526 research outputs found
The ultimate tactics of self-referential systems
Mathematics is usually regarded as a kind of language. The essential behavior
of physical phenomena can be expressed by mathematical laws, providing
descriptions and predictions. In the present essay I argue that, although
mathematics can be seen, in a first approach, as a language, it goes beyond
this concept. I conjecture that mathematics presents two extreme features,
denoted here by {\sl irreducibility} and {\sl insaturation}, representing
delimiters for self-referentiality. These features are then related to physical
laws by realizing that nature is a self-referential system obeying bounds
similar to those respected by mathematics. Self-referential systems can only be
autonomous entities by a kind of metabolism that provides and sustains such an
autonomy. A rational mind, able of consciousness, is a manifestation of the
self-referentiality of the Universe. Hence mathematics is here proposed to go
beyond language by actually representing the most fundamental existence
condition for self-referentiality. This idea is synthesized in the form of a
principle, namely, that {\sl mathematics is the ultimate tactics of
self-referential systems to mimic themselves}. That is, well beyond an
effective language to express the physical world, mathematics uncovers a deep
manifestation of the autonomous nature of the Universe, wherein the human brain
is but an instance.Comment: 9 pages. This essay received the 4th. Prize in the 2015 FQXi essay
contest: "Trick or Truth: the Mysterious Connection Between Physics and
Mathematics
Negative time delay for wave reflection from a one-dimensional semi-harmonic well
It is reported that the phase time of particles which are reflected by a
one-dimensional semi-harmonic well includes a time delay term which is negative
for definite intervals of the incoming energy. In this interval, the absolute
value of the negative time delay becomes larger as the incident energy becomes
smaller. The model is a rectangular well with zero potential energy at its
right and a harmonic-like interaction at its left.Comment: 6 pages, 5 eps figures. Talk presented at the XXX Workshop on
Geometric Methods in Physics, Bialowieza, Poland, 201
Three-body scattering in Poincar\'e invariant quantum mechanics
The relativistic three-nucleon problem is formulated by constructing a
dynamical unitary representation of the Poincar\'e group on the three-nucleon
Hilbert space. Two-body interactions are included that preserve the Poincar\'e
symmetry, lead to the same invariant two-body S-matrix as the corresponding
non-relativistic problem, and result in a three-body S-matrix satisfying
cluster properties. The resulting Faddeev equations are solved by direct
integration, without partial waves for both elastic and breakup reactions at
laboratory energies up to 2 Gev.Comment: 4 pages - no figures - contribution to the 20-th European Few-Body
Conferenc
Insolubility Theorems and EPR Argument
I wish to thank in particular Arthur Fine for very perceptive comments on a previous draft of this paper. Many thanks also to Theo Nieuwenhuizen for inspiration, to Max Schlosshauer for correspondence, to two anonymous referees for shrewd observations, and to audiences at Aberdeen, Cagliari and Oxford (in particular to Harvey Brown, Elise Crull, Simon Saunders, Chris Timpson and David Wallace) for stimulating questions. This paper was written during my tenure of a Leverhulme Grant on âThe Einstein Paradoxâ: The Debate on Nonlocality and Incompleteness in 1935 (Project Grant nr. F/00 152/AN), and it was revised for publication during my tenure of a Visiting Professorship in the Doctoral School of Philosophy and Epistemology, University of Cagliari (Contract nr. 268/21647).Peer reviewedPostprin
The Wasteland of Random Supergravities
We show that in a general \cal{N} = 1 supergravity with N \gg 1 scalar
fields, an exponentially small fraction of the de Sitter critical points are
metastable vacua. Taking the superpotential and Kahler potential to be random
functions, we construct a random matrix model for the Hessian matrix, which is
well-approximated by the sum of a Wigner matrix and two Wishart matrices. We
compute the eigenvalue spectrum analytically from the free convolution of the
constituent spectra and find that in typical configurations, a significant
fraction of the eigenvalues are negative. Building on the Tracy-Widom law
governing fluctuations of extreme eigenvalues, we determine the probability P
of a large fluctuation in which all the eigenvalues become positive. Strong
eigenvalue repulsion makes this extremely unlikely: we find P \propto exp(-c
N^p), with c, p being constants. For generic critical points we find p \approx
1.5, while for approximately-supersymmetric critical points, p \approx 1.3. Our
results have significant implications for the counting of de Sitter vacua in
string theory, but the number of vacua remains vast.Comment: 39 pages, 9 figures; v2: fixed typos, added refs and clarification
Information-theoretic postulates for quantum theory
Why are the laws of physics formulated in terms of complex Hilbert spaces?
Are there natural and consistent modifications of quantum theory that could be
tested experimentally? This book chapter gives a self-contained and accessible
summary of our paper [New J. Phys. 13, 063001, 2011] addressing these
questions, presenting the main ideas, but dropping many technical details. We
show that the formalism of quantum theory can be reconstructed from four
natural postulates, which do not refer to the mathematical formalism, but only
to the information-theoretic content of the physical theory. Our starting point
is to assume that there exist physical events (such as measurement outcomes)
that happen probabilistically, yielding the mathematical framework of "convex
state spaces". Then, quantum theory can be reconstructed by assuming that (i)
global states are determined by correlations between local measurements, (ii)
systems that carry the same amount of information have equivalent state spaces,
(iii) reversible time evolution can map every pure state to every other, and
(iv) positivity of probabilities is the only restriction on the possible
measurements.Comment: 17 pages, 3 figures. v3: some typos corrected and references updated.
Summarizes the argumentation and results of arXiv:1004.1483. Contribution to
the book "Quantum Theory: Informational Foundations and Foils", Springer
Verlag (http://www.springer.com/us/book/9789401773027), 201
Digraph Complexity Measures and Applications in Formal Language Theory
We investigate structural complexity measures on digraphs, in particular the
cycle rank. This concept is intimately related to a classical topic in formal
language theory, namely the star height of regular languages. We explore this
connection, and obtain several new algorithmic insights regarding both cycle
rank and star height. Among other results, we show that computing the cycle
rank is NP-complete, even for sparse digraphs of maximum outdegree 2.
Notwithstanding, we provide both a polynomial-time approximation algorithm and
an exponential-time exact algorithm for this problem. The former algorithm
yields an O((log n)^(3/2))- approximation in polynomial time, whereas the
latter yields the optimum solution, and runs in time and space O*(1.9129^n) on
digraphs of maximum outdegree at most two. Regarding the star height problem,
we identify a subclass of the regular languages for which we can precisely
determine the computational complexity of the star height problem. Namely, the
star height problem for bideterministic languages is NP-complete, and this
holds already for binary alphabets. Then we translate the algorithmic results
concerning cycle rank to the bideterministic star height problem, thus giving a
polynomial-time approximation as well as a reasonably fast exact exponential
algorithm for bideterministic star height.Comment: 19 pages, 1 figur
Probability representation and quantumness tests for qudits and two-mode light states
Using tomographic-probability representation of spin states, quantum behavior
of qudits is examined. For a general j-qudit state we propose an explicit
formula of quantumness witnetness whose negative average value is incompatible
with classical statistical model. Probability representations of quantum and
classical (2j+1)-level systems are compared within the framework of quantumness
tests. Trough employing Jordan-Schwinger map the method is extended to check
quantumness of two-mode light states.Comment: 5 pages, 2 figures, PDFLaTeX, Contribution to the 11th International
Conference on Squeezed States and Uncertainty Relations (ICSSUR'09), June
22-26, 2009, Olomouc, Czech Republi
- âŠ