Classical Topology and Quantum States

Abstract

Any two infinite-dimensional (separable) Hilbert spaces are unitarily isomorphic. The sets of all their self-adjoint operators are also therefore unitarily equivalent. Thus if all self-adjoint operators can be observed, and if there is no further major axiom in quantum physics than those formulated for example in Dirac's 'Quantum Mechanics', then a quantum physicist would not be able to tell a torus from a hole in the ground. We argue that there are indeed such axioms involving observables with smooth time evolution: they contain commutative subalgebras from which the spatial slice of spacetime with its topology (and with further refinements of the axiom, its C K − and C ∞ − structures) can be reconstructed using Gel'fand -Naimark theory and its extensions. Classical topology is an attribute of only certain quantum observables for these axioms, the spatial slice emergent from quantum physics getting progressively less differentiable with increasingly higher excitations of energy and eventually altogether ceasing to exist. After formulating these axioms, we apply them to show the possibility of topology change and to discuss quantized fuzzy topologies. Fundamental issues concerning the role of time in quantum physics are also addressed.

The spin-statistics connection in quantum gravity

“…It is clear that elements belonging to the subgroup Dif f ∞ 0 (Σ), the component connected to identity, act trivially on π 1 (Σ) 1 and hence on (a, b). Therefore what matters is the action of the so-called mapping class group M Σ [20,21], defined as…”

“…Let us specialize to (2 + 1)d. Let R ∞ (Σ) be the space of all Riemannian metrics on the space manifold Σ which are equal to some conical metric in a neighborhood N of infinity 1 . We quotient this space by the group Dif f ∞ (Σ) of diffeomorphisms of the spatial 2-manifold Σ to obtain Q [21]. Therefore we may denote the Q as…”

“…To see this we resort to the so-called covering space quantization, which we briefly review here (for a more complete account, see Ref. [21]). In this approach, wave functions (defining domains of operators like the Hamiltonian) are taken to be functions on the universal covering spaceQ of Q with certain specific properties.…”

“…This recipe obviously depends on which representation ρ we take. Actually, it can be shown [21] that there are at least as many inequivalent quantizations as there are unitary irreducible representations of M Σ .…”

“…A very useful construction of the universal covering spaceQ is as follows [21]. Let us assume that the space Q is connected (if this is not so we can always choose a connected component).…”