The Canonical Approach to Quantum Gravity: General Ideas and Geometrodynamics

We give an introduction to the canonical formalism of Einstein's theory of general relativity. This then serves as the starting point for one approach to quantum gravity called quantum geometrodynamics. The main features and applications of this approach are briefly summarized.Comment: 21 pages, 6 figures. Contribution to E. Seiler and I.-O. Stamatescu (editors): `Approaches To Fundamental Physics -- An Assessment Of Current Theoretical Ideas' (Springer Verlag, to appear

On the origin of probability in quantum mechanics

I give a brief introduction to many worlds or "no wavefunction collapse" quantum mechanics, suitable for non-specialists. I then discuss the origin of probability in such formulations, distinguishing between objective and subjective notions of probability.Comment: 7 pages, 2 figures. This version to appear as a Brief Review in Modern Physics Letter

An Analysis of the Representations of the Mapping Class Group of a Multi-Geon Three-Manifold

It is well known that the inequivalent unitary irreducible representations (UIR's) of the mapping class group $G$ of a 3-manifold give rise to ``theta sectors'' in theories of quantum gravity with fixed spatial topology. In this paper, we study several families of UIR's of $G$ and attempt to understand the physical implications of the resulting quantum sectors. The mapping class group of a three-manifold which is the connected sum of $\R^3$ with a finite number of identical irreducible primes is a semi-direct product group. Following Mackey's theory of induced representations, we provide an analysis of the structure of the general finite dimensional UIR of such a group. In the picture of quantized primes as particles (topological geons), this general group-theoretic analysis enables one to draw several interesting qualitative conclusions about the geons' behavior in different quantum sectors, without requiring an explicit knowledge of the UIR's corresponding to the individual primes.Comment: 52 pages, harvmac, 2 postscript figures, epsf required. Added an appendix proving the semi-direct product structure of the MCG, corrected an error in the characterization of the slide subgroup, reworded extensively. All our analysis and conclusions remain as befor

Experimental determination of the Berry phase in a superconducting charge pump

We present the first measurements of the Berry phase in a superconducting Cooper pair pump. A fixed amount of Berry phase is accumulated to the quantum-mechanical ground state in each adiabatic pumping cycle, which is determined by measuring the charge passing through the device. The dynamic and geometric phases are identified and measured quantitatively from their different response when pumping in opposite directions. Our observations, in particular, the dependencies of the dynamic and geometric effects on the superconducting phase bias across the pump, agree with the basic theoretical model of coherent Cooper pair pumping.Comment: 4 pages, 3 figure

Consistency of Semiclassical Gravity

We discuss some subtleties which arise in the semiclassical approximation to quantum gravity. We show that integrability conditions prevent the existence of Tomonaga-Schwinger time functions on the space of three-metrics but admit them on superspace. The concept of semiclassical time is carefully examined. We point out that central charges in the matter sector spoil the consistency of the semiclassical approximation unless the full quantum theory of gravity and matter is anomaly-free. We finally discuss consequences of these considerations for quantum field theory in flat spacetime, but with arbitrary foliations.Comment: 12 pages, LATEX, Report Freiburg THEP-94/2

Group averaging in the (p,q) oscillator representation of SL(2,R)

We investigate refined algebraic quantisation with group averaging in a finite-dimensional constrained Hamiltonian system that provides a simplified model of general relativity. The classical theory has gauge group SL(2,R) and a distinguished o(p,q) observable algebra. The gauge group of the quantum theory is the double cover of SL(2,R), and its representation on the auxiliary Hilbert space is isomorphic to the (p,q) oscillator representation. When p>1, q>1 and p+q == 0 (mod 2), we obtain a physical Hilbert space with a nontrivial representation of the o(p,q) quantum observable algebra. For p=q=1, the system provides the first example known to us where group averaging converges to an indefinite sesquilinear form.Comment: 34 pages. LaTeX with amsfonts, amsmath, amssymb. (References added; minor typos corrected.

Symplectic Cuts and Projection Quantization

The recently proposed projection quantization, which is a method to quantize particular subspaces of systems with known quantum theory, is shown to yield a genuine quantization in several cases. This may be inferred from exact results established within symplectic cutting.Comment: 12 pages, v2: additional examples and a new reference to related wor

All (qubit) decoherences: Complete characterization and physical implementation

We investigate decoherence channels that are modelled as a sequence of collisions of a quantum system (e.g., a qubit) with particles (e.g., qubits) of the environment. We show that collisions induce decoherence when a bi-partite interaction between the system qubit and an environment (reservoir) qubit is described by the controlled-U unitary transformation (gate). We characterize decoherence channels and in the case of a qubit we specify the most general decoherence channel and derive a corresponding master equation. Finally, we analyze entanglement that is generated during the process of decoherence between the system and its environment.Comment: 10 pages, 3 figure

Refined Algebraic Quantization in the oscillator representation of SL(2,R)

We investigate Refined Algebraic Quantization (RAQ) with group averaging in a constrained Hamiltonian system with unreduced phase space T^*R^4 and gauge group SL(2,R). The reduced phase space M is connected and contains four mutually disconnected `regular' sectors with topology R x S^1, but these sectors are connected to each other through an exceptional set where M is not a manifold and where M has non-Hausdorff topology. The RAQ physical Hilbert space H_{phys} decomposes as H_{phys} = (direct sum of) H_i, where the four subspaces H_i naturally correspond to the four regular sectors of M. The RAQ observable algebra A_{obs}, represented on H_{phys}, contains natural subalgebras represented on each H_i. The group averaging takes place in the oscillator representation of SL(2,R) on L^2(R^{2,2}), and ensuring convergence requires a subtle choice for the test state space: the classical analogue of this choice is to excise from M the exceptional set while nevertheless retaining information about the connections between the regular sectors. A quantum theory with the Hilbert space H_{phys} and a finitely-generated observable subalgebra of A_{obs} is recovered through both Ashtekar's Algebraic Quantization and Isham's group theoretic quantization.Comment: 30 pages, REVTeX v3.1 with amsfonts. (v4: Published version.