[Paper Review] Quantum Mechanics on Manifolds

Shogo Tanimura|ArXiv.org|Jun 28, 1993

Homotopy and Cohomology in Algebraic Topology3 references3 citations

TL;DR

This paper proposes a framework for quantum mechanics on manifolds using unitary representations of the isometry group G on vector bundle-valued functions over homogeneous spaces M = G/H. It demonstrates that when H is non-trivial, there exist infinitely many inequivalent quantum realizations, as shown in examples like S^n, T^n, and RP^n.

ABSTRACT

A definition of quantum mechanics on a manifold $ M $ is proposed and a method to realize the definition is presented. This scheme is applicable to a homogeneous space $ M = G / H $. The realization is a unitary representation of the transformation group $ G $ on the space of vector bundle-valued functions. When $ H e \{ e \} $, there exist a number of inequivalent realizations. As examples, quantum mechanics on a sphere $ S^n $, a torus $ T^n $ and a projective space $ \RP $ are studied. In any case, it is shown that there are an infinite number of inequivalent realizations.

Motivation & Objective

• To establish a general definition of quantum mechanics on curved manifolds, particularly homogeneous spaces.
• To address the challenge of defining quantum systems on non-trivial topological spaces where standard Hilbert space quantization fails.
• To provide a systematic method for constructing quantum theories on manifolds using group representation theory.
• To explore the existence and classification of inequivalent quantum realizations on symmetric spaces.
• To demonstrate the richness of quantum structures on manifolds through explicit examples like spheres, tori, and real projective spaces.

Proposed method

• Define quantum mechanics on a manifold M = G/H via unitary representations of the group G acting on the space of sections of a vector bundle over M.
• Utilize the group structure of G to induce quantum dynamics and observables through representation theory.
• Construct quantum states as square-integrable sections of associated vector bundles over M, equipped with G-invariant inner products.
• Employ induced representations to generate inequivalent quantum realizations when H ≠ {e}, corresponding to different choices of the stabilizer subgroup.
• Apply the method to specific manifolds: S^n (as SO(n+1)/SO(n)), T^n (as U(1)^n), and RP^n (as SO(n+1)/O(n)).
• Use the theory of induced representations and group characters to classify the resulting unitary representations and their inequivalence.

Experimental results

Research questions

• RQ1 How can quantum mechanics be consistently defined on a general manifold, particularly a homogeneous space?
• RQ2 What determines the existence of multiple inequivalent quantum realizations on the same manifold?
• RQ3 How do the topological and group-theoretic properties of M = G/H influence the structure of quantum states?
• RQ4 What role does the stabilizer subgroup H play in generating distinct quantum theories on the same manifold?
• RQ5 Can the framework be systematically applied to standard symmetric spaces like spheres, tori, and projective spaces?

Key findings

• The proposed framework successfully defines quantum mechanics on any homogeneous manifold M = G/H using unitary representations of G.
• When the stabilizer H is non-trivial, there exist infinitely many inequivalent unitary representations, leading to an infinite number of distinct quantum theories on the same manifold.
• On the n-sphere S^n, the method yields an infinite family of inequivalent quantum realizations, corresponding to different spinor or tensorial structures.
• For the n-torus T^n, the construction leads to an infinite set of inequivalent quantum systems, reflecting the abelian nature of its isometry group.
• In the case of real projective space RP^n, the method again produces infinitely many inequivalent quantum realizations due to the non-simply connected topology.
• The classification of these realizations is determined by the representation theory of the group G and the choice of induced representation from the subgroup H.