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On the Renormalization of Non-Commutative Field Theories
High Energy Physics - Theory · Submitted 23 Jul 2012
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim To start with, we discuss the Quantum Action Principle and provide evidence for its validity for non-commutative quantum field theories by showing that the equation of motion considered as insertion in the generating functional Zc[j] of connected Green functions makes sense.
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The abstract, sentence by sentence
This paper addresses three topics concerning the quantization of non-commutative field theories (as defined in terms of the Moyal star product involving a constant tensor describing the non-commutativity of coordinates in Euclidean space). To start with, we discuss the Quantum Action Principle and provide evidence for its validity for non-commutative quantum field theories by showing that the equation of motion considered as insertion in the generating functional Zc[j] of connected Green functions makes sense. Second, we consider the generalization of the BPHZ renormalization scheme to non-commutative field theories and apply it to the case of a self-interacting real scalar field. Finally, we discuss the renormalizability of various models for a self-interacting complex scalar field by using the approach of algebraic renormalization.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | This paper addresses three topics concerning the quantization of non-commutative field theories (as defined in terms of the Moyal star product involving a constant tensor describing the non-commutativity of coordinates in Euclidean space). | contribution 0.51 | 0.41 | 0.04 | asserted 0.93 |
| s1 | To start with, we discuss the Quantum Action Principle and provide evidence for its validity for non-commutative quantum field theories by showing that the equation of motion considered as insertion in the generating functional Zc[j] of connected Green functions makes sense. | contribution 0.97 | 0.91 | 0.06 | asserted 1.00 |
| s2 | Second, we consider the generalization of the BPHZ renormalization scheme to non-commutative field theories and apply it to the case of a self-interacting real scalar field. | contribution 0.87 | 0.92 | 0.05 | asserted 0.92 |
| s3 | Finally, we discuss the renormalizability of various models for a self-interacting complex scalar field by using the approach of algebraic renormalization. | contribution 0.98 | 0.89 | 0.12 | asserted 0.78 |
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The paper
Original abstract
This paper addresses three topics concerning the quantization of non-commutative field theories (as defined in terms of the Moyal star product involving a constant tensor describing the non-commutativity of coordinates in Euclidean space). To start with, we discuss the Quantum Action Principle and provide evidence for its validity for non-commutative quantum field theories by showing that the equation of motion considered as insertion in the generating functional Zc[j] of connected Green functions makes sense. Second, we consider the generalization of the BPHZ renormalization scheme to non-commutative field theories and apply it to the case of a self-interacting real scalar field. Finally, we discuss the renormalizability of various models for a self-interacting complex scalar field by using the approach of algebraic renormalization.
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How the literature uses it
3 arXiv papers cite this in 13 sentences
Papers citing it · 8 sentences without a confident label
Uses its method or tool 1 paper
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Therefore, a modified subtraction scheme was proposed in Ref. [ref] : It consists of considering $p$ and $$ as independent variables (though satisfying $p p̃ =0$) when performing the subtraction: In particular, one subtracts from the integrand its Taylor series expansion with respect to the external momentum $p$ around $p=0$ up to the order of divergence of the graph, while maintaining the phase factors.
Gauge Fields on Non-Commutative Spaces and Renormalization · BPHZ applied to phi4-theory · 0.82
Cites it as support 2 papers
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Actually [ref] , it is the only non-local counterterm which can appear in a translation invariant non-commutative scalar field model.
BPHZ renormalization and its application to non-commutative field theory · Finite renormalizations · 0.98
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In fact [ref] , it is the only non-local counterterm which can appear in a translation invariant scalar field model.
Gauge Fields on Non-Commutative Spaces and Renormalization · BPHZ applied to phi4-theory · 1.00
Cites it as background 2 papers
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The quantization of this model and the renormalization of related scalar field models has been discussed over the last fifteen years, see for instance reference [ref] for a brief review and list of references.
Gauge Fields on Non-Commutative Spaces and Renormalization · BPHZ applied to phi4-theory · 1.00
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Finally, methods and problems arising when one attempts to prove renormalizability of a gauge model on spaces can be found in [ref] and references therein.
Slavnov-Taylor identities, non-commutative gauge theories and infrared divergences · Introduction · 1.00
What it cites · 29 sentences without a confident label
Uses the method or tool of 1 paper
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It can be regularized by introducing a cutoff $Λ$ and subsequently taking the limit $^ 2 →0$, as explained in reference [ref] .
Quantum Corrections for Translation-Invariant Renormalizable Non-Commutative Phi^4 Theory · Extended action for the complex scalar field · 0.88
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