Given a symmetry group acting on a principal fibre bundle, symmetric states of the quantum theory of a diffeomorphism invariant theory of connections on this fibre bundle are defined. These symmetric states, equipped with a scalar product derived from the Ashtekar-Lewandowski measure for loop quantum gravity, form a Hilbert space of their own. Restriction to this Hilbert space yields a quantum symmetry reduction procedure in the framework of spin network states the structure of which is analyzed in detail.Three illustrating examples are discussed: Reduction of 3 + 1 to 2 + 1 dimensional quantum gravity, spherically symmetric quantum electromagnetism and spherically symmetric quantum gravity.

We show that the quantization of spherically symmetric pure gravity can be carried out completely in the framework of Ashtekar's self-dual representation. Consistent operator orderings can be given for the constraint functionals yielding two kinds of solutions for the constraint equations, corresponding classically to globally nondegenerate or degenerate metrics. The physical state functionals can be determined by quadratures and the reduced Hamiltonian system possesses 2 degrees of freedom, one of them corresponding to the classical Schwarzschild mass squared and the canonically conjugate one representing a measure for the deviation of the nonstatic field configurations from the static Schwarzschild one. There is a natural choice for the scalar product making the 2 fundamental observables self-adjoint. Finally, a unitary transformation is performed in order to calculate the triad-representation of the physical state functionals and to provide for a solution of the appropriately regularized Wheeler-DeWitt equation.

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The question how to quantize a classical system where an angle ϕ is one of the basic canonical variables has been controversial since the early days of quantum mechanics. The problem is that the angle is a multivalued or discontinuous variable on the corresponding phase space. The remedy is to replace ϕ by the smooth periodic functions cos ϕ and sin ϕ. In the case of the canonical pair (ϕ, p ϕ ) , p ϕ : orbital angular momentum (OAM), the phase space S ϕ,pϕ = {ϕ ∈ R mod 2π, p ϕ ∈ R} has the global topological structure S 1 × R of a cylinder on which the Poisson brackets of the three functions cos ϕ, sin ϕ and p ϕ obey the Lie algebra of the euclidean group E(2) in the plane. This property provides the basis for the quantization of the system in terms of irreducible unitary representations of the group E(2) or of its covering groups. A crucial point is that -due to the fact that the subgroup SO(2) ∼ = S 1 is multiply connected -these representations allow for fractional OAM l = (n + δ), n ∈ Z, δ ∈ [0, 1). Such δ = 0 have already been observed in cases like the Aharonov-Bohm and the fractional quantum Hall effects and they correspond to the quasi-momenta of Bloch waves in ideal crystals. The proposal of the present paper is to look for fractional OAM in connection with the quantum optics of Laguerre-Gaussian laser modes in external magnetic fields. The quantum theory of the phase space S ϕ,pϕ in terms of unitary representations of E(2) allows for two types of "coherent" states the properties of which are discussed in detail: Non-holomorphic minimal uncertainty states and holomorphic ones associated with Bargmann-Segal Hilbert spaces.PACS number(s): 03.65. Fd, 78.20.Ls

Dedicated to the memory of Julius Wess (1934Wess ( -2007, colleague and friend for many years.The historical developments of conformal transformations and symmetries are sketched: Their origin from stereographic projections of the globe, their blossoming in two dimensions within the field of analytic complex functions, the generic role of transformations by reciprocal radii in dimensions higher than two and their linearization in terms of polyspherical coordinates by Darboux, Weyl's attempt to extend General Relativity, the slow rise of finite dimensional conformal transformations in classical field theories and the problem of their interpretation, then since about 1970 the rapid spread of their acceptance for asymptotic and structural problems in quantum field theories and beyond, up to the current AdS/CFT conjecture. The occasion for the present article: hundred years ago Bateman and Cunningham discovered the form invariance of Maxwell's equations for electromagnetism with respect to conformal space-time transformations.