Space-time Noncommutative Field Theories Research Articles

Nonlocal Wess-Zumino model on nilpotent noncommutative superspace

We investigate the theory of the bosonic-fermionic noncommutativity, $[x^{\mu},\theta^{\alpha}] = i \lambda^{\mu \alpha}$, and the Wess-Zumino model deformed by the noncommutativity. Such noncommutativity links well-known space-time noncommutativity to superspace non-anticommutativity. The deformation has the nilpotency. We can explicitly evaluate noncommutative effect in terms of new interactions between component fields. The interaction terms that have Grassmann couplings are induced. The noncommutativity does completely break full $\mathcal{N}=1$ supersymmetry to $ \mathcal{N} = 0 $ theory in Minkowski signature. Similar to the space-time noncommutativity, this theory has higher derivative terms and becomes non-local theory. However this non-locality is milder than the space-time noncommutative field theory. Due to the nilpotent feature of the coupling constants, we find that there are only finite number of Feynman diagrams that give noncommutative corrections at each loop order.

Path integral for space-time noncommutative field theory

The path integral for space-time noncommutative theory is formulated by means of Schwinger's action principle which is based on the equations of motion and a suitable ansatz of asymptotic conditions. The resulting path integral has essentially the same physical basis as the Yang-Feldman formulation. It is first shown that higher derivative theories are neatly dealt with by the path integral formulation, and the underlying canonical structure is recovered by the Bjorken-Johnson-Low (BJL) prescription from correlation functions defined by the path integral. A simple theory which is non-local in time is then analyzed for an illustration of the complications related to quantization, unitarity and positive energy conditions. From the view point of BJL prescription, the naive quantization in the interaction picture is justified for space-time noncommutative theory but not for the simple theory non-local in time. We finally show that the perturbative unitarity and the positive energy condition, in the sense that only the positive energy flows in the positive time direction for any fixed time-slice in space-time, are not simultaneously satisfied for space-time noncommutative theory by the known methods of quantization.

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.

No UV/IR mixing in unitary space-time non-commutative field theory

In this article we calculate several divergent amplitudes in phi^4-theory on non-commutative space-time in the framework of Interaction Point Time Ordered Perturbation Theory (IPTOPT), continuing work done in hep-th/0209253. On the ground of these results we find corresponding Feynman rules which allow for a much easier diagrammatic calculation of amplitudes. The most important feature of the present theory is the lack of the UV/IR mixing problem in all amplitudes calculated so far. Although we are not yet able to give a rigorous proof, we provide a strong argument for this result to hold in general. Together with the found Feynman rules this opens promising vistas towards the systematic renormalization of non-commutative field theories.

Noncommutative spacetime, stringy spacetime uncertainty principle, and density fluctuations

We propose a variation of spacetime noncommutative field theory to realize the stringy spacetime uncertainty relation without breaking any of the global symmetries of the homogeneous isotropic universe. We study the spectrum of metric perturbations in this model for a wide class of accelerating background cosmologies. Spacetime noncommutativity leads to a coupling between the fluctuation modes and the background cosmology which is nonlocal in time. For each mode, there is a critical time at which the spacetime uncertainty relation is saturated. This is the time when the mode is generated. These effects lead to a spectrum of fluctuations whose spectral index is different from what is obtained for commutative spacetime in the infrared region, but is unchanged in the ultraviolet region. In the special case of an exponentially expanding background, we find a scale-invariant spectrum. but with a different magnitude than in the context of commutative spacetime if the Hubble constant is above the string scale.

Hermitian analyticity, IR/UV mixing and unitarity of noncommutative field theories

The IR/UV mixing and the violation of unitarity are two of the most intriguing aspects of noncommutative quantum field theories. In this paper the relation between these two phenomena is explained and established in an explicit form. We start out by showing that the S-matrix of noncommutative field theories is hermitian analytic. As a consequence, a noncommutative field theory is unitary if the discontinuities of its Feynman diagram amplitudes agree with the expressions calculated using the Cutkosky formulae. These unitarity constraints relate the discontinuities of amplitudes with physical intermediate states and allow us to see how the IR/UV mixing may lead to a breakdown of unitarity. Specifically, we show that the IR/UV singularity does not lead to the violation of unitarity in the space–space noncommutative case, but it does lead to its violation in a space–time noncommutative field theory. As a corollary, noncommutative field theory without IR/UV mixing will be unitary in both the space–space and space–time noncommutative case. To illustrate this, we introduce and analyse the noncommutative Lee model—an exactly solvable quantum field theory. We show that the model is free from the IR/UV mixing in both the space–space and space–time noncommutative cases. Our analysis is exact. Due to absence of the IR/UV mixing one can expect that the theory is unitary. We present some checks supporting this claim. Our analysis provides a counter example to the generally held belief that field theories with space–time noncommutativity are nonunitary.

Perturbative approach to higher derivative and nonlocal theories

We review a perturbative approach to deal with Lagrangians with higher or infinite order time derivatives. It enables us to construct a consistent Poisson structure and Hamiltonian with only first time derivatives order by order in coupling. We show that, to the lowest order, the Hamiltonian is bounded from below whenever the potential is. We consider spacetime noncommutative field theory as an example.

SPACE-TIME UNCERTAINTY AND NONCOMMUTATIVITY IN STRING THEORY

We analyze the nature of space-time nonlocality in string theory. After giving a brief overview on the conjecture of the space-time uncertainty principle, a (semi-classical) reformulation of string quantum mechanics, in which the dynamics is represented by the noncommutativity between temporal and spatial coordinates, is outlined. The formalism is then compared to the space-time noncommutative field theories associated with nonzero electric B-fields.

Space–time noncommutative field theories and unitarity

We study the perturbative unitarity of noncommutative scalar field theories. Field theories with space-time noncommutativity do not have a unitary S-matrix. Field theories with only space noncommutativity are perturbatively unitary. This can be understood from string theory, since space noncommutative field theories describe a low energy limit of string theory in a background magnetic field. On the other hand, there is no regime in which space-time noncommutative field theory is an appropriate description of string theory. Whenever space-time noncommutative field theory becomes relevant massive open string states cannot be neglected.