Field theories based on non-commutative spacetimes exhibit very distinctive non-local effects which mix the ultraviolet with the infrared in bizarre ways. In particular if the time coordinate is involved in the non-commutativity the theory seems to be seriously acausal and inconsistent with conventional hamiltonian evolution. To illustrate these effects we study the scattering of wave packets i...

Non-commutative differential geometry allows a scalar field to be regarded as a gauge connection, albeit on a discrete space. We explain how the underlying gauge principle corresponds to the independence of physics on the choice of vacuum state, should it be non-unique. A consequence is that Yang–Mills–Higgs theory can be reformulated as a generalised Yang– Mills gauge theory on Euclidean space...

The Kodaira-Spencer map is a component of the connection ∇. In particular, this implies that if κs 6= 0 then the connection∇ is nontrivial with respect to the Hodge decomposition. Various Hodge-theory facts imply that the global monodromy must be nontrivial in this case. We can be a bit more precise: if u ∈ V p,q is a vector such that κs(v)(u) 6= 0 for some tangent vector v ∈ T (S)s, then u can...

We calculate the long-range Vanderwaals force and the photoelectric cross section in a noncommutative set up. While we argue that non-commutativity effects could not be discerned for the Vanderwaals interactions, the result for the photoelectric effect shows deviation from the usual commutative one, which in principle can be used to put bounds on the space-space non-commutativity parameter.

We show that the validity of the non-commutative Khintchine inequality for some q with 1 < q < 2 implies its validity (with another constant) for all 1 ≤ p < q. We prove this for the inequality involving the Rademacher functions, but also for more general " lacunary " sequences, or even non-commutative analogues of the Rademacher functions. For instance, we may apply it to the " Z(2)-sequences ...

Looking for the universal covering of the smooth non-commutative torus leads to a curve of associative multiplications on the space O M (R) ∼= OC(R ) of Laurent Schwartz which is smooth in the deformation parameter ~. The Taylor expansion in ~ leads to the formal Moyal star product. The non-commutative torus and this version of the Heisenberg plane are examples of smooth *-algebras: smooth in t...

We rst exhibit in the commutative case the simple algebraic relations between the algebra of functions on a manifold and its in nitesimal length element ds. Its unitary representations correspond to Riemannian metrics and Spin structure while ds is the Dirac propagator ds = | = D 1 where D is the Dirac operator. We extend these simple relations to the non commutative case using Tomita's involut...

Recent theoretical results have demonstrated that non-commutative geometries naturally appear within the context of string/M-theory. One consequence of this possibility is that QED takes on a non-abelian nature due to the introduction of 3and 4-point functions. In addition, each QED vertex acquires a momentum dependent phase factor. We parameterize the effects of non-commutative space-time co-o...