arXiv subjects

André L. G. Mandolesi

Publications and source records attributed to André L. G. Mandolesi.

15 recordsLinked to original sources

arXiv2026

Hermitian Geometry of Complex Multivectors, Determinants and Orientations

Two geometric interpretations for complex multivectors and determinants are presented: a little known one in terms of square roots of volumes, and a new one using fractions of volumes. The fraction is determined by a new holomorphy index, which measures the lack of holomorphy of real subspaces of $\mathbb{C}^n$ via generalized Kähler angles or a disjointness angle, and relates real and complex exterior products. The interpretations are completed with a natural but uncommon concept of complex orientation, related to elementary complex transformations. We also propose graphical representations for complex blades (decomposable multivectors) as fractions of parallelotopes, and discuss how Clifford algebras relate (or not) to Hermitian geometry.

math.CV↗

arXiv2025

Enumeration of the New Plücker-like Equations

We give a more detailed description of the new system of Plücker-like equations from [4], discuss how it relates to the usual Plücker equations, and correct a mistake in that article.

math.AG↗

arXiv2025

Asymmetric Geometry of Total Grassmannians

Metrics in Grassmannians, or distances between subspaces of same dimension, have many uses, and extending them to the Total Grassmannian of subspaces of different dimensions is an important problem, as usual extensions lack good properties or give little information. Dimensional asymmetries call for the use of asymmetric metrics, and we present a natural method to obtain them, extending all the main Grassmannian metrics (geodesic, projection Frobenius, Fubini-Study, gap, etc.). Their geometry adequately reflects containment relations of subspaces, continuous paths link subspaces of distinct dimensions, and we describe minimal geodesics, shortest paths to move a subspace onto another. In particular, the Fubini-Study metric extends as an asymmetric angle that is easily computed, has many useful properties, and a nice geometric interpretation.

math.MG↗

arXiv2024

The Thousand Faces of Pythagoras (As Mil Faces de Pitágoras)

The Pythagorean Theorem is one of the oldest, more famous and more useful theorems of Mathematics, and possibly the one that has had the most impact in the evolution of this and other sciences. In this article, we look at it from different perpectives, some of them uncommon. We recall some of its history, some well known applications and generalizations, other less known ones, and show it still has many surprising facets which are usually ignored. (O Teorema de Pitágoras (TP) é um dos mais antigos, famosos e úteis teoremas da Matemática, e possivelmente o que maior impacto teve na evolução desta e outras ciências. Neste artigo, vamos olhar para este velho conhecido de diferentes perspectivas, algumas pouco usuais. Iremos lembrar um pouco da sua história, algumas aplicações e generalizações bem conhecidas, outras nem tanto, e ver que ele guarda muitas facetas surpreendentes e geralmente ignoradas.)

math.HO↗

arXiv2024

Multivector Contractions Revisited, Part I

We reorganize, simplify and expand the theory of contractions or interior products of multivectors, and related topics like Hodge star duality. Many results are generalized and new ones are given, like: geometric characterizations of blade contractions and regressive products, higher-order graded Leibniz rules, determinant formulas, improved complex star operators, etc. Different contractions and conventions found in the literature are discussed and compared, in special those of Clifford Geometric Algebra. Applications of the theory are developed in a follow-up paper.

math.GM↗

arXiv2024

Multivector Contractions Revisited, Part II

The theory of contractions of multivectors, and star duality, was reorganized in a previous article, and here we present some applications. First, we study inner and outer spaces associated to a general multivector $M$ via the equations $v \wedge M = 0$ and $v \lrcorner M=0$. They are then used to analyze special decompositions, factorizations and `carvings' of $M$, to define generalized grades, and to obtain new simplicity criteria, including a reduced set of Plücker-like relations. We also discuss how contractions are related to supersymmetry, and give formulas for supercommutators of multi-fermion creation and annihilation operators.

math.GM↗

arXiv2022

Asymmetric Metrics on the Full Grassmannian of Subspaces of Different Dimensions

Metrics on Grassmannians have a wide array of applications: machine learning, wireless communication, computer vision, etc. But the available distances between subspaces of distinct dimensions present problems, and the dimensional asymmetry of the subspaces calls for the use of asymmetric metrics. We extend the Fubini-Study metric as an asymmetric angle with useful properties, and whose relations to products of Grassmann and Clifford geometric algebras make it easy to compute. We also describe related angles that provide extra information, and a method to extend other Grassmannian metrics to asymmetric metrics on the full Grassmannian.

math.AG↗

arXiv2021

Blade Products and the Angle Bivector of Subspaces

Principal angles are used to define an angle bivector of subspaces, which fully describes their relative inclination. Its exponential is related to the Clifford geometric product of blades, gives rotors connecting subspaces via minimal geodesics in Grassmannians, and decomposes giving Plücker coordinates, projection factors and angles with various subspaces. This leads to new geometric interpretations for this product and its properties, and to formulas relating other blade products (scalar, inner, outer, etc., including those of Grassmann algebra) to angles between subspaces. Contractions are linked to an asymmetric angle, while commutators and anticommutators involve hyperbolic functions of the angle bivector, shedding new light on their properties.

math.MG↗

arXiv2021

Grassmann angles between real or complex subspaces

The Grassmann angle improves upon similar angles between subspaces that measure volume contraction in orthogonal projections. It works in real or complex spaces, with important differences, and is asymmetric, what makes it more efficient when dimensions are distinct. It can be seen as an angle in Grassmann algebra, being related to its products and those of Clifford algebra, and gives the Fubini-Study metric on Grassmannians, an asymmetric metric on the full Grassmannian, and Hausdorff distances between full sub-Grassmannians. We give formulas for computing it in arbitrary bases, and identities for angles with certain families of subspaces, some of which are linked to real and complex Pythagorean theorems for volumes and quantum probabilities. Unusual features of the angle with an orthogonal complement, or the angle in complex spaces, are examined.

math.MG↗

arXiv2020

Grassmann angle formulas and identities

Grassmann angles improve upon similar concepts of angle between subspaces that measure volume contraction in orthogonal projections, working for real or complex subspaces, and being more efficient when dimensions are different. Their relations with contractions, inner and exterior products of multivectors are used to obtain formulas for computing these or similar angles in terms of arbitrary bases, and various identities for the angles with certain families of subspaces. These include generalizations of the Pythagorean trigonometric identity $\cos^2θ+\sin^2θ=1$ for high dimensional and complex subspaces, which are connected to generalized Pythagorean theorems for volumes, quantum probabilities and Clifford geometric product.

math.GM↗

arXiv2020

Quantum fractionalism: the Born rule as a consequence of the complex Pythagorean theorem

Everettian Quantum Mechanics, or the Many Worlds Interpretation, lacks an explanation for quantum probabilities. We show that the values given by the Born rule equal projection factors, describing the contraction of Lebesgue measures in orthogonal projections from the complex line of a quantum state to eigenspaces of an observable. Unit total probability corresponds to a complex Pythagorean theorem: the measure of a subset of the complex line is the sum of the measures of its projections on all eigenspaces. To show that projection factors can work as probabilities, we postulate the existence of a continuum infinity of identical quantum universes, all with the same quasi-classical worlds. In a measurement, these factors give the relative amounts of worlds with each result, which we associate to frequentist and Bayesian probabilities. This solves the probability problem of Everett's theory, allowing its preferred basis problem to be solved as well, and may help settle questions about the nature of probability.

quant-ph↗

arXiv2020

Projection factors and generalized real and complex Pythagorean theorems

Projection factors describe the contraction of Lebesgue measures in orthogonal projections between subspaces of a real or complex inner product space. They are connected to Grassmann's exterior algebra and the Grassmann angle between subspaces, and lead to generalized Pythagorean theorems, relating measures of subsets of real or complex subspaces and their orthogonal projections on certain families of subspaces. The complex Pythagorean theorems differ from the real ones in that the measures are not squared, and this may have important implications for quantum theory. Projection factors of the complex line of a quantum state with the eigenspaces of an observable give the corresponding quantum probabilities. The complex Pythagorean theorem for lines corresponds to the condition of unit total probability, and may provide a way to solve the probability problem of Everettian quantum mechanics.

math.GM↗

arXiv2018

Everettian Decoherent Histories and Causal Histories

D. Wallace has tried to use decoherence to solve the preferred basis problem of Everettian Quantum Mechanics, and this solution lays the foundation for his proof of the Born rule. But this is a circular argument, as approximations used in decoherence usually rely on the probabilistic interpretation of the Hilbert space norm. He claims the norm can measure approximations even without probabilities, but this assumption has not been properly justified. Without it, the combination of the Everettian and decoherent histories formalisms leads to strange consequences, such as a proliferation of small amplitude histories with lots of macroscopic quantum jumps. Still, this erratic behavior may provide a way to justify the approximations, in a new histories formalism, in which macroscopic causal relations play a central role. Small histories, suffering too much interference, may lose causality, being thus discarded as invalid. The remaining branches can present some small interference, opening the possibility of experimental verification.

quant-ph↗

arXiv2018

Analysis of Wallace's Proof of the Born Rule in Everettian Quantum Mechanics II: Concepts and Axioms

Having analyzed the formal aspects of Wallace's proof of the Born rule, we now discuss the concepts and axioms upon which it is built. Justification for most axioms is shown to be problematic, and at times contradictory. Some of the problems are caused by ambiguities in the concepts used. We conclude the axioms are not reasonable enough to be taken as mandates of rationality in Everettian Quantum Mechanics. This invalidates the interpretation of Wallace's result as meaning it would be rational for Everettian agents to decide using the Born rule.

quant-ph↗

arXiv2017

Analysis of Wallace's Proof of the Born Rule in Everettian Quantum Mechanics: Formal Aspects

To solve the probability problem of the Many Worlds Interpretation of Quantum Mechanics, D.Wallace has presented a formal proof of the Born rule via decision theory, as proposed by D.Deutsch. The idea is to get subjective probabilities from rational decisions related to quantum measurements, showing the non-probabilistic parts of the quantum formalism, plus some rational constraints, ensure the squared modulus of quantum amplitudes play the role of such probabilities. We provide a new presentation of Wallace's proof, reorganized to simplify some arguments, and analyze it from a formal perspective. Similarities with classical decision theory are made explicit, to clarify its structure and main ideas. A simpler notation is used, and details are filled in, making it easier to follow and verify. Some problems have been identified, and we suggest possible corrections.

quant-ph↗