Artículos con la etiqueta ‘Mathematical Physics (math-ph)’

The changing concept of matter in H. Weyl’s thought, 1918 -1930

Por • 3 abr, 2014 • Category: Educacion

During the “long decade” of transformation of mathematical physics between 1915 and 1930, H. Weyl interacted with physics in two highly productive phases and contributed to it, among others, by his widely read book on Space – Time – Matter (Raum- Zeit – Materie), (1918-1923) and on Group Theory and Quantum Mechanics (Gruppentheorie und Quantenmechanik) (1928-1931). In this time Weyl’s understanding of the constitution of matter and its mathematical description changed considerably. At the beginning of the period he started from a «dynamistic» and geometrical conception of matter, following and extending the Mie-Hilbert approch, which he gave up during the year 1920. After transitional experiments with a singularity (and in this sense topological) approach in 1921/22, he developed an open perspective of what he called an “agency theory” of matter. The idea for it was formulated already before the advent of the “new” quantum mechanics in 1925/26.

Solving the Hard Problem of Bertrand’s Paradox

Por • 19 mar, 2014 • Category: Leyes

Bertrand’s paradox is a famous problem of probability theory, pointing to a possible inconsistency in Laplace’s principle of insufficient reason. In this article we show that Bertrand’s paradox contains two different problems: an «easy» problem and a «hard» problem. The easy problem can be solved by formulating Bertrand’s question in sufficiently precise terms, so allowing for a non ambiguous modelization of the entity subjected to the randomization. We then show that once the easy problem is settled, also the hard problem becomes solvable, provided Laplace’s principle of insufficient reason is applied not to the outcomes of the experiment, but to the different possible «ways of selecting» an interaction between the entity under investigation and that producing the randomization.

«Information-Friction» and its implications on minimum energy required for communication

Por • 10 ene, 2014 • Category: Ambiente

Just as there are frictional losses associated with moving masses on a surface, what if there were frictional losses associated with moving information on a substrate? Indeed, many methods of communication suffer from such frictional losses. We propose to model these losses as proportional to «bit-meters,» i.e., the product of mass of information (i.e., the number of bits) and the distance of information transport. We use this «information-friction» model to understand fundamental energy requirements on encoding and decoding in communication circuitry. First, for communication across a binary input AWGN channel, we arrive at limits on bit-meters (and thus energy consumption) for decoding implementations that have a predetermined input-independent lengths of messages.

The problem of space in the light of relativity: the views of H. Weyl and E. Cartan

Por • 14 nov, 2013 • Category: Opinion

Starting from a short review of the «classical» space problem in the sense of the 19th century (Helmholtz — Lie — Klein) it is discussed how the challenges posed by special and general relativity to the classical analysis were taken up by Hermann Weyl and Elie Cartan. Both mathematicians reconsidered the space problem from the point of view of transformations operating in the infinitesimal neighbourhoods of a manifold (spacetime). In a short outlook we survey further developments in mathematics and physics of the second half of the 20th century, in which core ideas of Weyl’s and/or Cartan’s analysis of the space problem were further investigated (mathematics) or incorporated into basic theories (physics).

When periodicities enforce aperiodicity

Por • 25 sep, 2013 • Category: Ciencia y tecnología

Aperiodic tilings are non-periodic tilings defined by local rules. They are widely used to model quasicrystals, and a central question is to understand which of the non-periodic tilings are actually aperiodic. Among tilings, those by rhombi can be easily seen as approximations of surfaces in higher dimensional spaces. In particular, those which approximate irrational planes are non-periodic. But which ones are also aperiodic? This paper introduces the notion of subperiod, which links algebraic properties of a plane with geometric properties of the tilings that approximate it. A necessary and sufficient condition is obtained for tilings that can be seen in the four dimensional Euclidean space. This result is then applied to some examples in higher codimensions, notably tilings with n-fold rotational symmetry

Why do the relativistic masses and momenta of faster-than-light particles decrease as their speeds increase?

Por • 19 sep, 2013 • Category: Crítica

It has recently been shown within a formal axiomatic framework using a definition of four-momentum based on the St\»uckelberg-Feynman-Sudarshan «switching principle» that Einstein’s relativistic dynamics is logically consistent with the existence of interacting faster-than-light inertial particles. Our results here show, using only basic natural assumptions on dynamics, that this definition is the only possible way to get a consistent theory of such particles moving within the geometry of Minkowskian spacetime.
We present a strictly formal proof from a streamlined axiom system that given any slow or fast inertial particle, all inertial observers agree on the value of (m . sqrt{|1-v^2|}), where m is the particle’s relativistic mass (or energy) and v its speed. This confirms formally the widely held belief that the relativistic mass and momentum of a positive-mass particle must decrease as its speed increases.

Contextuality: Wheeler’s universal regulating principle

Por • 20 jul, 2013 • Category: Ciencia y tecnología

In this essay I develop quantum contextuality as a potential candidate for Wheeler’s universal regulating principle, arguing — \textit{contrary} to Wheeler — that this ultimately implies that `bit’ comes from `it.’ In the process I develop a formal definition of physical determinism in the languages of domain theory and category theory.

Three Merry Roads to T-Violation

Por • 30 jun, 2013 • Category: Leyes

This paper is a tour of how the laws of nature can distinguish between the past and the future, or be T-violating. I argue that, in terms of the basic argumentative structure, there are really just three approaches currently being explored. I show how each is characterized by a symmetry principle, which provides a template for detecting T-violating laws even without knowing the laws of physics themselves. Each approach is illustrated with an example, and the prospects of each are considered in extensions of particle physics beyond the standard model.

Consensus time and conformity in the adaptive voter model

Por • 20 abr, 2013 • Category: sociologia

Tim Rogers, Thilo Gross Abstract: The adaptive voter model is a paradigmatic model in the study of opinion formation. Here we propose an extension for this model, in which conflicts are resolved by obtaining another opinion, and analytically study the time required for consensus to emerge. Our results shed light on the rich phenomenology of […]

Commuting and noncommuting infinitesimals

Por • 10 abr, 2013 • Category: Filosofía

Infinitesimals are natural products of the human imagination. Their history goes back to the Greek antiquity. Their role in the calculus and analysis has seen dramatic ups and downs. They have stimulated strong opinions and even vitriol. Edwin Hewitt developed hyperreal fields in the 1940s. Abraham Robinson’s infinitesimals date from the 1960s. A noncommutative version of infinitesimals, due to Alain Connes, has been in use since the 1990s. We review some of the hyperreal concepts, and compare them with some of the concepts underlying noncommutative geometry.