The Connes-Lott program on the sphere

We describe the classical Schwinger model as a study of the projective modules over the algebra of complex-valued functions on the sphere. On these modules, classified by $\pi_2(S^2)$, we construct hermitian connections with values in the universal differential envelope which leads us to the Schwinger model on the sphere. The Connes-Lott program is then applied using the Hilbert space of complexified inhomogeneous forms with its Atiyah-Kaehler structure. It splits in two minimal left ideals of the Clifford algebra preserved by the Dirac-Kaehler operator D=i(d-delta). The induced representation of the universal differential envelope, in order to recover its differential structure, is divided by the unwanted differential ideal and the obtained quotient is the usual complexified de Rham exterior algebra over the sphere with Clifford action on the "spinors" of the Hilbert space. The subsequent steps of the Connes-Lott program allow to define a matter action, and the field action is obtained using the Dixmier trace which reduces to the integral of the curvature squared.Comment: 34 pages, Latex, submitted for publicatio

Connes-Lott model building on the two-sphere

In this work we examine generalized Connes-Lott models on the two-sphere. The Hilbert space of the continuum spectral triple is taken as the space of sections of a twisted spinor bundle, allowing for nontrivial topological structure (magnetic monopoles). The finitely generated projective module over the full algebra is also taken as topologically non-trivial, which is possible over $S^2$. We also construct a real spectral triple enlarging this Hilbert space to include "particle" and "anti-particle" fields.Comment: 57 pages, LATE

Non-commutative Quantum Mechanics in Three Dimensions and Rotational Symmetry

We generalize the formulation of non-commutative quantum mechanics to three dimensional non-commutative space. Particular attention is paid to the identification of the quantum Hilbert space in which the physical states of the system are to be represented, the construction of the representation of the rotation group on this space, the deformation of the Leibnitz rule accompanying this representation and the implied necessity of deforming the co-product to restore the rotation symmetry automorphism. This also implies the breaking of rotational invariance on the level of the Schroedinger action and equation as well as the Hamiltonian, even for rotational invariant potentials. For rotational invariant potentials the symmetry breaking results purely from the deformation in the sense that the commutator of the Hamiltonian and angular momentum is proportional to the deformation.Comment: 21 page

Noncommutative Configuration Space. Classical and Quantum Mechanical Aspects ∗

In this work we examine noncommutativity of position coordinates in classical symplectic mechanics and its quantisation. In coordinates {q i,pk} the canonical symplectic two-form is ω0 = dq i ∧ dpi. It is well known in symplectic mechanics [5, 6, 7] that the interaction of a charged particle with a magnetic field can be described without a choice of a potential in a Lagrangian formalism. This is done introducing a modified symplectic two-form ω = ω0 − eF, where e is the charge and the (time-independent) magnetic field F is closed: dF = 0. With this symplectic structure, the canonical momentum variables acquire non-vanishing Poisson brackets: {pk,pl} = eFkl(q). Similarly we introduce a dual magnetic field G, which is a closed two-form in p-space interacting with the particle’s dual charge r. A new modified symplectic two-form ω = ω0 − eF + rG is then defined. Now, both p- and q-variables will cease to Poisson commute and upon quantisation they become noncommuting operators. In the particular case of a linear phase space R 2N, it makes sense to consider constant F and G fields. It is then possible to define global Darboux coordinates through a linear transformation. These can then be quantised in the usual way. Quadratic Hamiltonians are examined with some detail in the two- and three-dimensional cases