Quantum field theory on noncommutative spaces

Wulkenhaar, Raimar

doi:10.1007/978-3-030-29597-4_11

Cited by 9 publications

(22 citation statements)

References 246 publications

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“…Even standard local field theory is covered if one considers trivial 2-graphs with S = ∅. The paradigmatic example in mind, though, are theories of tensor fields, in particular matrix field theories [18,19] (related to noncommutative field theory) with combinatorial maps as diagrams and theories with tensorial interactions of arbitrary rank whose diagrams are coloured graphs [22,23]:…”

Section: Combinatorial Basis: 2-graphsmentioning

confidence: 99%

“…spectral dimension of an external matrix (the kinetic operator in the action) [18,19] and relates to the spacetime dimension D of a corresponding non-commutative field theory for even D [17]. Let the weights be ω| G e 1 = 1 and ω| G v…”

Section: The Hopf Algebra Of 2-graphs and Renormalization In Cnlftmentioning

confidence: 99%

“…Example 4 (Hopf algebra of matrix field theory/Grosse-Wulkenhaar model) The Feynman diagrams of matrix field theories [17][18][19] are combinatorial maps which are 2-graphs with bivalent edges, d h = 2 (Example 1). Thus, the interaction vertex graphs can be polygons with n vertices and disjoint unions thereof (multi-trace vertices).…”

Section: Definition 17 (Renormalizability and The Hopf Algebra Of Feynman 2-graphs)mentioning

confidence: 99%

“…But there are also examples of renormalizable field theories with certain "non-local" interactions for which a Connes-Kreimer type Hopf algebra can be found [11][12][13][14]. More specifically, these are combinatorially non-local field theories such as non-commutative quantum field theory and matrix field theory [15][16][17][18][19], and its generalization to field theories of higher rank tensor fields [20][21][22], including group field theory [23][24][25][26]. It is thus a natural question whether this is just a coincidence in specific examples or a more general feature, and whether the (Hopf algebra) mechanisms are different or the same.…”

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confidence: 99%

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Renormalization in Combinatorially Non-Local Field Theories: The Hopf Algebra of 2-Graphs

Thürigen

2021

Math Phys Anal Geom

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Renormalization in perturbative quantum field theory is based on a Hopf algebra of Feynman diagrams. A precondition for this is locality. Therefore one might suspect that non-local field theories such as matrix or tensor field theories cannot benefit from a similar algebraic understanding. Here I show that, on the contrary, perturbative renormalization of a broad class of such field theories is based in the same way on a Hopf algebra. Their interaction vertices have the structure of graphs. This gives the necessary concept of locality and leads to Feynman diagrams defined as “2-graphs” which generate the Hopf algebra. These results set the stage for a systematic study of perturbative renormalization as well as non-perturbative aspects, e.g. Dyson-Schwinger equations, for a number of combinatorially non-local field theories with possible applications to random geometry and quantum gravity.

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Section: Combinatorial Basis: 2-graphsmentioning

confidence: 99%

Section: The Hopf Algebra Of 2-graphs and Renormalization In Cnlftmentioning

confidence: 99%

Section: Definition 17 (Renormalizability and The Hopf Algebra Of Feynman 2-graphs)mentioning

confidence: 99%

mentioning

confidence: 99%

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Renormalization in Combinatorially Non-Local Field Theories: The Hopf Algebra of 2-Graphs

Thürigen

2021

Math Phys Anal Geom

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“…A classical example is BPHZ momentum subtraction as used for the general proof of renormalization [7,27,55]. In [53] I have shown that the Connes-Kreimer Hopf algebra describes renormalization not only in field theories with point-like interactions but also with full generality in combinatorially non-local field theories (cNLFT) such as non-commutative and matrix field theory [23,28,54], tensorial field theories [2,3] and group field theory [13,20,37].…”

Section: Introductionmentioning

confidence: 99%

Renormalization in Combinatorially Non-Local Field Theories: the BPHZ Momentum Scheme

Thürigen

2021

SIGMA

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Various combinatorially non-local field theories are known to be renormalizable. Still, explicit calculations of amplitudes are very rare and restricted to matrix field theory. In this contribution I want to demonstrate how the BPHZ momentum scheme in terms of the Connes-Kreimer Hopf algebra applies to any combinatorially non-local field theory which is renormalizable. This algebraic method improves the understanding of known results in noncommutative field theory in its matrix formulation. Furthermore, I use it to provide new explicit perturbative calculations of amplitudes in tensorial field theories of rank r>2.

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Noncommutative Geometry, the Spectral Standpoint

Connes¹

2021

New Spaces in Physics

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Quantum field theory on noncommutative spaces

Cited by 9 publications

References 246 publications

Renormalization in Combinatorially Non-Local Field Theories: The Hopf Algebra of 2-Graphs

Renormalization in Combinatorially Non-Local Field Theories: The Hopf Algebra of 2-Graphs

Renormalization in Combinatorially Non-Local Field Theories: the BPHZ Momentum Scheme

Noncommutative Geometry, the Spectral Standpoint

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