More on complexity of operators in quantum field theory

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.Comment: Modified the Sec. 4.1, where we offered a powerful proof: if (1) the ket vector and bra vector in quantum mechanics contain same physics, or (2) adding divergent terms to a Lagrangian will not change underlying physics, then complexity in quantum mechanics must be bi-invariant

Principles and symmetries of complexity in quantum field theory

Based on general and minimal properties of the {\it discrete} circuit complexity, we define the complexity in {\it 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 Schr\"{o}dinger's equation in isolated systems - the quantum state evolves by the process of minimizing "computational cost."Comment: Published version; added a short introduction on Finsler geometr

Thermodynamics and evaporation of perfect fluid dark matter black hole in phantom background

We present a novel interpretation of the thermodynamics of perfect fluid dark matter (PFDM) black hole based on Misner-Sharp energy, and then investigate its evaporation behavior. We find that the ratio between dark sector initial density and black hole horizon radius significantly influences black hole evaporation behaviors. We demonstrate that the presence of the dark sector can significantly extend the lifetime of a black hole which is similar to the Reissner-Nordstrom case. Our work reformulates the thermodynamics of PFDM black holes and points out the existence of long-lived black holes in the presence of the dark sector.Comment: 14 pages, 2 figures, comments are welcomed. v2: typos correcte

Investigating shadow images and rings of the charged Horndeski black hole illuminated by various thin accretions

In this paper, we investigate the shadows and rings of the charged Horndeski black hole illuminated by accretion flow that is both geometrically and optically thin. We consider two types of accretion models: spherical and thin-disk accretion flow. We find that in both types of models, the size of the charged Horndeski black hole shadow decreases with the increase of the charge, and it decreases more slowly for the Reissner-Nordstr\"om (RN) black hole. In the spherical accretion flow model, we find that the increase of the charge of Horndeski black hole brightens the light ring around it, and it brightens more significantly in comparison with RN black hole. Due to the Doppler effect, the charged Horndeski black holes with accretion flow of radial motion have darker shadows than those with the static accretion flow, but the size of the shadow is not affected by accretion flow motion. In the thin disk-shaped accretion flow model, we find that the brightness of the light ring around the charged Horndeski black hole is dominated by the direct emission from the accretion flow, and the contribution from lensed rings is relatively small, and that from the photon rings is negligible. We also find that the ring brightness decreases as the charge of Horndeski black hole increases, and the decrease is more significant than that in the RN black hole case. Moreover, the radiation position of the accretion flow can affect the shadow size and the ring brightness of the charged Horndeski black hole.Comment: 21 Pages, 16 Figures, 1 Table, accepted for publication in Eur. Phys. J.

‘Taoyuan No. 3-Spring Red’ and ‘Taoyuan No. 4-Red Glory’, the New Cultivars of Cherry Blossom in Subtropical Area

Cherry blossoms belong to the genus Prunus within the family Rosaceae. There are ∼100 species of indigenous plants, mainly distributed in temperate regions of the Northern Hemisphere, and there are more than 200 cultivated varieties (Obha et al. 2007). For blooming successfully in spring, it is necessary for those deciduous trees to undergo a period of chill beforehand. During the flowering period, the entire tree is devoid of leaves while the flowers blossom profusely on the branches, making it more aesthetically valuable. Cherry blossom viewing holds significant cultural importance in many temperate countries and plays a crucial role in international tourism activities (McClellan 2005). Taiwan is located at the junction of the subtropical and tropical climatic zones in the Northern Hemisphere, with the Tropic of Cancer, at the latitude of 23.5°, passing through the southern regions of Taiwan’s main island. The central and northern regions of Taiwan are characterized by a subtropical climate. According to the Flora of Taiwan (Editorial Committee of the Flora of Taiwan 1993), there are 11 native species of the Prunus genus in Taiwan. Among them, Taiwan cherry (Prunus campanulata Maxim) is distributed in southern China, Ryukyu Islands, and Japan, as well as in broad-leaved forests at high elevations (500–2000 m) throughout Taiwan (Editorial Committee of the Flora of Taiwan 1993). The native Taiwan cherry, displaying a range of flower colors from light pink to deep purple red, naturally blooms from January to March. Due to the mild winter climate in lower altitude areas of Taiwan, the selection of cherry blossom cultivars for landscape applications has been limited. Seed-propagated seedlings have been widely used for landscape purposes in urban and suburban areas, particularly in low-altitude regions (<500 m) (Newsroom Compilation 2006) in northern Taiwan. Within the population of Taiwan cherry trees, prolonged hybridization has contributed to a significant level of genetic diversity, resulting in individual specimens with variations in flower color, flower morphology, and heat and cold tolerance (Liou 2018). Previous studies and evidence have indicated the essentiality of the chill accumulation during the winter for the budbreak of deciduous woody plants (Arora et al. 2003; Erez et al. 1979). Insufficient chill accumulation has been found to result in incomplete development of floral bud organs (Fadón et al. 2021; Wang et al. 2016; Yamane et al. 2011), as well as a reduction in the germination of dormant buds (Erez and Couvillon 1987; Ferlito et al. 2021). The smaller quantity of flowers or variations in flowering periods among landscape Taiwan cherry trees in the same area was often observed. The new cultivars, Taoyuan No. 3-Spring Red and Taoyuan No. 4-Red Glory, were selected from local races in the central and northern regions of Taiwan. Through single plant selection, line observation, and comparative trial, these two cultivars were selected for their low chilling requirement for earlier flowering habits, good ratio of tree height to width, and high adaptability to local terroir, making them suitable for landscape applications in subtropical areas