In this paper we present a new theory of exterior calculus over k-dimensional domains in a smooth n-manifold, unifying the discrete and continuum theories. The discrete theory carries a full set of classical operators, products, and relations of calculus and geometry and is free from the standard problems encountered in other theories of discrete exterior calculus. It is based on the Koszul com...

We prove that Wigner functions contain a symplectic connection. The latter covariantises the symplectic exterior derivative on phase space. We analyse the role played by this connection and introduce the notion of local symplectic covariance of quantum–mechanical states. This latter symmetry is at work in the Schroedinger equation on phase space.

A fermionic analogue of the Hodge star operation is shown to have an explicit operator representation in models with fermions, in spacetimes of any dimension. This operator realizes a conjugation (pairing) not used explicitly in field-theory, and induces a metric in the space of wave-function(al)s just as in exterior calculus. If made real (Hermitian), this induced metric turns out to be identi...

• Connections in Rn . We have by now defined a number of different notions of differentiation: the differential of maps between manifolds, the exterior derivative of differential forms, and the Lie derivative. To begin to see the need for a new notion of differentiation (which will turn out to be the most fundamental and will apply to general tensor fields) consider first the problem of definin...

Several approaches to Hawking radiation on the Schwarzschild spacetime rely in some way or another on the fact that the Kruskal manifold has two causally disconnected exterior regions. To assess the physical input implied by the presence of the second exterior region, we investigate the Hawking(-Unruh) effect for a real scalar field on the RP geon: an inextendible, globally hyperbolic, space an...

We establish the existence of a stationary weak solution of a generalized Boussinesq model for thermally driven convection in exterior domains. We use the fact that the exterior domain can be approximated by interior domains.

1 Exterior Algebras 2 1.1 The exterior algebra of a module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Free modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Determinants done right . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 The trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....