Multisymplectic effective general boundary field theory

Arjang, Mona; Zapata, José A.

doi:10.1088/0264-9381/31/9/095013

Cited by 9 publications

(13 citation statements)

References 45 publications

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“…I will define observable currents within a version for multisymplectic field theory over a discrete spacetime [2]. Even when this paper is entirely classical, this study is relevant for quantum field theory because a quantization of the mentioned framework leads to a spin foam model formulation of lattice field theory.…”

Section: Objective and Resultsmentioning

confidence: 99%

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Observable currents in lattice field theories

Zapata

2016

Preprint

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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.

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Section: Objective and Resultsmentioning

confidence: 99%

“…The formalism exists also for gauge theories and extended gauge theories (including general relativity). For a more detailed exposition of the framework for all types of fields, see [2].…”

Section: Review Of a Geometric Framework For Classical Lattice Field ...mentioning

confidence: 99%

Observable currents in lattice field theories

Zapata

2016

Preprint

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“…In particular, we expect that in this setting, the discrete Cartan form would only involve integration over ∂U , since boundary variations can be localized to codimension-one simplices, unlike for conforming finite element spaces. Furthermore, we aim to investigate how the discrete variational structures presented in this paper (in the conforming setting and extended to the discontinuous Galerkin setting) can be used to provide a geometric variational framework for studying lattice field theories, building on the discrete variational framework for lattice field theories initiated in Arjang and Zapata [4].…”

Section: Discussionmentioning

confidence: 99%

Variational Structures in Cochain Projection Based Variational Discretizations of Lagrangian PDEs

Tran¹,

Leok²

2021

Preprint

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Compatible discretizations, such as finite element exterior calculus, provide a discretization framework that respect the cohomological structure of the de Rham complex, which can be used to systematically construct stable mixed finite element methods. Multisymplectic variational integrators are a class of geometric numerical integrators for Lagrangian and Hamiltonian field theories, and they yield methods that preserve the multisymplectic structure and momentumconservation properties of the continuous system. In this paper, we investigate the synthesis of these two approaches, by constructing discretization of the variational principle for Lagrangian field theories utilizing structure-preserving finite element projections. In our investigation, compatible discretization by cochain projections plays a pivotal role in the preservation of the variational structure at the discrete level, allowing the discrete variational structure to essentially be the restriction of the continuum variational structure to a finite-dimensional subspace. The preservation of the variational structure at the discrete level will allow us to construct a discrete Cartan form, which encodes the variational structure of the discrete theory, and subsequently, we utilize the discrete Cartan form to naturally state discrete analogues of Noether's theorem and multisymplecticity, which generalize those introduced in the discrete Lagrangian variational framework by Marsden et al. [29]. We will study both covariant spacetime discretization and canonical spatial semi-discretization, and subsequently relate the two in the case of spacetime tensor product finite element spaces.

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“…One may also relate this to the multisymplectic approach to field theory. For a discrete spacetime version, see [41].…”

Section: Axiomatizationmentioning

confidence: 99%

A local and operational framework for the foundations of physics

Oeckl

2016

Preprint

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We discuss a novel framework for physical theories that is based on the principles of locality and operationalism. It generalizes and unifies previous frameworks, including the standard formulation of quantum theory, the convex operational framework and Segal's approach to quantum field theory. It is capable of encoding both classical and quantum (field) theories, implements spacetime locality in a manifest way and contains the complete modern notion of measurement in the quantum case. Its mathematical content can be condensed into a set of axioms that are similar to those of Atiyah and Segal. This is supplemented by two basic rules for extracting probabilities or expectation values for measurement processes. The framework, called the positive formalism, is derived in three completely different ways. One derivation is from first principles, one starts with classical field theory and one with quantum field theory. The latter derivation arose previously in the programme of the general boundary formulation of quantum theory. As in the convex operational framework, the difference between classical and quantum theories essentially arises from certain partially ordered vector spaces being either lattices or anti-lattices. If we add the ad hoc ingredient of imposing anti-lattice structures, the derivation from first principles may be seen as a reconstruction of quantum theory. Among other things, the positive formalism suggests a statistical approach to classical field theories with dynamical metric, provides a common ground for quantum information theory and quantum field theory, introduces a notion of local measurement into quantum field theory, and suggests a new perspective on quantum gravity by removing the incompatibility with general relativistic principles. The positive formalism as a framework for quantum theory is in conflict with various interpretations or modifications of quantum theory, including physical collapse theories, manyworlds interpretations, and non-local hidden variable theories.

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Multisymplectic effective general boundary field theory

Cited by 9 publications

References 45 publications

Observable currents in lattice field theories

Observable currents in lattice field theories

Variational Structures in Cochain Projection Based Variational Discretizations of Lagrangian PDEs

A local and operational framework for the foundations of physics

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