The Mathematical Structure of the Quantum BRST Constraint Method

Costello, Patrick

doi:10.48550/arxiv.0905.3570

Cited by 4 publications

(5 citation statements)

References 74 publications

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“…It also implies that these algebras contain multiplicative mollifiers for unbounded operators which appear in the algebraic treatment of supersymmetric models [4] or of constraint systems [1,6]. Thus the resolvent algebras provide in many respects a natural and convenient mathematical setting for the discussion of finite and infinite quantum systems.…”

Section: Discussionmentioning

confidence: 99%

“…The resolvent algebras already have proved to be useful in several applications to quantum physics such as the representation theory of abelian Lie algebras of derivations [3], the study of constraint systems and of the BRST method in a C*-algebraic setting [1,6], the treatment of supersymmetric models on non-compact spacetimes and the rigorous construction of corresponding JLOK-cocycles [4]. Their virtues also came to light in the formulation and analysis of the dynamics of finite and infinite quantum systems [1,5].…”

Section: Introductionmentioning

confidence: 99%

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Quantum Systems and Resolvent Algebras

Buchholz

Grundling

2015

Lecture Notes in Physics

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This survey article is concerned with the modeling of the kinematical structure of quantum systems in an algebraic framework which eliminates certain conceptual and computational difficulties of the conventional approaches. Relying on the Heisenberg picture it is based on the resolvents of the basic canonically conjugate operators and covers finite and infinite quantum systems. The resulting C*-algebras, the resolvent algebras, have many desirable properties. On one hand they encode specific information about the dimension of the respective quantum system and have the mathematically comfortable feature of being nuclear, and for finite dimensional systems they are even postliminal.This comes along with a surprisingly simple structure of their representations. On the other hand, they are a convenient framework for the study of interacting as well as constrained quantum systems since they allow the direct application of C*-algebraic methods which often simplify the analysis. Some pertinent facts are illustrated by instructive examples.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Quantum Systems and Resolvent Algebras

Buchholz

Grundling

2015

Lecture Notes in Physics

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“…Here we want to obtain the algebra of physical observables from our chosen field algebra F e = (A Λ ⊕C)⋊ α (Gau d Λ) by enforcing the local Gauss law constraint, and by imposing gauge invariance in an appropriate form. There is a wide range of methods in the literature for enforcing quantum constraints, not all equivalent [6]. For lattice QCD, so far constraint methods include the method developed by Kijowski and Rudolph in [22], the direct construction of explicit gauge invariant quantities through either Wilson loops of Fermi bilinears connected with a flux line (cf.…”

Section: Enforcement Of Local Gauss Law Constraintsmentioning

confidence: 99%

QCD on an Infinite Lattice

Grundling

Rudolph

2013

Commun. Math. Phys.

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We construct a mathematically well-defined framework for the kinematics of Hamiltonian QCD on an infinite lattice in R 3 , and it is done in a C*-algebraic context. This is based on the finite lattice model for Hamiltonian QCD developed by Kijowski, Rudolph e.a. (cf. [22,23]). To extend this model to an infinite lattice, we need to take an infinite tensor product of nonunital C*-algebras, which is a nonstandard situation. We use a recent construction for such situations, developed in [10]. Once the field C*-algebra is constructed for the fermions and gauge bosons, we define local and global gauge transformations, and identify the Gauss law constraint. The full field algebra is the crossed product of the previous one with the local gauge transformations. The rest of the paper is concerned with enforcing the Gauss law constraint to obtain the C*-algebra of quantum observables. For this, we use the method of enforcing quantum constraints developed by Grundling and Hurst (cf. [11]). In particular, the natural inductive limit structure of the field algebra is a central component of the analysis, and the constraint system defined by the Gauss law constraint is a system of local constraints in the sense of [15]. Using the techniques developed in that area, we solve the full constraint system by first solving the finite (local) systems and then combining the results appropriately. We do not consider dynamics.

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“…Only more recently it has found applications to problems of physical interest, cf. for example [5,8,10,11,16].…”

Section: Introductionmentioning

confidence: 99%

Trapped bosons, thermodynamic limit and condensation: a study in the framework of resolvent algebras

Bahns,

Buchholz

2020

Preprint

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The virtues of resolvent algebras, compared to other approaches for the treatment of canonical quantum systems, are exemplified by infinite systems of non-relativistic bosons. Within this framework, equilibrium states of trapped and untrapped bosons are defined on a fixed C*-algebra for all physically meaningful values of the temperature and chemical potential. Moreover, the algebra provides the tools for their analysis without having to rely on ad hoc prescriptions for the test of pertinent features, such as the appearance of Bose-Einstein condensates. The method is illustrated in case of non-interacting systems in any number of spatial dimensions and sheds new light on the appearance of condensates. Yet the framework also covers interactions and thus provides a universal basis for the analysis of bosonic systems.

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The Mathematical Structure of the Quantum BRST Constraint Method

Cited by 4 publications

References 74 publications

Quantum Systems and Resolvent Algebras

Quantum Systems and Resolvent Algebras

QCD on an Infinite Lattice

Trapped bosons, thermodynamic limit and condensation: a study in the framework of resolvent algebras

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