Symmetries of noncommutative scalar field theory

Abstract

We investigate symmetries of the scalar field theory with harmonic term on the Moyal space with euclidean scalar product and general symplectic form. The classical action is invariant under the orthogonal group if this group acts also on the symplectic structure. We find that the invariance under the orthogonal group can be restored also at the quantum level by restricting the symplectic structures to a particular orbit. * Work supported by the Belgian Interuniversity Attraction Pole (IAP) within the framework "Nonlinear systems, stochastic processes, and statistical mechanics" (NOSY). arXiv:0911.2645v2 [math-ph] 6 Jan 2011In the past few years, there has been a growing interest in the noncommutative quantum fields theories (for a review, see [1]). These theories on "spaces" coming from noncommutative geometry [2] are indeed strong candidates for new physics behind the Standard Model of particle physics. Moreover, fields theories defined on the Moyal space [3], one of the simplest example of noncommutative space, can be seen as an effective regime of string theory [4] and matrix theory [5].The simplest generalization of the ϕ 4 commutative scalar theory gives rise to a new type of divergence, the ultraviolet-infrared (UV/IR) mixing [6], which is responsible for the nonrenormalizability of this model. The first solution to this problem of UV/IR mixing, due to Grosse and Wulkenhaar, was to add some harmonic term in the action [7], and the resulting theory is then renormalizable up to all orders in perturbation [7,8,9]. The renormalizability of this model seems to be related to a new symmetry thanks to this harmonic term, the Langmann-Szabo duality [10], which exchanges positions and impulsions at the level of the quadratic terms of the action. This Langmann-Szabo duality has also been interpreted in the framework of superalgebras as a grading symmetry [11,12] and an adapted differential calculus has been exhibited (see also [13,14]). Note that another interpretation of the harmonic term has been given in [15]. Moreover, the vacuum solutions of this theory have been studied in [16], and it has been proved that the beta function vanishes up to irrelevant terms [17,18]. Associated to this scalar field theory with harmonic term, a gauge invariant action, candidate to renormalizability, has been exhibited in [19] (see also [20,21]), but its vacua are always non-trivial [22].However, since a symplectic structure Σ is necessary to define the Moyal product, the group of rotations is no longer a symmetry group for this theory on the Moyal space as soon as the dimension D > 2. The question of rotational invariance of the theory is important since it corresponds to the Lorentz invariance, necessary for a physical theory, in the Minkowskian framework. This problem is related to the fact that a symplectic structure is not natural on a configuration (or position) space, contrary to a phase space. A simple standard way to restore the rotational invariance of the classical action is to consider a family of actions labeled by the s...

Closed star product on noncommutative ℝ 3 and scalar field dynamics

“…This combined with (2.16) yields 20) for any function f, g ∈ S(R 3 ). It turns out that expressing the closed product ⋆ D under the form (2.20) will simplify the ensuing computations.…”

Metrics and causality on Moyal planes

“…We now introduce another family of Moyal spectral triples that appeared in the mathematical physics literature in the general context of field theories and gauge theories built on noncommutative spaces (for a complete review including essential aspects of noncommutative differential geometry underlying these field theories together with a list of essential references see [25]) including the case of Moyal spaces [26], [27], [28]. This triple advocated rather recently (see [29] and related references therein) occurred in attempts to extend desirable perturbative properties (namely renormalisability) showing up in a certain class of scalar field theories on Moyal space [30], [31], [32] to the more difficult situation of gauge theories on Moyal spaces [33], [34], [35]. The use of spectral action principle leads to a gauge-invariant action with interesting properties but whose quantum properties are difficult [36] to study and are still under investigations.…”

Involutive representations of coordinate algebras and quantum spaces

“…Informally, this framework amounts (among other tasks) to represent the abstract involutive algebras of operators stemming from the above mentioned coordinate algebras on well chosen involutive algebras of functions equipped with a deformed product, i.e star-product, which can be achieved through the introduction of some suitable (invertible) quantization map. One well-known example heavily used in earlier studies of quantum properties of NCFT on Moyal spaces [31][32][33] (for a review on gauge theories on Moyal spaces see [34]; families of star products on the Moyal space R 4 θ have been constructed in [35]) is the Weyl quantization map, linked to the Wigner-Weyl transform, giving rise to the Moyal product. Other star-products related to κ-Minkowski spaces as well as deformations of R 3 with su(2) noncommutativity have also appeared and used to construct and study NCFT on these spaces [36][37][38][39][40][41][42][43][44][46][47][48] (for a general construction see [45]).…”