Given a unitary operator A representing a physical quantity of interest, we employ concepts from group representation theory to deene two natural signal energy densities for A. The rst is invariant to A and proves useful when the eeect of A is to be ignored; the second is covariant to A and measures the \A" content of signals. The construction is quite general and is also easily extended to the...

The underlying mathematics of the wavelet formalism is a representation of the inhomogeneous Lorentz group or the affine group. Within the framework of wavelets, it is possible to define the " window " which allows us to introduce a Lorentz-covariant cutoff procedure. The window plays the central role in tackling the problem of photon localization. It is possible to make a transition from light...

The coherent states associated to the discrete series representations D(E o , s) of SO(3, 2) are constructed in terms of (spin-)tensor fields on D = 4 anti-de Sitter space. For E o > s + 5 the linear space H Eo,s spanned by these states is proved to carry the unitary irreducible representation D(E o , s). The SO(3, 2)-covariant generalized Fourier transform in this space is exhibited. The quasi...

We show that the grading of fields by conformal weight, when built into the initial group symmetry, provides a discrete, non-central conformal extension of any group containing dilatations. We find a faithful vector representation of the extended conformal group and show that it has a scale-invariant scalar product and satisfies a closed commutator algebra. The commutator algebra contains the i...

Using Poincaré duality in K-theory, we state and prove a Lefschetz fixed point formula for endomorphisms of crossed product C-algebras C0(X)⋊ G coming from covariant pairs. Here G is assumed countable, X a manifold, and X ⋊ G cocompact and proper. The formula in question expresses the graded trace of the map on rationalized K-theory of C0(X) ⋊ G induced by the endomorphism, i.e. the Lefschetz n...

We derive a new expression for the supersymmetric Schur polynomials sλ(x/y). The origin of this formula goes back to representation theory of the Lie superalgebra gl(m|n) and gives rise to a determinantal formula for sλ(x/y). In the second part, we use this determinantal formula to derive new expressions for the dimension and superdimension of covariant representations Vλ of the Lie superalgebr...

This paper describes a relationship between essentially finite groupoids and 2-vector spaces. In particular, we show to construct 2-vector spaces of Vect-valued presheaves on such groupoids. We define 2-linear maps corresponding to functors between groupoids in both a covariant and contravariant way, which are ambidextrous adjoints. This is used to construct a representation— a weak functor—fro...

| Given a unitary operator A representing a physical quantity of interest, we employ concepts from group representation theory to deene two natural signal energy densities for A. The rst is invariant to A and proves useful when the eeect of A is to be ignored; the second is covariant to A and measures the \A" content of signals. We also consider joint densities for multiple operators. In the pr...

In this work we apply the Lie group representation method introduced in the real time formalism for finite-temperature quantum-field theory, thermofield dynamics, to derive a spinorial density matrix equation. Symmetry properties of such equation are analysed, and as a basic result it is shown that one solution is the generalised density matrix operator proposed by Heinz, to deal with gauge cov...

We developed the formal connection of the field theoretical Bethe-Salpeter equation including the ladder approximation with its representation on the light-front for a bosonic model. We use the light-front Green's function for the N-particle system for two-particles plus N-2 intermediate bosons. We derived an infinite set of coupled hierarchy equations or iterated resolvent from which the Green...