Quantized equations of motion in non-commutative theories

Physics Letters B, 2001

The unitarity condition for noncommutative field theories on compact space-times is considered. Although such theories are free of ultraviolet divergences, the compactness of space-like coordinates implies discreteness of the time variable which leads to appearance of unphysical modes and violation of unitarity even in absence of the star-product in the interaction terms. Thus, this conclusion holds also for other quantum field theories with discrete time. We also discuss violation of causality in terms of commutators for observables and possibility for a violation of the spin-statistics theorem.

2006

We discuss a functional integral approach to construction of Lorentz-covariant quantum gauge theories on a noncommutative space-time. There have been quite a number of work in this direction, mostly using various Moyal-type star products to construct Lagrangians. One of the most influential works was that of Seiberg and Witten [8], where they, among many other things, noted that simple problems of evolution of a string in a background force field invariably leads to some kind of noncommutativity of space-time coordinates. The type of noncommutativity they were using led to violation of the Lorentz covariance. There was a lot of works discussing these violations and attempting to fix this problem. In our paper [3] we proposed a version of the Moyal-type approach based on a group earlier used by Doplicher-Fredenhagen-Roberts [4] for other reasons. We came to the this group by contracting the group SO(4, 1) used by Snyder [9] to treat noncommutative space-time. Though our approach allo...

Journal of High Energy Physics, 2004

In this work quantum physics in noncommutative spacetime is developed. It is based on the work of Doplicher et al. which allows for time-space noncommutativity. The Moyal plane is treated in detail. In the context of noncommutative quantum mechanics, some important points are explored, such as the formal construction of the theory, symmetries, causality, simultaneity and observables. The dynamics generated by a noncommutative Schrodinger equation is studied. We prove in particular the following: suppose the Hamiltonian of a quantum mechanical particle on spacetime has no explicit time dependence, and the spatial coordinates commute in its noncommutative form (the only noncommutativity being between time and a space coordinate). Then the commutative and noncommutative versions of the Hamiltonian have identical spectra.

2003

We start by reviewing the formulation of noncommutative quantum mechanics as a constrained system. Then, we address to the problem of field theories defined on a noncommutative space-time manifold. The Moyal product is introduced and the appearance of the UV/IR mechanism is exemplified. The emphasis is on finding and analyzing noncommutative quantum field theories which are renormalizable and free of nonintegrable infrared singularities. In this last connection we give a detailed discussion of the quantization of the noncommutative Wess-Zumino model as well as of its low energy behavior. Lectures delivered at the XII Sumer School Jorge Andre Swieca. Section Particles and Fields,

The European Physical Journal C, 2001

Violation of unitarity for noncommutative field theory on compact spacetimes is considered. Although such theories are free of ultraviolet divergences, they still violate unitarity while in a usual field theory such a violation occurs when the theory is nonrenormalizable. The compactness of space-like coordinates implies discreteness of the time variable which leads to appearance of unphysical modes and violation of unitarity even in the absence of a star-product in the interaction terms. Thus, this conclusion holds also for other quantum field theories with discrete time. Violation of causality, among others, occurs also as the nonvanishing of the commutation relations between observables at space-like distances with a typical scale of noncommutativity. While this feature allows for a possible violation of the spin-statistics theorem, such a violation does not rescue the situation but makes the scale of causality violation as the inverse of the mass appearing in the considered model, i.e., even more severe. We also stress the role of smearing over the noncommutative coordinates entering the field operator symbols.

Symmetry, Integrability and Geometry: Methods and Applications, 2010

In the present work we review the twisted field construction of quantum field theory on noncommutative spacetimes based on twisted Poincaré invariance. We present the latest development in the field, in particular the notion of equivalence of such quantum field theories on a noncommutative spacetime, in this regard we work out explicitly the inequivalence between twisted quantum field theories on Moyal and Wick-Voros planes; the duality between deformations of the multiplication map on the algebra of functions on spacetime F (R 4 ) and coproduct deformations of the Poincaré-Hopf algebra HP acting on F (R 4 ); the appearance of a nonassociative product on F (R 4 ) when gauge fields are also included in the picture. The last part of the manuscript is dedicated to the phenomenology of noncommutative quantum field theories in the particular approach adopted in this review. CPT violating processes, modification of two-point temperature correlation function in CMB spectrum analysis and Pauli-forbidden transition in Be 4 are all effects which show up in such a noncommutative setting. We review how they appear and in particular the constraint we can infer from comparison between theoretical computations and experimental bounds on such effects. The best bound we can get, coming from Borexino experiment, is 10 24 TeV for the energy scale of noncommutativity, which corresponds to a length scale 10 −43 m. This bound comes from a different model of spacetime deformation more adapted to applications in atomic physics. It is thus model dependent even though similar bounds are expected for the Moyal spacetime as well as argued elsewhere.

Physics Letters B, 2011

We revisit the problem of quantizing field theories on noncommutative Moyal spacetime with light-like noncommutativity. To tackle the issues arising from noncommuting and hence nonlocal time, we argue that for this case light-front quantization procedure should be employed. In this appropriate quantization scheme we perform the non-planar loop analysis for the light-like noncommutative field theories. One of the important and peculiar features of light-front quantization is that the UV cutoff of the light-cone Hamiltonian manifests itself as an IR cutoff for the light-cone momentum, p +. Due to this feature, the naive results of covariant quantization for the light-like case allude to the absence of the UV/IR mixing in the light-front quantization. However, by a careful analysis of non-planar loop integrals we show that this is not the case and the UV/IR mixing persists. In addition, we argue in favour of the perturbative unitarity of light-like noncommutative field theories in the light-front quantization scheme.

Fortschritte der Physik, 2003

This is the written version of a talk I gave at the 35th Symposium Ahrenshoop in Berlin, Germany, August 2002. It is an exposition of joint work with S. Doplicher, K. Fredenhagen, and Gh. Piacitelli [1]. The violation of unitarity found in quantum field theory on noncommutative spacetimes in the context of the so-called modified Feynman rules is linked to the notion of time ordering implicitely used in the assumption that perturbation theory may be done in terms of Feynman propagators. Two alternative approaches which do not entail a violation of unitarity are sketched. An outlook upon our more recent work is given.

Physics Letters B, 2002

It is shown that the violation of unitarity observed in space/time noncommutative field theories is due to an improper definition of quantum field theory on noncommutative spacetime. * Supported by the Graduiertenkolleg "Zukünftige Entwicklungen in der Teilchenphysik".

Physics Letters B, 2004