Noncommutative Scalar Field Research Articles

Wilson loops and area-preserving diffeomorphisms in twisted noncommutative gauge theory

We use twist deformation techniques to analyze the behavior under area-preserving diffeomorphisms of quantum averages of Wilson loops in Yang-Mills theory on the noncommutative plane. We find that while the classical gauge theory is manifestly twist covariant, the holonomy operators break the quantum implementation of the twisted symmetry in the usual formal definition of the twisted quantum field theory. These results are deduced by analyzing general criteria which guarantee twist invariance of noncommutative quantum field theories. From this a number of general results are also obtained, such as the twisted symplectic invariance of noncommutative scalar quantum field theories with polynomial interactions and the existence of a large class of holonomy operators with both twisted gauge covariance and twisted symplectic invariance.

NONCOMMUTATIVE SCALAR FIELD MINIMALLY COUPLED TO GRAVITY

A model for noncommutative scalar fields coupled to gravity based on the generalization of the Moyal product is proposed. Solutions compatible with homogeneous and isotropic flat Robertson-Walker spaces to first non-trivial order in the perturbation of the star-product are presented. It is shown that in the context of a typical chaotic inflationary scenario, at least in the slow-roll regime, noncommutativity yields no observable effect.

A non-perturbative approach to non-commutative scalar field theory

Non-commutative Euclidean scalar field theory is shown to have an eigenvalue sector which is dominated by a well-defined eigenvalue density, and can be described by a matrix model. This is established using regularizations of R^{2n}_\theta via fuzzy spaces for the free and weakly coupled case, and extends naturally to the non-perturbative domain. It allows to study the renormalization of the effective potential using matrix model techniques, and is closely related to UV/IR mixing. In particular we find a phase transition for the \phi^4 model at strong coupling, to a phase which is identified with the striped or matrix phase. The method is expected to be applicable in 4 dimensions, where a critical line is found which terminates at a non-trivial point, with nonzero critical coupling. This provides evidence for a non-trivial fixed-point for the 4-dimensional NC \phi^4 model.

Trouble with space-like noncommutative field theory

It is argued that the one-loop effective action for space-like noncommutative (i) λφ4 scalar field theory and (ii) U(1) gauge theory does not exist. This indicates that such theories are not renormalizable already at one-loop order and suggests supersymmetrization and reinvestigating other types of noncommutativity.

Perturbation theory of the space-time noncommutative real scalar field theories

The perturbative framework of the space-time noncommutative real scalar field theory is formulated, based on the unitary S matrix. Unitarity of the S matrix is explicitly checked order by order using the Heisenberg picture of Lagrangian formalism of the second quantized operators, with emphasis on the so-called minimal realization of the time-ordering step function and of the importance of the $\ensuremath{\star}$-time ordering. The Feynman rule is established and is presented using ${\ensuremath{\varphi}}^{4}$ scalar field theory. It is shown that the divergence structure of space-time noncommutative theory is the same as the one of space-space noncommutative theory, while there is no UV-IR mixing problem in this space-time noncommutative theory.

Renormalization of the energy–momentum tensor in noncommutative complex scalar field theory

We study the renormalization of dimension-four composite operators and the energy–momentum tensor in noncommutative complex scalar field theory. The proper operator basis is defined and it is proved that the bare composite operators are expressed via renormalized ones with the help of an appropriate mixing matrix which is calculated in the one-loop approximation. The number and form of the operators in the basis and the structure of the mixing matrix essentially differ from those in the corresponding commutative theory and in noncommutative real scalar field theory. We show that the energy–momentum tensor in the noncommutative complex scalar field theory is defined up to six arbitrary constants. The canonically defined energy–momentum tensor is not finite and must be replaced by the “improved” one, in order to provide finiteness. Suitable “improving” terms are found. Renormalization of dimension-four composite operators at zero momentum transfer is also studied. It is shown that the mixing matrices are different for the cases of arbitrary and zero momentum transfer. The energy–momentum vector, unlike the energy–momentum tensor, is defined unambiguously and does not require “improving”, in order to be conserved and finite, at least in the one-loop approximation.

Space-time noncommutativity tends to create bound states

We study the spectrum of fluctuations about static solutions in 1+1 dimensional non-commutative scalar field models. In the case of soliton solutions non-commutativity leads to creation of new bound states. In the case of static singular solutions an infinite tower of bound states is produced whose spectrum has a striking similarity to the spectrum of confined quark states.

Unitarity in space–time noncommutative field theories

In noncommutative field theories conventional wisdom is that the unitarity is noncompatible with the perturbation analysis when time is involved in the noncommutative coordinates. However, as suggested by Bahns et al. recently, the root of the problem lies in the improper definition of the time-ordered product. In this article, functional formalism of S-matrix is explicitly constructed for the noncommutative φp scalar field theory using the field equation in the Heisenberg picture and proper definition of time-ordering. This S-matrix is manifestly unitary. Using the free spectral (Wightmann) function as the free field propagator, we demonstrate the perturbation obeys the unitarity, and present the exact two particle scattering amplitude for (1+1)-dimensional noncommutative nonlinear Schrödinger model.

Remarks on 't Hooft's brick wall model

A semi-classical reasoning leads to the non-commutativity of the space and time coordinates near the horizon of Schwarzschild black hole. This non-commutativity in turn provides a mechanism to interpret the brick wall thickness hypothesis in 't Hooft's brick wall model as well as the boundary condition imposed for the field considered. For concreteness, we consider a noncommutative scalar field model near the horizon and derive the effective metric via the equation of motion of noncommutative scalar field. This metric displays a new horizon in addition to the original one associated with the Schwarzschild black hole. The infinite red-shifting of the scalar field on the new horizon determines the range of the noncommutativ space and explains the relevant boundary condition for the field. This range enables us to calculate the entropy of black hole as proportional to the area of its original horizon along the same line as in 't Hooft's model , and the thickness of the brick wall is found to be proportional to the thermal average of the noncommutative space-time range. The Hawking temperature has been derived in this formalism. The study here represents an attempt to reveal some physics beyond the brick wall model.

Renormalization of the energy–momentum tensor in non-commutative scalar field theory

We consider the one-loop renormalization of dimension four composite operators and the energy–momentum tensor in non-commutative φ 4 scalar field theory. Proper operator bases are constructed and it is proved that the bare composite operators are expressed via renormalized ones, with the help of a mixing matrix, whose explicit form is calculated. The corresponding matrix elements turn out to differ from the commutative theory. The canonically defined energy–momentum tensor is not finite and must be replaced by the “improved” one, in order to provide finiteness. The suitable “improving” terms are found.

The renormalization of non-commutative field theories in the limit of large non-commutativity

We show that renormalized non-commutative scalar field theories do not reduce to their planar sector in the limit of large non-commutativity. This follows from the fact that the RG equation of the Wilson–Polchinski type which describes the genus zero sector of non-commutative field theories couples generic planar amplitudes with non-planar amplitudes at exceptional values of the external momenta. We prove that the renormalization problem can be consistently restricted to this set of amplitudes. In the resulting renormalized theory non-planar divergences are treated as UV divergences requiring appropriate non-local counterterms. In 4 dimensions the model turns out to have one more relevant (non-planar) coupling than its commutative counterpart. This non-planar coupling is “evanescent”: although in the massive (but not in the massless) case its contribution to planar amplitudes vanishes when the floating cut-off equals the renormalization scale, this coupling is needed to make the Wilsonian effective action UV finite at all values of the floating cut-off.

A singular perturbation problem and the existence of non-commutative non-rotationally symmetric scalar solitons

The existence of solitons in non-commutative scalar field theories is proved for large values of the non-commutativity parameter using functional analysis. In the case of even phase-space dimensions greater than or equal to four, the solitons include some that are not rotationally symmetric.

On Noncommutative Multi-Solitons

We find the moduli space of multi-solitons in noncommutative scalar field theories at large θ, in arbitrary dimension. The existence of a non-trivial moduli space at leading order in 1/θ is a consequence of a Bogomolnyi bound obeyed by the kinetic energy of the θ=∞ solitons. In two spatial dimensions, the parameter space for k solitons is a Kähler de-singularization of the symmetric product (ℝ2) k /S k . We exploit the existence of this moduli space to construct solitons on quotient spaces of the plane: ℝ2/ℤ k , cylinder, and T 2 . However, we show that tori of area less than or equal to 2πθ do not admit stable solitons. In four dimensions the moduli space provides an explicit Kähler resolution of (ℝ4) k /S k . In general spatial dimension 2d, we show it is isomorphic to the Hilbert scheme of k points in ℂ d , which for d>2 (and k>3) is not smooth and can have multiple branches.

Noncommutative scalar field coupled to gravity

A model for a noncommutative scalar field coupled to gravity is proposed via an extension of the Moyal product. It is shown that there are solutions compatible with homogeneity and isotropy to first non-trivial order in the perturbation of the star-product, with the gravity sector described by a flat Robertson-Walker metric. We show that in the slow-roll regime of a typical chaotic inflationary scenario, noncommutativity has negligible impact.

Stripes from (non-commutative) stars

We show that lattice regularization of non-commutative field theories can be used to study non-perturbative vacuum phases. Specifically we provide evidence for the existence of a striped phase in two-dimensional non-commutative scalar field theory.

A note on noncommutative scalar multisolitons

We prove that there do not exist multisoliton solutions of noncommutative scalar field theory in the Moyal plane which interpolate smoothly between n overlapping solitons and n solitons with an infinite separation.

One-loop noncommutative U(1) gauge theory from bosonic worldline approach

We develop a method to compute the one-loop effective action of noncommutative U(1) gauge theory based on the bosonic worldline formalism, and derive compact expressions for N-point 1PI amplitudes. The method, resembling perturbative string computations, shows that open Wilson lines emerge as a gauge invariant completion of certain terms in the effective action. The terms involving open Wilson lines are of the form reminiscent of closed string exchanges between the states living on the two boundaries of a cylinder. They are also consistent with recent matrix theory analysis and the results from noncommutative scalar field theories with cubic interactions.

Noncommutative cohomological field theory and GMS soliton

We show that it is possible to construct a quantum field theory that is invariant under the translation of the noncommutative parameter θμν. This is realized in a noncommutative cohomological field theory. As an example, a noncommutative cohomological scalar field theory is constructed, and its partition function is calculated. The partition function is the Euler number of Gopakumar, Minwalla, and Strominger (GMS) soliton space.

Open Wilson lines as closed strings in non-commutative field theories

Open Wilson line operators and the generalized star product have been studied extensively in non-commutative gauge theories. We show that they also show up in non-commutative scalar field theories as universal structures. We first point out that the dipole picture of non-commutative geometry provides an intuitive argument for robustness of the open Wilson lines and generalized star products therein. We calculate the one-loop effective action of the non-commutative scalar field theory with cubic self-interaction and show explicitly that the generalized star products arise in the non-planar part. It is shown that, in the low-energy, large non-commutativity limit, the non-planar part is expressible solely in terms of the scalar open Wilson line operator and descendants, the latter being interpreted as composite operators representing a closed string.

Anatomy of one-loop effective action in non-commutative scalar field theories

One-loop effective action of noncommutative scalar field theory with cubic self-interaction is studied. Utilizing worldline formulation, both planar and nonplanar part of the effective action are computed explicitly. We find complete agreement of the result with Seiberg-Witten limit of string worldsheet computation and standard Feynman diagrammatics. We prove that, at low-energy and large noncommutativity limit, nonplanar part of the effective action is simplified enormously and is resummable into a quadratic action of scalar open Wilson line operators.