Exact solution of noncommutative field theory in background magnetic fields

We obtain the exact non-perturbative solution of a scalar field theory defined on a space with noncommuting position and momentum coordinates. The model describes non-locally interacting charged particles in a background magnetic field. It is an exactly solvable quantum field theory which has non-trivial interactions only when it is defined with a finite ultraviolet cutoff. We propose that small perturbations of this theory can produce solvable models with renormalizable interactions. ∗Email: langmann@theophys.kth.se †Email: R.J.Szabo@ma.hw.ac.uk ‡Also at ITEP, Moscow, Russia. Email: Konstantin.Zarembo@teorfys.uu.se Quantum field theories on noncommutative spaces have received a surge of interest in recent years, primarily because they can be obtained as limits of string theory with background magnetic fields in which the massive string modes decouple (see [1] for reviews and exhaustive lists of references). They capture many of the non-local effects possessed by string theory but in a much simpler setting, and have attained a fundamental level of interest as examples of non-local field theories which may be well-defined. Various versions of them have also been proposed as effective field theory descriptions of some planar condensed matter systems in strong magnetic fields, such as quantum Hall models. Because of their embedding into string theory, these models are sometimes believed to be unitary and renormalizable.§ However, they possess several unusual aspects which continue to challenge the conventional wisdom of quantum field theory, and question the renormalizability and overall consistency of these field theories. On a canonical noncommutative space, the usual pointwise product of fields is replaced by the star-product Φ ? Φ′(x) = ∫ dk dq (2π)D Φ̃(k) Φ̃′(q) e i kμ θ μν qν e i (k+q)·x , (1) where the tildes denote Fourier transforms and θ is a constant antisymmetric matrix. The noncommutativity of space is encoded in the fact that the commutators of coordinates computed with this product are non-vanishing, as x ? x = x x + i θ . In perturbation theory, the phases in (1) produce momentum dependent vertices in Feynman diagrams which affect the interactions of the quantum field theory at energy scales below the scale 1/ √ |θ| set by the dimensionful noncommutativity parameter θ . The most drastic example of this is known as ultraviolet/infrared (UV/IR) mixing. If one uses Fourier expansion of fields in a basis of plane waves e i p·x, as in (1), then the natural regularization of the quantum field theory is the restriction of momenta p to an annulus Λ0 < |p| < Λ, where Λ0 is an IR cutoff and Λ a UV cutoff. Removing the cutoffs amounts to taking the limits Λ0 → 0 and Λ → ∞. Planar diagrams essentially coincide with those of ordinary quantum field theory, while non-planar graphs are modified by phases containing internal and external line momenta and are generically convergent. The rapid phase oscillations in (1) imply that a high-momentum cutoff Λ generates an effective IR cutoff Λ0 = 1/|θ|Λ. This appears to ruin the usual Wilsonian renormalization procedure which would require a clear separation of high and low momentum scales. However, the puzzling UV/IR mixing properties may simply be an artifact of perturbation theory which disappears when summed to all orders. This is a non-perturbative issue which is in general difficult to address. In this letter we will formulate a noncommutative scalar field theory which is exactly solvable and obtain its non-perturbative solution explicitly. The model describes charged scalar particles in a background magnetic field with a four-point interaction defined by the star-product [2]. We will circumvent the problems set in by UV/IR mixing by using a basis for the expansion of fields on R which differs from the more conventional plane wave basis and which will allow us to make sense of the field theory at a fully non-perturbative level. This expansion provides a natural non-perturbative regularization of the quantum field theory, producing both a short-distance and low-momentum cutoff simultaneously. We will show how to extract from this the exact expressions for Green’s functions of the quantum field theory. §Disclaimer: Views and opinions mentioned in this letter do not necessarily reflect those of the authors.