On the existence of bi-invariant Finsler metrics on Lie groups

Latifi, Dariush; Toomanian, Megerdich

doi:10.1186/2251-7456-7-37

Cited by 9 publications

(12 citation statements)

References 7 publications

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“…The bi-invariance means the geometry is both right and left invariant. It has been shown that, in bi-invariant Finsler geometry, the curve c(s) is a geodesic if and only if there is a constant generator H(s) =H such that [34,35] c(s) =Hc(s) or c(s) = exp(sH) . The minimization 'min' in (3.9) in the sense of 'geodesic' is already taken care of in (5.9).…”

Section: Geodesics and Complexity Of Su(n) Operatorsmentioning

confidence: 99%

Principles and symmetries of complexity in quantum field theory

Yang

Niu

et al. 2019

Eur. Phys. J. C

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Based on general and minimal properties of the discrete circuit complexity, we define the complexity in continuous systems in a geometrical way. We first show that the Finsler metric naturally emerges in the geometry of the complexity in continuous systems. Due to fundamental symmetries of quantum field theories, the Finsler metric is more constrained and consequently, the complexity of SU(n) operators is uniquely determined as a length of a geodesic in the Finsler geometry. Our Finsler metric is bi-invariant contrary to the right-invariance of discrete qubit systems. We clarify why the bi-invariance is relevant in quantum field theoretic systems. After comparing our results with discrete qubit systems we show most results in k-local right-invariant metric can also appear in our framework. Based on the bi-invariance of our formalism, we propose a new interpretation for the Schrödinger's equation in isolated systems -the quantum state evolves by the process of minimizing "computational cost." arXiv:1803.01797v5 [hep-th]

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Section: Geodesics and Complexity Of Su(n) Operatorsmentioning

confidence: 99%

Principles and symmetries of complexity in quantum field theory

Yang

Niu

et al. 2019

Eur. Phys. J. C

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“…(2.10). It has been shown in Refs [58,59]. that, if the Finsler metric is bi-invariant, the curve c(s) is a geodesic if and only if there is a constant generator H(s) =H such thatċ (s) =Hc(s) or c(s) = exp(sH) .…”

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confidence: 99%

“…10). It has been shown that, in bi-invariant Finsler geometry, the curve c(s) is a geodesic if and only if there is a constant generator H(s) =H such that[58,59] c(s) =Hc(s) or c(s) = exp(sH) . (5.13) With the boundary conditionÔ = c(1) = exp(H),H = lnÔ, which is the same as Ref.…”

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confidence: 99%

More on complexity of operators in quantum field theory

Yang

Niu

et al. 2019

J. High Energ. Phys.

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Recently it has been shown that the complexity of SU(n) operator is determined by the geodesic length in a bi-invariant Finsler geometry, which is constrained by some symmetries of quantum field theory. It is based on three axioms and one assumption regarding the complexity in continuous systems. By relaxing one axiom and an assumption, we find that the complexity formula is naturally generalized to the Schatten p-norm type. We also clarify the relation between our complexity and other works. First, we show that our results in a bi-invariant geometry are consistent with the ones in a right-invariant geometry such as k-local geometry. Here, a careful analysis of the sectional curvature is crucial. Second, we show that our complexity can concretely realize the conjectured pattern of the time-evolution of the complexity: the linear growth up to saturation time. The saturation time can be estimated by the relation between the topology and curvature of SU(n) groups. arXiv:1809.06678v2 [hep-th]

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“…Bi-invariant Finsler functions on Lie groups have been extensively investigated in the literature (see e.g. [9,6]). However, explicit examples for such functions -that are non-Riemannianare rarely calculated.…”

Section: Introductionmentioning

confidence: 99%

Homogeneous nonlinear splittings and Finsler submersions

Hajdu

Mestdag

2021

Preprint

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A nonlinear splitting on a fibre bundle is a generalization of an Ehresmann connection. An example is given by the homogeneous nonlinear splitting of a Finsler function on the total manifold of a fibre bundle. We show how homogeneous nonlinear splittings and nonlinear lifts can be used to construct submersions between Euclidean, Minkowski and Finsler spaces. As an application we consider a semisimple Lie algebra and use our methods to give new examples of Finsler functions on a reductive homogeneous space.

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On the existence of bi-invariant Finsler metrics on Lie groups

Abstract: In this paper, we prove some common fixed point results for four mappings satisfying generalized contractive condition in S-metric space. Our results extend and improve several previous works.

Cited by 9 publications

References 7 publications

Principles and symmetries of complexity in quantum field theory

Principles and symmetries of complexity in quantum field theory

More on complexity of operators in quantum field theory

Homogeneous nonlinear splittings and Finsler submersions

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