Noncommutative Scalar Field Research Articles

Noncommutative Complex Scalar Field and Casimir Effect

Using the noncommutative deformed canonical commutation relations proposed by Carmona et al. [J. M. Carmona, J. L. Cort\'es, J. Gamboa, and F. Mendez, J. High Energy Phys. 03 (2003) 058.][J. Gamboa, J. Lop\'ez-Sarrion, and A. P. Polychronakos, Phys. Lett. B 634, 471 (2006).][J. M. Carmona, J. L. Cort\'es, Ashok Das, J. Gamboa, and F. Mendez, Mod. Phys. Lett. A 21, 883 (2006).], a model describing the dynamics of the noncommutative complex scalar field is proposed. The noncommutative field equations are solved, and the vacuum energy is calculated to the second order in the parameter of noncommutativity. As an application to this model, the Casimir effect, due to the zero-point fluctuations of the noncommutative complex scalar field, is considered. It turns out that in spite of its smallness, the noncommutativity gives rise to a repulsive force at the microscopic level, leading to a modified Casimir potential with a minimum at the point ${a}_{\mathrm{min}}=\sqrt{\frac{5}{84}}\ensuremath{\pi}\ensuremath{\theta}$.

Quantum field theory on quantized Bergman domain

We present an oscillator realization of discrete series representations of group SU(2, 2). We give formulas for the coherent state star-product quantization of a Bergman domain D. A formulation of a (regularized) noncommutative scalar field on a quantized D is given.

FUNCTIONAL RENORMALIZATION OF NONCOMMUTATIVE SCALAR FIELD THEORY

In this paper we apply the Functional Renormalization Group Equation (FRGE) to the noncommutative scalar field theory proposed by Grosse and Wulkenhaar. We derive the flow equation in the matrix representation and discuss the theory space for the self-dual model. The features introduced by the external dimensionful scale provided by the noncommutativity parameter, originally pointed out in R. Gurau and O. J. Rosten, J. High Energy Phys.0907, 064 (2009), are discussed in the FRGE context. Using a technical assumption, but without resorting to any truncation, it is then shown that the theory is asymptotically safe for suitably small values of the ϕ4coupling, recovering the result of M. Disertori et al., Phys. Lett. B649, 95 (2007). Finally, we show how the FRGE can be easily used to compute the one-loop beta-functions of the duality covariant model.

SCALAR FIELD THEORY ON κ-MINKOWSKI SPACE–TIME AND TRANSLATION AND LORENTZ INVARIANCE

We investigate the properties of κ-Minkowski space–time by using representations of the corresponding deformed algebra in terms of undeformed Heisenberg–Weyl algebra. The deformed algebra consists of κ-Poincaré algebra extended with the generators of the deformed Weyl algebra. The part of deformed algebra, generated by rotation, boost and momentum generators, is described by the Hopf algebra structure. The approach used in our considerations is completely Lorentz covariant. We further use an advantage of this approach to consistently construct a star product, which has a property that under integration sign, it can be replaced by a standard pointwise multiplication, a property that was since known to hold for Moyal but not for κ-Minkowski space–time. This star product also has generalized trace and cyclic properties, and the construction alone is accomplished by considering a classical Dirac operator representation of deformed algebra and requiring it to be Hermitian. We find that the obtained star product is not translationally invariant, leading to a conclusion that the classical Dirac operator representation is the one where translation invariance cannot simultaneously be implemented along with hermiticity. However, due to the integral property satisfied by the star product, noncommutative free scalar field theory does not have a problem with translation symmetry breaking and can be shown to reduce to an ordinary free scalar field theory without nonlocal features and tachyonic modes and basically of the very same form. The issue of Lorentz invariance of the theory is also discussed.

Symmetries of noncommutative scalar field theory

We investigate symmetries of the scalar field theory with a harmonic term on the Moyal space with the 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 also be restored at the quantum level by restricting the symplectic structures to a particular orbit.

Noncommutative double scalar fields in FRW cosmology as cosmical oscillators

We investigate the effects caused by noncommutativity of the phase space generated by two scalar fields that have non-minimal conformal couplings to the background curvature in the FRW cosmology. We restrict deformation of the minisuperspace to noncommutativity between the scalar fields and between their canonical conjugate momenta. Then, the investigation is carried out by means of a comparative analysis of the mathematical properties (supplemented with some diagrams) of the time evolution of variables in a classical model and the wavefunction of the universe in a quantum perspective, both in the commutative and noncommutative frames. We find that the imposition of noncommutativity causes more ability in tuning time solutions of the scalar fields and, hence, has important implications in the evolution of the Universe. We find that the noncommutative parameter in the momenta sector is the only responsible parameter for the noncommutative effects in the spatially flat universes. A distinguishing feature of the noncommutative solutions of the scalar fields is that they can be simulated with the well-known three harmonic oscillators depending on three values of the spatial curvature, namely the free, forced and damped harmonic oscillators corresponding to the flat, closed and open universes, respectively. In this respect, we call them cosmical oscillators. In particular, in closed universes, when the noncommutative parameters are small, the cosmical oscillators have an analogous effect with the familiar beating effect in the sound phenomena. Some of the special solutions in the classical model and the allowed wavefunctions in the quantum model make bounds on the values of the noncommutative parameters. The existence of a non-zero constant potential (i.e. a cosmological constant) does not change time evolutions of the scalar fields, but modifies the scale factor. An interesting feature of the well-behaved solutions of the wavefunctions is that the functional form of the radial part is the same as the commutative ones within a given replacement of constants caused by the noncommutative parameters. Furthermore, the Noether theorem has been employed to explore the effects of noncommutativity on the underlying symmetries in the commutative frame. Two of the six Noether symmetries of spatially flat universes, in general, are retained in the noncommutative case, and one out of the three in non-flat universes.

Noncommutative solitons of gravity

We investigate a three-dimensional gravitational theory on a noncommutative space which has a cosmological constant term only. We found various kinds of nontrivial solutions by applying a similar technique which was used to seek noncommutative solitons in noncommutative scalar field theories. Some of those solutions correspond to bubbles of spacetimes or represent dimensional reduction. The solution which interpolates Gμν = 0 and the Minkowski metric is also found. All solutions we obtained are non-perturbative in the noncommutative parameter θ, therefore they are different from solutions found in other contexts of noncommutative theory of gravity and would have a close relation to quantum gravity.

Wilsonian renormalization of noncommutative scalar field theory

Drawing on analogies with the commutative case, the Wilsonian picture of renormalization is developed for noncommutative scalar field theory. The dimensionful noncommutativity parameter, θ, induces several new features. Fixed-points are replaced by `floating-points' (actions which are scale independent only up to appearances of θ written in cutoff units). Furthermore, it is found that one must use correctly normalized operators, with respect to a new scalar product, to define the right notion of relevance and irrelevance. In this framework it is straightforward and intuitive to reproduce the classification of operators found by Grosse & Wulkenhaar, around the Gaussian floating-point. The one-loop β-function of their model is computed directly within the exact renormalization group, reproducing the previous result that it vanishes in the self-dual theory, in the limit of large cutoff. With the link between this methodology and earlier results made, it is discussed how the vanishing of the β-function to all loops, as found by Disertori et al., should be interpreted in a Wilsonian framework.

The Cutkosky rule of three dimensional noncommutative field theory in Lie algebraic noncommutative spacetime

We investigate the unitarity of three dimensional noncommutative scalar field theory in the Lie algebraic noncommutative spacetime [x^i,x^j]=2i kappa epsilon^{ijk}x_k. This noncommutative field theory possesses a SL(2,R)/Z_2 group momentum space, which leads to a Hopf algebraic translational symmetry. We check the Cutkosky rule of the one-loop self-energy diagrams in the noncommutative phi^3 theory when we include a braiding, which is necessary for the noncommutative field theory to possess the Hopf algebraic translational symmetry at quantum level. Then, we find that the Cutkosky rule is satisfied if the mass is less than 1/(2^(1/2)kappa).

Duality covariant quantum field theory on noncommutative Minkowski space

We prove that a scalar quantum field theory defined on noncommutative Minkowski spacetime with noncommuting momentum coordinates is covariant with respect to the UV/IR duality which exchanges coordinates and momenta. The proof is based on suitable resonance expansions of charged noncommutative scalar fields in a background electric field, which yields an effective description of the field theory in terms of a coupled complex two-matrix model. The two independent matrix degrees of freedom ensure unitarity and manifest CT-invariance of the field theory. The formalism describes an analytic continuation of the renormalizable Grosse-Wulkenhaar models to Minkowski signature.

Noncommutative corrections to classical black holes

We calculate leading long-distance noncommutative corrections to the classical Schwarzschild black hole sourced by a massive noncommutative scalar field. The energy-momentum tensor is taken O(l{sup 4}) in the noncommutative parameter l and is treated in the semiclassical (tree-level) approximation. These noncommutative corrections dominate classical post-post-Newtonian corrections if l>1/M{sub P}. However, they are still very small to be observable in present-day experiments.

Stability conditions for a noncommutative scalar field coupled to gravity

We consider a noncommutative scalar field with a covariantly constant noncommutative parameter in a curved space–time background. For a potential as a noncommutative polynomial it is shown that the stability conditions are unaffected by the noncommutativity, a result that is valid irrespective whether space–time has horizons or not.

Twisted noncommutative field theory with the Wick-Voros and Moyal products

We study the noncommutative scalar field theory in the presence of the Wick-Voros product (or normally ordered product), a variant of the more studied Moyal product. We discuss both the classical and the quantum field theory in the quartic potential case, and calculate the Green's functions up to one loop, for the two and four points cases.

Noncommutative radial waves

We study radial waves in a (2+1)-dimensional noncommutative scalar field theory, using operatorial methods. The waves propagate along a discrete radial coordinate and are described by finite series deformations of Bessel-type functions. At radius much larger than the noncommutativity scale , one recovers the usual commutative behaviour. At small distances, classical divergences are smoothed out by noncommutativity.

Noncommutative scalar fields from symplectic deformation

This paper is concerned with the quantum theory of noncommutative scalar fields in two dimensional space-time. It is shown that the noncommutativity originates from the the deformation of symplectic structures. The quantization is performed and the modes expansions of the fields, in the presence of an electromagnetic background, are derived. The Hamiltonian of the theory is given and the degeneracies lifting, induced by the deformation, is also discussed.

On the effective action of noncommutative Yang-Mills theory

We compute here the Yang-Mills effective action on Moyal space by integrating over the scalar fields in a noncommutative scalar field theory with harmonic term, minimally coupled to an external gauge potential. We also explain the special regularisation scheme chosen here and give some links to the Schwinger parametric representation. Finally, we discuss the results obtained: a noncommutative possibly renormalisable Yang-Mills theory.

Noncommutative harmonic analysis, sampling theory and the Duflo map in 2+1 quantum gravity

We show that the ⋆-product for U(su2), group Fourier transform and effective action arising in Freidel and Livine (2005 Preprint hep-th/0502106) in an effective theory for the integer spin Ponzano–Regge quantum gravity model are compatible with the noncommutative bicovariant differential calculus, quantum group Fourier transform and noncommutative scalar field theory previously proposed for the 2+1 Euclidean quantum gravity using quantum group methods in Batista and Majid (2003 J. Math. Phys. 44 107–37). The two are related by a classicalization map which we introduce. We show, however, that noncommutative spacetime has a richer structure which already sees the half-integer spin information. We argue that the anomalous extra ‘time’ dimension seen in the noncommutative geometry should be viewed as the renormalization group flow visible in the coarse-graining in going from SU2 to SO3. Combining our methods we develop practical tools for noncommutative harmonic analysis for the model including radial quantum delta-functions and Gaussians, the Duflo map and elements of ‘noncommutative sampling theory’. This allows us to understand the bandwidth limitation in the 2+1 quantum gravity arising from the bounded SU2 momentum and to interpret the Duflo map as noncommutative compression. Our methods also provide a generalized twist operator for the ⋆-product.

Spontaneous breaking of translational invariance in noncommutativeλϕ4theory in two dimensions

The spontaneous breaking of translational invariance in noncommutative self-interacting scalar field theory in two dimensions is investigated by effective action techniques. The analysis confirms the existence of the stripe phase, already observed in lattice simulations, due to the nonlocal nature of the noncommutative dynamics.

Noncommutative real scalar field theory in 2+1 dimensions at finite temperature

We study thermal effects for a noncommutative real scalar field in 2+1 dimensions including a Grosse-Wulkenhaar term. Using a perturbative expansion for the free energy, we deduce some general properties of the corresponding contributions, in the thermodynamic limit. We show that the model can be consistently interpreted as defined on a finite volume, which is naturally determined by the noncommutativity scale.

One-loop effects in a self-dual planar noncommutative theory

We study the UV properties, and derive the explicit form of the one-loop effective action, for a noncommutative complex scalar field theory in 2+1 dimensions with a Grosse-Wulkenhaar term, at the self-dual point. We also consider quantum effects around non-trivial minima of the classical action which appear when the potential allows for the spontaneous breaking of the U(1) symmetry. For those solutions, we show that the one-loop correction to the vacuum energy is a function of a special combination of the amplitude of the classical solution and the coupling constant.