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Optimal Paths for Symmetric Actions in the Unitary Group

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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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Authors and Affiliations

  1. Departamento de Matemática, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, Esquina 50 y 115 s/n (B1900), La Plata, Argentina

    Jorge Antezana

  2. Instituto Argentino de Matemática “Alberto P. Calderón”, CONICET (Consejo Nacional de Investigaciones Científicas y Técnicas, Argentina), Saavedra 15 3er piso, C1083ACA, Buenos Aires, Argentina

    Jorge Antezana, Gabriel Larotonda & Alejandro Varela

  3. Instituto de Ciencias, Universidad Nacional de General Sarmiento, J.M. Gutiérrez 1150, Los Polvorines, B1613GSX, Buenos Aires, Argentina

    Gabriel Larotonda & Alejandro Varela

Authors
  1. Jorge Antezana
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  2. Gabriel Larotonda
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  3. Alejandro Varela
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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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  • DOI: https://doi.org/10.1007/s00220-014-2041-x

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