Quantum Theory of Noncommutative Fields (original) (raw)
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Noncommutative Quantum Field Theories ∗
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,
Noncommutative field theory from quantum mechanical space–space noncommutativity
Physics Letters A, 2007
We investigate the incorporation of space noncommutativity into field theory by extending to the spectral continuum the minisuperspace action of the quantum mechanical harmonic oscillator propagator with an enlarged Heisenberg algebra. In addition to the usual ⋆-product deformation of the algebra of field functions, we show that the parameter of noncommutativity can occur in noncommutative field theory even in the case of free fields without self-interacting potentials.
Reviews of Modern Physics, 2001
We review the generalization of field theory to space-time with noncommuting coordinates, starting with the basics and covering most of the active directions of research. Such theories are now known to emerge from limits of M theory and string theory, and to describe quantum Hall states. In the last few years they have been studied intensively, and many qualitatively new phenomena have been discovered, both on the classical and quantum level. Submitted to Reviews of Modern Physics.
Noncommutative Quantum Mechanics from Noncommutative Quantum Field Theory
Physical Review Letters, 2002
We derive noncommutative multi-particle quantum mechanics from noncommutative quantum field theory in the nonrelativistic limit. Paricles of opposite charges are found to have opposite noncommutativity. As a result, there is no noncommutative correction to the hydrogen atom spectrum at the tree level. We also comment on the obstacles to take noncommutative phenomenology seriously, and propose a way to construct noncommutative SU (5) grand unified theory.
Construction of a Noncommutative Quantum Field Theory
Spectral Analysis, Differential Equations and Mathematical Physics: A Festschrift in Honor of Fritz Gesztesy’s 60th Birthday
We review our recent successful attempt to construct the planar sector of a nonlocal scalar field model in four dimensional Euclidean deformed space-time, which needs 4 (instead of 3) relevant/marginal operators in the defining Lagrangian. As we have shown earlier, this model is renormalizable up to all orders in pertubation theory. In addition a new fixed point appears, at which the beta function for the coupling constant vanishes. This way, we were able to tame the Landau ghost. We next discuss Ward identities and Schwinger-Dyson equations and derive integral equations for the renormalized N-point functions. They are the starting point of a nonperturbative construction of the model. Dear Fritz! I (H.G.) almost cannot believe, that you become 60! I still remember the time, when you came from Graz to Vienna in the early 80's. I enjoyed our long standing interactions, our discussions on spectral concentration, how we handled the non-relativistic limit of the Dirac equation and especially our treatment of index problems and their connection to scattering theory. The last subject became of particular interests through the developments connected to noncommutative geometry and we enjoyed a recent Workshop at ESI on that subject together. Here I review another outcome of using ideas from noncommutative geometry. I hope you will enjoy reading that a four dimensional quantum field theory model can be constructed on such a deformed space. I wish you many new results for your interesting work and many happy years to come and hope for your visits to Vienna.
Heisenberg Evolution in a Quantum Theory of Noncommutative Fields
Journal of High Energy Physics, 2004
A quantum theory of noncommutative fields was recently proposed by Carmona, Cortez, Gamboa and Mendez (hep-th/0301248). The implications of the noncommutativity of the fields, intended as the requirements [φ, φ + ] = θδ 3 (x − x ′), [π, π + ] = Bδ 3 (x − x ′), were analyzed on the basis of an analogy with previous results on the so-called "noncommutative harmonic oscillator construction". Some departures from Lorentz symmetry turned out to play a key role in the emerging framework. We first consider the same hamiltonian proposed in hep-th/0301248, and we show that the theory can be analyzed straightforwardly within the framework of Heisenberg evolution equation without any need of making reference to the "noncommutative harmonic oscillator construction". We then consider a rather general class of alternative hamiltonians, and we observe that violations of Lorentz invariance are inevitably encountered. These violations must therefore be viewed as intrinsically associated with the proposed type of noncommutativity of fields, rather than as a consequence of a specific choice of Hamiltonian.
Quantum Fields on Noncommutative Spacetimes: Theory and Phenomenology
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.
Toward an axiomatic formulation of noncommutative quantum field theory
Journal of Mathematical Physics, 2011
We propose new Wightman functions as vacuum expectation values of products of field operators in the noncommutative space-time. These Wightman functions involve the ⋆-product. In the case of only space-space noncommutativity (θ0i = 0), we prove the CPT theorem using the noncommutative form of the Wightman functions. As a byproduct, one arrives at the general conclusion of the following theorem that the violation of CPT invariance implies the violation of not only Lorentz invariance, but also its subgroup of symmetry SO(1, 1)×SO(2). We also show that the spin-statistics theorem, for the simplest case of a scalar field, holds within this formalism.