bulk geometry leading to actions for matter couplings that split as a sum over 4-simplices.

gr-qc↗

arXiv2021

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

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 non-trivial topological charges, and we study their properties both analytically and via Monte Carlo simulations.

hep-lat↗

arXiv2020

Homotopy classes of gauge fields and the lattice

For a smooth manifold $M$, possibly with boundary and corners, and a Lie group $G$, we consider a suitable description of gauge fields in terms of parallel transport, as groupoid homomorphisms from a certain path groupoid in $M$ to $G$. Using a cotriangulation $\mathscr{C}$ of $M$, and collections of finite-dimensional families of paths relative to $\mathscr{C}$, 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 further local topological information sufficient to reconstruct a principal $G$-bundle on $M$ up to equivalence. The space of ELG fields of a given pair $(M,\mathscr{C})$ is a covering for the space of fields in Lattice Gauge Theory, whose connected components parametrize equivalence classes of principal $G$-bundles on $M$. We give a criterion to determine when ELG fields over different cotriangulations define equivalent bundles.

math-ph↗

arXiv2019

Gauge from holography and holographic gravitational observables

In a spacetime divided into two regions $U_1$ and $U_2$ by a hypersurface $Σ$, a perturbation of the field in $U_1$ is coupled to perturbations in $U_2$ 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 non local) gauge invariant conserved currents on hypersurfaces such that $\partial Σ\subset \partial U$. The set of observables of this type is sufficient to distinguish gauge inequivalent solutions. The integral of a conserved current on a hypersurface is sensitive only to its homology class $[Σ]$, and if $U$ is homeomorphic to a four ball the homology class is determined by its boundary $S = \partial Σ$. We will see that a result of Anderson and Torre implies that for a class of theories including vacuum General Relativity all local observables are holographic in the sense that they can be written as integrals of over the two dimensional surface $S$. However, non holographic observables are needed to distinguish between gauge inequivalent solutions.

gr-qc↗

arXiv2018

Observable currents and a covariant Poisson algebra of physical observables

Observable currents are locally defined gauge invariant conserved currents; physical observables may be calculated integrating them on appropriate hypersurfaces. Due to the conservation law the hypersurfaces become irrelevant up to homology, and the main objects of interest become the observable currents them selves. Gauge inequivalent solutions can be distinguished by means of observable currents. With the aim of modeling spacetime local physics, we work on spacetime domains $U\subset M$ which may have boundaries and corners. Hamiltonian observable currents are those satisfying ${\sf d_v}F=-ι_VΩ_L+{\sf d_h}σ^F$ and a certain boundary condition. The family of Hamiltonian observable currents is endowed with a bracket that gives it a structure which generalizes a Lie algebra in which the Jacobi relation is modified by the presence of a boundary term. If the domain of interest has no boundaries the resulting algebra of observables is a Lie algebra. In the resulting framework algebras of observable currents are associated to bounded domains, and the local algebras obey interesting gluing properties. These results are due to considering currents that defined only locally in field space and to a revision of the concept of gauge invariance in bounded spacetime domains. A perturbation of the field on a bounded spacetime domain is regarded as gauge if: (i) the "first order holographic imprint" that it leaves in any hypersurface locally splitting a spacetime domain into two subdomains is negligible according to the linearized gluing field equation, and (ii) the perturbation vanishes at the boundary of the domain. A current is gauge invariant if the variation in them induced by any gauge perturbation vanishes up to boundary terms.

gr-qc↗

arXiv2016

Observable currents in lattice field theories

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 bracket gives the space of observable currents the structure of a Lie algebra. Peierls bracket for bulk observables gives an algebra homomorphism mapping equivalence classes of bulk observables to weak observable currents. The study covers scalar fields, nonlinear sigma models and gauge theories (including gauge theory formulations of general relativity) on the lattice. Even when this paper is entirely classical, this study is relevant for quantum field theory because a quantization of the framework leads to a spin foam model formulation of lattice field theory.

math-ph↗

arXiv2014

Multisymplectic effective General Boundary Field Theory

The transfer matrix in lattice field theory connects the covariant and the initial data frameworks; in spin foam models, it can be written as a composition of elementary cellular amplitudes/propagators. We present a framework for discrete spacetime classical field theory in which solutions to the field equations over elementary spacetime cells may be amalgamated if they satisfy simple gluing conditions matching the composition rules of cellular amplitudes in spin foam models. Furthermore, the formalism is endowed with a multisymplectic structure responsible for local conservation laws. Some models within our framework are effective theories modeling a system at a given scale. Our framework allows us to study coarse graining and the continuum limit.

math-ph↗

arXiv2005

Loop quantization as a continuum limit

We present an implementation of Wilson's renormalization group and a continuum limit tailored for loop quantization. The dynamics of loop quantized theories is constructed as a continuum limit of dynamics of effective theories. After presenting the general formalism we show as first explicit example the 2d Ising field theory. It is an interacting relativistic quantum field theory with local degrees of freedom quantized by loop quantization techniques.

hep-th↗

arXiv2005

Geometry of C-flat connections, coarse graining and the continuum limit

A notion of effective gauge fields which does not involve a background metric is introduced. The role of scale is played by cellular decompositions of the base manifold. Once a cellular decomposition is chosen, the corresponding space of effective gauge fields is the space of flat connections with singularities on its codimension two skeleton, ${\cal A}_{C-flat} \subset \bar{\cal A}_M$. If cellular decomposition $C_2$ is finer than cellular decomposition $C_1$, there is a coarse graining map $π_{C_2 \to C_1}: {\cal A}_{C_2-flat} \to {\cal A}_{C_1-flat}$. We prove that the triple $({\cal A}_{C_2-flat}, π_{C_2 \to C_1}, {\cal A}_{C_1-flat})$ is a principal fiber bundle with a preferred global section given by the natural inclusion map $i_{C_1 \to C_2}: {\cal A}_{C_1-flat} \to {\cal A}_{C_2-flat}$. Since the spaces ${\cal A}_{C-flat}$ are partially ordered (by inclusion) and this order is directed in the direction of refinement, we can define a continuum limit, $C \to M$. We prove that, in an appropriate sense, $\lim_{C \to M} {\cal A}_{C-flat} = \bar{\cal A}_M$. We also define a construction of measures in $\bar{\cal A}_M$ as the continuum limit (not a projective limit) of effective measures.

hep-th↗

arXiv2004

Loop quantization from a lattice gauge theory perspective

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: (1) The values of the coupling constants need to be those of an UV-attractive fixed point (2) The kinematics of canonical loop quantization and embedded spin foam models are compatible (3) The weights assigned to embedded spin foams are independent of the 2-polyhedron used to regularize the path integral, $|J|_x = |J|_{x'}$ (4) An area spectrum with edge contributions proportional to $l_{\rm PL}^2 (j+1 / 2)$ is not compatible with embedded spin foam models and/or canonical loop quantization

gr-qc↗

arXiv2002

Continuum spin foam model for 3d gravity

An example illustrating a continuum spin foam framework is presented. This covariant framework induces the kinematics of canonical loop quantization, and its dynamics is generated by a {\em renormalized} sum over colored polyhedra. Physically the example corresponds to 3d gravity with cosmological constant. Starting from a kinematical structure that accommodates local degrees of freedom and does not involve the choice of any background structure (e. g. triangulation), the dynamics reduces the field theory to have only global degrees of freedom. The result is {\em projectively} equivalent to the Turaev-Viro model.

gr-qc↗

arXiv2000

A proposal for analyzing the classical limit of kinematic loop gravity

We analyze the classical limit of kinematic loop quantum gravity in which the diffeomorphism and hamiltonian constraints are ignored. We show that there are no quantum states in which the primary variables of the loop approach, namely the SU(2) holonomies along {\em all} possible loops, approximate their classical counterparts. At most a countable number of loops must be specified. To preserve spatial covariance, we choose this set of loops to be based on physical lattices specified by the quasi-classical states themselves. We construct ``macroscopic'' operators based on such lattices and propose that these operators be used to analyze the classical limit. Thus, our aim is to approximate classical data using states in which appropriate macroscopic operators have low quantum fluctuations. Although, in principle, the holonomies of `large' loops on these lattices could be used to analyze the classical limit, we argue that it may be simpler to base the analysis on an alternate set of ``flux'' based operators. We explicitly construct candidate quasi-classical states in 2 spatial dimensions and indicate how these constructions may generalize to 3d. We discuss the less robust aspects of our proposal with a view towards possible modifications. Finally, we show that our proposal also applies to the diffeomorphism invariant Rovelli model which couples a matter reference system to the Hussain Kucha{\v r} model.

gr-qc↗