Optimal Paths for Symmetric Actions in the Unitary Group

Abstract

Given a positive and unitarily invariant Lagrangian \({\mathcal{L}}\) defined in the algebra of matrices, and a fixed time interval \({[0,t_0]\subset\mathbb R}\), we study the action defined in the Lie group of \({n\times n}\) unitary matrices \({\mathcal{U}(n)}\) by

$$\mathcal{S}(\alpha)=\int_0^{t_0} \mathcal{L}(\dot\alpha(t))\,dt, $$

where \({\alpha:[0,t_0]\to\mathcal{U}(n)}\) is a rectifiable curve. We prove that the one-parameter subgroups of \({\mathcal{U}(n)}\) are the optimal paths, provided the spectrum of the exponent is bounded by π. Moreover, if \({\mathcal{L}}\) is strictly convex, we prove that one-parameter subgroups are the unique optimal curves joining given endpoints. Finally, we also study the connection of these results with unitarily invariant metrics in \({\mathcal{U}(n)}\) as well as angular metrics in the Grassmann manifold.

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Correspondence to Gabriel Larotonda.

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Communicated by A. Connes

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Antezana, J., Larotonda, G. & Varela, A. Optimal Paths for Symmetric Actions in the Unitary Group. Commun. Math. Phys. 328, 481–497 (2014). https://doi.org/10.1007/s00220-014-2041-x

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Keywords

  • Optimal Path
  • Unitary Group
  • Unitary Matrice
  • Geodesic Segment
  • Grassmann Manifold