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Register a new userA Poincare Covariant Noncommutative Spacetime?
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We interpret, in the realm of relativistic quantum field theory, the tangential operator given by Coleman, Mandula as an appropriate coordinate operator. The investigation shows that the operator generates a Snyder-like noncommutative spacetime with a minimal length that is given by the mass. By using this operator to define a noncommutative spacetime, we obtain a Poincare invariant noncommutative spacetime and in addition solve the soccer-ball problem. Moreover, from recent progress in deformation theory we extract the idea how to obtain, in a physical and mathematical well-defined manner, an emerging noncommutative spacetime. This is done by a strict deformation quantization known as Rieffel deformation (or warped convolutions). The result is a noncommutative spacetime combining a Snyder and a Moyal-Weyl type of noncommutativity that in addition behaves covariant under transformations of the textbf{whole} Poincare group.
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We introduce a framework in noncommutative geometry consisting of a $*$-algebra $mathcal A$, a bimodule $Omega^1$ endowed with a derivation $mathcal Ato Omega^1$ and with a Hermitian structure $Omega^1otimes bar{Omega}^1to mathcal A$ (a noncommutative Kahler form), and a cyclic 1-cochain $mathcal Ato mathbb C$ whose coboundary is determined by the previous structures. These data give moment map equations on the space of connections on an arbitrary finitely-generated projective $mathcal A$-module. As particular cases, we obtain a large class of equations in algebra (Kings equations for representations of quivers, including ADHM equations), in classical gauge theory (Hermitian Yang-Mills equations, Hitchin equations, Bogomolny and Nahm equations, etc.), as well as in noncommutative gauge theory by Connes, Douglas and Schwarz. We also discuss Nekrasovs beautiful proposal for re-interpreting noncommutative instantons on $mathbb{C}^nsimeq mathbb{R}^{2n}$ as infinite-dimensional solutions of Kings equation $$sum_{i=1}^n [T_i^dagger, T_i]=hbarcdot ncdotmathrm{Id}_{mathcal H}$$ where $mathcal H$ is a Hilbert space completion of a finitely-generated $mathbb C[T_1,dots,T_n]$-module (e.g. an ideal of finite codimension).
It is noted that the Poincare sphere for polarization optics contains the symmetries of the Lorentz group. The sphere is thus capable of describing the internal space-time symmetries dictated by Wigners little groups. For massive particles, the little group is like the three-dimensional rotation group, while it is like the two-dimensional Euclidean group for massless particles. It is shown that the Poincare sphere, in addition, has a symmetry parameter corresponding to reducing the particle mass from a positive value to zero. The Poincare sphere thus the gives one unified picture of Wigners little groups for massive and massless particles.
We study the Harmonic and Dirac Oscillator problem extended to a three-dimensional noncom- mutative space where the noncommutativity is induced by a shift of the dynamical variables with generators of SL(2;R) in a unitary irreducible representation. The Hilbert space gets the structure of a direct product with the representation space as a factor, where there exist operators which realize the algebra of Lorentz transformations. The spectrum of these models are considered in perturbation theory, both for small and large noncommutativity parameters, finding no constraints between coordinates and momenta noncom- mutativity parameters. Since the representation space of the unitary irreducible representations SL(2;R) can be realized in terms of spaces of square-integrable functions, we conclude that these models are equivalent to quantum mechanical models of particles living in a space with an additional compact dimension. PACS: 03.65.-w; 11.30.Cp; 02.40.Gh
We present a physical interpretation of the doubling of the algebra, which is the basic ingredient of the noncommutative spectral geometry, developed by Connes and collaborators as an approach to unification. We discuss its connection to dissipation and to the gauge structure of the theory. We then argue, following t Hoofts conjecture, that noncommutative spectral geometry classical construction carries implicit in its feature of the doubling of the algebra the seeds of quantization.
We investigate a new property of nets of local algebras over 4-dimensional globally hyperbolic spacetimes, called punctured Haag duality. This property consists in the usual Haag duality for the restriction of the net to the causal complement of a point $p$ of the spacetime. Punctured Haag duality implies Haag duality and local definiteness. Our main result is that, if the theory is locally covariant in the sense of Brunetti, Fredenhagen and Verch, then also the converse holds. The free Klein-Gordon field provides an example in which this property is verified.
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