On a Semiclassical Limit of Loop Space Quantum Mechanics

Abstract

Following earlier work, we view two-dimensional nonlinear sigma model as single particle quantum mechanics in the free loop space of the target space. In a natural semiclassical limit of this model, the wavefunction localizes on the submanifold of vanishing loops. One would expect that the semiclassical expansion should be related to the tubular expansion of the theory around the submanifold and effective dynamics on the submanifold is obtainable using Born-Oppenheimer approximation. We develop a framework to carry out such an analysis at the leading order. In particular, we show that the linearized tachyon effective equation is correctly reproduced up to divergent terms all proportional to the Ricci scalar. The steps are as follows: first we define a finite dimensional analogue of the loop space quantum mechanics (LSQM) where we discuss its tubular expansion and how that is related to a semiclassical expansion of the Hamiltonian. Then we study an explicit construction of the relevant tubular neighborhood in loop space using exponential maps. Such a tubular geometry is obtained from a Riemannian structure on the tangent bundle of target space which views the zero-section as a submanifold admitting a tubular neighborhood. Using this result and exploiting an analogy with the toy model, we arrive at the final result for LSQM.

A cut-off tubular geometry of loop space

“…One can rewrite NLSM as a single particle model in LM such that the complete geometry of the configuration space is manifest [3,4,5,6,7]. For small loops which are entirely contained in a single normal neighbourhood [8] in M, the configuration space can be described as a tubular neighbourhood [9,10,11] around ∆ → LM (see [7]).…”

“…For small loops which are entirely contained in a single normal neighbourhood [8] in M, the configuration space can be described as a tubular neighbourhood [9,10,11] around ∆ → LM (see [7]). Here…”

“…constant maps from S 1 to M. It was demonstrated in [7] that in a semi-classical limit (α → 0) of the loop space quantum mechanics (LSQM) the string wavefunction localises on ∆. This in turn implies that the semi-classical expansion is to be derived from tubular expansion of geometric quantities of LM.…”

“…There are two obstacles in performing the computation described in [7], 1. Lack of a detailed understanding of the tubular geometry in LM.…”

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AbstractMotivated by the computation of loop space quantum mechanics as indicated in [7], here we seek a better understanding of the tubular geometry of loop space LM corresponding to a Riemannian manifold M around the submanifold of vanishing loops. Our approach is to first compute the tubular metric of (M 2N +1 ) C around the diagonal submanifold, where (M N ) C is the Cartesian product of N copies of M with a cyclic ordering.

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