Dynamical noncommutativity and noether theorem in twisted \({\phi^{\star 4}}\) theory. (English) Zbl 1170.53073
The authors propose a generalization of the usual \(*\)-Moyal product in order to consider a noncommutative charged scalar field theory. This generalized Moyal product presents many advantages with respect to the usual one. First of all it is of dynamic nature, and does not produce anomaly on the shell for the energy-momentum tensor, (i.e., \(T^{\alpha\beta}_{/\beta}=0\)). Furthermore, it allows to obtain a noncommutative Noether theorem, similar to the classic field theory.
Remark. The non-commutative product, considered in this beautiful paper, can be considered a serious improvement with respect to usual rigid \(*\)-products introduced under the philosophy that wants to see quantized field theories like (Moyal) deformed classical ones. However, it is just this aspect that shows its limit. In fact, it insists on the general belief that in order to describe a quantum field theory it is necessary to perform a type of non-commutative product before. Instead, working in the geometric theory of quantum (super) PDE’s, one understands how to go beyond this obsolete point of view.
Remark. The non-commutative product, considered in this beautiful paper, can be considered a serious improvement with respect to usual rigid \(*\)-products introduced under the philosophy that wants to see quantized field theories like (Moyal) deformed classical ones. However, it is just this aspect that shows its limit. In fact, it insists on the general belief that in order to describe a quantum field theory it is necessary to perform a type of non-commutative product before. Instead, working in the geometric theory of quantum (super) PDE’s, one understands how to go beyond this obsolete point of view.
Reviewer: Agostino Prástaro (Roma)
MSC:
| 53D55 | Deformation quantization, star products |
| 81T75 | Noncommutative geometry methods in quantum field theory |
| 70S10 | Symmetries and conservation laws in mechanics of particles and systems |
Keywords:
deformation quantization; star-Moyal product; noncommutative geometry; noncommutative Noether theoremReferences:
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