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Journal of Mathematical Physics

Geometric derivation of the kinetic energy in collective models

DAVID J. ROWE, George Rosensteel

Published Mar 1, 1979• 23 citations
Publisher

Abstract

A separation of the many-particle kinetic energy into collective and intrinsic components is shown to result simply from a general form of the Laplace–Beltrami operator. The geometric structure of the decomposition is thereby clearly exhibited and the intricate computations previously necessary are eliminated.
DOI
10.1063/1.524097

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Direct-sum decompositions of the Euclidean space of particle momenta into collective and intrinsic subspaces are achieved using projectors on so(3) and gl(3,R) algebra spaces. The separation of the N-particle kinetic energy into the corresponding collective and intrinsic components is then simply obtained. In the case of gl(3,R) the resulting intrinsic kinetic energy is expressed in terms of an appropriate subset of the generators of the direct product group SO(N)*SO(N). A detailed comparison of these results with those of other authors is given.
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The most general transformations of coordinates that allow for the exact separation of the kinetic energy operator of a quantum many-body system into total center of mass kinetic energy and internal kinetic energy are found and discussed. It is found (i) that the suitable transformations, depending on the number of particles, have a certain number of free parameters and this allows for the generalization of the Jacobi coordinates to a much larger class of coordinates with the same properties and (ii) that there is a new, uncommon, additive group structure hidden in the transformation matrices that is connected to certain geometric properties of the set of coordinates.
Abstract In quantum theory on curved backgrounds, Heisenberg’s uncertainty principle is usually discussed in terms of ensemble variances and flat-space commutators. Here we take a different, preparation-based viewpoint tailored to sharp position measurements on spacelike hypersurfaces in general relativity. A projective localization is modeled as a von Neumann–Lüders projection onto a geodesic ball B Σ ( r ) of radius r on a Cauchy slice ( Σ , h ) , with the post-measurement state described by Dirichlet data. Using DeWitt-type momentum operators adapted to an orthonormal frame, we construct a geometric, coordinate-invariant momentum standard deviation σ p and show that strict confinement to B Σ ( r ) enforces an intrinsic kinetic-energy floor. The lower bound is set by the first Dirichlet eigenvalue λ 1 of the Laplace–Beltrami operator on the ball, σ p ⩾ ℏ λ 1 , and is manifestly invariant under changes of coordinates and foliation. A variance decomposition separates the contribution of the modulus | ψ | from phase-gradient fluctuations and clarifies how the spectral geometry of ( Σ , h ) controls momentum uncertainty. Assuming only minimal geometric information, weak mean-convexity of the boundary yields a universal, scale-invariant Heisenberg-type product bound, σ p r ⩾ π ℏ / 2 , depending only on the proper radius r .
In its geometric form, the Maupertuis Principle states that the movement of a classical particle in an external potential V(x) can be understood as a free movement in a curved space with the metric gμν(x) = 2M[V(x) - E]δμν. We extend this principle to the quantum regime by showing that the wavefunction of the particle is governed by a Schrödinger equation of a free particle moving through curved space. The kinetic operator is the Weyl-invariant Laplace–Beltrami operator. On the basis of this observation, we calculate the semiclassical expansion of the particle density.