Paulo Teotonio-Sobrinho

Unitary quantum physics with time-space noncommutativity

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 H of a quantum mechanical particle on spacetime N−1 × 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 noncommutative version of H and H have identical spectra.

Time-space noncommutativity: quantised evolutions

In previous work [7], we developed quantum physics on the Moyal plane with time-space noncommutativity, basing ourselves on the work of Doplicher et al. [1]. Here we extend it to certain noncommutative versions of the cylinder, 3 and × S3. In all these models, only discrete time translations are possible, a result known before in the first two cases [2]–[6]. One striking consequence of quantised time translations is that even though a time independent hamiltonian is an observable, in scattering processes, it is conserved only modulo 2π/θ, where θ is the noncommutative parameter. (In contrast, on a one-dimensional periodic lattice of lattice spacing a and length L = Na, only momentum mod 2π/L is observable (and can be conserved).) Suggestions for further study of this effect are made. Scattering theory is formulated and an approach to quantum field theory is outlined.

Quantum fields with noncommutative target spaces

Quantum field theories (QFTs) on noncommutative spacetimes are currently under intensive study. Usually such theories have world sheet noncommutativity. In the present work, instead, we study QFTs with commutative world sheet and noncommutative target space. Such noncommutativity can be interpreted in terms of twisted statistics and is related to earlier work of Oeckl [R. Oeckl, Commun. Math. Phys. 217, 451 (2001).], and others [A. P. Balachandran, G. Mangano, A. Pinzul, and S. Vaidya, Int. J. Mod. Phys. A 21, 3111 (2006); A. P. Balachandran, A. Pinzul, and B. A. Qureshi, Phys. Lett. B 634, 434 (2006); A. P. Balachandran, A. Pinzul, B. A. Qureshi, and S. Vaidya, arXiv:hep-th/0608138; A. P. Balachandran, T. R. Govindarajan, G. Mangano, A. Pinzul, B. A. Qureshi, and S. Vaidya, Phys. Rev. D 75, 045009 (2007); A. Pinzul, Int. J. Mod. Phys. A 20, 6268 (2005); G. Fiore and J. Wess, Phys. Rev. D 75, 105022 (2007); Y. Sasai and N. Sasakura, Prog. Theor. Phys. 118, 785 (2007)]. The twisted spectra of their free Hamiltonians has been found earlier by Carmona et al. [J. M. Carmona, J. L. Cortes, J. Gamboa, and F. Mendez, Phys. Lett. B 565, 222 (2003); J. M. Carmona, J. L. Cortes, J. Gamboa, and F. Mendez, J. High Energy Phys. 03 (2003) 058]. We review their derivation and then compute the partition function of one such typical theory. It leads to a deformed blackbody spectrum, which is analyzed in detail. The difference between the usual and the deformed blackbody spectrum appears in the region of high frequencies. Therefore we expect that the deformed blackbody radiation may potentially be used to compute a Greisen-Zatsepin-Kuzmin cutoff which will depend on the noncommutative parameter

Unitary quantum physics with time-space non-commutativity

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QUANTUM TOPOLOGY CHANGE IN (2 + 1)d

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2D quantum double models from a 3D perspective

In these lectures4 quantum physics in noncommutative spacetime is developed. It is based on the work of Doplicher et al. which allows for time-space noncommutativity. 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. The theory is further extended to certain noncommutative versions of the cylinder, 3 and × S3. In all these models, only discrete time translations are possible. One striking consequence of quantised time translations is that even though a time independent Hamiltonian is an observable, in scattering processes, it is conserved only modulo 2π/θ, where θ is the noncommutative parameter. Scattering theory is formulated and an approach to quantumfield theory is outlined.

Dirac operator on the q-deformed fuzzy sphere and its spectrum

The topology of orientable (2+1) space–times can be captured by certain lumps of nontrivial topology called topological geons. They are the topological analogs of conventional solitons. We give a description of topological geons where the degrees of freedom related to topology are separated from the complete theory that contain metric (dynamical) degrees of freedom. The formalism also allows us to investigate processes of quantum topology change. They correspond to creation and annihilation of quantum geons. Selection rules for such processes are derived.

Topological low-temperature limit of Z(2) spin-gauge theory

In this paper we look at three dimensional (3D) lattice models that are generalizations of the state sum model used to define the Kuperberg invariant of 3-manifolds. The partition function is a scalar constructed as a tensor network where the building blocks are tensors given by the structure constants of an involutory Hopf algebra . These models are very general and are hard to solve in its entire parameter space. One can obtain familiar models, such as ordinary gauge theories, by letting be the group algebra of a discrete group G and staying on a certain region of the parameter space. We consider the transfer matrix of the model and show that quantum double Hamiltonians are derived from a particular choice of the parameters. Such a construction naturally leads to the star and plaquette operators of the quantum double Hamiltonians, of which the toric code is a special case when . This formulation is convenient to study ground states of these generalized quantum double models where they can naturally be interpreted as tensor network states. For a surface ?, the ground state degeneracy is determined by the Kuperberg 3-manifold invariant of . It is also possible to obtain extra models by simply enlarging the allowed parameter space but keeping the solubility of the model. While some of these extra models have appeared before in the literature, our 3D perspective allows for an uniform description of them.

The spin-statistics connection in quantum gravity

The q-deformed fuzzy sphere SqF2(N) is the algebra of (N+1) × (N+1) dim. matrices, covariant with respect to the adjoint action of Uq(su(2)) and in the limit q→1, it reduces to the fuzzy sphere SF2(N). We construct the Dirac operator on the q-deformed fuzzy sphere-SqF2(N) using the spinor modules of Uq(su(2)). We explicitly obtain the zero modes and also calculate the spectrum for this Dirac operator. Using this Dirac operator, we construct the Uq(su(2)) invariant action for the spinor fields on SqF2(N) which are regularised and have only finite modes. We analyse the spectrum for both q being root of unity and real, showing interesting features like its novel degeneracy. We also study various limits of the parameter space (q, N) and recover the known spectrum in both fuzzy and commutative sphere.

A NONASSOCIATIVE QUATERNION SCALAR FIELD THEORY

We study Z(2) lattice gauge theory on triangulations of compact d-manifolds. We reformulate the theory algebraically, describing it in terms of the structure constants of a bidimensional vector space H equipped with algebra and coalgebra structures, and prove that in the low-temperature limit H reduces to a Hopf algebra, in which case the theory becomes equivalent to a topological field theory. The degeneracy of the ground state is shown to be a topological invariant. This fact is used to compute the zeroth- and first-order terms in the low-temperature expansion of Z for arbitrary triangulations. In finite temperatures, the algebraic reformulation gives rise to new duality relations among classical spin models, related to changes of basis of H.