Quantum field theory and Hopf algebra cohomology

Brouder, Christian; Fauser, Bertfried; Frabetti, Alessandra; Oeckl, Robert

doi:10.1088/0305-4470/37/22/014

Cited by 42 publications

(68 citation statements)

References 36 publications

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“…To represent internal edges, we define the formal elements R i,j ∈ S(V ) ⊗v with 1 ≤ i ≤ j ≤ v using the inverse Feynman propagator (4). 2 For i = j the definition is (5) with the field operators φ(x) and φ(y) inserted at the i th and j th positions, respectively. For i = j the definition is…”

Section: Algebraic Representation Of Graphsmentioning

confidence: 99%

“…We refer the reader to [5] for a classical treatment of Hopf algebras and to [6] for the Hopf algebra structure of the symmetric algebra. The significance of this Hopf algebra structure for quantum field theory was developed in [2]. (Note, however, that the product taken there is the normal product and not the time-ordered one.)…”

Section: The Field Operator Algebra As a Hopf Algebramentioning

confidence: 99%

“…In particular, using the Hopf algebra and its cohomology it was shown (among other things) how different products of the algebra of field operators are related by Drinfeld twists and how interactions correspond to 2-cocycles [2].…”

Section: Introductionmentioning

confidence: 99%

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Generating loop graphs via Hopf algebra in quantum field theory

Mestre

Oeckl

2006

Journal of Mathematical Physics

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We use the Hopf algebra structure of the time-ordered algebra of field operators to generate all connected weighted Feynman graphs in a recursive and efficient manner. The algebraic representation of the graphs is such that they can be evaluated directly as contributions to the connected n-point functions. The recursion proceeds by loop order and vertex number.The combinatorics of perturbative quantum field theory is traditionally dealt with via functional methods and generating functions. However, it is possible to use a more intrinsic algebraic approach instead, rooted directly at the level of n-point functions and pioneered in the 1960's, see [1].More recently, it was realized that the Hopf algebra structure of the algebra of field operators (with the normal or with the time-ordered product) can be fruitfully exploited. In particular, using the Hopf algebra and its cohomology it was shown (among other things) how different products of the algebra of field operators are related by Drinfeld twists and how interactions correspond to 2-cocycles [2].Relations between different types of n-point functions and the associated combinatorics of Feynman graphs via the Hopf algebra structure of the timeordered algebra of field operators was established in [3]. More precisely, the *

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Section: Algebraic Representation Of Graphsmentioning

confidence: 99%

Section: The Field Operator Algebra As a Hopf Algebramentioning

confidence: 99%

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Generating loop graphs via Hopf algebra in quantum field theory

Mestre

Oeckl

2006

Journal of Mathematical Physics

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“…The approach was developed in part by applying methods borrowed from quantum field theory [7,5], in a simplified group theoretical setting. In group theory terms, this earlier symmetric function work concerns the characters of the general linear group.…”

Section: Motivationmentioning

confidence: 99%

The Hopf algebra structure of the character rings of classical groups

Fauser

Jarvis

King

2012

J. Phys. A: Math. Theor.

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The character ring Char-GL of covariant irreducible tensor representations of the general linear group admits a Hopf algebra structure isomorphic to the Hopf algebra Symm-Λ of symmetric functions. Here we study the character rings Char-O and Char-Sp of the orthogonal and symplectic subgroups of the general linear group within the same framework of symmetric functions. We show that Char-O and Char-Sp also admit natural Hopf algebra structures that are isomorphic to that of Char-GL , and hence to Symm-Λ . The isomorphisms are determined explicitly, along with the specification of standard bases for Char-O and Char-Sp analogous to those used for Symm-Λ . A major structural change arising from the adoption of these bases is the introduction of new orthogonal and symplectic Schur-Hall scalar products. Significantly, the adjoint with respect to multiplication no longer coincides, as it does in the Char-GL case, with a Foulkes derivative or skew operation. The adjoint and Foulkes derivative now require separate definitions, and their properties are explored here in the orthogonal and symplectic cases. Moreover, the Hopf algebras Char-O and Char-Sp are not self-dual. The dual Hopf algebras Char-O * and Char-Sp * are identified. Finally, the Hopf algebra of the universal rational character ring Char-GLrat of mixed irreducible tensor representations of the general linear group is introduced and its structure maps identified.

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“…the pointwise product of distributions on the right side of (2.6) exists and the 'coefficients' of (F ⋆ m G) satisfy again (2.2). ⋆ m corresponds to a ⋆-product in the sense of deformation quantization [1] (see also [29,5]), and it may be interpreted as Wick's Theorem for 'off-shell fields' (i.e. fields which are not restricted by any field equation).…”

Section: ) Which Induces a Product ⋆ M : F (O) × F (O) → F (O) The mentioning

confidence: 99%

Causal Perturbation Theory in Terms of Retarded Products, and a Proof of the Action Ward Identity

2004

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In the framework of perturbative algebraic quantum field theory a local construction of interacting fields in terms of retarded products is performed, based on earlier work of Steinmann [42]. In our formalism the entries of the retarded products are local functionals of the off shell classical fields, and we prove that the interacting fields depend only on the action and not on terms in the Lagrangian which are total derivatives, thus providing a proof of Stora's 'Action Ward Identity ' [45]. The theory depends on free parameters which flow under the renormalization group. This flow can be derived in our local * Work supported by the Deutsche Forschungsgemeinschaft.1 framework independently of the infrared behavior, as was first established by Hollands and Wald [32]. We explicitly compute non-trivial examples for the renormalization of the interaction and the field.

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Quantum field theory and Hopf algebra cohomology

Cited by 42 publications

References 36 publications

Generating loop graphs via Hopf algebra in quantum field theory

Generating loop graphs via Hopf algebra in quantum field theory

The Hopf algebra structure of the character rings of classical groups

Causal Perturbation Theory in Terms of Retarded Products, and a Proof of the Action Ward Identity

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