Symmetries of projective spaces and spheres

Let $H$ be either a complex inner product space of dimension at least two, or a real inner product space of dimension at least three, and let us fix an $\alpha\in\left(0,\tfrac{\pi}{2}\right)$. The purpose of this paper is to characterise all bijective transformations on the projective space $P(H)$ which preserve the quantum angle $\alpha$ (or Fubini-Study distance $\alpha$) between lines in both directions. (Let us emphasise that we do not assume anything about the preservation of other quantum angles). For real inner product spaces and when $H=\mathbb{C}^2$ we do this for every $\alpha$, and when $H$ is a complex inner product space of dimension at least three we describe the structure of such transformations for $\alpha\leq\tfrac{\pi}{4}$. Our result immediately gives an Uhlhorn-type generalisation of Wigner's theorem on quantum mechanical symmetry transformations, that is considered to be a cornerstone of the mathematical foundations of quantum mechanics. Namely, under the above assumptions, every bijective map on the set of pure states of a quantum mechanical system that preserves the transition probability $\cos^2\alpha$ in both directions is a Wigner symmetry (thus automatically preserves all transition probabilities), except for the case when $H=\mathbb{C}^2$ and $\alpha = \tfrac{\pi}{4}$ where an additional possibility occurs. (Note that the classical theorem of Uhlhorn is the solution for the $\alpha = \tfrac{\pi}{2}$ case). Usually in the literature, results which are connected to Wigner's theorem are discussed under the assumption of completeness of $H$, however, here we shall remove this unnecessary hypothesis in our investigation. Our main tool is a characterisation of bijective maps on unit spheres of real inner product spaces which preserve one spherical angle in both directions

Evolutionary Psychology is Not Evil! (… and Here’s Why …)

Evolutionary psychology has faced ‘implacable hostility’ (Dawkins, 2005) from a number of intellectual fronts. Critics of evolutionary psychology have tried to paint this perspective variously as reductionist and overly deterministic, at best, and as sexist, racist, and downright evil at worst. The current paper argues that all psychological frameworks which assume that human beings are the result of the organic evolutionary forces of natural and sexual selection are, essentially, evolutionary in nature (regardless of whether they traditionally fall under the label of evolutionary psychology). In other words, the perspective presented here argues that all psychology is evolutionary psychology. Two specific mis-characterizations of evolutionary psychology ((a) that it is eugenicist in nature and (b) that it is a fully non-situationist, immutable perspective on behavior) are addressed here with an eye toward elaborating on how these distorted conceptions of evolutionary psychology are non-constructive and non-progressive. A final section focuses on how the social sciences in general could benefit from being evolutionized. well that it is actually a bit unsettling

Isometries of Grassmann spaces, II

Botelho, Jamison, and Molnár [1], and Gehér and Šemrl [4] have recently described the general form of surjective isometries of Grassmann spaces of all projections of a fixed finite rank on a Hilbert space H. As a straightforward consequence one can characterize surjective isometries of Grassmann spaces of projections of a fixed finite corank. In this paper we solve the remaining structural problem for surjective isometries on the set of all projections of infinite rank and infinite corank when H is separable. The proof technique is entirely different from the previous ones and is based on the study of geodesics in the Grassmannian . However, the same method gives an alternative proof in the case of finite rank projections

Quid pro quo? - Gaben und Geschenke in mittelalterlicher irischer Literatur

Die vorliegende Untersuchung befasst sich mit der Funktion von Gaben und Geschenken in der mittelalterlichen, irischen Literatur und deren Bedeutung für die historische Mittelalterforschung. Analysiert wird der kommunikative Wert der Gabe in der mittelalterlichen irischen Literatur und deren Zusammenhang mit der frühmittelalterlichen, irischen Sozialstruktur. Dabei soll untersucht werden, unter welchen methodischen Voraussetzungen die Literatur als Quelle für die historische Mittelalterforschung herangezogen werden kann. Der interdisziplinäre Vergleich bietet dabei die Möglichkeit, die aus der Literatur gewonnenen Konzepte in Beziehung zu „realen“ Gesellschaftsstrukturen und deren Kommunikationsmöglichkeiten zu setzen. Als Grundlage werden dabei die theoretischen Konzepte von Jürgen Link und Gert Althoff verwendet. Basierend auf Michel Foucaults theoretischem Gebäude der Diskurstheorie, versteht Jürgen Link die vormoderne Literatur („vormodern“ im Sinne von: bis 1700) als „Interdiskurs“, der sich dadurch auszeichnet, im Gegensatz zu sogenannten „Spezialdiskursen“, verschiedene Elemente aus anderen Diskursen zu übernehmen und somit in den „Interdiskurs“ Literatur zu integrieren. Die dergestalt gewonnenen interdiskursiven Elemente können in der Literatur vom Leser auf diese Weise wiedererkannt und sinnkonstituierend eingeordnet werden. Elemente anderer Spezialdiskurse, z. B. des Rechts, der Politik, der Medizin etc., finden somit ihren Eingang in den „Interdiskurs“ Literatur, wo sie für den Leser als wiedererkennbare Symbole sinnkonstituierend wirken. Gert Althoff prägte den Begriff des Rituals, das er als Kommunikation der Beteiligten einer sozialen Interaktion definiert. Solche Rituale dienen somit der Verständigung über verschiedene Inhalte wie Sozialstatus, Hierarchie, Freundschaft/Feindschaft, (politischen) Konsens oder Dissens. In dieser Untersuchung soll die Literatur nach Links Vorbild als Interdiskurs verstanden werden. Gaben und Geschenke werden dabei als Rituale in Althoffs Sinn definiert, die als interdiskursive Elemente aus der „realen“ Welt der übrigen Spezialdiskurse importiert werden, um innerhalb der Erzählung ihren Kommunikationswert zu erhalten. Grundlagen der Untersuchungen sind hierbei mittelalterliche irische, sowie walisische Sagentexte, irische und walisische mittelalterliche Rechtstexte, Hagiographien, Bußbücher. Modelle der soziologischen Forschung dienen dazu, die allgemeine Funktion der Gabe in archaischen und modernen Kulturen zu evaluieren. Für die vorliegende Analyse kann Folgendes festgestellt werden: - Mittelalterliche irische Literatur kann unter den genannten theoretischen Voraussetzungen für die mediävistische Forschung als Quelle benutzt werden. - Die Darreichung der Gabe als symbolische Kommunikation hat eine „reale“ Dimension in der historischen mittelalterlichen irischen Gesellschaft. - Der symbolische Wert der Gabe liegt in ihrer kommunikativen Wirkung; die Gabe ist imstande, eine Aussage über verschiedene Bereiche des sozialen Lebens zu machen. Sie ermöglicht es, auf symbolischem Wege Informationen über Sozialstatus, Hierarchie, Konsens/Dissens oder diplomatische Beziehungen zu transportieren. - Die Funktion der Gabe kann nur im interdisziplinären Vergleich erarbeitet werden, der die interdiskursive Literatur in Beziehung zu anderen Bereichen der Mittelalterforschung zu setzen imstande ist.Based on the theories of Jürgen Link and Gert Althoff, this paper will explore the function and meaning of gift-giving in early Irish literature. Link suggests that literature should be understood as an “interdiscourse”, which imports elements of special discourses, where these discursive elements are used as communally understood symbols. In this context, gift-giving can be seen as symbolic action (a “ritual” in Althoff’s sense of the word) that follows the pattern of the do-ut-des-principle. In epic literature, scenes of gift-giving are often difficult to interpret - without further examination of the social context. This paper, therefore, takes an interdisciplinary approach, comparing various medieval sources on the subject, such as: Early Irish (and Welsh) epic literature, early Irish and Welsh law texts, hagiographic texts and Penitentials. Sociological models then serve as a basis to evaluate the overall human perspectives on the meaning of gift-giving in archaic and modern societies. The main aim of this paper is to extract information about formalized behaviour from the literary texts - under the assumption that they originated from within early medieval Irish society; in order to gain greater understanding of the medieval Irish mind-set concerning gift-giving and its social message. It will thus be demonstrated that - early medieval Irish literature is a valid source for historical research, - gift-giving is used as a symbolic action in early medieval Ireland, - gift-giving as a symbolic action was used as a tool for communicative purposes (e.g. to transport information about status or about diplomatic relations) in early medieval Irish society, and that - its functions can only be fully understood in its social and historical context by using interdisciplinary methods

Evolutionary Psychology is Not Evil! (… and Here’s Why …)

Evolutionary psychology has faced ‘implacable hostility’ (Dawkins, 2005) from a number of intellectual fronts. Critics of evolutionary psychology have tried to paint this perspective variously as reductionist and overly deterministic, at best, and as sexist, racist, and downright evil at worst. The current paper argues that all psychological frameworks which assume that human beings are the result of the organic evolutionary forces of natural and sexual selection are, essentially, evolutionary in nature (regardless of whether they traditionally fall under the label of evolutionary psychology). In other words, the perspective presented here argues that all psychology is evolutionary psychology. Two specific mis-characterizations of evolutionary psychology ((a) that it is eugenicist in nature and (b) that it is a fully non-situationist, immutable perspective on behavior) are addressed here with an eye toward elaborating on how these distorted conceptions of evolutionary psychology are non-constructive and non-progressive. A final section focuses on how the social sciences in general could benefit from being evolutionized. well that it is actually a bit unsettling

Coexistency on Hilbert space effect algebras and a characterisation of its symmetry transformations

The Hilbert space effect algebra is a fundamental mathematical structure which is used to describe unsharp quantum measurements in Ludwig's formulation of quantum mechanics. Each effect represents a quantum (fuzzy) event. The relation of coexistence plays an important role in this theory, as it expresses when two quantum events can be measured together by applying a suitable apparatus. This paper's first goal is to answer a very natural question about this relation, namely, when two effects are coexistent with exactly the same effects? The other main aim is to describe all automorphisms of the effect algebra with respect to the relation of coexistence. In particular, we will see that they can differ quite a lot from usual standard automorphisms, which appear for instance in Ludwig's theorem. As a byproduct of our methods we also strengthen a theorem of Molnar

On isometric embeddings of Wasserstein spaces - the discrete case

The aim of this short paper is to offer a complete characterization of all (not necessarily surjective) isometric embeddings of the Wasserstein space \mathcal{W}_p(\X), where \X is a countable discrete metric space and $0<p<\infty$ is any parameter value. Roughly speaking, we will prove that any isometric embedding can be described by a special kind of \X\times(0,1]-indexed family of nonnegative finite measures. Our result implies that a typical non-surjective isometric embedding of \ws splits mass and does not preserve the shape of measures. In order to stress that the lack of surjectivity is what makes things challenging, we will prove alternatively that \ws is isometrically rigid for all $0<p<\infty$