Local Quantum Constraints

Grundling, Hendrik; Lledó, Fernando

doi:10.1142/s0129055x00000459

Cited by 22 publications

(75 citation statements)

References 27 publications

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“…in quantum electromagnetism, cf. [12,14,16]. Now in a representation π of R(X, σ ) for which Ker π(R(λ, f )) = {0} we have by Theorem 4.2(vi) that π(R(λ, f ))φ π (f ) = iλπ(R(λ, f )) − 1 on Dom φ π (f ).…”

Section: Constraint Theorymentioning

confidence: 97%

The resolvent algebra: A new approach to canonical quantum systems

Buchholz¹,

Grundling²

2008

Journal of Functional Analysis

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149

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The standard C * -algebraic version of the algebra of canonical commutation relations, the Weyl algebra, frequently causes difficulties in applications since it neither admits the formulation of physically interesting dynamical laws nor does it incorporate pertinent physical observables such as (bounded functions of) the Hamiltonian. Here a novel C * -algebra of the canonical commutation relations is presented which does not suffer from such problems. It is based on the resolvents of the canonical operators and their algebraic relations. The resulting C * -algebra, the resolvent algebra, is shown to have many desirable analytic properties and the regularity structure of its representations is surprisingly simple. Moreover, the resolvent algebra is a convenient framework for applications to interacting and to constrained quantum systems, as we demonstrate by several examples.

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Section: Constraint Theorymentioning

confidence: 97%

The resolvent algebra: A new approach to canonical quantum systems

Buchholz¹,

Grundling²

2008

Journal of Functional Analysis

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149

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“…(32) in the proof of the preceding lemma. The intertwining property is proved by direct computation.…”

Section: Proposition the Mapping φmentioning

confidence: 88%

“…In particular, if it is possible to describe at the C*-algebraic level in M cont the choice of nonfaithful representation of E(2) needed to define discrete helicity. Techniques of local quantum constraints (see [30,32]) may possibly be applied to O → M cont (O) in order to consider this question. (Here, the use of abstract C*-algebras in a first step can be relevant.)…”

Section: Discussionmentioning

confidence: 99%

“…[67]). For a detailed treatment of the nets associated to the electromagnetic vector potential (including a general analysis of the localised constraints) see [32].…”

Section: Remarkmentioning

confidence: 99%

“…for short). For a detailed treatment of quantum electromagnetism in terms of the vector potential (including constraints) we refer to [32]. An alternative and systematic analysis of relativistic particles in the context of geometric quantisation is given in [23,24].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Massless Relativistic Wave Equations and Quantum Field Theory

Lledó

2004

Ann. Henri Poincaré

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We give a simple and direct construction of a massless quantum field with arbitrary discrete helicity that satisfies Wightman axioms and the corresponding relativistic wave equation in the distributional sense. We underline the mathematical differences to massive models. The construction is based on the notion of massless free net (cf. Section 3) and the detailed analysis of covariant and massless canonical (Wigner) representations of the Poincaré group. A characteristic feature of massless models with nontrivial helicity is the fact that the fibre degrees of freedom of the covariant and canonical representations do not coincide. We use massless relativistic wave equations as constraint equations reducing the fibre degrees of freedom of the covariant representation. They are characterised by invariant (and in contrast with the massive case non reducing) one-dimensional projections. The definition of one-particle Hilbert space structure that specifies the quantum field uses distinguished elements of the intertwiner space between E(2) (the two-fold cover of the 2-dimensional Euclidean group) and E(2).We conclude with a brief comparison between the free nets constructed in Section 3 and a recent alternative construction that uses the notion of modular localisation.

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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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Local Quantum Constraints

Cited by 22 publications

References 27 publications

The resolvent algebra: A new approach to canonical quantum systems

The resolvent algebra: A new approach to canonical quantum systems

Massless Relativistic Wave Equations and Quantum Field Theory

QCD on an Infinite Lattice

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